Equivariant Seiberg-Witten Floer Homology

Matilde Marcolli, Bai-Ling Wang

Contents

1 Introduction 2 1.1 Three dimensional Seiberg-Witten theory . . . . . . . . . . . . . 9

2 Non-equivariant Floer theory 13 2.1 Chamber structure . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2 Perturbation of flow lines . . . . . . . . . . . . . . . . . . . . . . 18 2.3 Transversality of M(a, b) . . . . . . . . . . . . . . . . . . . . . . 22 2.4 Floer homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3 Morse-Bott theory 30 3.1 Framed moduli space . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.2 Gradient flow lines . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.3 Relative Morse Index . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.4 Decay estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.5 Transversality of M(O_{a}, O_{b}) . . . . . . . . . . . . . . . . . . . . . 42

4 Boundary structure 43 4.1 Convergence theorem . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.2 Gluing theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.3 Multiple gluing theorem and corner structure . . . . . . . . . . . 76

5 Equivariant Homology 83 5.1 The equivariant complex . . . . . . . . . . . . . . . . . . . . . . . 84 5.2 Equivariant Floer Homology . . . . . . . . . . . . . . . . . . . . . 85 5.3 Intermezzo: the coefficient m_{ac} . . . . . . . . . . . . . . . . . . . 89

6 Topological invariance 95 6.1 The boundary structure of M(O_{a}, O_{a′}) . . . . . . . . . . . . . . . 102 6.2 Obstruction bundle and gluing theorems . . . . . . . . . . . . . . 114 6.3 Proof of topological invariance . . . . . . . . . . . . . . . . . . . 127 7 Wall crossing formula for the Casson-type invariant 140

7.1 Comparison with the non-equivariant Floer Homology . . . . . . 141 7.2 Wall-crossing formula: the algebraic picture . . . . . . . . . . . . 147 7.3 Wall-crossing formula: the geometric picture . . . . . . . . . . . . 152

1 Introduction

und das hat mit ihrem Singen die Loreley getan.

(H. Heine)

Floer homology, as a gauge theoretic invariant of three-manifolds, made its first appearance in the seminal work of Floer [24] on what is now referred to as the Yang–Mills, or instanton, Floer homology. Instanton Floer homology came to play an essential conceptual role in Atiyah’s Topological Quantum Field Theory formulation of Donaldson theory [1], and in the definition of relative Donaldson invariants of 4-manifolds with boundary and their gluing formulae, see for instance the work of Taubes [60]. Soon after the introduction of the new Seiberg–Witten gauge theory, it became clear that an analogue of Floer homology existed in the Seiberg–Witten context. In the last four years, there has been a number of significant papers using Seiberg–Witten Floer homology and gluing theorems for Seiberg–Witten invariants. It is now known (see e.g. [29]) that a good understanding of the relation between the instanton and the Seiberg–Witten Floer homology can lead to very striking topological consequences. However, in the literature (both in the instanton and in the Seiberg–Witten context) there seems to be a fundamental need for a satisfactory foundational work which presents a detailed and careful construction of the Floer homology. It is important to mention that, perhaps surprisingly, the technical aspects involved in the Seiberg–Witten gauge theory are substantially different from the Yang–Mills case and require different techniques. An important issue, which has no analogue in the Yang–Mills case and to which much of this paper is dedicated, is the metric dependence of the Seiberg–Witten Floer homology in the case of homology 3-spheres. Providing a metric independent version of Floer homology is essential in all the important applications which involve surgery formulae and gluing of relative invariants. We develop an equivariant version of Seiberg–Witten Floer homology precisely to the purpose of avoiding the metric dependence problem. The main technical issues involved center around the fine structure of the compactification of the moduli spaces of flow lines of the Chern–Simons–Dirac functional, and around the use of Taubes’ obstruction bundles in the course of the proof of the topological invariance of the equivariant Seiberg–Witten Floer homology.

It is not easy to give the skeptical reader an introduction to this paper that justifies the more than hundred and fifty pages to come. To be perfectly honest, it took a long time to convince ourselves as well of the real need to unravel all the detailed technical issues that appear in the corpus of this paper. Nonetheless, we hope the introductory paragraph above gives sufficiently clear an idea of why it is genuinely useful to write a paper that covers the material presented here. This paper has been under preparation since 1995. It circulated in different draft versions in 1996 and 1997. Regrettably, more than two years passed between the time when the paper was submitted and the time when a referee report was finally made available to us. We should signal to the reader the appearance, in the meanwhile, of other papers that have a more or less extended overlap with the present one, most notably [27] and [63]. In that respect, we are certainly guilty of not having circulated, through the customary electronic distribution, the last revision of this paper dating back to November 1997. We trust that this paper, in the present and hopefully last of its long series of avatars, will soon see the light as a printed form incarnation.

We summarize, in the rest of the Introduction, the main results of the various sections of this paper, pointing to precise references to Theorems, Sections, etc. where the various statements are presented and proved.

The dimensional reduction of the Seiberg-Witten equations on 4-manifolds leads to equations on a compact oriented 3-manifold Y, obtained by considering translation invariant solutions of the original equations in a temporal gauge on an infinite cylinder Y ×ℝ with translation invariant metric. The solutions modulo gauge transformations of the reduced equations on Y can be regarded as the critical points of a Chern–Simons–Dirac functional, defined on the configuration space of U(1)-connections and spinors. There is an associated Floer-type homology, which depends on the choice of the Spin_{c}-structure. Some of the properties of this Seiberg-Witten Floer homology have been discussed in [11], [13], [27], [38], [40], [52] [61], [62], [63]. The properties are different according to whether the manifold Y is a rational homology sphere or a manifold with b^{1}(Y) > 0. In particular, if Y is a rational homology sphere, there is always a reducible point in the moduli space, namely solutions with vanishing spinor and a non-trivial stabilizer of the gauge group action. Unlike the case with non-zero Betti number, the reducible cannot be perturbed away just by adding a co-closed 1-form to the curvature equation. This gives rise to an interesting metric dependence phenomenon. The Seiberg-Witten Floer homology for manifolds with b^{1}(Y) > 0 presents a periodicity of the grading and is graded over the integers only after passing to a cover. In this case, however, with the formulation used in this paper, it is no longer finitely generated. There are different ways of defining the Floer homology when b^{1}(Y) > 1, some of which still give rise to a finitely generated complex. Some of these different constructions are summarized briefly in Section 4, in Remark 4.5, an a more detailed account will be given in [41]. For the purpose of this paper, the important issue is that, for b^{1}(Y) > 1, the Floer homology is independent of the metric and perturbation. Since we consider perturbations that are cohomologically trivial, the metric and perturbation independence is achieved in the case b^{1}(Y) = 1 as well. Thus, we have Seiberg-Witten Floer homologies with substantially different properties depending on the underlying three-manifold Y. The Floer homology always depends on a choice of the Spin_{c}-structure, as does the construction of Seiberg– Witten invariants. Thus, we really obtain, over a 3-manifold Y, a family of Floer homologies SW F_{∗}(Y, s), parameterized by the set of Spin_{c}-structures s ∈ S(Y). We shall always consider a fixed Spin_{c}-structure, hence, for simplicity of notation, we shall not explicitly mention the s dependence in the following.

We give a unifying approach to Seiberg-Witten Floer theory, by introducing a version of Seiberg-Witten Floer homology that is defined for all three-manifolds and is always metric independent. In the case of manifolds with b^{1}(Y) > 0 we shall only consider Spin_{c}-structures with c_{1}(L) ≠ 0. The case b^{1}(Y) > 0 and c_{1}(L) = 0 has a more subtle behavior with respect to the choice of the perturbation and the compactification of the moduli spaces of flow lines (cf. Remark 2.16). Since this case arises in important problems connected to surgery formulae, we are going to deal with it separately in [41]. Along the lines of the theory Austin and Braam [6] developed for instanton homology, we construct an equivariant version of the Seiberg-Witten Floer homology. In the case of rational homology spheres, this approach counts all the contributions, both from the reducible point and from the irreducibles. The equivariant Floer homology also depends on the choice of the Spin_{c}-structure. We prove the invariance of the equivariant Floer homology with respect to the metric and perturbation. The argument involves Taubes’ obstruction bundle technique and an appropriate gluing theorem. In the two distinct cases of rational homology spheres and of manifolds with non-trivial Betti number, we compare the equivariant theory with the constructions of non-equivariant Seiberg-Witten Floer theories, as presented in the work of K. Iga, M. Marcolli, B.L. Wang, and R.G. Wang. In the case of manifolds with non-trivial Betti number, we obtain an isomorphism of the equivariant and the non-equivariant theories (both considered for a fixed Spin_{c}-structure) and in the case of rational homology spheres we construct some exact sequences that relate the equivariant and the non-equivariant theory. The results are obtained by means of the spectral sequences associated to a filtration of the complexes. The metric independence of the equivariant Floer homology together with the exact sequences lead to an algebraic proof of the wall crossing formula for the associated Casson-type invariant. We also provide a geometric proof of the same formula, through the analysis of the local structure of the moduli spaces.

The paper is organized as follows. Section 1 and 2 present preliminary material, starting, in Propositions 2.1, 2.2, and 2.3, with a brief account of Seiberg-Witten gauge theory on three-manifolds and a brief overview of the properties and the local structure of the moduli space M of critical points of the Chern–Simons–Dirac functional, and then continuing in Sections 2.2 and 2.3 with an account of the properties of the moduli spaces M(a, b) of flow lines connecting critical points. These are the moduli spaces that appear in the construction of the non-equivariant versions of Seiberg-Witten Floer theory, as used for instance in [13], [38], [61], [62]. The main result in Sections 2.2 and 2.3 is the construction of a suitable class of perturbations of the flow lines equations, in Definition 2.8, and the proof of transversality in Proposition 2.12. We also discuss the notion of relative Morse index of critical points of the Chern– Simons–Dirac functional in Proposition 2.10 and the orientation of the moduli spaces M(a, b) in Proposition 2.13. The construction and properties of the nonequivariant Seiberg-Witten Floer theory are briefly summarized in Section 2.4, the main result being the gluing formula of Lemma 2.14, which is essential in establishing the property ∂ ∘ ∂ = 0 for the boundary of the Floer complex. We do not present a complete treatment of the non-equivariant Floer homology, since all the necessary results follow from the equivariant case of Sections 4, 5, and 6, and the comparison results of Section 7. However, we discuss some essential aspects of the non-equivariant theory, for instance the phenomenon of metric dependence in the case of homology spheres. In the proof of Theorem 3.4 it is shown that, in the case of homology spheres, the unique reducible point is isolated whenever the metric and perturbation are chosen so that the twisted Dirac operator has trivial kernel. This condition on the choice of metric and perturbation determines a chamber structure with codimension one wall which is analyzed in Section 2.1. We prove in Theorem 2.5 that the condition on the Dirac operator is generic and that the space of metrics and perturbations breaks into chambers with codimension one walls. We analyze the structure of the walls in Theorem 2.6.

In Section 3 we introduce the framed moduli space M^{0} in the framed configuration space B^{0}. The results of Lemma 3.2, Lemma 3.3, and Theorem 3.4 can be summarized as follows: the framed moduli space M^{0} is a smooth manifold with a U(1)-action: the action is free in the case of manifolds with b^{1}(Y) > 0. When b^{1}(Y) = 0, there is a unique fixed point in M^{0}, which corresponds to the reducible point in M. In the case of rational homology spheres, the finitely many inequivalent flat U(1) connections correspond to the unique fixed point in M^{0} for all the different possible choices of the Spin_{c}-structure. In Theorem 3.4 we also show that the critical orbits are all isolated and the Hessian is non-degenerate in the directions orthogonal to the orbits. That is, the Chern– Simons–Dirac functional satisfies the Morse-Bott condition.

In the remaining of Section 3 we analyze the moduli spaces M(O_{a}, O_{b}) of gradient flow lines connecting critical orbits O_{a} and O_{b}. In Section 3.3, Theorem 3.8, we discuss the relative Morse index of critical orbits, which requires introducing the relevant Fredholm theory. The analysis of Section 3.4 is aimed at proving that all finite energy solutions of the Seiberg–Witten equations on the manifold Y × ℝ decay asymptotically to critical orbits, with an exponential weight which is determined by the smallest absolute value of the non-trivial eigenvalues of the Hessian. Thus, every finite energy solution lies in some moduli space M(O_{a}, O_{b}). The exponential decay to the endpoints of solutions in a moduli space M(O_{a}, O_{b}) is proved in Theorem 3.12, using the result of Lemma 3.10 (a Palais–Smale condition), and the estimate of Lemma 3.13. The argument of Theorem 3.12 is basically a “finite energy implies finite length” type result (cf. [56]). Finally, we give the transversality result for the moduli spaces M(O_{a}, O_{b}) in Proposition 3.14.

Section 4 is dedicated to the existence and properties of the compactification of the moduli spaces M̂(O_{a}, O_{b}) of unparameterized flow lines. We prove that a compactification can be obtained by adding boundary strata of broken trajectories. The codimension k strata in the boundary are of the form

⋃_{c_{1},···c_{k}} M̂(O_{a}, O_{c_{1}}) ×_{O_{c_{1}}} M̂(O_{c_{1}}, O_{c_{2}}) × ⋯ ×_{O_{c_{k}}} M̂(O_{c_{k}}, O_{b}),  (1)

where the union is over all possible sequences of the critical points c_{1}, ⋯ c_{k} with decreasing indices. We also show that the compactification has a fine structure of a smooth manifold with corners. The analysis in Section 4.1 shows that a compactification exists, and that the points in the ideal boundary consist of a certain subset of the set (1) of broken trajectories through intermediate critical points. This is the main result of Theorem 4.1. Unlike other problems of compactification in gauge theory, the construction of Floer homology requires a more detailed analysis of the compactification. In fact, the property the the boundary D of the (equivariant) Floer complex satisfies D ∘ D = 0 requires non only to know that a compactification of the moduli spaces M̂(O_{a}, O_{b}) exists, but also that all the broken trajectories (1) in the ideal boundary occur in the actual boundary of the compactification. Moreover, since the argument in the proof of D ∘ D = 0 is based on a version of Stokes’ theorem for manifolds with corners, it is essential to analyze the fine structure of the compactification at the boundary strata (1).

In Section 4.2 we prove that in fact all broken trajectories through intermediate critical points listed in (1) actually occur in the compactification, the main result being the gluing theorem 4.9. This theorem only deals with the codimension one boundary, namely with the trajectories that break through one intermediate critical orbit. The gluing construction which proves Theorem 4.9 involves several technical aspects. The main idea is to introduce a pre-gluing procedure, in which an approximate solution to the Seiberg–Witten equations is obtained by splicing together with cutoff functions a pair of solutions in the fibered product

M̂(O_{a}, O_{b}) ×_{O_{b}} M̂(O_{b}, O_{c}).

The gluing construction then takes place by proving, via a fixed point argument (Remark 4.20), that close enough to the approximate solution there is a unique actual solution in M̂(O_{a}, O_{c}). The central technical issues connected to this argument revolve around the analysis of the eigenspace of small eigenvalues of the linearization of the Seiberg–Witten equations at the approximate solutions. Lemma 4.7 and Lemma 4.8 deal with the slices of the gauge action. These are necessary in order to introduce the pre-gluing construction in Lemma 4.10. The analysis of the small eigenvalues is developed essentially in Lemma 4.12, Lemma 4.13, and Lemma 4.15, and then recalled, in Section 6, in Lemma 6.13. In Lemma 4.15 we also provide an estimate of the rate of decay of the small eigenvalues. The eigenspaces of small eigenvalues, which give the normal bundle for the gluing construction, are introduced in Definition 4.16 and in Proposition 4.17, cf. also Corollary 4.18 on the splitting of the index, and Remark 4.22.

In Section 4.3 we extend the results of Section 4.2, in order to deal with the strata of higher codimension in the compactification, and to show the fine structure of the compactified moduli spaces, namely the fact that the moduli spaces M̂(O_{a}, O_{b}) compactify to smooth manifolds with corners, in the sense of Melrose [42]. The main result on the corner structure, Theorem 4.23, is based on Proposition 4.25, which is an inductive generalization of Theorem 4.9, and shows the existence of a smooth atlas of charts with corners.

Throughout all the gluing construction in Section 4, we make essential use of the transversality result, namely of the fact that the linearizations at the solutions in M̂(O_{a}, O_{b}) and M̂(O_{b}, O_{c}) have trivial cokernel. We say in such cases that the gluing is “unobstructed”. The spectral analysis in Lemma 4.12, Lemma 4.13, and Lemma 4.15, however, is formulated in more general terms that adapt to the case where cokernels are present. This gluing theory “with obstructions” is elaborated in Section 6, with the purpose of proving the topological invariance of the equivariant Floer homology.

As we discuss in the beginning of Section 4, the fine structure of the compactification of the moduli spaces of flow lines is necessary in order to establish the existence of the Floer complex, namely the fact that the boundary operator D of the equivariant Floer complex satisfies D^{2} = 0. Section 5 introduces the equivariant complex, the boundary operator, and the equivariant Floer homology. With the essential use of the results of Section 4, we prove the property D^{2} = 0 in Theorem 5.1. We then give an explicit description of the boundary operator of the equivariant Floer complex in Proposition 5.3.

Section 6 contains the proof of the invariance of the equivariant Seiberg– Witten Floer homology with respect to the choice of the metric and perturbation. The proof of the invariance is obtained by defining a chain map I connecting the equivariant Floer complexes associated to choices (g_{0}, ν_{0}) and (g_{1}, ν_{1}), a similar chain map J in the opposite direction, and then showing that there is a chain homotopy H, satisfying id − JI = DH + HD, that induces an isomorphism on the level of cohomology. The chain map is constructed by means of moduli spaces M(O_{a}, O_{a′}) of solutions of Seiberg-Witten equations on the cylinder Y × ℝ endowed with a metric g_{t} + dt^{2} that varies between g_{0} + dt^{2} and g_{1} + dt^{2} along the cylinder. The main theorem, Theorem 6.1 is presented at the beginning of Section 6, followed by a “model” proof of the easiest case of metrics and perturbation in the same chamber. The proof of the general statement of Theorem 6.1 will only be given in Section 6.3, after the necessary technical tools have been introduced.

The construction of the chain map, in the general case of Theorem 6.1, requires a careful analysis of the boundary structure of the moduli spaces M(O_{a}, O_{a′}). In Section 6.1 we present the properties of the moduli spaces M(O_{a}, O_{a′}). In Theorem 6.4, we prove the existence of a compactification for the moduli spaces M(O_{a}, O_{a′}), obtained by adding fibered products of lower dimensional moduli spaces. This is the analogue of the results of Section 4.1 in the case of flow lines. We then give a transversality result, in Lemma 6.5 and Lemma 6.6, under the hypothesis that at least one of the critical orbits O_{a} and O_{a′} has a free U(1) action, or that the relative Morse index μ(θ_{0}) − μ(θ_{1}) is non-negative. In the case of the moduli space M(θ_{0}, θ_{1}), with μ(θ_{0})−μ(θ_{1}) < 0, we show in Corollary 6.7 and in Lemma 6.8 that the transversality result fails. The moduli space M(θ_{0}, θ_{1}) consists in this case of a unique reducible solution and the linearization has a non-trivial cokernel of dimension equal to the index |μ(θ_{0}) − μ(θ_{1})|. We prove in Theorem 6.9 the gluing theorem for all the nonsingular boundary strata in M(O_{a}, O_{a′})^{∗}, namely those with trivial Cokernels. Similarly, we prove in Theorem 6.10 the gluing theorem for all the non-singular boundary strata in M^{P} (O_{a}, O_{b})^{∗}.

Section 6.2 contains the general theory needed in order to deal with the gluing theorems in the presence of obstructions. It relies heavily on Taubes’ technique developed in [58] and [57], together with Donaldson’s results in [16]. We recall in Lemma 6.13 the necessary eigenvalue splitting for the Laplacians at the approximate solution, as proved in Lemma 4.12, Lemma 4.13, and Lemma 4.15 of Section 4.2. In Proposition 6.17 we introduce the obstruction bundle with the canonical obstruction section. In Lemma 6.14 we derive the fixed point argument which we use to complete the proof of the non-obstructed gluing of Theorem 4.9 and to formulate the gluing with obstruction in Proposition 6.17. In the remaining of Section 6.2 we analyze the modified boundary strata of M(O_{a}, O_{a′})^{∗} and M^{P} (O_{a}, O_{b})^{∗}. In Theorem 6.19 and Theorem 6.21 we identify the extra boundary components in M(O_{a}, O_{a′})^{∗} due to the zeroes of the obstruction section, and we define in Proposition 6.20 the additional gluing maps. Similarly, in Theorem 6.22, we identify the extra boundary components in M^{P} (O_{a}, O_{b})^{∗} due to the zeroes of the obstruction sections, and we introduce the additional gluing maps.

In Section 6.3, we finally complete the proof of the topological invariance. The modification of the boundary structure discussed in Section 6.2 prescribes correction terms for the maps I and H (but not J), so that the identities ID − DI = 0 and id−JI = DH+HD can still be satisfied. We first discuss, in Lemma 6.25 and Lemma 6.26, some more properties of the zeroes of the obstruction sections and some identities obtained by counting these zeroes, which are useful in checking the identities satisfied by the coefficients of the maps I and H. We then introduce, in Definition 6.27, the modified maps I and H, and we prove, in Lemma 6.28, Lemma 6.29, Lemma 6.30, and Lemma 6.31, that the necessary identities hold, thus completing the proof of Theorem 6.1 on the topological invariance of the equivariant Seiberg–Witten Floer homology. It is clear, from the way the proof is structured, that the argument breaks down for the non-equivariant Floer homology. While the invariance within the same chamber is still verified (nothing changes in that part of the argument), in the case of metrics and perturbations in different chambers we need essentially the contribution of the reducible points in order to construct the chain map I and chain homotopy H, as one can see form Definition 6.27, and the Lemmata 6.28, 6.29, 6.30, and 6.31.

Section 7 deals with the wall crossing formula for the Casson-type invariant of homology spheres obtained as the Euler characteristic of the (non-equivariant) Seiberg-Witten Floer homology. In Section 7.1 we derive relations between the equivariant and the non-equivariant Floer homologies. We prove in Theorem 7.1 that, when we have b^{1}(Y) > 0, the equivariant and the non-equivariant Floer homologies are isomorphic. This is not a surprising result: in fact, it is conceptually like considering the equivariant homology of a manifold with a free group action, for which it is well known that one recovers the homology of the quotient. In the case of rational homology spheres, we prove in Theorem 7.3 that there is an exact sequence relating the equivariant Floer homology with the non-equivariant and with the polynomial algebra H_{∗}(BU(1), ℝ). The results of both Theorem 7.1 and 7.3 are derived by considering filtrations of the complexes and the associated spectral sequences, as in Lemma 7.2 and Lemma 7.4. We also give an explicit expression of the connecting homomorphism in the exact sequence in Proposition 7.5. The exact sequence that relates the metric independent equivariant Floer homology with the metric dependent nonequivariant one is the tool we exploit in Section 7.2, in order to derive, in a purely algebraic way, the wall crossing formula for the Euler characteristic in Theorem 7.7. The argument is based on the topological invariance of the equivariant Floer homology proved in Theorem 6.1, and the result of Proposition 7.6 on the counting of the ranks of the equivariant Floer groups. The result is generalized to multiple wall crossings, and to the case of the J-invariant perturbations of [14], in Propositions 7.8, 7.9, and in Corollary 7.10. In Section 7.3 we show how the same wall crossing formula can be derived geometrically by considering the local model of the parameterized moduli space of critical points M(g, ν) along with a deformation of the metric and perturbation (g, ν).

Given the length of this work, we have thought it useful to add an index of notation at the end of the paper.

1.1 Three dimensional Seiberg-Witten theory

Any three–manifold admits a Spin-structure. A choice of the metric determines a (non-canonical) choice of a “trivial” Spin-structure with spinor bundle S. A Spin_{c}-structure is therefore obtained by twisting S with a line bundle L. Suppose given a three-manifold Y with a Spin_{c} structure S ⊗ L. Consider the space A_{k} of pairs (A, ψ), where A is a U(1) connection on the line bundle L and ψ is a section of S ⊗ L, endowed with a fixed L_{k}^{2}-Sobolev completion.

The group G̃_{k+1} is the gauge group of maps of Y in U(1) locally modeled on the Lie algebra

Lie(G̃_{k+1}) = L_{k+1}^{2}(Ω^{0}(Y, iℝ)),

acting on pairs (A, ψ) by

λ : (A, ψ) ↦ (A − 2λ^{−1}dλ, λψ).

The group G_{k+1} is the subgroup of G̃_{k+1} of gauge transformations λ satisfying the condition

\frac{i}{2π} ∫_{Y} c_{1}(L) ∧ λ^{−1}dλ = 0.  (2)

Here c_{1}(L) represents the Chern class of the line bundle L that defines the Spin_{c}- structure on Y. Clearly, the group G coincides with the full gauge group G̃ in the case when c_{1}(L) = 0 rationally, hence in particular for rational homology spheres.

The functional

C(A, ψ) = \frac{−1}{2} ∫_{Y} (A − A_{0}) ∧ (F_{A} + F_{A_{0}}) + \frac{1}{2} ∫_{Y} < ψ, ∂_{A}ψ > dv  (3)

was first introduced by Kronheimer and Mrowka in the proof of the Thom conjecture [30]. It is defined on the space A of connections and sections and it is invariant under the action of the identity component G_{k+1}. Thus, it descends to a real-valued functional on the space B_{k} = A_{k}/G_{k+1}.

The first order increment of this functional defines a 1-form on the L_{k}^{2}- tangent space T A,

F |_{(A,ψ,ρ)} (α, ϕ) = ∫_{Y} −α ∧ (F_{A} − ∗σ(ψ, ψ)) + ∫_{Y} < ϕ, ∂_{A}ψ > .  (4)

Thus, the gradient flow of the functional (3), with respect to the L^{2}-inner product, is given by the paths of connections and sections (A(t), ψ(t)) that satisfy the equations

\frac{d}{dt}ψ = −∂_{A}ψ,  (5)

and

\frac{d}{dt}A = − ∗ F_{A} + σ(ψ, ψ),  (6)

where the 1-form σ(ψ, ψ) is given in local coordinates by \frac{1}{2} < e_{i}ψ, ψ > e^{i}.

These equations can be thought of as the three-dimensional reduction of the Seiberg-Witten equations on four-manifolds introduced in [64] (see also [4], [5], [13], [38], [61], [62]). In fact, we can consider the four-manifold Y × ℝ with a cylindrical metric g + dt^{2}, and with Spin_{c} structure determined by the pullback of S ⊗ L via the projection π : Y × ℝ → Y. Thus we have S^{+} ⊗ L ≅ π^{∗}(S ⊗ L). Over Y we have S^{+} ⊗ L ≅ S^{−} ⊗ L under Clifford multiplication by dt, and the identifications Ω^{2+}(Y × ℝ, iℝ) ≅ π^{∗}(Ω^{1}(Y, iℝ)) and Ω^{1}(Y × ℝ, iℝ) ≅ π^{∗}(Ω^{0}(Y, iℝ) ⊕ Ω^{1}(Y, iℝ)), that is π^{∗}(ρ(t)) = ∗ρ(t) + ρ(t) ∧ dt and π^{∗}(f(t), α(t)) = α(t) + f(t)dt on Y × ℝ.

Consider a pair (𝔸, Ψ) on Y × ℝ, where 𝔸 is a U(1) connection on the determinant line bundle of S^{+} ⊗ L and Ψ is a section of the spinor bundle S^{+} ⊗ L. An element (𝔸, Ψ) is in a temporal gauge if the dt-component of 𝔸 vanishes identically. Thus, a path (A(t), ψ(t)) corresponds to an element (𝔸, Ψ) in a temporal gauge.

Lemma 1.1 The Seiberg-Witten equations (5) and (6) on Y ×ℝ are equivalent to

F_{𝔸}^{+} = Ψ · Ψ̄,  (7)

and

D_{𝔸}Ψ = 0,  (8)

where Ψ · Ψ̄ is the self-dual two-form given in local coordinates by Ψ · Ψ̄ = ⟨e_{i}e_{j}Ψ, Ψ⟩e^{i} ∧ e^{j} and D_{𝔸} is the Dirac operator twisted with the connection 𝔸 acting on sections of S^{+} ⊗ L.

\underline{Proof.} The Dirac operator D_{𝔸} : S^{+} ⊗ L → S^{−} ⊗ L on the four-manifold, twisted with the connection 𝔸, has the form

D_{𝔸} = ∂_{t} + ∂_{A(t)},

where ∂_{A} is the self-adjoint Dirac operator on Y twisted with the time dependent connection A(t). For the curvature equation (7), write F_{𝔸}^{+} = \frac{1}{2}(F_{𝔸} + ∗_{4}F_{𝔸}). Since F_{𝔸}^{−} acts trivially on the positive spinors, the action of F_{𝔸}^{+} corresponds precisely to the action of \frac{dA}{dt} + ∗_{3}F_{A}. Here we have introduced the notation ∗_{4} and ∗_{3} to distinguish the Hodge ∗-operator on the 4-manifold Y × ℝ and on the 3-manifold Y. In the following, we shall drop the subscript, since it will be clear which ∗-operator is being used. Thus, given the expression of the 2-form Ψ · Ψ̄ and of the one form σ(ψ(t), ψ(t)), we can write equation (7) as

\frac{1}{2}(F_{it} + ϵ^{itjk}F_{jk})e^{i} ∧ dt = \frac{1}{4} < e_{i}e_{t}ψ, ψ > e^{i} ∧ dt,

with an implicit sum over repeated indices and the symbol ϵ denoting the sign of the permutation {itjk}. Upon applying the ∗_{3} operator and using the identification of S^{+} ⊗ L and S^{−} ⊗ L on Y × ℝ under Clifford multiplication by dt, we can identify this equation with the equation (6). ⋄

The Seiberg–Witten equations on the 4-manifold Y ×ℝ are preserved by the action of the gauge group G̃_{Y ×ℝ} of maps of Y × ℝ in U(1). Any element (𝔸, Ψ) can be transformed into a temporal gauge by effect of a gauge transformation. Suitable Sobolev completions of these spaces will be introduced in Proposition 2.7. For a general overview of the Seiberg-Witten theory on four-manifold see [15], [39], [43], [54].

The critical points of the functional C are pairs (A, ψ) that satisfy

\begin{matrix}∂_{A}ψ = 0, \\ ∗ F_{A} = σ(ψ, ψ). \end{matrix}  (9)

Let M_{k} be the moduli space of solutions of (9) in B_{k} and M̃_{k} be the moduli space of solutions in B̃_{k} = A_{k}/G̃_{k+1}.

Lemma 1.2 By elliptic regularity, M_{k} can be represented by smooth elements.

Lemma 1.2 follows from the Sobolev embedding theorems. In the following we drop the subscript k and just write M for the moduli space of critical points.

The deformation complex that determines the virtual dimension of M is given by

0 → Ω^{0}(Y, iℝ) ⊕ Ω^{1}(Y, iℝ) ⊕ Γ(S ⊗ L) →^{L} Ω^{0}(Y, iℝ) ⊕ Ω^{1}(Y, iℝ) ⊕ Γ(S ⊗ L) → 0,

with an index zero Fredholm operator

L |_{(A,ψ)} (f, α, ϕ) = \{ \begin{matrix}T |_{(A,ψ)} (α, ϕ) + G |_{(A,ψ)} (f) \\ G^{∗} |_{(A,ψ)} (α, ϕ) \end{matrix}  (10)

defined between the L_{k}^{2} and the L_{k−1}^{2}-Sobolev completions of the spaces above, where the operator

G |_{(A,ψ)} (f) = (−df, fψ)  (11)

is the infinitesimal action of the gauge group and G^{∗} is the adjoint with respect to the L^{2} pairing. The map T is the linearization of the equation (9) at a pair (A, ψ),

T |_{(A,ψ)} (α, ϕ) = \{ \begin{matrix}∗dα − 2iImσ(ψ, ϕ) \\ ∂_{A}ϕ + αψ. \end{matrix}  (12)

The Hessian of the functional C is given by a quadratic form in the increment (α, ϕ) ∈ T A,

∇F |_{(A,ψ)} (α, ϕ) =< α, ∗dα − 2iImσ(ψ, ϕ) > + < ϕ, ∂_{A}ϕ + αψ > . (13)

This descends to the same operator T on the L_{k}^{2}-tangent space T_{[A,ψ]}B_{k}, when (A, ψ) is a solution of (9), since the condition

G_{(A,ψ)}^{∗}(T_{(A,ψ)}(α, ϕ)) = 0

is satisfied. The operator T is essentially self-adjoint.

As in the case of Donaldson theory [1], the linearization is a first order elliptic operator, hence its spectrum is not in general bounded from below, and this affects the definition of the index of critical points, as we are going to discuss in Proposition 2.10.

2 Non-equivariant Floer theory

There are two disjoint situations in which the non-equivariant Seiberg-Witten Floer homology can be defined. They have been considered in [13], [38], [62] and are summarized in this section.

We consider the functional C perturbed with a co-closed 1-form ρ,

C_{ρ}(A, ψ) = C(A, ψ) − 2i ∫_{Y} (A − A_{0}) ∧ ∗ρ.  (14)

We have the corresponding perturbed critical point equations

\begin{matrix}∂_{A}ψ = 0, \\ ∗ F_{A} = σ(ψ, ψ) + 2iρ. \end{matrix}  (15)

Notice that the first integral cohomology group of Y counts the homotopy classes of gauge transformations, namely H^{1}(Y, ℤ) ≅ π_{0}(G̃) under the identification

λ → h = [\frac{i}{2π} λ^{−1}dλ] ∈ H^{1}(Y, ℤ).

We denote with H the subgroup of the classes h ∈ H^{1}(Y, ℤ) that satisfy

⟨c_{1}(L) ∪ h, [Y]⟩ = \frac{i}{2π} ∫_{Y} c_{1}(L) ∧ λ^{−1}dλ = 0.

We have the following compactness result for the set of critical points of the Seiberg-Witten functional.

Proposition 2.1 The moduli space M̃_{ρ} of solutions of (15) modulo the action of the full gauge group G̃ is compact. The space M_{ρ} of solutions, modulo the subgroup G ⊂ G̃ of gauge transformations satisfying (2), consists of a copy of M̃_{ρ} for each class in H^{1}(Y, ℤ)/H. M_{ρ} is compact iff H^{1}(Y, ℤ)/H is finite. In particular, M_{ρ} is compact if b^{1}(Y) = 0.

Notice that the condition that H^{1}(Y, ℤ)/H is finite corresponds to c_{1}(L) = 0 rationally.

Proposition 2.2 If the manifold Y has b^{1}(Y) > 0, then, for an open set of small perturbations, the perturbed equations (15) do not admit reducible solutions. Moreover, by the Sard-Smale theorem, for a generic choice of ρ the corresponding moduli space M̃_{ρ} is a smooth manifold that is cut out transversely by the equations. M̃_{ρ} is compact and zero-dimensional, hence it consists of a finite set of points.

All the moduli spaces come with a natural orientation defined by the determinant line bundle of the Fredholm linearization, as in [17].

When Y is a homology sphere the perturbation ρ can be written as ρ = ∗dν and the equations (15) admit one gauge class of reducible solutions [ν, 0].

Proposition 2.3 If Y is a homology sphere, then there is a unique gauge class of reducible solutions θ = [ν, 0] of (15) with ρ = ∗dν. If the metric on Y is such that Ker(∂_{ν}^{g}) = 0, where ∂_{ν}^{g} is the self-adjoint Dirac operator on Y twisted with the U(1)-connection ν, then the analysis of the local Kuranishi model shows that the reducible point is isolated and non-degenerate.

Propositions 2.1, 2.2, and 2.3 are proven in [62], [38], and [13] respectively. See also the more recent [35].

2.1 Chamber structure

In this subsection we prove that Ker(∂_{ν}^{g}) = 0 is satisfied for generic metrics. The condition Ker(∂_{ν}^{g}) ≠ 0 defines a chamber structure in the space of metrics. The walls form a stratified space with a codimension one top stratum and higher codimensional strata, in the sense described in Theorem 2.6.

Notice that there is a more abstract approach [28] describing a stratification of the space of Fredholm operators by their index and kernel dimension. However, the result we are interested in does not follow directly by simply applying the results of [28]. In fact, we know from the general result of Section 1 of [28] that the space F_{0}(S, Y) of index zero Fredholm operators, acting on the bundle S (or S ⊗ L) over Y, has the structure of a real analytic manifold, with a stratification given by the sets F_{0,n}(S, Y) of Fredholm operators of index zero with the dimension of the kernel equal to n. In our specific problem, however, we are considering the particular map from the space Met×Z^{1}(Y, ℝ) of metrics and perturbations (g, ∗dν) to the space F_{0}(S, Y) given by

\begin{matrix}X : Met × Z^{1}(Y, ℝ) → F_{0}(S, Y) \\ (g, ∗dν) ↦ ∂_{ν}^{g} =: X (g, ν). \end{matrix}  (16)

Thus, we need a specific result that shows how the image of the map Ξ lies in F_{0}(S, Y) with respect to the stratification of [28]. This is the purpose of our Theorem 2.5 and Theorem 2.6. Although it is quite possible that a “proof by library search” of these results may be provided combining the circle of ideas in [28], [37], and the more recent [21], we prefer to give a direct proof that covers our specific case. A note should be added: during the long period between the initial submission of this paper and the completion of its refereeing process, a discussion of the chamber structure has also appeared in [35] and [53].

Remark 2.4 Suppose Y is a homology sphere. There is a quaternion structure on the spinor bundle S, J : S → S̄ locally given by (z_{1}, z_{2}) ↦ (−z̄_{2}, z̄_{1}). The action of J extends to the configuration space A as (A, ψ) ↦ (A^{∗}, Jψ). This means that J acts on ν by Jν = −ν, hence

J(∂_{ν}^{g}ψ) = ∂_{−ν}^{g}Jψ.

In other words, the Dirac operator ∂^{g} on a three-manifold Y is quaternion linear, but the twisted Dirac operator ∂_{ν}^{g} is only complex linear.

A J-invariant perturbation of the Seiberg-Witten equations has been constructed by W. Chen [14], where the Dirac operator is perturbed with a smooth real function f on Y. This choice of perturbation leads to a different chamber structure in the space of metrics which lies inside one chamber of the more general perturbation

∂_{A}ψ = fψ,
∗F_{A} = σ(ψ, ψ) + ∗dν.

In this setting, transversality can be achieved by adding a perturbation by a function of the holonomy. This choice of perturbations lead to a different wall crossing formula. However, we are going to show in Section 7, Proposition 7.9 and Corollary 7.10, that the wall crossing formula for the J-invariant perturbations can be derived with the same technique that we employ in the case on perturbations ν.

The following theorems discuss the chamber structure in the space of metrics and perturbations.

Theorem 2.5 Let Met be the space of all Riemannian metrics on a homology 3-sphere Y. Consider the twisted Dirac operator ∂_{ν}^{g} associated with the chosen metric g and the connection ν. The condition Ker∂_{ν}^{g} ≠ 0 determines a real codimension one subset in the space of Met × Z^{1}(Y, iℝ).

\underline{Proof.} Let g_{0} be a metric on Y such that Ker∂_{ν_{0}}^{g_{0}} ≠ 0 for the connection ν_{0}. We can decompose the spinor space as H ⊕ H^{⊥}, under the L^{2} inner product, where H = Ker∂_{ν_{0}}^{g_{0}}, equipped with a Hermitian metric from the Spin_{c} structure. Consider the Dirac operator ∂_{ν}^{g} for (g, ν) sufficiently close to (g_{0}, ν_{0}) in the C^{∞}- topology. Under the isometry identification of the spinor spaces for g_{0} and g, the Dirac operator for g can be considered to act on the spinor space of g_{0}, still denoted by ∂_{ν}^{g}. \underline{Claim 1 :} ∂_{ν}^{g} acting on the spinor space S_{g_{0}} is self-adjoint if and only if the metrics g_{0} and g define the same volume element.

Suppose dvol_{g} = f dvol_{g_{0}} for a positive function f on Y, then with a direct calculation we get

< ∂_{ν}^{g}ψ, ϕ >_{g_{0}}=< ψ, f∂_{ν}^{g}(f^{−1}ϕ) >_{g_{0}} .

Claim 1 is then immediate. Denote by Met^{0} the space of metrics which have the same volume element as the metric g_{0}. \underline{Claim 2 :} If two metrics g_{1} and g_{2} are conformal, that is, g_{1} = e^{2u}g_{2} for a real function on Y, then the multiplication by e^{−u} defines an isomorphism between Ker∂_{ν}^{g_{1}} and Ker∂_{ν}^{g_{2}}.

This is the consequence of the following relation: under the isometry identification of the spinor spaces for g_{1} and g_{2}, we have (see [26] or [33] Theorem II.5.24)

∂_{ν}^{g_{1}} = e^{−u}∂_{ν}^{g_{2}}e^{u}.

Notice that, in the formula for the variation of the Dirac operator under conformal changes in the metric, different conventions are used in the literature. Here we are following the convention and notation of [26].

Therefore, we only need to prove that the condition Ker∂_{ν}^{g} ≠ 0 determines a real codimension one subset in the space of Met^{0} × Z^{1}(Y, iℝ).

We want to reduce the problem of the existence of solutions of the equation ∂_{ν}^{g}ψ = 0 to a finite dimensional problem on H.

As a map from Met^{0} × Z^{1}(Y, iℝ) × Γ(S) to Γ(S), the linearization of the equation ∂_{ν}^{g}ψ = 0 at (g_{0}, ν_{0}, 0) is invertible when restricted to H^{⊥}. Thus, the implicit function theorem provides a unique map q : U → H^{⊥} defined on a neighborhood U of (g_{0}, ν_{0}, 0) in Met^{0} × Z^{1}(Y, iℝ) × H, such that

(1 − Π)∂_{ν}^{g}(ϕ + q(g, ν, ϕ)) = 0

for all (g, ν, ϕ) ∈ U, where Π is the projection onto H.

Therefore, the operator ∂_{ν}^{g} has non-trivial kernel if and only if the equation

Π∂_{ν}^{g}(ϕ + q(g, ν, ϕ)) = 0

admits solutions in H. This is a finite dimensional problem. Define a map

L : (Met^{0} × Z^{1}(Y, iℝ)) ∩ U ⟶ U(H)

by setting

L(g, ν)(ϕ) = Π∂_{ν}^{g}(ϕ + q(g, ν, ϕ))

Direct calculation implies that L(g, ν) is a Hermitian transformation of the space H, that L(g, ν) ∈ U(H), where U(H) is the Hermitian transformation group on H. The kernel of ∂_{ν}^{g} is non-trivial if and only if the kernel of L(g, ν) is non-trivial. The determinant is a real-valued function on U(H). Thus, we have a real-valued function f(g, ν) = det(L(g, ν)) on the neighborhood of (g_{0}, ν_{0}) in Met^{0} × Z^{1}(Y, iℝ). Those (g, ν) with non-trivial kernel have value 0 for this function.

Now we only need to check that the derivative of f(g, ν) at (g_{0}, ν_{0}) is surjective, then the condition Ker∂_{ν}^{g} ≠ 0 determines a real codimension one subset in the space of Met^{0} × Z^{1}(Y, iℝ) by Morse theory. It can be checked by differentiating f(g, ν) at (g_{0}, ν_{0}) along (0, α)-direction, for α ∈ Ω^{1}(Y, iℝ). Since

Df_{(g_{0},ν_{0})}(0, α) = T r(ϕ ↦ Π(\frac{1}{2}α.ϕ)),

which is non-zero for suitable choice of α.

Theorem 2.5 now follows from Claim 2 and the fact that any metric is conformally equivalent to a metric in Met^{0}. ⋄

Let W denote the wall in the space of metrics and perturbations Met × Z^{1}(Y, iℝ),

W = {(g, ∗dν)| Ker∂_{ν}^{g} ≠ 0}.

Motivated by Chen’s work [14], we analyze more carefully the structure of W. This analysis will be useful in Section 7, in establishing the results on the wall crossing formulae.

Theorem 2.6 W is a stratified space with the top stratum W_{1} consisting of those (g, ν) with Ker∂_{ν}^{g} ≅ ℂ. In general, the set of pairs (g, ν) with Ker∂_{ν}^{g} ≅ ℂ^{n} is a codimension 2n − 1 subset W_{n} in Met × Z^{1}(Y, iℝ).

\underline{Proof:} As in the proof of Theorem 2.5, we only need to prove the result for (g, ν) in Met^{0} × Z^{1}(Y, iℝ) (see Claim 1 and Claim 2 in the proof of Theorem 2.5). Consider a real Hilbert bundle L over

Met^{0} × Z^{1}(Y, iℝ) × (L_{1}^{2}(S) − {0})

whose fiber over (g, ν, ψ) is

L_{(g,ν,ψ)} = {ϕ ∈ L_{1}^{2}(S)|Re⟨ϕ, iψ⟩_{g} = 0}

Define a section ζ of L by assigning to (g, ν, ψ) the element ∂_{ν}^{g}ψ. \underline{Claim:} ζ is transverse to the zero section of L.

We need to prove that the differential map of ζ is surjective at zeroes of ζ. Suppose that (g_{0}, ν_{0}, ψ_{0}) (with ψ ≠ 0) satisfies ∂_{ν_{0}}^{g_{0}}ψ_{0} = 0. Differentiating ζ with respect to the directions tangent to Z^{1}(Y, iℝ) × (L_{1}^{2}(S) − {0}) only, we see that the differential map is

\begin{matrix}Dζ : Ω_{L_{2}^{2}}^{1}(Y, iℝ) ⊕ L_{1}^{2}(S) & ⟶ \{ϕ ∈ L_{1}^{2}(S)|Re⟨ϕ, iψ⟩_{g} = 0\} \\ (ν_{1}, ψ_{1}) & ↦ ∂_{ν_{0}}^{g_{0}}ψ_{1} + \frac{1}{2}ν_{1}.ψ_{0}. \end{matrix}

If ϕ ∈ L_{(g_{0},ν_{0},ψ_{0})} is orthogonal to the image of Dζ, then ϕ satisfies:

\{ \begin{matrix}(1). Re⟨ϕ, iψ_{0}⟩_{g_{0}} = 0, \\ (2). Re⟨ϕ, ν_{1}.ψ_{0}⟩_{g_{0}} = 0,  for any ν_{1} ∈ Ω^{1}(Y, iℝ). \\ (3). ∂_{ν_{0}}^{g_{0}}ϕ = 0. \end{matrix}

From the second equation we see that there exists a function f : Y → ℝ such that ϕ = ifψ_{0}. Substitute this into equation (3), using ∂_{ν_{0}}^{g_{0}}ψ_{0} = 0. We obtain df = 0 which implies f = C is a constant function. Then

Re⟨iCψ_{0}, iψ_{0}⟩_{g_{0}} = C|ψ_{0}|^{2}

implies C = 0. Therefore, ϕ = 0, that means, Dζ is surjective at (g_{0}, ν_{0}, ψ_{0}). It is easy to see that the index of Dζ is the index of ∂_{ν_{0}}^{g_{0}}, which is 1, since iψ_{0} is orthogonal to the image of ∂_{ν_{0}}^{g_{0}}.

From the Claim, ζ^{−1}(0) is a Banach manifold and the projection

Π : ζ^{−1}(0) → Met^{0} × Ω^{1}(Y, iℝ)

is a Fredholm operator of index 1. Note that for any (g, ν) ∈ Met^{0} × Ω^{1}(Y, iℝ) we have Π^{−1}(g, ν) = Ker∂_{ν}^{g}.

Moreover, at (g_{0}, ν_{0}, ψ_{0}) we have

\{ \begin{matrix}Ker(Π_{∗}) = \{ϕ|∂_{ν_{0}}^{g_{0}}ϕ = 0\} \\ dimKer(Π_{∗}) − dimCoker(Π_{∗}) = 1. \end{matrix}

Therefore, dimCoker(Π_{∗}) = dimKer(Π_{∗}) − 1. Then the Theorem follows, with the top stratum of codimension one described in Theorem 2.5, and the stratification given by

W_{n} = X^{−1}(F_{0,2n}(S, Y) ∩ X (Met × Z^{1}(Y, ℝ))),

where X is the map of (16). Thus, the structure of stratified set on W is induced by the structure on Image(X) inside the stratified set F_{0}(S, Y). ⋄

Notice that on any three-manifold it is possible to find special metrics for which the dimension of the space of harmonic spinors is arbitrarily large. The result for S^{3} was proved by Hitchin [26] and recently generalized to all manifolds of dimension 3 mod 4 by Bär [9].

2.2 Perturbation of flow lines

We introduce suitable moduli spaces of gradient flow lines connecting critical points. We prove in the Section 2.3, Proposition 2.12 that, generically, these are smooth manifolds that are cut out transversely, hence with the dimension prescribed by the index theorem. This property depends on an accurate choice of a class of perturbations for the gradient flow equations.

Consider the space of connections and spinor sections (𝔸, Ψ) on Y × ℝ with the product metric g + dt^{2}, topologized with the weighted Sobolev norms [24] [36]. Here we choose the weight e_{δ}(t) = e^{δ̃t}, where δ̃ is a smooth function with bounded derivatives, δ̃ : ℝ → [−δ, δ] for some fixed positive number δ, such that δ̃(t) ≡ −δ for t ≤ −1 and δ̃(t) ≡ δ for t ≥ 1. The L_{k,δ}^{2} norm is defined as ∥f∥_{2,k,δ} = ∥e_{δ}f∥_{2,k}, where the Sobolev norms are defined with respect to a fixed reference connection 𝔸_{0} in L_{k,loc}^{2}. By the Sobolev multiplication theorem of Proposition 2.7 below, the spaces L_{k,δ,𝔸_{0}}^{2} and L_{k,δ,𝔸_{1}}^{2} are equal whenever we have 𝔸_{0} − 𝔸_{1} = α_{0} + α_{1}, with α_{0} in L_{ℓ,δ,𝔸_{0}}^{2} and α_{1} is C^{ℓ} bounded, with ℓ ≥ k − 1, cf. [44]. The weight e_{δ} imposes an exponential decay as asymptotic condition along the cylinder. A proof of the following Proposition can be found, for instance, in Section 9 of [20].

Proposition 2.7 Let Y be a compact oriented three-manifold endowed with a fixed Riemannian metric g_{0}. Consider the cylinder Y ×ℝ with the metric g_{0}+dt^{2}. The weighted Sobolev spaces L_{k,δ}^{2} on the manifold Y × ℝ satisfy the following Sobolev embeddings.

(i) The embedding L_{k,δ}^{2} ↪ L_{k−1,δ}^{2} is compact for all k ≥ 1.

(ii) If k > m + 2 we have a continuous embedding L_{k,δ}^{2} ↪ C^{m}.

(iii) If k > m + 3 the embedding L_{k,δ}^{2} ↪ C^{m} is compact.

(iv) If 2 < k′ and k ≤ k′ the multiplication map L_{k,δ}^{2} ⊗ L_{k,δ}^{2} →^{m} L_{k,2δ}^{2} is continuous. Consider a metric g_{t} + dt^{2} on the cylinder Y × ℝ such that for a fixed T we have g_{t} ≡ g_{0} for t ≥ T and g_{t} ≡ g_{1} for t ≤ −T and g_{t} varies smoothly when t ∈ [−1, 1]. The same Sobolev embedding theorems hold for the L_{k,δ}^{2} spaces on (Y × ℝ, g_{t} + dt^{2}).

Choose smooth representatives (A_{0}, ψ_{0}) and (A_{1}, ψ_{1}) of a and b in M. Choose a smooth path (A(t), ψ(t)) such that for t ≤ 0 it satisfies (A(t), ψ(t)) ≡ (A_{0}, ψ_{0}) and for t ≥ 1 it is (A(t), ψ(t)) ≡ (A_{1}, ψ_{1}). The configuration space A_{k,δ}(a, b) is given by the space of pairs (𝔸, Ψ) on Y × ℝ satisfying

(𝔸, Ψ) ∈ (A(t), ψ(t)) + L_{k,δ}^{2}(Ω^{1}(Y × ℝ) ⊕ Γ(S^{+} ⊗ L)).  (17)

Consider the group G_{k+1,δ}(a, b) of gauge transformations in G̃_{Y ×ℝ}, locally modeled on L_{k+1,δ}^{2}(Ω^{0}(Y × ℝ, iℝ)), that approach elements λ_{±∞} in the stabilizers G_{a} and G_{b} of (A_{0}, ψ_{0}) and (A_{1}, ψ_{1}) as t → ±∞. This gauge group acts on A_{k,δ}(a, b) and we can form the quotient B_{k,δ}(a, b). There is an action of ℝ by translations on B_{k,δ}(a, b).

We consider the perturbed gradient flow equations for a path (A(t), ψ(t)),

\frac{d}{dt}ψ(t) = −∂_{A(t)}ψ(t)  (18)

and

\frac{d}{dt}A(t) = σ(ψ(t), ψ(t)) − ∗F_{A(t)} + 2iρ + 2q_{(𝔸,Ψ)}(t).  (19)

Equations (18) and (19) can be rewritten in terms of pairs (𝔸, Ψ) in the form

D_{𝔸}Ψ = 0  (20)

and

F_{𝔸}^{+} = Ψ · Ψ̄ + iμ + P_{(𝔸,Ψ)},  (21)

as proved in Lemma 1.1.

The perturbation P = ∗q + q ∧ dt is a function of B(a, b) to π^{∗}(Ω^{1}(Y, iℝ)) ≅ Ω^{2+}(Y ×ℝ, iℝ), such that the corresponding equations in a temporal gauge (18) and (19) are preserved under the action of ℝ by reparameterization of the path (A(t), ψ(t)). The class of such perturbations is described as follows.

Definition 2.8 The space of perturbations P is the space of maps

P : B_{k,δ}(a, b) → L_{k,δ}^{2}Ω^{2+}(Y × ℝ, iℝ),

that satisfy the following conditions.

(1) P_{(𝔸,Ψ)} = ∗q_{(𝔸,Ψ)}(t) + q_{(𝔸,Ψ)}(t) ∧ dt, where q_{(𝔸,Ψ)}(t) satisfies

q_{(𝔸,Ψ)^{T}} (t) = q_{(𝔸,Ψ)}(t + T),

for any T ∈ ℝ, where (𝔸, Ψ)^{T} is the T -translate of (𝔸, Ψ).

(2) the L_{k,δ}^{2}-norm of the perturbation P_{(𝔸,Ψ)} is bounded uniformly with respect to (𝔸, Ψ);

(3) the linearization DP_{(𝔸,Ψ)} is a pseudodifferential operator of order < 1. In particular DP_{(𝔸,Ψ)} is a compact operator from the L_{k,δ}^{2} to the L_{k−1,δ}^{2} tangent spaces.

(4) for all l ≤ k − 1, we have

∥q_{(𝔸,Ψ)}(t)∥_{L_{l}^{2}} ≤ C_{l}∥∇C_{ρ}(A(t), ψ(t))∥_{L_{l}^{2}}

in the L_{l}^{2}-norm on Y × {t}, where

∇C_{ρ}(A(t), ψ(t)) = (−∂_{A(t)}ψ(t), σ(ψ(t), ψ(t)) − ∗F_{A(t)} + 2iρ)  (22)

is the gradient flow of the functional C_{ρ}, and 0 < C_{l} < 1.

(5) The inequality

∥Dq_{(𝔸,Ψ)}(α(t), ϕ(t))∥ ≤ C_{(𝔸,Ψ)}∥∇C_{ρ}(A(t), ψ(t))∥ · ∥(α(t), ψ(t))∥

holds for t ≥ T_{0}. Here Dq_{(𝔸,Ψ)} is the linearization of the perturbation q_{(𝔸,Ψ)}.

With a perturbation in the class P the equations (18) and (19) are invariant with respect to the action of ℝ by translations along the gradient flow lines, that is if (A(t), ψ(t)) is a solution of (18) and (19), then (A(t + T), ψ(t + T)) is also a solution for any T ∈ ℝ.

An example of perturbation with these properties has been constructed by Froyshov [25].

Proposition 2.9 The class of perturbations introduced by Froyshov in [25] is contained in our class P.

According to Froyshov’s construction, for fixed smooth compactly supported functions η_{1}, η_{2}, with supp(η_{1}) ⊂ [−1, 1] and η_{2}|_{I}(t) = t on an interval I containing all the critical values of C_{ρ}, a function h : B_{k,δ}(a, b) → C^{m}(ℝ) is defined as

h_{(𝔸,Ψ)}(T) = ∫_{ℝ} η_{1}(s − T)η_{2}(∫_{ℝ} η_{1}(t − s)C_{ρ}(A(t), ψ(t))dt)ds.

Let Ω_{Ξ}^{2}(Y × ℝ) be the set of C^{m} 2-forms ω that are compactly supported in Y × Ξ, where Ξ is the complement of a union of small intervals centered at the critical values of C_{ρ}. Froyshov’s perturbation is constructed by setting

P_{(𝔸,Ψ)} = (h_{(𝔸,Ψ)}^{∗}(ω))^{+},

where h_{(𝔸,Ψ)}^{∗}(ω) is the pullback of ω along the map Id_{Y} ×h_{(𝔸,Ψ)} : Y ×ℝ → Y ×ℝ.

As shown in [25], the function h_{(𝔸,Ψ)} is bounded with all derivatives, uniformly with respect to (𝔸, Ψ). Moreover, by the choice of Ξ, the perturbation h_{(𝔸,Ψ)}^{∗}(ω) is smooth and compactly supported, hence in L_{k,δ}^{2}.

Condition (1) holds, since the function h_{(𝔸,Ψ)}(t) satisfies

h_{(𝔸,Ψ)}(t + τ) = h_{(𝔸,Ψ)^{τ}} (t),

where (𝔸, Ψ)^{τ} is the τ-reparameterized solution represented in a temporal gauge by (A(t + τ), ψ(t + τ)). In fact,

h_{(𝔸,Ψ)^{τ}} (T) = ∫_{ℝ} η_{1}(s − T)η_{2}(∫_{ℝ} η_{1}(t − s)C_{ρ}(A(t + τ), ψ(t + τ))dt)ds =

= ∫_{ℝ} η_{1}(s − T)η_{2}(∫_{ℝ} η_{1}(u − s − τ)C_{ρ}(A(u), ψ(u))du)ds =

= ∫_{ℝ} η_{1}(v − T − τ)η_{2}(∫_{ℝ} η_{1}(u − v)C_{ρ}(A(u), ψ(u))du)dv = h_{(𝔸,Ψ)}(T + τ).

Condition (2) holds: in fact, it is shown in [25] that the function h_{(𝔸,Ψ)} is bounded with all derivatives, uniformly with respect to (𝔸, Ψ). The Sobolev embeddings of Proposition 2.7 provide the uniform bound in the L_{k,δ}^{2}-norms.

Condition (3) and (5): we can write the function h_{(𝔸,Ψ)} with the notation

h_{(𝔸,Ψ)} = η_{1} ∗ (η_{2}(η_{1} ∗ C_{ρ}(A, ψ))),

where ∗ denotes the convolution product on ℝ. We obtain the variation with respect to (𝔸, Ψ) of the form

v_{(𝔸,Ψ)}(α, ϕ) = η_{1} ∗ (η′_{2}(η_{1} ∗ C_{ρ}(A, ψ))η_{1} ∗ ⟨∇C_{ρ}(A, ψ), (α, ϕ)⟩) .

Thus, as shown in Froyshov ([25], Prop.5), for ω a C^{m} form, the linearization of the perturbation h_{(𝔸,Ψ)}^{∗}(ω) at the point (ω, 𝔸, Ψ) is a bounded operator K_{(ω,𝔸,Ψ)} : L_{k,δ}^{2} → C^{m} with

supp (K_{(ω,𝔸,Ψ)}(α, Φ)) ⊂ h_{(𝔸,Ψ)}^{−1}(Ξ) × Y.

Thus Condition (3) follows, since η_{1} and η_{2} are smooth compactly supported functions. In particular, if ω is a smooth form, the linearization DP_{(𝔸,Ψ)} is a smoothing operator. The expression of the variation v_{(𝔸,Ψ)} also shows that Condition (5) holds.

Condition (4) follows from an estimate on ∥dh_{(𝔸,Ψ)}∥,

∥dh_{(𝔸,Ψ)}(t)∥ ≤ C∥∇C_{ρ}(A(t), ψ(t))∥,

where the constant C only depends on the asymptotic values a and b of (𝔸, Ψ).

Other perturbations of the functional C can be used to achieve transversality of the moduli space of flow lines. For instance see the discussion in [29], and the class of perturbations introduced in [11]. The perturbations introduced in [11] have the advantage of being defined directly as perturbations of the Chern– Simons–Dirac functional C, instead of being perturbations of the 4-dimensional Seiberg–Witten equations, as in the case of the class considered here.

2.3 Transversality of M(a, b)

Let L_{(𝔸,Ψ)} be the linearization of equations (20) and (21) on B_{k,δ}(a, b). The operator L is of the form

L_{(𝔸,Ψ,P)}(α, Φ) = \{ \begin{matrix}D_{𝔸}Φ + αΨ \\ d^{+}α − \frac{1}{2}Im(Ψ · Φ̄) + DP_{(𝔸,Ψ)}(α, Φ) \\ G_{(𝔸,Ψ)}^{∗}(α, Φ) \end{matrix}  (23)

mapping

L_{k,δ}^{2}(Ω^{1}(Y × ℝ, iℝ) ⊕ Γ(S^{+} ⊗ L)) → L_{k−1,δ}^{2}(Ω^{0}(Y × ℝ, iℝ) ⊕ Ω^{2+}(Y × ℝ, iℝ)).

The operator G^{∗} is the adjoint, in the L_{0,δ}^{2}-pairing, of the linearization of the gauge group action G_{(𝔸,Ψ)}(f) = (−df, fΨ).

As the following proposition shows, the operator L_{(𝔸,Ψ,P)} is obtained by adding the compact perturbation DP_{(𝔸,Ψ)} to a Fredholm map from L_{k,δ}^{2} to L_{k−1,δ}^{2}, hence it is Fredholm. Therefore we have a well defined relative Morse index of two critical points a and b in M.

Proposition 2.10 Suppose a and b are irreducible critical points for the functional C_{ρ}. Let {λ_{a}} and {λ_{b}} be the eigenvalues of the Hessian T at the points a and b. Assume that the positive number δ satisfies the condition

δ < min{|λ_{a}|, |λ_{b}|}.

Let (𝔸, Ψ) be a solution of (20) and (21) in B_{k,δ}(a, b). Then the linearization L_{(𝔸,Ψ)} is a Fredholm operator of index

Ind(L_{(𝔸,Ψ)}) = σ(a, b).

The right hand side σ(a, b) is the spectral flow of the operator ∇F along a path (A(t), ψ(t)) in A that corresponds to (𝔸, Ψ) under π^{∗}. The quantity σ(a, b) is independent of the path, hence

σ(a, b) = μ(a) − μ(b)

defines a relative Morse index of a and b, where μ(a) is the spectral flow of ∇F on a path joining a to a fixed [A_{0}, ψ_{0}] in M.

Remark 2.11 In the case with Y a homology sphere, Proposition 2.10 holds for a reducible point a = [ν, 0] under the assumption that, for the chosen metric g on Y, the condition Ker(∂_{ν}^{g}) = 0 is satisfied.

We state and prove an analogue of Proposition 2.10 in the context of framed moduli spaces and equivariant theory in Theorem 3.8. We also prove, in Theorem 3.8, that the relative Morse index of points in M̃ is well defined.

Consider the moduli space M(a, b) of solutions of the equations (20) and (21) in B_{k,δ}(a, b).

Proposition 2.12 When a and b are irreducible critical points of C_{ρ}, for a generic choice of the perturbation P ∈ P, the moduli space M(a, b) of gradient flow lines is a smooth oriented manifold, cut out transversely by the equations, of dimension

dim(M(a, b)) = μ(a) − μ(b),

where μ(a) − μ(b) is the relative Morse index of the critical points.

\underline{Proof:} It is first necessary to know that there are no reducible flow lines connecting the critical points a and b. This fact is an easy consequence of the definition of the configuration space (17), since the exponential weight in the Sobolev norm forces elements in (17) to decay at the ends to the asymptotic values, which are irreducible by assumption. On the convergence of flow lines to the endpoints a and b, see the results of Section 3.4. Thus, M(a, b) lies entirely in the irreducible component B′_{k,δ}(a, b), provided that at least one of the endpoints a and b is irreducible. The statement then follows via the implicit function theorem, upon showing that, for a generic choice of the perturbation P, the linearization L is surjective.

Consider the operator

L̂_{(𝔸,Ψ,P)}(α, Φ, p) = L_{(𝔸,Ψ,P)}(α, Φ) + p_{(𝔸,Ψ,P)}(α, Φ),

where we vary the perturbation by an element p_{(𝔸,Ψ,P)} of T_{P} P. This corresponds to varying the parameter ω ∈ Ω_{Ξ}^{2}(Y × ℝ) in Froyshov’s class of perturbations.

The operator L is Fredholm, therefore L̂ has a closed range. We show that L̂ is surjective by proving that it has dense range.

Suppose given an element (β, ξ, g) in L_{−k−1,−δ}^{2} that is L^{2}-orthogonal to the range of the operator L̂. Here β is an element in Ω^{2+}(Y × ℝ, iℝ)), ξ is a spinor, and g is a zero-form. The element (β, ξ, g) is in the kernel of the adjoint L̃^{∗}, which is an elliptic operator with L_{−k,−δ}^{2} coefficients, thus (β, ξ) lives in L_{−k,−δ}^{2} by elliptic regularity. In fact, the perturbation satisfies Condition (3) of Definition 2.8. If we consider the L^{2}-pairing of L_{k,δ}^{2} and L_{−k,−δ}^{2}, we get

⟨β, d^{+}α − \frac{1}{2}Im(Ψ · Φ̄)) + DP_{(𝔸,Ψ)}(α, Φ) + p_{(𝔸,Ψ,P)}(α, Φ)⟩+

+⟨ξ, D_{𝔸}Φ + αΨ⟩ + ⟨g, G_{(𝔸,Ψ)}^{∗}(α, Φ)⟩ = 0.

By varying p ∈ P we force β ≡ 0. The remaining inner product

⟨ξ, D_{𝔸}Φ + iαΨ⟩ + ⟨g, G_{(𝔸,Ψ)}^{∗}(α, Φ)⟩ = 0

gives the following equations (a) (e_{−δ}de_{δ})g = \frac{1}{2}ξ · Ψ̄ and

(b) D_{A}ξ − gΨ = 0. We assume that Ψ is not identically zero. Applying d^{∗} to (a) and using (b) we obtain d^{∗}(e_{−δ}de_{δ}g) + g|Ψ|^{2} = 0. Equivalently, we get

(e_{δ/2}d^{∗}e_{−δ/2})(e_{−δ/2}de_{δ/2})e_{δ/2}g + |Ψ|^{2}e_{δ/2}g = 0.

The equation

Δ_{δ/2}e_{δ/2}g + e_{δ/2}g|Ψ|^{2} = 0,

with

Δ_{δ} = e_{−δ}Δe_{δ},

implies that g ≡ 0, since g decays at ±∞ and the maximum principle applies. Then, by varying α alone in ⟨ξ, D_{𝔸}Φ + αΨ⟩ = 0, we force ξ to vanish on some arbitrary open set. We obtain (β, ξ) ≡ 0.

Thus the operator L̃ is surjective. This implies that zero is a regular value for the map defined by the equations (20) and (21). Therefore the moduli space Mod of triples ([𝔸, Ψ], P) in B′_{L_{k,δ}^{2}}(a, b)⊕P that satisfy the equations is a smooth

(infinite dimensional) manifold with tangent space Ker(L̂).

The projection Π : Mod → P given by Π([𝔸, Ψ], P) = P linearizes to a surjective Fredholm operator DΠ : Ker(L̃) → T_{P} P. The kernel of DΠ is Ker(DΠ_{(𝔸,Ψ,P)}) = Ker(L_{(𝔸,Ψ,P)}). The infinite dimensional Sard theorem implies that the moduli space M(a, b), for a generic perturbation P ∈ P, is the inverse image under the projection map from Mod to P of a regular value. Thus M(a, b) is a smooth manifold which is cut out transversely by the equations. Equivalently, the linearization L with a fixed generic q is surjective.

The virtual dimension of the moduli space M(a, b) equals the index of the Fredholm operator L. According to Proposition 2.10, this is the relative Morse index μ(a) − μ(b).

The orientation of M(a, b) is given by a trivialization of the determinant line bundle of the operator L. This is obtained given a choice of an orientation of

H_{δ}^{0}(Y × ℝ) ⊕ H_{δ}^{2+}(Y × ℝ) ⊕ H_{δ}^{1}(Y × ℝ),

the cohomology groups of δ-decaying forms, as discussed in the following Proposition. ⋄

Proposition 2.13 The manifold M(a, b) is oriented by a trivialization of the determinant line bundle of the operator L. This is obtained from an orientation of

H_{δ}^{0}(Y × ℝ) ⊕ H_{δ}^{2+}(Y × ℝ) ⊕ H_{δ}^{1}(Y × ℝ),

the cohomology groups of δ-decaying forms.

\underline{Proof:} Suppose given [x] = [𝔸, Ψ] ∈ M(a, b). Let x = (A(t), ψ(t)) be a temporal gauge representative such that

lim_{t→∞}(A(t), ψ(t)) = (A_{b}, ψ_{b}),

and

lim_{t→−∞}(A(t), ψ(t)) = (A_{a}, ψ_{a}).

Consider the family of operators

L_{x}(α, Φ) = \{ \begin{matrix}d^{+}α − \frac{1}{2}Im(Ψ · Φ̄) \\ D_{𝔸}Φ + αΨ \\ G_{(𝔸,Ψ)}^{∗}(α, Φ) \end{matrix}

acting on the space of L_{k,δ}^{2}-decaying 1-forms and spinor sections on Y × ℝ.

An orientation of the moduli space M(a, b) is determines by a trivialization of the determinant line bundle of the family of operators L_{x}. We can separate L_{x} in the first order term and a perturbation,

L_{x} = L_{x}^{1} + L_{x}^{0},

with

L_{x}^{1} = d^{+} + d_{δ}^{∗} + D_{𝔸}

and

L_{x}^{0} = ( \begin{matrix}−\frac{1}{2}Im(·Ψ̄) \\ i < Ψ, · > \end{matrix} ) .

The operators induced by L_{x} on the asymptotic ends are

L(±∞) = L^{1}(±∞) + L^{0}(±∞),

where we have

L^{1}(∞) = ( \begin{matrix}∗d & d^{∗} & 0 \\ d & 0 & 0 \\ 0 & 0 & ∂_{A_{b}} \end{matrix} ) ,

L^{1}(−∞) = ( \begin{matrix}∗d & d^{∗} & 0 \\ d & 0 & 0 \\ 0 & 0 & ∂_{A_{a}} \end{matrix} ) ,

and

L^{0}(∞) = ( \begin{matrix}0 & 0 & −2iImσ(·, ψ_{b}) \\ 0 & 0 & −i < ·, ψ_{b} > \\ ·ψ_{b} & ψ_{b} & 0 \end{matrix} ) ,

L^{0}(−∞) = ( \begin{matrix}0 & 0 & −2iImσ(·, ψ_{a}) \\ 0 & 0 & −i < ·, ψ_{a} > \\ ·ψ_{a} & ψ_{a} & 0 \end{matrix} ) ,

acting on Ω^{0}(Y, iℝ) ⊕ Ω^{1}(Y, iℝ) ⊕ Γ(S ⊗ L).

We can consider a deformation L_{x}^{ϵ} of the family L_{x} obtained as in [43] Section 6.6,

L_{x}^{ϵ} = L_{x}^{1} + (1 − ϵ)L_{x}^{0}.

The deformation changes the asymptotic operators in the form

L^{ϵ}(±∞) = L^{1}(±∞) + (1 − ϵ)L^{0}(±∞),

We can guarantee that this is a deformation via Fredholm operators provided that the weight δ is chosen such that δ/2 is not in the spectrum of L^{ϵ}(±∞) for all ϵ ([36] Theorem 6.2, and [45] Lemma 8.3.1). That is, if the spectrum of the operators L^{ϵ}(±∞) is uniformly bounded away from zero.

If this is the case, then a trivialization of the determinant line of the family L_{x}^{ϵ} is obtained by a trivialization at ϵ = 1. This induces a trivialization of L_{x}. The trivialization at ϵ = 1 is a trivialization of the determinant line of the operator d^{+} + d_{δ}^{∗} + D_{𝔸}.

The Dirac operator is complex linear and it preserves the orientation induced by the complex structure on the spinor bundle S^{+} ⊗ L. Thus a trivialization is obtained by an orientation of

H_{δ}^{0}(Y × ℝ) ⊕ H_{δ}^{2+}(Y × ℝ) ⊕ H_{δ}^{1}(Y × ℝ),

the cohomology groups of δ-decaying forms, [43], [46].

However, the condition on the spectrum of L^{ϵ}(±∞) may not always be satisfied: the deformation L_{x}^{ϵ} may not be through Fredholm operators. It is still possible to obtain an orientation of M(a, b): the following argument was suggested to us by L. Nicolaescu [49].

We can change the family L_{x} by a deformation such that the new family H_{x} satisfies

H_{x}|_{Y ×[−1,1]} ≡ L_{x}|_{Y ×[−1,1]}.

On Y × (−∞, −2] ∪ Y × [2, ∞) it satisfies

H_{x}|_{Y ×(−∞,−2]} = d^{+} + d_{δ}^{∗} + D_{A_{a}} + ( \begin{matrix}−\frac{1}{2}Im(·ψ̄_{a}) \\ i < ψ_{a}, · > \end{matrix} ) ,

H_{x}|_{Y ×[2,∞)} = d^{+} + d_{δ}^{∗} + D_{A_{b}} + ( \begin{matrix}−\frac{1}{2}Im(·ψ̄_{b}) \\ i < ψ_{b}, · > \end{matrix} ) .

A trivialization of the determinant line of H_{x} induces a trivialization of the determinant of L_{x}.

The index of the family Ind(H_{x}) on Y × ℝ equals the index on Y × [−3, 3] with APS boundary conditions [3], [51].

Now we can consider the family of operators H̃_{x} that satisfies

H̃_{x}|_{Y ×[−3,3]} ≡ H_{x}|_{Y ×[−3,3]}.

On Y × (−∞, −4] ∪ Y × [4, ∞) it satisfies

H̃_{x}|_{Y ×(−∞,−4]} = d^{+} + d_{δ}^{∗} + D_{A_{a}},
H̃_{x}|_{Y ×[4,∞)} = d^{+} + d_{δ}^{∗} + D_{A_{b}}.

On the cylinder Y × [−4, −3] and Y × [3, 4] the index of the operator H̃_{x} can be obtained as Ind(\frac{∂}{∂t} + H(t)), where H(t) is independent of x (but depends on the asymptotic values a or b). Thus, Ind(H̃_{x}) on Y × [−4, −3] and Y × [3, 4] is the spectral flow SF (H(t)) of H(t).

The indices of H_{x} and H̃_{x} are related by the excision formula

Ind(H_{x}) − SF (H(t)) = Ind(H̃_{x}),

generalizing the excision formula of [50].

Thus the relative orientation of H_{x} and by H̃_{x} is exactly (−1)^{SF (H)}. Notice that [50] provides examples where this spectral flow is computed explicitly and is odd.

Finally we can introduce a deformation H̃_{x}^{ϵ} with a homotopy that shrinks to zero the spinor part. In this case the asymptotic operators H(±∞) remain constant, hence the deformation is through Fredholm operators. This implies that the orientation determined by H̃_{x} is the same as the one determined by d^{+} + d_{δ}^{∗} + D_{𝔸}, that is by an orientation of

H_{δ}^{0}(Y × ℝ) ⊕ H_{δ}^{2+}(Y × ℝ) ⊕ H_{δ}^{1}(Y × ℝ).

2.4 Floer homology

The Floer complex has generators

C_{q} = {b ∈ M′ | μ(b) = q},

where M′ is the irreducible part of the moduli space of critical points. The Morse index μ(b) is computed with respect to a fixed element [A_{0}, ψ_{0}] in M, [A_{0}, ψ_{0}] = [ν, 0] in the case of a homology sphere.

The boundary operator is given by

∂a = ∑_{b|μ(a)−μ(b)=1} ϵ(a, b)b,  (24)

where ϵ(a, b) is the algebraic sum over the paths joining a and b of the signs given by the orientation,

ϵ(a, b) = ∑_{γ∈M̂(a,b)} ϵ_{γ}.

Here M(a, b) is the moduli space of flow lines on Y × ℝ with asymptotic values a and b. A description of M(a, b) will be given in the next section. The space M̂(a, b) is the quotient of M(a, b) by the action of ℝ by translations. A compactness result for M̂(a, b) is needed in order to make sense of ϵ(a, b). This result will follow from the more general result proved in Section 4 in the equivariant setup, in Theorem 4.1, Theorem 4.9, and Proposition 4.25.

The property that ∂ ∘ ∂ = 0 relies on the gluing formula of Lemma 2.14 that follows from an accurate analysis of the properties of the gradient flow moduli space M(a, b). In our setting, again, this result will follow from the more general results in the equivariant context, see Theorem 4.9.

In the case where Y is a homology sphere, we need to ensure that in the expression of ∂^{2} there is no contribution coming from trajectories that break through the unique reducible solution. In other words, no component of the form M̂(a, θ) × M̂(θ, b) can appear in the boundary of M̂(a, b) for μ(a) − μ(b) = 2. This has been proved in [13]. In fact, the following gluing formula holds.

Lemma 2.14 Suppose given a, b and c in M′, irreducible critical points with μ(a) > μ(b) > μ(c). Then, for large enough T, there is a local diffeomorphism

M̂(a, b) × M̂(b, c) × [T, ∞) → M̂(a, c).

If Y is a homology sphere and θ = [ν, 0] is the unique reducible critical point, there is a local diffeomorphism

M̂(a, θ) × M̂(θ, c) × U(1) × [T, ∞) → M̂(a, c).

In this case U(1) is the stabilizer of the reducible solution θ = [ν, 0].

In Theorem 4.9, we prove the gluing formula in the equivariant setup. A proof of Lemma 2.14 can be found in [38] and [13], and it follows from our equivariant result, as discussed at the end of Section 4.2. The result has an immediate corollary.

Corollary 2.15 Suppose Y is a homology sphere. Let θ be the unique reducible solution. If a and c are irreducible critical points such that μ(a) − μ(c) = 2, then generically there will be no boundary strata of the form M̂(a, θ) × M̂(θ, c). In fact for dimensional reasons the moduli space M̂(θ, c) of gradient flow lines is generically empty if μ(θ) − μ(c) = 1.

The property that ∂^{2} = 0 in the Floer complex follows then from the fact that the matrix elements

< ∂∂a, c >= ∑_{b} < ∂a, b >< ∂b, c >,

of the operator ∂^{2} are the algebraic sum of the points of the oriented zerodimensional manifold

∪_{b∈M′} M̂(a, b) × M̂(b, c) = ∂M̂(a, c).

In the case with non-trivial b^{1}(Y), we construct the Floer homology under the assumption that c_{1}(L) ≠ 0 rationally and that the perturbation is restricted to the trivial cohomology class [∗ρ] = 0.

Remark 2.16 For a 3-manifold Y with b_{1} > 0 and a Spin_{c}-structure with c_{1}(L) = 0 rationally, there is no nice way at this stage to formulate the corresponding monopole homology. In fact, the Chern–Simons–Dirac functional is ℝ-valued on the configuration space as long as the perturbation term represents a trivial de Rham cohomology class. In this case, however, the condition c_{1}(L) = 0 implies the existence of a reducible set of critical points that is a torus T^{b_{1}(Y)}. These can be degenerate, in the sense of Morse-Bott, even in the framed configuration space. Perturbing the functional with a 1-form ρ that is non-trivial cohomologically can destroy this reducible set, but the functional would no longer be ℝ-valued. There is then no uniform energy bound on the space of flow lines of a fixed virtual dimension. This creates a problem in the compactification by broken trajectories (see Theorem 4.1 and Theorem 4.9). The right framework for this bad case seems to be a Novikov type complex, where trajectories with the same virtual dimension but with different energies are counted separately as coefficients of a power series. Since this case has important applications in the gluing formulae, we deal with it separately in [41].

Under our assumptions, when we have b^{1}(Y) > 1 and a nontrivial c_{1}(L), the Floer homology groups can be proved to be independent of the metric and of the perturbation. In fact a chain map and a chain homotopy are constructed by a cobordism argument between moduli spaces for two different metrics and perturbations. This result follows from the topological invariance of the equivariant Floer homology proved in Theorem 6.1, in Section 6, and the equivalence of equivariant and non-equivariant Floer theories in the case of manifolds with b^{1}(Y) > 1 and a nontrivial c_{1}(L), proved in Section 7, Theorem 7.1.

In the case with b^{1}(Y) = 1 one expects to find a dependence on the choice of the perturbation, see [35], however, since we are only considering perturbations that are cohomologically trivial, we obtain independence of the metric and perturbation as in the b^{1}(Y) > 1 case. A similar dependence was detected in [5] in the case of the invariant of three-manifolds obtained by counting points in M̃ with the orientation. In [38] it is proved in the case b^{1}(Y) > 0 that this invariant is in fact the Euler characteristic of the Floer homology. The same invariant was introduced in [12] following the Quantum Field Theory formulation of Seiberg-Witten theory.

In the case of a homology sphere the metric dependence problem is more complicated. In fact due to the reducible solution a cobordism argument does not work and more generally the construction of a chain homotopy can fail due to the presence of moduli spaces of gradient flow lines that connect the irreducibles to the reducible critical point. The space of metrics and perturbations breaks into chambers with codimension-one walls, so that the Floer groups are isomorphic for metrics that belong to the same chamber and are in general nonisomorphic when the metric crosses a wall. We shall discuss the wall-crossing phenomenon in Section 7.

3 Morse-Bott theory

We are now going to introduce the equivariant Floer complex. This can be constructed for all three-manifolds. Clearly in the case of an integral homology sphere there will be no question of different Spin_{c}-structures. In the case of a rational homology sphere, we have finitely many choices of Spin_{c}-structures and there is a reducible point corresponding to each of these. In the case of manifolds with b^{1}(Y) > 0 there are infinitely many possible choices of Spin_{c}- structures. This gives rise to a family of Floer complexes corresponding to the different choices of the Spin_{c}-structures. The relation between these requires further investigation and is analyzed elsewhere (cf. [11]). In all the following we always assume to work with a fixed choice of the Spin_{c}-structure.

In order to consider reducible as well as irreducible generators, we introduce a framed configuration space with a U(1)-action, where the functional C_{ρ} is defined as a U(1)-invariant real valued functional. In order to apply the analogue of the finite dimensional equivariant Morse theory [7], we need C to be a Morse-Bott function. That is, we have to ensure that the Hessian is non-degenerate on the normal bundle to the critical U(1)-orbits.

3.1 Framed moduli space

Definition 3.1 Let x_{0} be a fixed base point in Y. We define the space B^{0} to be the quotient of A with respect to the action of the subgroup G^{0} ⊂ G of gauge transformations λ that act as the identity on the fiber of S ⊗ L over x_{0} and that satisfy the condition (2). The space B^{0} is the framed configuration space.

The action of the group G^{0} on A is free, therefore the space B^{0} is an infinite dimensional Banach manifold (using a fixed L_{k}^{2} norm) that carries a residual U(1) action. There is a fibration B^{0} → B over the unframed configuration space with fiber U(1). The solutions of the three dimensional Seiberg-Witten equations (5) and (6) in B^{0} form the framed moduli space M^{0}, that is the critical set of the functional (3) modulo based gauge transformations.

As in the case of Donaldson theory [16], an equivalent description of the framed configuration space can be given as the triples (A, ψ, ϕ) with (A, ψ) ∈ A and ϕ a unit vector in the fiber S ⊗ L|_{x_{0}}. The full gauge group acts freely on this space. Solutions of the Seiberg-Witten equations in this configuration space modulo the full gauge group provide another model of framed moduli space. This has been used in [4] and [5]. We use the description given in definition 3.1, since M^{0} has an explicit U(1) action which allows us to work equivariantly.

Since the action of the base point preserving gauge transformations on A is free, the reducible solutions with ψ ≡ 0 now have trivial stabilizer, hence they are smooth points in M^{0}. The reducible part of the unframed moduli space M corresponds exactly to the fixed point set for the U(1) action on M^{0}.

Lemma 3.2 Consider the unperturbed equations (9) in B^{0}. Let [A, 0] be a solution that is a fixed point of the U(1)-action. Then the virtual tangent space of M^{0} at the point [A, 0] is H^{1}(Y, ℝ) ⊕ Ker(∂_{A}). Moreover, the set of fixed points in M^{0} is identified with the torus H^{1}(Y, ℝ)/H^{1}(Y, ℤ) together with one point determined by the choice of the Spin_{c}-structure, out of a discrete set given by the torsion part of H^{1}(Y, ℤ).

\underline{Proof:} Fixed points are flat U(1)-connections modulo gauge: these are the representations of π_{1}(Y) into U(1). The linearization T at a point [A, 0] is of the form (− ∗ d, ∂_{A}). For the element in the torsion part of H^{1}(Y, ℤ) see Theorem 3.4 below. ⋄

We need to perturb the equations in some generic way in order to have M^{0} cut out transversely. For the perturbed equations (15), the fixed point set is described by the equation ∗F_{A} = 2iρ.

Lemma 3.3 If Y has non-trivial b^{1}(Y) and the functional C is perturbed with a generic co-closed 1-form ρ, then the set of critical orbits M^{0} contains no fixed point and is cut out transversely by the equations. The Hessian of the perturbed functional C_{ρ} is non-degenerate in the directions normal to the critical orbits.

\underline{Proof:} If the Chern class c_{1}(L) is non-trivial, choose a perturbation ρ with [∗ρ] ≠ iπc_{1}(L), or perturb with a harmonic form if c_{1}(L) = 0. This implies that there are no solutions of the equation F_{A} = 2i ∗ ρ.

Consider the linearization L̂_{(A,ψ,ρ)} of the equations (15), where we allow the perturbation to vary,

L̂_{(A,ψ,ρ)}(α, ϕ, η) = L_{(A,ψ,ρ)}(α, ϕ) − 2iη.

The operator L̂ has closed range, since L is Fredholm. We show that L̂ is surjective. Let (β, ξ, g) be an element that is L^{2}-orthogonal to the range of L̂. Then (β, ξ, g) is in the kernel of the adjoint, hence by elliptic regularity we can consider the L^{2} pairing of L_{k}^{2} and L_{−k}^{2},

⟨β, − ∗ dα − df + 2σ(ψ, ϕ) − 2iη⟩ + ⟨ξ, ∂_{A}ϕ + αψ + fψ⟩+
+⟨g, G^{∗}(α, ϕ)⟩ = 0.

The argument is analogous to the proof of Proposition 2.12. By varying η we force β ≡ 0. The vanishing of

⟨ξ, ∂_{A}ϕ + αψ + fψ⟩ + ⟨g, G^{∗}(α, ϕ)⟩

gives an equation Δg + 1/2g|ψ|^{2} = 0 which implies g ≡ 0 by the maximum principle. Then by varying ϕ and α we get ∂_{A}ξ = 0 and σ(ξ, ψ) = 0. The latter is satisfied if ξ is an imaginary multiple of ψ, ξ = iλψ, where neither of the two vanishes. Both ξ and ψ are in the kernel of ∂_{A}, thus if either of them vanishes on an open set it has to vanish identically (and we know that ψ is not identically zero). If we have ξ = iλψ, we obtain that ξ is identically zero as a consequence of the vanishing of the inner product ⟨ξ, fψ⟩ for arbitrary smooth compactly supported functions f.

This is enough to show that for a generic perturbation ρ the moduli space M (and therefore also M^{0}) is cut out transversely, as in the analogous proof of Proposition 2.12. ⋄

When b^{1}(Y) = 0 the virtual tangent space at a solution of ∗F_{A} = 2i ∗ dν is identified with Ker(∂_{A}) and the perturbation ρ = ∗dν is not enough to ensure that M^{0} is cut out transversely and that the fixed point set is separated from the other components of M^{0}.

Theorem 3.4 Let Y be a rational homology sphere. Suppose we choose a perturbation by a co-closed 1-form ρ and a generic metric. Then the framed moduli space M^{0} consists of a disjoint union of finitely many circles (corresponding to the irreducible part of the unframed M) and finitely many points (the reducibles of M). These correspond to different choices of the Spin_{c}-structure: for a fixed Spin_{c}-structure we have a unique fixed point of the U(1)-action. Moreover, the Hessian of the functional C_{ρ} is non-degenerate in the normal directions to the critical orbits.

\underline{Proof:} The choice of a perturbation ρ = ∗dν makes the fixed point set into the finitely many solutions modulo gauge of

F_{A} = dν,  (25)

namely flat U(1) connections modulo gauge. These are representations of the π_{1}(Y) into U(1), hence they are identified with the finitely many elements in the group H^{1}(Y, ℤ). Each of these elements specifies one choice of the Spin^{c}– structure. Thus, for fixed Spin^{c}–structure, there is a unique fixed point θ = [A_{0}+ν, 0] of the U(1)-action in the moduli space M^{0}. If the metric on Y satisfies Ker(∂_{A}) = 0 at a solution [A, 0] of (25), then the virtual tangent space at this fixed point is zero-dimensional, hence the Hessian of C_{ρ} is non-degenerate at the fixed points. The condition is satisfied for a generic metric because of Lemma 2.5.

The irreducible component of M is a zero dimensional manifold if the perturbation ρ is generic. The moduli space M is compact, since b^{1}(Y) = 0, and therefore so is M^{0}. However, we have to show that the unique fixed point is separated from the other components of M^{0}. We want to show that no sequence of irreducible solutions can converge to a reducible solution. We can apply the same perturbative argument used in [13], based on the local Kuranishi model. In fact if (A, ψ) is an irreducible solution which is sufficiently close to a reducible solution (A_{0} + ν, 0), then we have an expansion

A = A_{0} + ν + ϵα_{1} + ϵ^{2}α_{2} + ⋯,

and

ψ = ϵψ_{1} + ϵ^{2}ψ_{2} + ϵ^{3}ψ_{3} + · · · .

Using the equation (15) and the condition that Ker(∂_{A_{0}+ν}) = 0 we get that ψ_{i} ≡ 0 for all i, in contradiction with the assumption that (A, ψ) is an irreducible solution. Thus, for a fixed choice of the Spin^{c}–structure, the framed moduli space M^{0} consists of finitely many circles and a unique point fixed by the U(1) action. ⋄

Corollary 3.5 Under the choice of a generic perturbation (and of a generic metric in the case with b^{1}(Y) = 0), the functional C_{ρ} satisfies the Morse-Bott property.

Remark 3.6 The condition that ∂_{A} satisfies Ker(∂_{A}) = 0 at the reducible critical point breaks the space of metrics and perturbations into chambers, as discussed in Theorem 2.6. The Floer groups can be expected to change when crossing a wall corresponding to metrics with non-trivial Ker(∂_{A}). The problem of possible dependence of the metric in this case was already addressed by Donaldson in [15]. In Section 7, we prove that indeed the non-equivariant Floer groups do change when crossing a wall. In fact, we derive explicit wall crossing formulae for the Euler characteristic of the Floer homology.

3.2 Gradient flow lines

Denote by O_{a} the critical orbit in M^{0} that corresponds to a critical point a ∈ M. From Proposition 3.4 we know that M^{0} is a U(1)-fibration over M, where O_{a} is a circle if a is irreducible and O_{a} is the point a itself otherwise.

Let us introduce the configuration space A_{k,δ}(O_{a}, O_{b}). Consider the space A_{k,δ}((A_{0}, ψ_{0}), (A_{1}, ψ_{1})) of elements (𝔸, Ψ) of the form

(𝔸, Ψ) ∈ (A(t), ψ(t)) + L_{k,δ}^{2}(Ω^{1}(Y × ℝ) ⊕ Γ(S^{+} ⊗ L)),  (26)

where (A(t), ψ(t)) is a smooth path such that (A(t), ψ(t)) ≡ (A_{0}, ψ_{0}) for t ≤ 0 and (A(t), ψ(t)) ≡ (A_{1}, ψ_{1}) for t ≥ 1, with G^{0}-gauge classes [A_{0}, ψ_{0}] ∈ O_{a} and [A_{1}, ψ_{1}] ∈ O_{b}.

The space A_{k,δ}(O_{a}, O_{b}) is given by

A_{k,δ}(O_{a}, O_{b}) = ⋃_{λ∈U(1)} A_{k,δ}(λ(A_{0}, ψ_{0}), λ(A_{1}, ψ_{1})).  (27)

The space is endowed with a U(1)-action.

Let B_{kδ}^{0}(O_{a}, O_{b}) be the quotient of A_{k,δ}(O_{a}, O_{b}) modulo the free action of the gauge group G_{k+1,δ}^{0}(O_{a}, O_{b}) of based gauge transformations, modeled on the L_{k+1,δ}^{2} completion of the Lie algebra, that decay, as t → ±∞, to elements λ_{±∞} in the stabilizers G_{a} and G_{b}. The quotient has an induced U(1)-action and endpoint maps

e_{a}^{+} : B_{kδ}^{0}(O_{a}, O_{b}) → O_{a}

and

e_{b}^{−} : B_{kδ}^{0}(O_{a}, O_{b}) → O_{b}.

We denote by M(O_{a}, O_{b}) the moduli space of solutions of equations (20) and (21) in B_{kδ}^{0}(O_{a}, O_{b}). These will be our moduli spaces of flow lines connecting the orbits O_{a} and O_{b}.

Remark 3.7 Suppose one of O_{a} and O_{b} is not the fixed point θ. Then no reducible solution arises among the flow lines M(O_{a}, O_{b}).

In fact, either (A_{a}, ψ_{a}) or (A_{b}, ψ_{b}) has non-trivial spinor. Thus ψ(t) has to be non-trivial, in fact, the exponential weight on the Sobolev norms forces the elements in (26) to decay to the endpoints, at least one of which has nonvanishing spinor. Thus, the space A_{k,δ}(O_{a}, O_{b}) only contains irreducible lines. The action of G_{k+1,δ}^{0}(O_{a}, O_{b}) is free and B_{k,δ}^{0}(O_{a}, O_{b}) is a manifold with a free U(1)-action.

3.3 Relative Morse Index

We can rephrase Proposition 2.10 in the case of framed moduli spaces as follows.

Theorem 3.8 Suppose O_{a} and O_{b} are critical orbits of C_{ρ} in B^{0}. If Y is a rational homology sphere, assume that the metric and perturbation are chosen generically so that C_{ρ} satisfies the Morse-Bott condition. Let {λ_{a}} and {λ_{b}} be the eigenvalues of the Hessian operators Q_{a}, Q_{b} (that is the Hessian in B^{0} restricted to the directions orthogonal to the U(1)-orbits). Choose a weight δ satisfying 0 < δ < min{|λ_{a}|, |λ_{b}|}. Then the linearization L_{(𝔸,Ψ)} at a solution [𝔸, Ψ] ∈ B_{k,δ}^{0}(O_{a}, O_{b}) of (20) and (21) is a Fredholm operator from L_{k,δ}^{2} to L_{k−1,δ}^{2}. The virtual dimension of the moduli space M(O_{a}, O_{b}) is given by the index of L_{(𝔸,Ψ)} and is obtained as

dim M(O_{a}, O_{b}) = σ(O_{a}, O_{b}) + 1 − dim G_{a},

where σ(O_{a}, O_{b}) is the spectral flow of the operator ∇F on a path (A(t), ψ(t)) in A corresponding to (𝔸, Ψ). The quantity σ(O_{a}, O_{b}) is independent of the path in B_{k}^{0}; by additivity of the spectral flow, it can be written as

σ(O_{a}, O_{b}) = μ(O_{a}) − μ(O_{b}),

where μ(O_{a}) is the flow of ∇F on a path connecting the orbit O_{a} to a fixed orbit in M^{0}.

\underline{Proof:} The fact that the linearization L_{(𝔸,Ψ)} on the spaces B_{k,δ}(a, b) and on B_{k,δ}(O_{a}, O_{b}) is Fredholm follows from Theorem 6.2 of [36], or Lemma 8.3.1 of [45] (cf. the previous discussion of this point in Proposition 2.13), provided that the operator T has trivial kernel at the points a and b and the weight δ is smaller than the least eigenvalue of T, see also [59].

We shall write Ind_{e}(L_{(𝔸,Ψ)}) for the index formula in B_{kδ}^{0}(O_{a}, O_{b}). The subscript denotes the fact that we are computing

Ind_{e}(L_{(𝔸,Ψ)}) = dim Ker_{e}(L_{(𝔸,Ψ)}) − dim Coker(L_{(𝔸,Ψ)}),

where Ker_{e}(L_{(𝔸,Ψ)}) is the extended kernel, which consists of solutions (α, Φ) of L_{(𝔸,Ψ)}(α, Φ) = 0, with (α, Φ) − (α_{−}, ϕ_{−}) in L_{k,δ}^{2}, where (α_{−}, ϕ_{−}) is a solution of L_{A_{a},ψ_{a}}(α_{−}, ϕ_{−}) = 0, namely a tangent vector to the orbit O_{a} in T B^{0}. Notice that this Ker_{e}(L_{(𝔸,Ψ)}) is the correct space representing the tangent space of M(O_{a}, O_{b}) in B_{kδ}^{0}(O_{a}, O_{b}). The cokernel is simply given by L_{k−1,δ}^{2} solutions of L_{(𝔸,Ψ)}^{∗}(f, β, ξ) = 0, as in the setting of [3] I, pg. 58. The formula for the index of L_{(𝔸,Ψ)} on B_{k,δ}^{0}(a, b),

Ind_{e}(L_{(𝔸,Ψ)}) = σ(O_{a}, O_{b}) + 1 − dim G_{a},

then follows from the splitting of the index that will be proven in Corollary 4.18, together with the additivity of the spectral flow, and [3] III pg.95.

Thus we obtain the virtual dimension of M(O_{a}, O_{b}),

dim M(O_{a}, O_{b}) = σ(O_{a}, O_{b}) + 1 − dim G_{a},

where the +1 contribution depends upon the presence of the U(1)-action and dim G_{a} is the dimension of the stabilizer of the point a.

The relative Morse index is well defined. To see this, we have to examine the spectral flow of the operator ∇F_{(A(t),ψ(t))} on a path (A(t), ψ(t)) in A with endpoints (A, ψ) and λ(A, ψ), where λ is a gauge transformation satisfying

\frac{i}{2π} ∫_{Y} c_{1}(L) ∧ λ^{−1}dλ = 0.

According to [3] III pg.95, and [62], this spectral flow is just the index of L on the manifold Y × S^{1}. The index on a closed four-manifold is

Ind(L) = c_{1}(L)^{2} − \frac{2χ + 3σ}{4}

(cf. Corollary 4.6.2 of [43], or Theorem 2.3.8 of [39]), and this quantity vanishes in the case of a manifold of the form Y × S^{1}. ⋄

Given the relative Morse index, we can define the Morse index of a critical orbit up to fixing arbitrarily the index of a particular solution. In the case with b^{1}(Y) > 0 there is no canonical choice, hence the grading of the Floer complex is only defined up to an integer. When b^{1}(Y) = 0 we can remove this ambiguity by fixing the trivial solution θ = [A_{0} + ν, 0] to have index zero.

Notice that, in the case with b^{1}(Y) > 0, c_{1}(L) ≠ 0, and [∗ρ] = 0, the relative Morse index would be defined only up to a periodicity if we considered solutions modulo the full gauge group G̃. This is related to the fact that the functional C is well defined on A/G, but is only defined as a circle valued functional on A/G̃. The relation between ℤ-graded and ℤ_{l}-graded Seiberg-Witten Floer theory will be sketched briefly in Section 4.1, in Remark 4.5.

3.4 Decay estimate

In this subsection we introduce some analytic properties of the functional C_{ρ}. We show that finite energy solutions of the flow equations necessarily converge to asymptotic values that lie on some critical orbit. Moreover, we show that, if O_{a} or O_{b} is an irreducible orbit, then the flow lines in the moduli space M(O_{a}, O_{b}) decay exponentially towards the endpoints. Notice that in this case the moduli space M(O_{a}, O_{b}) only contains irreducible flow lines.

We give the following preliminary definition.

Definition 3.9 A smooth path (A(t), ψ(t)) in A is of finite energy if the integral

∫_{−∞}^{∞} ∥∇C_{ρ}(A(t), ψ(t))∥_{L^{2}}^{2}dt < ∞  (28)

is finite.

Notice that any solution of (18) and (19) with asymptotic values in O_{a} and O_{b} is of finite energy, in fact in this case the total variation of the functional C_{ρ} along the path (A(t), ψ(t)) is finite and (28) satisfies

∫_{−∞}^{∞} ∥∇C_{ρ}(A(t), ψ(t))∥_{L^{2}}^{2}dt ≤ C(C_{ρ}(a) − C_{ρ}(b)),

because of the assumptions on the perturbation q_{(𝔸,Ψ)}. Finite energy solutions of the flow equations have nice properties: they necessarily decay to asymptotic values that are critical points of C_{ρ} as we prove in Corollary 3.11 and in Theorem 3.12. We begin by introducing some analytic properties of the functional C_{ρ} (see also [25], [46], [13]).

Lemma 3.10 Let M_{ρ} be the moduli space of critical points of C_{ρ}, with ρ a sufficiently small perturbation. For any ϵ > 0 there is a λ > 0 such that, if the L_{1}^{2}-distance of a point [A, ψ] of B to all the points in M_{ρ} is at least ϵ, then

∥∇C_{ρ}(A, ψ)∥_{L^{2}} > λ.

\underline{Proof:} For a sequence [A_{i}, ψ_{i}] of elements of B with a distance at least ϵ from all the critical points, such that

∥∇C_{ρ}(A_{i}, ψ_{i})∥_{L^{2}} → 0,

as i → ∞, we would have

∥ ∗ F_{A_{i}} − σ(ψ_{i}, ψ_{i}) − iρ∥ + ∥∂_{A_{i}}ψ_{i}∥ → 0.

Thus, there is a constant C such that

∫_{Y} | ∗ F_{A_{i}} − σ(ψ_{i}, ψ_{i}) − iρ|^{2} + |∂_{A_{i}}ψ_{i}|^{2}dv < C.

If the perturbation ρ is sufficiently small, the Weizenböck formula implies that

∫_{Y} |F_{A_{i}}|^{2} + |σ(ψ_{i}, ψ_{i})|^{2} + \frac{κ}{2}|ψ_{i}|^{2} + 2|∇_{A_{i}}ψ_{i}|^{2}dv < C.

Thus we have a uniform bound on the norms ∥ψ_{i}∥_{L^{4}}, ∥F_{A_{i}}∥_{L^{2}}, and ∥∇_{A_{i}}ψ_{i}∥_{L^{2}}. An elliptic estimate shows that there is a subsequence that converges in the L_{1}^{2} norm to a solution of the critical point equations (15), and this contradicts the assumption. ⋄ Corollary 3.11 Let (A(t), ψ(t)) be a smooth finite energy solution of equations (18) and (19) with a smooth perturbation q. Then there exist critical points a and b of C_{ρ}, such that the lim_{t→±∞}(A(t), ψ(t)) are in the gauge classes of a and b.

\underline{Proof:} The finite energy condition (28) implies that

∥∇C_{ρ}(A(t), ψ(t))∥ → 0

as t → ±∞. The Palais-Smale condition of Lemma 3.10 implies that there exist T large, such that for |t| > T, (A(t), ψ(t)) lies in a very small ϵ-neighborhood of critical points of C_{ρ}. ⋄

Now we prove the exponential decay property.

Theorem 3.12 Let O_{a} and O_{b} be non-degenerate critical orbits in B_{0}. There exists a weight δ > 0 such that the following holds. Suppose given any solution [𝔸, Ψ] of (20) and (21) that is represented by a smooth pair (A(t), ψ(t)) in a temporal gauge, with asymptotic values (A_{a}, ψ_{a}) and (A_{b}, ψ_{b}) in O_{a} and O_{b}. Then there exists a constant K such that, for t outside an interval [−T, T], the distance in any fixed C^{l}-topology of (A(t), ψ(t)) from the endpoints is

dist_{C^{l}}((A(t), ψ(t)), (A_{i}, ψ_{i})) < K exp(−δ|t|),

with i = a if t < −T and i = b if t > T.

\underline{Proof:} The proof consists of a few steps. Let us consider the decaying as t → ∞; the other case with t → −∞ is analogous.

For simplicity of notation we shall prove the Theorem in the case of flow lines in B, with the action of the gauge group G of gauge maps that satisfy (2), and perturbation ρ satisfying [∗ρ] = 0 in cohomology. This ensures that the functional C_{ρ} is ℝ-valued. All the claims and the proofs extend directly to the case of the based space with the G^{0}-action and a non-degenerate critical orbit O_{b} in the quotient space B_{0}. In this case, the distance of (A, Ψ) from the orbit O_{b} is the minimal distance from points on the orbit.

Lemma 3.13 Suppose b is a non-degenerate critical point of C_{ρ}. Then there exists a constant C_{b} such that if the L^{2}-distance from [A, ψ] to b is sufficiently small, then we have the following estimate of the L^{2}-distance from [A, ψ] to b:

dist_{L^{2}}([A, ψ], b) ≤ C_{b}∥∇C_{ρ}(A, ψ)∥_{L^{2}}.

\underline{Proof:} Consider the Hessian operator T_{b} acting as an unbounded operator on the space of L^{2} connections and sections. Since b is a non-degenerate critical point, we have that

h_{b} = max\{\frac{1}{|λ_{i}|}| λ_{i} is a eigenvalue of the Hessian operator T_{b} at b\}

exists and is bounded. We know that [A, ψ] ↦ ∇C_{ρ}(A, ψ) defines a L^{2}-tangent section, which is smooth and transverse to zero at b. Thus, for any ϵ > 0, we may choose a small neighborhood U_{b} of b which may be identified with a small neighborhood of 0 in the L^{2}-tangent space of B at b, such that for all (A, ψ) = b + (α, ψ) in U_{b} we have

∥∇C_{ρ}(A, ψ) − T_{b}(α, ϕ)∥_{L^{2}} ≤ ϵ,

where we write ∇C_{ρ}(A, ψ) as the sum of the linear and a non-linear term,

∇C_{ρ}(A, ψ) = T_{b}(α, ϕ) + N(α, ϕ),

with N(α, ϕ) = (σ(ϕ, ϕ), α · ϕ).

When the neighborhood U_{b} is small enough, we can ensure that

∥N(α, ϕ)∥ ≤ 1/2∥∇C_{ρ}(A, ψ)∥,

so that we have

∥T_{b}(α, ϕ)∥ ≤ ∥∇C_{ρ}(A, ψ)∥ + ∥N(α, ϕ)∥
≤ 3/2∥∇C_{ρ}(A, ψ)∥.

Thus, we get the following estimate:

dist_{L^{2}}([A, ψ], b)
= ∥(α, ϕ)∥_{L^{2}}
≤ h_{b}∥(T_{b}(α, ϕ))∥_{L^{2}}
≤ \frac{3}{2}h_{b}∥(∇C_{ρ}([A, ψ]))∥_{L^{2}(Y)}
≤ \frac{3}{2}h_{b}∥∇C_{ρ}([A, ψ])∥_{L^{2}}.

The Lemma follows upon choosing the constant C_{b} with

C_{b} > \frac{3}{2}h_{b}.

Claim 1: Let (A(t), ψ(t)) be a representative of a path in B, such that

lim_{t→∞}(A(t), ψ(t)) = (A_{b}, ψ_{b})

where b = [A_{b}, ψ_{b}] is a non-degenerate critical point in M_{ρ}. Then there is a T_{0} >> 0 and a constant K_{b}, such that the inequality

|C_{ρ}(A(t), ψ(t)) − C_{ρ}(A_{b}, ψ_{b})| ≤ K_{b}∥∇C_{ρ}(A(t), ψ(t))∥_{L^{2}}^{2}

holds for all t ≥ T_{0}. Proof of Claim 1:

Choose T_{0} such that for all t ≥ T_{0} the path (A(t), ψ(t)) lies in a neighborhood U_{b} of b for which the result of Lemma 3.13 holds. Notice that such a T_{0} depends in general on which path (A(t), ψ(t)) is considered.

With the notation of (4) and (13), we have

C_{ρ}(A, ψ) ∼ C_{ρ}(A_{b}, ψ_{b}) + F_{b}(α, ϕ) + ∇F_{b}(α, ϕ),

where (A, ψ) = b + (α, ϕ) in U_{b}. Since b is a critical point, we have F_{b} ≡ 0. Thus we get

|C_{ρ}(A, ψ) − C_{ρ}(A_{b}, ψ_{b})| ≤ |⟨T_{b}(α, ϕ), (α, ϕ)⟩|.

Now applying Lemma 3.13 we obtain the estimate

|C_{ρ}(A, ψ) − C_{ρ}(A_{b}, ψ_{b})| ≤ ∥T_{b}(α, ϕ)∥ · ∥(α, ϕ)∥

≤ \frac{3}{2}C_{b}∥∇C_{ρ}(A, ψ)∥_{L^{2}}^{2}.

This completes the proof of Claim 1, with the constant K_{b} ≥ 3C_{b}/2. Claim 2: For a finite energy solution (A(t), ψ(t)) of (18) and (19), the inequality

\frac{1}{2} ∫_{t}^{∞} ∥∇C_{ρ}(A(s), ψ(s))∥_{L^{2}}^{2}ds ≤ C_{ρ}(A(t), ψ(t))−

−C_{ρ}(A_{b}, ψ_{b}) ≤ \frac{3}{2} ∫_{t}^{∞} ∥∇C_{ρ}(A(s), ψ(s))∥_{L^{2}}^{2}ds

holds for large t. \underline{Proof of Claim 2:} Without loss of generality we can assume that the perturbation in P satisfies Condition (4) of 2.8 with C_{0} < 1/2, so that

∥q_{(𝔸,Ψ)}(t)∥_{L^{2}} < \frac{1}{2}∥∇C_{ρ}(A(t), ψ(t))∥_{L^{2}}.

Thus, we can replace the equality

C_{ρ}(A(t), ψ(t)) − C_{ρ}((A_{b}, ψ_{b})) = ∫_{t}^{∞} −\frac{d}{ds}C_{ρ}(A(s), ψ(s))ds

= ∫_{t}^{∞} − < \frac{d}{ds}(A(s), ψ(s)), ∇C_{ρ}(A(s), ψ(s)) > ds =

= ∫_{t}^{∞} ∥∇C_{ρ}(A(s), ψ(s))∥^{2}ds,

that holds for solutions of the unperturbed equations with the inequality of Claim 2 for solutions of the perturbed equations.

Claim 3: Let (A(t)), ψ(t)) be a finite energy solution of the equations (18) and (19). The quantity

E(t) = ∫_{t}^{∞} ∥∇C_{ρ}(A(s), ψ(s))∥_{L^{2}}^{2}ds

decays exponentially as t → ∞. \underline{Proof of Claim 3:} In fact, the inequality of Claim 2 gives the first inequality in the following estimate:

E(t) ≤ 2(C_{ρ}(A(t), ψ(t)) − C_{ρ}(A_{b}, ψ_{b})) ≤

≤ K_{b}∥∇C_{ρ}(A(t), ψ(t))∥^{2} = −K_{b} \frac{d}{dt}E(t).

The second inequality follows from Claim 1. Claim 4: For large t we have the inequality

dist_{L^{2}}((A(t), ψ(t)), (A_{b}, ψ_{b})) ≤ K(∫_{t−1}^{∞} ∥∇C_{ρ}(A(s), ψ(s))∥_{L^{2}}^{2}ds)^{1/2},

when x(t) = (A(t), ψ(t)) is a finite energy solution of (18) and (19). \underline{Proof of Claim 4:} We can prove that the following inequality holds true for t ≥ T_{0}:

∫_{t}^{∞} ∥\frac{d}{ds}x(s)∥_{L^{2}}ds ≤ \frac{2K_{b}}{1 − C_{1}} (C_{ρ}(x(t)) − C_{ρ}(b))^{1/2},

where T_{0} is such that for all t ≥ T_{0} the perturbation q_{(𝔸,Ψ)} satisfies the inequality

∥q_{(A(t),ψ(t))}(t)∥_{L^{2}} ≤ C_{1}∥∇C_{ρ}(A(t), ψ(t))∥_{L^{2}},

with 0 < C_{1} < 1, as in property (4) of definition 2.8. Moreover, T_{0} is such that Lemma 3.13 holds. The proof of this inequality follows [56] Lemma 3.1 pg.542. We have

∥\frac{d}{dt}x(t)∥^{2} + ∥∇C_{ρ}(x(t))∥^{2} + 2⟨\frac{d}{dt}x(t), ∇C_{ρ}(x(t))⟩ =

= ∥q(t)∥^{2} ≤ C_{1}^{2}∥\frac{d}{dt}x(t)∥^{2} ≤ C_{1}∥\frac{d}{dt}x(t)∥^{2}.

Thus we get

−⟨\frac{d}{dt}x(t), ∇C_{ρ}(x(t))⟩ ≥ \frac{(1 − C_{1})}{2}(∥\frac{d}{dt}x(t)∥^{2} + ∥∇C_{ρ}(x(t))∥^{2})

≥ (1 − C_{1})∥\frac{d}{dt}x(t)∥ · ∥∇C_{ρ}(x(t))∥,

for all t ∈ [T_{0}, ∞). That is,

−\frac{d}{dt}C_{ρ}(x(t)) ≥ (1 − C_{1})∥\frac{d}{dt}x(t)∥ · ∥∇C_{ρ}(x(t))∥.

We obtain the inequality

−\frac{d}{dt}(C_{ρ}(x(t))−C_{ρ}(b))^{1/2} ≥ \frac{(1 − C_{1})}{2}(C_{ρ}(x(t))−C_{ρ}(b))^{−1/2}∥\frac{d}{dt}x(t)∥·∥∇C_{ρ}(x(t))∥

≥ \frac{(1 − C_{1})}{2K_{b}}∥\frac{d}{dt}x(t)∥.

The last inequality follows from Claim 1. We can now integrate both sides to obtain

∫_{t}^{∞} ∥\frac{d}{ds}x(s)∥_{L^{2}}ds ≤ \frac{2K_{b}}{(1 − C_{1})} ∫_{t}^{∞} −\frac{d}{ds}(C_{ρ}(x(s)) − C_{ρ}(b))^{1/2}ds

= \frac{2K_{b}}{(1 − C_{1})}(C_{ρ}(x(t)) − C_{ρ}(b))^{1/2}.

Thus by Claim 2 we have

dist_{L^{2}}((A(t), ψ(t)), (A_{b}, ψ_{b})) ≤ K(∫_{t}^{∞} ∥∇C_{ρ}(A(s), ψ(s))∥_{L^{2}}^{2}ds)^{1/2}.

The exponential decay of E(t) proves the claim of the Theorem for the case of L^{2}-topology. Smooth estimates then follow by a bootstrapping argument and elliptic regularity. This completes the proof of the Theorem. ⋄

Analogous exponential decay estimates have been proven in [46], cf. [4] and [13].

3.5 Transversality of M(O_{a}, O_{b})

We have the following transversality result for the moduli spaces M(O_{a}, O_{b}).

Proposition 3.14 Suppose given two orbits O_{a} and O_{b} in M^{0}. For a generic choice of the perturbation P ∈ P, the space M(O_{a}, O_{b}) is a smooth ℝ × U(1)- manifold with dimension given by μ(O_{a}) − μ(O_{b}) + 1 − dim G_{a}. M(O_{a}, O_{b}) is non-empty only if μ(O_{a}) − μ(O_{b}) ≥ 1. There are endpoint maps

e_{a}^{+} : M(O_{a}, O_{b}) → O_{a}

and

e_{b}^{−} : M(O_{a}, O_{b}) → O_{b}

that are smooth U(1)-equivariant maps.

The computation of the virtual dimension follows from Theorem 3.8. The transversality statement follows from Proposition 2.12, together with the Remark 3.7, to the effect that the smooth manifolds M(a, b) of Proposition 2.12 are just the quotient of M(O_{a}, O_{b}) with respect to the free U(1)-action. The properties of the endpoint maps follow from the discussion on the convergence to the endpoints of Section 3.4.

We have the analogue of Corollary 2.15 in this case.

Corollary 3.15 Suppose θ is the reducible critical point in M and b any point in M. Then, after a generic perturbation, M(θ, O_{b}) is a smooth ℝ × U(1)- manifold with dimension μ(θ) − μ(O_{b}). Moreover, M(θ, O_{b}) is non-empty only if μ(θ) − μ(O_{b}) ≥ 2.

4 Boundary structure

The purpose of this part of the work is to study the compactification of the moduli spaces M̂(O_{a}, O_{b}). The first step consists of showing the existence of a compactification, obtained by adding a certain set of broken trajectories, namely trajectories that break through other critical orbits of intermediate relative Morse index. We shall introduce the notation M̂(O_{a}, O_{b})^{∗} for the compactified moduli spaces. Proving the existence of the compactification by broken trajectories is dealt with in Section 4.1. Section 4.2 then deals with the gluing construction, and Section 4.3 analyzes the fine structure of the compactification.

It is perhaps useful to recall why in Floer theory it is necessary to develop the full gluing construction, as in Section 4.2. The results of Section 4.1 are sufficient to prove the existence of a compactification for the moduli spaces of flow lines, obtained by adding broken trajectories. The points of the ideal boundary are not completely identified, and the purpose of the gluing construction of Section 4.2 is precisely to identify all the broken trajectories that appear in the boundary. Recall also that, in the Floer complex, the boundary operator is obtained by counting elements in the oriented moduli spaces M̂(a, b)^{∗} for critical points a, b of relative index one. The statement that this is a boundary operator, namely that ∂ ∘∂ = 0, depends precisely upon having a complete description of the ideal boundary in the compactified moduli spaces M̂(a, c)^{∗} for critical points a and c of relative index two. In other words, proving that all the broken trajectories

∪_{μ(a)>μ(b)>μ(c)}M̂(a, b)^{∗} × M̂(b, c)^{∗}

appear in the compactification M̂(a, c)^{∗} is necessary in order to have a chain complex. In the equivariant setting we are considering here, there are components of the boundary operator D which are obtained by pullback and pushforward of forms on compactified moduli spaces M̂(O_{a}, O_{b})^{∗}, for orbits of relative index one, and on M̂(O_{a}, O_{c})^{∗} for orbits of relative index two (see the explicit description of the boundary operator given in Section 5, in (69). The proof that D ∘ D = 0 relies upon a version of Stokes’ theorem for manifolds with corners (as in [7]). For this reason, it is necessary to show that the compactification of the moduli spaces of flow lines of Section 4.1 has the structure of a smooth manifold with corners. This is proved in the case of the codimension one boundary in Section 4.2, and the generalization to the strata of higher codimension is considered in Section 4.3.

Thus, in Floer theory, a very elaborate analysis of the compactification of the moduli spaces of flow lines is required, in order to construct the Floer complex.

This situation is essentially different from other issues of compactification in gauge theory. For instance, in the case of Donaldson invariants [18], one only needs to show the existence of the Uhlenbeck compactification of the moduli space of anti-self-dual connections. The definition of the invariants and the proof of the diffeomorphism invariance do not require to show that every ideal antiself-dual connection actually appears in the Uhlenbeck compactification, nor they require the existence of a fine structure (such as that of smooth manifold with corners) on the compactification.

To start our analysis, we fix a unique way to identify the space of unparameterized trajectories

M̂(O_{a}, O_{b}) = M(O_{a}, O_{b})/ℝ

with a subset of the space of parameterized trajectories M(O_{a}, O_{b}). It is sufficient to choose the parameterization x(t) of x̂ ∈ M̂(O_{a}, O_{b}) which satisfies the equal energy condition

∫_{−∞}^{0} ∥∇C_{ρ}(A(t), ψ(t))∥_{L^{2}}^{2}dt = ∫_{0}^{∞} ∥∇C_{ρ}(A(t), ψ(t))∥_{L^{2}}^{2}dt.  (29)

This lifting of M̂(O_{a}, O_{b}) to M(O_{a}, O_{b}) is unique. In fact, in the family of gradient flows {[𝔸, Ψ]^{T} , T ∈ ℝ} that represent the class x ∈ M̂(O_{a}, O_{b}) there is a unique element which satisfies the equal energy condition (29). The lifting is often referred to as M^{bal}(O_{a}, O_{b}) ⊂ M(O_{a}, O_{b}), the balanced moduli space. We have M̂(O_{a}, O_{b}) ≅ M^{bal}(O_{a}, O_{b}). In the following, in order to avoid exceeding use of different notation, we shall always write M̂(O_{a}, O_{b}), unless we need to make explicit use of the equal energy condition (29), in which case we may recall that M̂(O_{a}, O_{b}) is realized by the balanced moduli space M^{bal}(O_{a}, O_{b}) ⊂ M(O_{a}, O_{b}), as in the proof of Theorem 4.9 in Section 4.2.

4.1 Convergence theorem

The following theorem describes convergence in the moduli space M̂(O_{a}, O_{b}), and proves the existence of a compactification, which we denote M̂(O_{a}, O_{b})^{∗}, obtained by adding broken trajectories.

Theorem 4.1 Consider the moduli space M̂(O_{a}, O_{b}), with μ(O_{a}) − μ(O_{b}) = k + 1, k ≥ 0. The space M̂(O_{a}, O_{b}) is precompact. Namely, any sequence [x̂_{i}] of elements in M̂(O_{a}, O_{b}) has a subsequence which either converges in norm to another solution [x̂] ∈ M̂(O_{a}, O_{b}), or converges to a broken trajectory. This means that there are critical orbits O_{c_{1}}, …, O_{c_{k}}, with μ(O_{a}) ≥ μ(O_{c_{1}}) > … > μ(O_{c_{k}}) ≥ μ(O_{b}), trajectories [y_{j}] ∈ M(O_{c_{j}}, O_{c_{j+1}}), and a sequence of real numbers T_{i_{k},j} ∈ ℝ such that the sequence of parameterized trajectories x_{i_{k}}^{T_{i_{k},j}} converges smoothly on compact sets

x_{i_{k}}^{T_{i_{k},j}} → y_{j}.

Here x_{i_{k}}^{T_{i_{k},j}} denotes the lifting of x̂_{i_{k}} to the space of parameterized trajectories

specified by the condition x_{i_{k}}^{T_{i_{k},j}}(0) = x_{i_{k}}(T_{i_{k},j}), where x_{i_{k}} is the equal energy lift of x̂_{i_{k}}.

\underline{Proof:} The proof of Theorem 4.1 consists of several steps. Suppose that the perturbation P_{(𝔸,Ψ)} is smooth. Given a sequence of unparameterized trajectories [x̂_{i}] in M̂(O_{a}, O_{b}), choose representatives x_{i} of the corresponding equal energy lift in M(O_{a}, O_{b}). By elliptic regularity these can be represented by smooth solutions.

Let (A_{i}(±∞), ψ_{i}(±∞)) be the asymptotic values of the elements x_{i} at the ends of the cylinder Y × ℝ. These represent elements [A_{i}(±∞), ψ_{i}(±∞)] in O_{a} and O_{b} respectively. Claim 1: There is a subsequence {x_{i_{k}}} that converges smoothly on compact sets to a solution y of the perturbed flow equations. \underline{Proof of Claim 1:} We first show that there is a subsequence that converges uniformly on compact sets. By the use of a bootstrapping argument is then possible to improve the convergence to C^{∞} on compact sets.

It is useful to recall the following weak version of Arzela’-Ascoli.

Proposition 4.2 Let K be a compact subset of ℝ and (X, ∥·∥) a normed metric space. S is a subset of continuous functions from K to X. Suppose the following conditions are satisfied

(i) pointwise bound:

sup_{f∈S} ∥f(t)∥ < ∞

for all t ∈ K; (ii) local equicontinuity: given ϵ > 0, for all t in K there exists a neighborhood V_{t} such that, for all τ ∈ V_{t}

∥f(τ) − f(t)∥ < ϵ

uniformly in f ∈ S.

Then the set S is uniformly bounded, that is

sup_{t∈K,f∈S} ∥f(t)∥ = M < ∞.

We show that the sequence x_{i} satisfies properties (i) and (ii) of Proposition 4.2. \underline{Equicontinuity:} Choose t ≤ t′. We have

dist_{L_{1}^{2}}(x_{i}(t), x_{i}(t′)) ≤ K ∫_{t}^{t′} ∥∇C_{ρ}(x_{i}(s))∥_{L_{1}^{2}}ds,

where we denote by dist_{L_{1}^{2}}(x(t), x(t′)) the quantity

dist_{L_{1}^{2}}(x(t), x(t′)) = | ∫_{t}^{t′} ∥\frac{d}{ds}x(s)∥_{L_{1}^{2}}ds|.

Since the perturbation satisfies condition (4) of Definition 2.8, we obtain

dist_{L_{1}^{2}}(x_{i}(t), x_{i}(t′)) ≤ K|t − t′|^{1/2}| ∫_{t}^{t′} ∥∇C_{ρ}(x_{i}(s))∥_{L_{1}^{2}}^{2}ds|^{1/2} ≤

K̃|t − t′|^{1/2}| ∫_{t−1}^{t′+1} ∥∇C_{ρ}(x_{i}(s))∥_{L^{2}}^{2}ds|^{1/2} ≤ K̃E|t − t′|^{1/2}.

Here the first step comes from the Hölder inequality and the second step from Lemma 6.14 of [46]. \underline{Pointwise bound:} Fix t_{0}. We can assume that

∥∇C_{ρ}(x_{i}(t_{0}))∥_{L^{2}} > λ

for all i >> 1, where λ is the constant of Lemma 3.10. In fact, if there is a subsequence x_{i_{k}} such that this condition is not satisfied, then by Lemma 3.10 the elements x_{i_{k}}(t_{0}) lie in an ϵ-neighborhood of a critical point hence their norms are bounded.

We can therefore choose a sequence t_{i} of real numbers such that

∥∇C_{ρ}(x_{i}(t_{i}))∥_{L^{2}} = λ

and

∥∇C_{ρ}(x_{i}(t))∥_{L^{2}} > λ

for t ∈ (t_{i}, t_{0}]. Lemma 3.10 then implies that there is an element (A_{a}, ψ_{a}) with [A_{a}, ψ_{a}] ∈ O_{a} and there are gauge transformations λ_{i} such that

dist_{L_{1}^{2}}(λ_{i}x_{i}(t_{i}), (A_{a}, ψ_{a})) ≤ ϵ

for all i, and therefore we can write

dist_{L_{1}^{2}}(λ_{i}x_{i}(t_{0}), (A_{a}, ψ_{a})) ≤ dist_{L_{1}^{2}}(λ_{i}x_{i}(t_{i}), (A_{a}, ψ_{a}))+dist_{L_{1}^{2}}(λ_{i}x_{i}(t_{i}), λ_{i}x_{i}(t_{0}))

≤ ϵ + K ∫_{t_{i}}^{t_{0}} ∥∇C_{ρ}(λ_{i}x_{i}(t))∥_{L_{1}^{2}}dt.

Using the inequality

∥∇C_{ρ}(λ_{i}x_{i}(t))∥_{L_{1}^{2}} ≤ \frac{∥∇C_{ρ}(λ_{i}x_{i}(t))∥_{L_{1}^{2}}^{2}}{∥∇C_{ρ}(λ_{i}x_{i}(t))∥_{L^{2}}} ≤ \frac{∥∇C_{ρ}(λ_{i}x_{i}(t))∥_{L_{1}^{2}}^{2}}{λ} ,

for t ∈ [t_{i}, t_{0}], and Lemma 6.14 of [46], we obtain

dist_{L_{1}^{2}}(λ_{i}x_{i}(t_{0}), (A_{a}, ψ_{a})) ≤ ϵ + \frac{K̃}{λ} ∫_{t_{i}−1}^{t_{0}+1} ∥∇C_{ρ}(λ_{i}x_{i}(t))∥_{L^{2}}^{2} ≤ ϵ + K̃E/λ.

Now Proposition 4.2 implies that, given any compact set K ⊂ ℝ, we have

sup_{t∈K} ∥λ_{i}x_{i}(t)∥_{L_{1}^{2}(Y ×{t})} = M < ∞  (30)

uniformly in i.

For simplicity of notation we refer in the following to the gauge transformed sequence λ_{i}x_{i} simply as x_{i}.

From the uniform estimate (30) and from the equations (18) and (19) we obtain a uniform bound of the L^{2} norms of \frac{d}{dt}x_{i}(t) on Y × K:

∥\frac{d}{dt}x_{i}(t)∥_{L^{2}(Y ×\{t\})} ≤ C∥x_{i}(t)∥_{L_{1}^{2}(Y ×\{t\})},

hence the left hand side is bounded uniformly with respect to t ∈ K and i. We use the fact that the perturbations P_{(𝔸_{i},Ψ_{i})} are uniformly bounded with respect to (𝔸_{i}, Ψ_{i}) according to condition (2) of Definition 2.8.

In the following we shall always consider the set K to be some large interval [−T, T]. The previous estimates provide a uniform bound of the norms ∥x_{i}∥_{L_{1}^{2}(Y ×[−T,T])}. In fact, we have

∥x_{i}∥_{L_{1}^{2}(Y ×[−T,T])}^{2} ≤ 2T M + 2T CM.

In order to bound the higher Sobolev norms we need the following gauge fixing condition. Up to gauge transformations λ_{i} in the group G̃_{k+1}(Y ×[−T, T]), it is possible to make 𝔸_{i} − 𝔸_{0} into a sequence of co-closed 1-forms.

We have elliptic estimates for the domain Y × [−T, T],

∥𝔸_{i} − 𝔸_{0}∥_{L_{k}^{2}(Y ×[−T,T])} ≤ C(∥(d^{∗} + d^{+})(𝔸_{i} − 𝔸_{0})∥_{L_{k−1}^{2}(Y ×[−T′,T′])}+

+∥𝔸_{i} − 𝔸_{0}∥_{L_{k−1}^{2}(Y ×[−T,T])})

and

∥Ψ_{i}∥_{L_{k}^{2}(Y ×[−T,T])} ≤ C (∥∇_{A}Ψ_{i}∥_{L_{k−1}^{2}(Y ×[−T′,T′])} + ∥Ψ_{i}∥_{L_{k−1}^{2}(Y ×[−T,T])}) ,

where [−T′, T′] is any strictly smaller interval contained in [−T, T]. We also have an estimate given by the curvature equation:

∥F_{𝔸_{i}}^{+}∥_{L_{k}^{2}(Y ×[−T,T])} ≤ ∥Ψ_{i} · Ψ̄_{i}∥_{L_{k}^{2}(Y ×[−T,T])} + ∥iμ + P_{(𝔸_{i},Ψ_{i})}∥_{L_{k}^{2}(Y ×[−T,T])}.

These provide a uniform bound on the L_{k}^{2}-norms of the (𝔸_{i}, Ψ_{i}) on a smaller Y × [−T′, T′]. By the Sobolev embeddings, this implies that on a smaller Y × [−T′′, T′′] there is a subsequence x_{i_{k}} = (𝔸_{i_{k}}, Ψ_{i_{k}}) that converges uniformly with all derivatives.

This completes the proof of Claim 1: there is a subsequence x_{i_{k}} that converges smoothly on compact sets to a solution y = (𝔸, Ψ) of (18) and (19) on Y × ℝ.

The limit y is of finite energy, hence it defines an element of some moduli space M(O_{c}, O_{d}) according to Corollary 3.11.

For simplicity of notation, assume that the sequence x_{i} itself converges to y smoothly on compact sets. Claim 2: If the limit y is an element of M(O_{a}, O_{b}) then the convergence x_{i} → y is strong in the L_{k,δ}^{2} norm. \underline{Proof of Claim 2:} There exists a T_{0} such that for all t ≤ −T_{0} we have

dist_{L_{1}^{2}}(y(t), (Ã_{a}, ψ̃_{a})) ≤ ϵ/2,

for some element (Ã_{a}, ψ̃_{a}) with [Ã_{a}, ψ̃_{a}] ∈ O_{a}. For i >> 1 we also have

dist_{L_{1}^{2}}(x_{i}(−T_{0}), y(−T_{0})) ≤ ϵ/2,

hence

dist_{L_{1}^{2}}(x_{i}(−T_{0}), (Ã_{a}, ψ̃_{a})) ≤ ϵ.

This implies that (Ã_{a}, ψ̃_{a}) is the same as the element (A_{a}, ψ_{a}). Moreover, the exponential decay property ensures that we have

dist_{L_{1}^{2}}(x_{i}(t), (A_{a}, ψ_{a})) ≤ ϵ  (31)

for all t ≤ −T_{0}. This, together with the uniform convergence on compact sets, implies that the convergence is uniform on all Y × ℝ. Thus, we have a uniform exponential decay towards the endpoints, hence convergence in the L_{1,δ}^{2} norm.

Notice that in this case we can ensure that there is a unique T_{0} such that (31) is satisfied for all t ≤ −T_{0}, whereas, in the proof of the exponential decay, the interval (−∞, T_{0}] depends on the solution (A(t), ψ(t)).

To improve the convergence to L_{k,δ}^{2} we need to choose suitable gauge transformations.

Lemma 4.3 There exist gauge transformations λ_{i} in G_{2,δ}^{0}(O_{a}, O_{b}) such that the forms λ_{i}(𝔸_{i} − 𝔸_{0}) are co-closed, that is

d_{δ}^{∗}λ_{i}(𝔸_{i} − 𝔸_{0}) = 0.

\underline{Proof of Lemma:} Let α_{i} = 𝔸_{i} − 𝔸_{0}. The element e_{δ/2}d^{∗}e_{−δ}α_{i} is L^{2}-orthogonal to the kernel of the Laplacian Δ_{δ/2}. Thus we can define the elements

g_{i} = e_{−δ/2}Δ_{δ/2}^{−1}(e_{δ/2}d^{∗}e_{−δ}α_{i}),

with

Δ_{δ/2}^{−1} : Ker(Δ_{δ/2})_{L_{1,δ}^{2}}^{⊥} → Ker(Δ_{δ/2})_{L_{2,δ}^{2}}^{⊥}.

These are elements in the L_{2,δ}^{2} completion of the Lie algebra hence they define gauge transformations λ_{i} ∈ G_{2,δ}^{0}(O_{a}, O_{b}). We have

λ_{i}(𝔸_{i} − 𝔸_{0}) = β_{i} = α_{i} − dg_{i}.

Thus, we can compute d_{δ}^{∗}β_{i} as

d_{δ}^{∗}β_{i} = d_{δ}^{∗}α_{i} − d_{δ}^{∗}de_{−δ/2}Δ_{δ/2}^{−1}(e_{δ/2}d^{∗}e_{−δ}α_{i})
= d_{δ}^{∗}α_{i} − e_{δ/2}(d_{δ/2}^{∗}d_{δ/2})Δ_{δ/2}^{−1}e_{δ/2}d^{∗}e_{δ}α_{i}
= d_{δ}^{∗}α_{i} − d_{δ}^{∗}α_{i} = 0.

This completes the proof of the Lemma.

Now we can use the elliptic estimates to improve the convergence to L_{k,δ}^{2}. This completes the proof of Claim 2.

Thus, we see that broken trajectories arise when the limit y is in some M(O_{c}, O_{d}) where either O_{c} ≠ O_{a} or O_{d} ≠ O_{b}. These are constrained by the condition

C_{ρ}(a) ≥ C_{ρ}(c) ≥ C_{ρ}(d) ≥ C_{ρ}(b).

Without loss of generality, assume that y is in M(O_{a}, O_{d}) and C_{ρ}(a) > C_{ρ}(d) > C_{ρ}(b).

There exists a sequence of real numbers T_{i} such that C_{ρ}(x_{i}(T_{i})) = α, with α satisfying C_{ρ}(d) > α > C_{ρ}(b). Consider the reparameterization x_{i}^{T_{i}}(t) = x_{i}(t+T_{i}). This is another possible lifting of x̂_{i} to the parameterized trajectories (which does not satisfy the equal energy condition). We can apply the result of Claim 1 to the sequence x_{i}^{T_{i}} and we obtain a subsequence that converges smoothly on compact sets to a solution ỹ of the equations (18) and (19) on Y × ℝ. Now the asymptotic values ỹ(±∞) are constrained by

C_{ρ}(d) ≥ C_{ρ}(ỹ(−∞)) ≥ C_{ρ}(ỹ(∞)) ≥ C_{ρ}(b).

The process can be repeated: at each step we either get strong convergence or another broken trajectory. If we start with relative index k + 1, the process ends at most at the order k, in fact the condition

μ(O_{a}) ≥ μ(O_{c}) > μ(O_{d}) ≥ μ(O_{b})

has to be satisfied for dimensional reasons. ⋄

Corollary 4.4 Notice that, in particular, if μ(O_{a}) − μ(O_{b}) = 1 with O_{a} a free orbit, or if μ(O_{a}) − μ(O_{b}) = 2, with O_{a} = θ the unique fixed point, the previous argument shows that M̂(O_{a}, O_{b}) is already compact, M̂(O_{a}, O_{b}) = M̂(O_{a}, O_{b})^{∗}.

A brief notational remark about the case b^{1}(Y) > 0:

Remark 4.5 In the case of a three-manifold with b^{1}(Y) > 0, the Seiberg-Witten Floer homology considered in this paper differs from the construction of [38], where the connected component of the full gauge group G̃(Y) is considered. The relation is illustrated in the following.

With our notation, we have G̃(Y), the full gauge group on Y, and G(Y), the group of gauge transformations on Y satisfying the condition (2). We have another possible choice of an even smaller subgroup, the identity component G_{c}(Y) of the gauge group. These all have subgroups of base point preserving gauge transformations, G̃^{0}(Y), G^{0}(Y), G_{c}^{0}(Y).

Thus, we have configuration spaces B̃^{0}, B^{0}, and B_{c}^{0}. The map B_{c}^{0} → B̃^{0} has fiber H^{1}(Y, ℤ), and the map B^{0} → B̃^{0} has fiber the group H

H = {h ∈ H^{1}(Y, ℤ)|⟨c_{1}(L) ∪ h, [Y]⟩ = 0}.

The group H is the kernel of the morphism

ξ : H^{1}(Y, ℤ) → ℤ,
ξ(h) = ⟨c_{1}(L) ∪ h, [Y]⟩.

The quotient H^{1}(Y, ℤ)/H is either trivial or ℤ. It is trivial only when c_{1}(L) = 0 rationally, that is in the case of Remark 2.16 which will be considered separately in [41]. Thus we have the following diagram

\xymatrix{ & B_{c}^{0}(Y) = A(Y)/G_{c}^{0}(Y) \ar@{-->}[dl]_{H} \ar[dd]^{H^{1}(Y,Z)} \\ B^{0}(Y) = A(Y)/G^{0}(Y) \ar@{-->}[dr]^{Z=H^{1}(Y,Z)/H} \\  & B̃^{0}(Y) = A(Y)/G̃^{0}(Y)}

As we are about to see, the choice of B̃^{0}(Y) gives rise to a periodic Floer homology of periodicity

l = g.c.d{⟨c_{1}(L) ∪ h, [Y]⟩, h ∈ H^{1}(Y, ℤ)}.

The choice of B^{0}(Y), that was suggested to us by R.G. Wang, represents the minimal cover of B̃^{0}(Y) that gives rise to a ℤ-graded Floer homology. The choice of B_{c}^{0}(Y) also gives rise to a ℤ-graded complex.

In [38] the Floer homology is constructed using the group G_{c}(Y). The relation to our construction here is as follows. Let {O_{a}^{h}}_{h∈H} be the family of distinct critical orbits in B_{c}^{0} that correspond to a critical orbit O_{a} in B^{0}. Notice that in B_{c}^{0} the relative Morse index satisfies

μ(O_{a}) − μ(O_{a}^{h}) = ∫_{Y} c_{1}(L) ∧ h = 0.

We have the configuration space of flow lines

A_{k,δ}(O_{a}^{h}, O_{b}^{h′}) = ⋃_{λ∈U(1)} A_{k,δ}(λ(A_{a}, ψ_{a}), λ(A_{b}, ψ_{b})),

with (A_{a}, ψ_{a}) and (A_{b}, ψ_{b}) representatives of elements [A_{a}, ψ_{a}] and [A_{b}, ψ_{b}] in O_{a}^{h} and O_{b}^{h′} respectively, in the space B_{c}^{0}(Y). The space A_{k,δ}(O_{a}^{h}, O_{b}^{h′}) is acted upon by the gauge group of gauge transformations in G̃(Y ×ℝ) that decay to elements of G_{a} and G_{b} respectively. We can form the quotient space B_{c}^{0}(O_{a}^{h}, O_{b}^{h′}).

We can consider the moduli spaces of flow lines M(O_{a}^{h}, O_{b}^{h′}) in the space B_{c}^{0}(O_{a}, O_{b}). In the analysis of the boundary structure of M(O_{a}^{h}, O_{b}^{h′}), there may be a question of whether a sequence of elements [x_{i}] of M(O_{a}^{h}, O_{b}^{h′}) can converge to a solution y that lives in another M(O_{a}^{h}, O_{b}^{h′}) (for different h and h′). Claim 2 of Theorem 4.1 rules out this possibility. Thus, a nice compactification of the moduli spaces M(O_{a}^{h}, O_{b}^{h′}) can be achieved in this setting as well, for all manifolds with b^{1}(Y) > 0 and c_{1}(L) ≠ 0. There is, therefore, a well defined associated Seiberg-Witten Floer homology which is finer than the one we are considering here.

On the other hand we can also consider critical orbits Õ_{a} in the configuration space B̃^{0}(Y). In this case the relative Morse index is not well defined, in fact the spectral flow between two gauge equivalent orbits satisfies

μ(λÕ_{a}) − μ(Õ_{a}) = ∫_{Y} c_{1}(L) ∧ h(λ),

with h(λ) = λ^{−1}dλ. Thus the grading of the Floer complex is defined only up to a periodicity of

l = g.c.d{⟨c_{1}(L) ∪ h(λ), [Y]⟩}.

Therefore in the configuration space B̃(Õ_{a}, Õ_{b}) obtained as a quotient of

A_{k,δ}(Õ_{a}, Õ_{b}) = ⋃_{λ∈U(1)} A_{k,δ}(λ(A_{a}, ψ_{a}), λ(A_{b}, ψ_{b})),

with (A_{a}, ψ_{a}) and (A_{b}, ψ_{b}) representatives of elements [A_{a}, ψ_{a}] and [A_{b}, ψ_{b}] in Õ_{a} and Õ_{b} respectively, in the space B̃^{0}(Y). The space A_{k,δ}(Õ_{a}, Õ_{b}) is acted upon by the gauge group G_{k+1,δ}(Õ_{a}, Õ_{b}) of gauge transformations in G̃(Y × ℝ) that decay to elements of G_{a} and G_{b}.

Now the space M(Õ_{a}, Õ_{b}) has infinitely many components of virtual dimensions

μ(O_{a}) − μ(O_{b}) + kl,

with k ∈ ℤ. Each component has uniformly bounded energy, hence it has a compactification by boundary strata of broken trajectories.

The Floer homology defined this way, namely with generators in M̃ and with the boundary operator defined by counting the components of minimal energy in M(Õ_{a}, Õ_{b}), is finitely generated and ℤ_{l}-graded. There are various other ways of defining the Floer homology for b^{1}(Y) > 0. For instance, a way of obtaining a ℤ–graded Floer homology which is also finitely generated (again for c_{1}(L) ≠ 0) is by a filtration as in [23] or [34]. The comparison between these different versions will be discussed in [41]. For the purpose of this paper we shall only consider the Seiberg-Witten Floer homology defined by the choice of the gauge group G.

4.2 Gluing theorem

Theorem 4.1 proves that lower dimensional moduli spaces appear naturally in the compactification of the spaces M̂(O_{a}, O_{b}). In the rest of this section we describe a gluing formula, thus proving that all the broken trajectories in the ideal boundary which break through one intermediate critical point actually occur as codimension one strata in M̂(O_{a}, O_{b})^{∗}. The case of multiply broken trajectories is analyzed in Section 4.3.

In order to show that the gluing construction is well defined over the gauge classes, we need a preliminary discussion of slices of the gauge action.

Definition 4.6 Let Γ = (𝔸, Ψ) be an element in A_{k,δ}(O_{a}, O_{b}). The slice S_{Γ} at Γ for the action of the gauge group G_{k+1,δ}^{0}(Y ×ℝ), is the set of elements (𝔸, Ψ)+ (α, ϕ) in the configuration space A_{k,δ}(O_{a}, O_{b}) with the following properties. The element (α, ϕ) satisfies

(α, ϕ) ∈ Ker(G_{(𝔸,Ψ)}^{∗}),

Where G_{(𝔸,Ψ)}^{∗} is the adjoint with respect to the L_{δ}^{2} norm of the linearization of the gauge group action. There is a T_{0} >> 0, such that, on Y × [T_{0}, ∞) we have

(α, ϕ) = (α̃, ϕ̃) + (e_{b}^{−})^{∗}(A_{1}, ψ_{1}),

and on Y × (−∞, −T_{0}] we have

(α, ϕ) = (α̃, ϕ̃) + (e_{a}^{+})^{∗}(A_{0}, ψ_{0}),

with (α̃, ϕ̃) is in L_{k,δ}^{2} on Y × ℝ. The elements (A_{0}, ψ_{0}) and (A_{1}, ψ_{1}) are representatives in the configuration space A over Y of elements in the orbits O_{a} and O_{b}.

The following Lemma shows that there is a uniform choice of slices for the gluing construction, cf. Lemma 2.1.4 and 2.1.5 of [44].

Lemma 4.7 Given Γ = (𝔸, Ψ) in the space A_{k,δ}(O_{a}, O_{b}), consider μ_{(𝔸,Ψ)}, the first positive eigenvalue of the operator G_{(𝔸,Ψ)}^{∗}G_{(𝔸,Ψ)}. There is a ball of radius r(μ) around Γ in the configuration space A_{k,δ}(O_{a}, O_{b}) such that the intersection of the ball with the slice S_{Γ} embeds into B_{k,δ}^{0}(O_{a}, O_{b}).

Moreover, there is a radius r such that we have an embedding

S_{(𝔸,Ψ)} ∩ B_{r}(𝔸, Ψ) ↪ B_{k,δ}^{0}(O_{a}, O_{b})

for all [𝔸, Ψ] in M(O_{a}, O_{b}).

\underline{Proof:} The claim that the radius can be taken to be r = r(μ) follows from Lemma 10.3.1 of [45] (cf. similar arguments in Lemma 2.1.5 of [44]). We need to show that there is a uniform lower bound for μ_{(𝔸,Ψ)} as [𝔸, Ψ] varies in M(O_{a}, O_{b}). Consider a sequence of elements x_{i} in M(O_{a}, O_{b}). Theorem 4.1 implies that a subsequence converges smoothly on compact sets to a limit y that defines an element in some M(O_{c}, O_{d}). Since we are considering the adjoint operator G_{(𝔸,Ψ)}^{∗} with respect to the L_{δ}^{2} norm, for a generic choice of metric and perturbation, we have μ_{y} > 0, as discussed in Lemma 3.13. We also have μ_{x_{i}} → μ_{y}, and this provides the uniform bound. In fact, according to Theorem 4.1, only finitely many different components M(O_{c}, O_{d}) can appear in the boundary of M(O_{a}, O_{b}). ⋄

The following result is also useful in constructing a gluing map at the level of gauge classes, cf. Lemma 2.1.8 of [44].

Lemma 4.8 Consider a compact subset K ⊂ M(O_{a}, O_{b}). Then there is a radius r, such that, for every [Γ] ∈ K, there is a U(1)-invariant submanifold V_{Γ} ⊂ A_{k,δ}(O_{a}, O_{b}) with the following properties. Suppose given any solution (𝔸, Ψ) of equations (20) and (21) that is in the ball of radius r around Γ. Then V_{Γ} contains an element gauge equivalent to (𝔸, Ψ). Moreover, the quotient by the action of the gauge group G_{k+1,δ}^{0}(Y × ℝ) induces a map

V_{Γ} → M(O_{a}, O_{b})

that is a diffeomorphism on a neighborhood of [Γ]. Moreover, there is a T_{0} such that, for all T ≥ T_{0} we can find families of gauge transformations

{λ_{x}^{±} | x ∈ V_{Γ}}

such that the elements in the set

W_{Γ}^{±} = {λ_{x}^{±}(x) | x ∈ V_{Γ}}

are in a temporal gauge on (−∞, −T] in the case of W_{Γ}^{−} and on [T, ∞) in the case of W_{Γ}^{+}. The set W_{Γ}^{±} has the same properties of V_{Γ}, namely the quotient with respect to the gauge action induces a diffeomorphism

W_{Γ}^{±} → M(O_{a}, O_{b})

on a neighborhood of Γ.

\underline{Proof:} It is sufficient to take V_{Γ} to be the intersection of the space of solutions to (20) and (21) in A_{k,δ}(O_{a}, O_{b}) with the slice S_{Γ}. According to Lemma 4.7, there is a radius r_{1} such that the intersection of this set with the ball of radius r_{1} around Γ embeds in M(O_{a}, O_{b}). In general, elements of V_{Γ} are not in a temporal gauge, however, following Corollary 3.1.7 of [45], there exists a constant C independent of x ∈ V_{Γ} such that we obtain ∥λ_{x}^{±}(x)∥ ≤ C∥x∥, and the elements in the set

W_{Γ}^{±} = {λ_{x}^{±}(x) | x ∈ V_{Γ}}

are in a temporal gauge on (−∞, −T] and on [T, ∞), respectively. This follows from the exponential decay of solutions towards the endpoints. Upon rescaling the radius r to r/C the properties stated for V_{Γ} hold for W_{Γ}^{±}. The constant C depends only on the set K, cf. Lemma 2.1.8 and Proposition 2.6.4 of [44]. ⋄

Now we can introduce the gluing construction. This identifies the codimension one boundary in the compactification M(O_{a}, O_{b})^{∗}.

Theorem 4.9 Suppose given O_{a}, O_{b} and O_{c} in M^{0} with μ(O_{a}) > μ(O_{b}) > μ(O_{c}). Assume that b is irreducible. Then, given a compact set

K ⊂ M(O_{a}, O_{b}) ×_{O_{b}} M(O_{b}, O_{c}),

there are a lower bound T_{0}(K) > 0 and a smooth map

# : K × [T_{0}, ∞) → M(O_{a}, O_{c})
((𝔸_{1}, Ψ_{1}), (𝔸_{2}, Ψ_{2}), T) ↦ (𝔸_{1}#_{T} 𝔸_{2}, Ψ_{1}#_{T} Ψ_{2}),

such that #_{T} is an embedding for all T > T_{0}(K). The gluing map # induces a smooth embedding

ˆ# : K̂ × [T_{0}, ∞) → M̂(O_{a}, O_{c}),

where

K̂ ⊂ M̂(O_{a}, O_{b}) ×_{O_{b}} M̂(O_{b}, O_{c}).

This local diffeomorphism is compatible with the orientation, where the orientation on K × [T_{0}, ∞) is induced by the product orientation on

M(O_{a}, O_{b}) × ℝ × M(O_{b}, O_{c}).

If b is reducible, then O_{b} is just a point and there is a similar orientation preserving local diffeomorphism

ˆ# : M̂(O_{a}, b) × M̂(b, O_{c}) × [T_{0}, ∞) → M̂(O_{a}, O_{c}).

Moreover, any sequence of trajectories in M̂(O_{a}, O_{c}) converging to a broken trajectory in M̂(O_{a}, O_{b})×_{O_{b}} M̂(O_{b}, O_{c}) lies eventually in the image of the gluing map.

In order to prove Theorem 4.9, we first define a pre-gluing map #_{T}^{0} . This provides an approximate solution via the following construction. Consider classes [x] = [𝔸_{1}, Ψ_{1}] and [y] = [𝔸_{2}, Ψ_{2}] in M(O_{a}, O_{b}) and M(O_{b}, O_{c}) respectively. Let

(A_{1}(t), ψ_{1}(t)) ∈ A_{k,δ}(O_{a}, O_{b}), (A_{2}(t), ψ_{2}(t)) ∈ A_{k,δ}(O_{b}, O_{c})

be temporal gauge representatives of [x] and [y].

Choose slices

S_{Γ_{ab}} ⊂ A_{k,δ}(O_{a}, O_{b})  (32)

and

S_{Γ_{bc}} ⊂ A_{k,δ}(O_{b}, O_{c}),  (33)

determined respectively by the elements

Γ_{ab} ∈ A_{k,δ}((A_{a}, ψ_{a}), (A_{b}, ψ_{b}))

and

Γ_{bc} ∈ A_{k,δ}((A_{b}, ψ_{b}), (A_{c}, ψ_{c})).

We choose them so that there is a representative x in the ball of radius r in A_{k,δ}(O_{a}, O_{b}) centered at Γ_{ab} and a representative y in the ball of radius r in A_{k,δ}(O_{b}, O_{c}) centered at Γ_{bc}. According to Lemma 4.8, there are gauge transformations λ_{1}^{+} and λ_{2}^{−} such that we have

λ_{1}^{+}x ∈ W_{Γ_{ab}}^{+}, λ_{2}^{−}y ∈ W_{Γ_{bc}}^{−}.  (34)

Thus, for |t| > T_{0} and for all T ≥ T_{0}, we can write λ_{1}^{+}x(t) = λ_{b}(A_{b}, ψ_{b}) + (α_{1}(t), ϕ_{1}(t)) and λ_{2}^{−}y(t) = λ_{b}(A_{b}, ψ_{b}) + (α_{2}(t), ϕ_{2}(t)). Here we have λ_{b} ∈ G(Y) and λ_{b}(A_{b}, ψ_{b}) is a representative of an element in the orbit O_{b} that satisfies

lim_{t→∞}(A_{1}(t), ψ_{1}(t)) = λ_{b}(A_{b}, ψ_{b}) = lim_{t→−∞}(A_{2}(t), ψ_{2}(t)).

The elements (α_{i}, ϕ_{i}) have exponentially decaying behavior, as in Proposition 3.12.

We construct an approximate solution x#_{T}^{0} y = (𝔸_{1}#_{T}^{0} 𝔸_{2}, Ψ_{1}#_{T}^{0} Ψ_{2}) of the form

x#_{T}^{0} y = \{ \begin{matrix}λ_{1}^{+}(𝔸_{1}^{2T} , Ψ_{1}^{2T}) & t ≤ −1 \\ λ_{b}(A_{b}, ψ_{b}) + ρ^{−}(t)(α_{1}(t + 2T), ϕ_{1}(t + 2T))+ \\ ρ^{+}(t)(α_{2}(t − 2T), ϕ_{2}(t − 2T)) & −1 ≤ t ≤ 1 \\ λ_{2}^{−}(𝔸_{2}^{−2T} , Ψ_{2}^{−2T}) & t ≥ 1. \end{matrix} (35)

Here ρ^{±}(t) are smooth cutoff functions with bounded derivative, such that ρ^{−}(t) is equal to one for t ≤ −1 and to zero for t ≥ 0 and ρ^{+}(t) is equal to zero for t ≤ 0 and to one for t ≥ 1.

Lemma 4.10 The pre-gluing map #_{T}^{0} descends to a map at the level of gauge classes. Namely, it is well defined with respect to the choice of slices of the gauge action.

\underline{Proof:} Consider a different choice of slices

SΓ̃_{ab} ⊂ A_{k,δ}(O_{a}, O_{b})

determined by

Γ̃_{ab} ∈ A_{k,δ}((Ã_{a}, ψ̃_{a}), (Ã_{b}, ψ̃_{b}))

and

SΓ̃_{bc} ⊂ A_{k,δ}(O_{b}, O_{c})

determined by

Γ̃_{bc} ∈ A_{k,δ}((Ã_{b}, ψ̃_{b}), (Ã_{c}, ψ̃_{c})).

Suppose that x and y are now in balls of radius r′ ≤ r centered at Γ̃_{ab} and Γ̃_{bc} respectively. Then, according to Lemma 4.8, there are gauge transformations λ̃_{1}^{+} and λ̃_{2}^{−} that conjugate x and y so that we get

λ̃_{1}^{+}x ∈ W + ; Γ̃_{ab}, λ̃_{2}^{−}y ∈ W − ; Γ̃_{bc}.

Consider the pre-glued element of the form

x̃#_{T}^{0} ỹ = \{ \begin{matrix}λ̃_{1}^{+}(𝔸_{1}^{2T} , Ψ_{1}^{2T}) & t ≤ −1 \\ λ̃_{b}(Ã_{b}, ψ̃_{b}) + ρ^{−}(t)(α̃_{1}(t + 2T), ϕ̃_{1}(t + 2T))+ \\ ρ^{+}(t)(α̃_{2}(t − 2T), ϕ̃_{2}(t − 2T)) & −1 ≤ t ≤ 1 \\ λ̃_{2}^{−}(𝔸_{2}^{−2T} , Ψ_{2}^{−2T}) & t ≥ 1. \end{matrix}

The temporal gauge representatives are defined up to an ambiguity given by the action gauge transformations that are constant in the ℝ direction, that is by the action of G(Y). Thus, on the interval [−1, 1], the gauge transformations λ̃_{i}^{±} ∘ λ_{i}^{±}^{−1} are constant with respect to t,

λ̃_{i}^{±} ∘ λ_{i}^{±}^{−1}|_{Y ×[−1,1]} = λ̃_{b}λ_{b}^{−1}.

Thus λ̃_{1}^{+} ∘ λ_{1}^{+}^{−1} and λ̃_{2}^{−} ∘ λ_{2}^{−}^{−1} define a gauge transformation λ in G(Y × ℝ) (depending on T) such that

x̃#_{T}^{0} ỹ = λ(x#_{T}^{0} y)

is satisfied. Thus, Lemma 4.7 and Lemma 4.8 imply that the pre-gluing map induces a well defined map #_{T}^{0} on compact subsets of broken trajectories,

#_{T}^{0} : K ⊂ M̂(O_{a}, O_{b}) ×_{O_{b}} M̂(O_{b}, O_{c}) → B̂_{k,δ}^{0}(O_{a}, O_{c}).

⋄

The proof of Theorem 4.9 will consist of showing that the pre-glued approximate solution x#_{T}^{0} y can be perturbed to an actual solution of the flow equations by a small perturbation.

We need a preliminary discussion of some properties of the linearization of the Seiberg–Witten equations at the pre-glued solution.

Consider the weight function e_{δ^{T}} (t) = e^{δ^{T} (t)}, with δ^{T} (t) a function that satisfies e_{δ^{T}} (t) = e^{δ(t+T)} for t < 0 and e_{δ^{T}} (t) = e^{δ(t−T)} for t > 0. Here e_{δ}(t) = e^{δ(t)} is the usual weight function introduced in Proposition 2.7. We define the norm L_{k,δ(T)}^{2} as ∥f∥_{L_{k,δ(T)}^{2}} = ∥fe_{δ^{T}} ∥_{L_{k}^{2}}. We can choose the weight function e_{δ^{T}} (t) as above, so that the weighted norms satisfy the estimates

C_{1}e^{δT} ∥f∥_{L_{k,δ}^{2}} ≤ ∥f∥_{L_{k,δ(T)}^{2}} ≤ C_{2}e^{δT} ∥f∥_{L_{k,δ}^{2}},  (36)

with constants C_{1} and C_{2} independent of T.

Suppose given classes [x] = [𝔸_{1}, Ψ_{1}] and [y] = [𝔸_{2}, Ψ_{2}] in M(O_{a}, O_{b}) and M(O_{b}, O_{c}) respectively. We can consider the operator

L_{x}(α, Φ) = \{ \begin{matrix}d^{+}α − \frac{1}{2}Im(Ψ_{1} · Φ̄) + DP_{(𝔸_{1},Ψ_{1})}(α, Φ) \\ D_{𝔸_{1}}Φ + αΨ_{1} \\ G_{Γ_{ab}}^{∗}(α, Φ) \end{matrix}  (37)

acting on the space of L_{1,δ}^{2}-decaying 1-forms and spinor sections on Y × ℝ. Analogously we have

L_{y}(α, Φ) = \{ \begin{matrix}d^{+}α − \frac{1}{2}Im(Ψ_{2} · Φ̄) + DP_{(𝔸_{2},Ψ_{2})}(α, Φ) \\ D_{𝔸_{2}}Φ + αΨ_{2} \\ G_{Γ_{bc}}^{∗}(α, Φ) \end{matrix}  (38)

acting on L_{1,δ}^{2} 1-forms and spinor sections on Y × ℝ. We also consider

L_{x#_{T}^{0} y}(α, Φ) = \{ \begin{matrix}d^{+}α − \frac{1}{2}Im(Ψ_{1}#_{T}^{0} Ψ_{2} · Φ̄) + DP_{(𝔸_{1}#_{T}^{0} 𝔸_{2},Ψ_{1}#_{T}^{0} Ψ_{2})}(α, Φ) \\ D_{𝔸_{1}#_{T}^{0} 𝔸_{2}}Φ + α · (Ψ_{1}#_{T}^{0} Ψ_{2}) \\ G_{Γ_{ac}}^{∗}(α, Φ) \end{matrix}

(39)

acting on L_{1,δ(T)}^{2} 1-forms and spinor sections on Y × ℝ. Moreover, if (A_{b}, ψ_{b}) is a representative of a point on O_{b}, with O_{b} a free U(1)-orbit, consider the operator

L_{b}(α, Φ) = \{ \begin{matrix}d^{+}α − \frac{1}{2}Im(ψ_{b} · Φ̄) \\ D_{A_{b}}Φ + αψ_{b} \\ G_{(A_{b},ψ_{b})}^{∗}(α, Φ) \end{matrix}  (40)

acting on L_{1}^{2} 1-forms and spinor sections on Y × ℝ. If O_{b} ≡ b is fixed by the U(1)-action, consider the operator

L_{b}(α, Φ) = \{ \begin{matrix}d^{+}α \\ D_{A_{b}}Φ \\ G_{(A_{b},ψ_{b})}^{∗}(α, Φ) \end{matrix}  (41)

acting on L_{1}^{2} 1-forms and spinor sections on Y × ℝ.

The central technique in the gluing construction is the study of the behavior of the eigenspaces of small eigenvalues of the Laplacians L_{x#_{T}^{0} y}^{∗}L_{x#_{T}^{0} y} and L_{x#_{T}^{0} y}L_{x#_{T}^{0} y}^{∗} of the linearization of the Seiberg–Witten equations at the approximate solutions. The general philosophy can be summarized as follows: we are trying to paste together solutions of a system of non-linear elliptic equations. If the equations were linear, we would simply encounter the obstruction to inverting the operator L_{x#_{T}^{0} y}^{∗}. This obstruction is represented by Coker(L_{x#_{T}^{0} y}). However, the equations being non-linear, the presence of eigenvectors with small eigenvalues of L_{x#_{T}^{0} y}L_{x#_{T}^{0} y}^{∗} also represents obstructions. (cf. the discussion of this phenomenon in [58], pg. 169). For this reason, we introduce the notation “approximate cokernel”, ApprCoker(L_{x#_{T}^{0} y}) (cf. Definition 4.16) to denote this eigenspace of small eigenvalues. The notation may be slightly confusing, as it may suggest a space of approximate eigenvectors, whereas it simply denotes the eigenspace of small eigenvalues, as explained in Definition 4.16. We trust that the reader will not be confused by our, perhaps unorthodox, choice of notation.

So the central technical issue in the gluing construction becomes relating the small eigenvalues eigenspace of the Laplacian L_{x#_{T}^{0} y}L_{x#_{T}^{0} y}^{∗} (we call this space ApprCoker(L_{x#_{T}^{0} y})) to the kernels of the Laplacians L_{x}L_{x}^{∗}, L_{y}L_{y}^{∗}, and L_{b}L_{b}^{∗}, and similarly for the other Laplacians L_{x}^{∗}L_{x} and L_{y}^{∗}L_{y} and the small eigenvalues eigenspace of L_{x#_{T}^{0} y}^{∗}L_{x#_{T}^{0} y} (which we call ApprKer(L_{x#_{T}^{0} y})).

First of all we can identify explicitly the kernel and cokernel of the operator L_{b} as in (40) and (41).

Lemma 4.11 The operator L_{b} defined as in (41), that is, for O_{b} ≡ b the fixed point of the U(1)-action, has trivial L_{1,δ}^{2}–kernel and trivial L_{0,δ}^{2}–cokernel. The operator L_{b} defined as in (40), with O_{b} a free U(1)-orbit, also has trivial kernel in L_{1,δ}^{2}, and trivial cokernel in L_{0,δ}^{2}.

\underline{Proof:} We prove the case of (41), where the critical orbit O_{b} ≡ b is the unique fixed point of the U(1) action b = θ. The irreducible case, with a free U(1) orbit O_{b} follows by an analogous argument.

Let (α, Φ) satisfy

\{ \begin{matrix}d^{+}α = 0 \\ D_{A_{b}}Φ = 0 \\ G_{A_{b},0}^{∗}(α, Φ) = 0, \end{matrix}

where the operator G_{A_{b},0}^{∗} is the adjoint of the infinitesimal gauge group action with respect to the L_{0,δ}^{2} inner product. In this case, this is simply given by the operator e_{δ}d^{∗}e_{−δ}. The Dirac equation D_{A_{b}}Φ = 0 can be rewritten in terms of the 3-dimensional Dirac operator, (∂_{t} + ∂_{A_{b}})Φ = 0. Upon expanding the spinor on a basis of eigenvectors for the 3-dimensional Dirac operator ∂_{A_{b}}, we obtain Φ = ∑ a_{k}(t)ϕ_{k} and the equation has solutions Φ(t) = ∑_{k} a_{k}(0)e^{λ_{k}t}ϕ_{k}. Since none of these solutions is in L_{1,δ}^{2}, we have obtained Φ ≡ 0. Now we consider the curvature equation. This gives d^{+}α = 0 and e_{δ}d^{∗}e_{−δ}α = 0. These two conditions imply that e_{−δ/2}α is a harmonic representative in H_{δ/2}^{2+}(Y × ℝ), with respect to the Laplacian Δ_{δ/2} = e_{δ/2}Δe_{−δ/2}. For a rational homology sphere Y (the only case where we have to consider the presence of the reducible point θ), this is necessarily trivial. The argument for the cokernel is similar. Consider an element in the cokernel, namely, suppose given a triple (β, f, ξ) of a self-dual 2-form, a 0-form, and a spinor section of the bundle of negative spinors over Y × ℝ, such that we have

⟨β, d^{+}α⟩ + ⟨f, e_{δ}d^{∗}e_{−δ}α⟩ + ⟨ξ, D_{A_{b}}Φ⟩ = 0

in the L^{2} pairing. We then have D_{A_{b}}ξ = 0 which implies ξ = 0 by the previous argument, and d^{∗}ω = e_{−δ}de_{δ}f. This equation implies d^{∗}ω = 0 and e_{−δ}de_{δ}f = 0. The first equation then implies that ω is a harmonic representative in H^{2+}(Y × ℝ). Again, this is trivial since the only case where the reducible θ appears is when Y is a rational homology sphere. The remaining equation gives f = ce_{−δ}. The condition f(x_{0}) = 0 at the base point x_{0} = (y_{0}, t_{0}) implies c = 0, hence f ≡ 0. ⋄

The following Lemma, together with Lemma 4.13 and Lemma 4.15, contains the key argument that will be used in Proposition 4.17 to relate the spaces ApprCoker(L_{x#_{T}^{0} y}) and ApprKer(L_{x#_{T}^{0} y}) of eigenvectors of small eigenvalues to the kernels and cokernels of L_{x} and L_{y} and in Corollary 4.18 for the formula on the splitting of the index.

Lemma 4.12 Let x and y be elements in M(O_{a}, O_{b}) and M(O_{b}, O_{c}) respectively. Suppose given a sequence ξ_{k} of L_{1,δ}^{2} 1-forms and spinor sections on Y ×ℝ. Assume these elements live in the tangent space, in the L_{1,δ}^{2}–norm, to the fiber (e_{a}^{+})^{−1}(x_{a}) of the endpoint map

e_{a}^{+} : S_{Γ_{ac}} → O_{a},

with x_{a} the asymptotic value at t → −∞ of x ∈ M(O_{a}, O_{b}), and for a given choice of slices S_{Γ_{ab}} and S_{Γ_{bc}} as in (32) and (33), and

S_{Γ_{ac}} ⊂ A_{1,δ}(O_{a}, O_{c}).

Suppose given a corresponding sequence of gluing parameters T_{k} with T_{k} → ∞. We assume that ξ_{k} and T_{k} satisfy e^{δT_{k}}∥ξ_{k}∥_{L_{1,δ}^{2}} = 1 and ∥L_{x#_{T_{k}}^{0} y}ξ_{k}∥_{L_{0,δ(T_{k})}^{2}} → 0,

with L_{x#_{T_{k}}^{0} y} as in (39). We have L_{1}^{2} convergence e^{δT_{k}}ξ_{k}|_{Y ×[−T_{k},T_{k}]} → 0. Then there exist L_{1,δ}^{2}-elements u and v in Ker(L_{x}) and Ker(L_{y}) respectively, with L_{x} and L_{y} as in (37) and (38), such that we have convergence

∥ρ_{1−T_{k}}^{−}ξ_{k}^{−T_{k}} − u∥_{L_{1,δ}^{2}} → 0

and

∥ρ_{−1+T_{k}}^{+}ξ_{k}^{T_{k}} − v∥_{L_{1,δ}^{2}} → 0.

Here we have ρ_{−1+T_{k}}^{+}(t) = ρ^{+}(t − 1 + T_{k}) and ρ_{1−T_{k}}^{−}(t) = ρ^{−}(t + 1 − T_{k}).

\underline{Proof:} We first want to prove that the supports of the ξ_{k} become more and more concentrated at the asymptotic ends as k → ∞. Consider the operator L_{b} as in (40) or (41). If ζ : Y × ℝ → [0, 1] is a smooth function which is equal to 1 on Y × [−1/2, 1/2] and equal to zero outside Y × (−1, 1). Moreover suppose that, for any fixed t, the function ζ is constant on Y × \{t\}. Let ζ_{k}(t) = ζ(\frac{t}{T_{k}}). Then, using the pointwise estimate |ζ_{k}| ≤ 1, we have

∥L_{b} ζ_{k}e^{δT_{k}}ξ_{k}∥_{L_{0,δ}^{2}} ≤ ∥ζ′_{k}e^{δT_{k}}ξ_{k}∥_{L_{0,δ}^{2}} + ∥ζ_{k}e^{δT_{k}}L_{b}ξ_{k}∥_{L_{0,δ}^{2}}

≤ \frac{1}{T_{k}} max |ζ′| + ∥ζ_{k}(L_{x#_{T_{k}}^{0} y} − L_{b})e^{δT_{k}}ξ_{k}∥_{L_{0,δ}^{2}} + ∥e^{δT_{k}}L_{x#_{T_{k}}^{0} y}ξ_{k}∥_{L_{0,δ}^{2}}

≤ \frac{1}{T_{k}} max |ζ′| + sup_{t∈[−T_{k},T_{k}]} ∥L_{x#_{T_{k}}^{0} y} − L_{b}∥ ∥e^{δT_{k}}ξ_{k}∥_{L_{1,δ}^{2}} + ∥L_{x#_{T_{k}}^{0} y}ξ_{k}∥_{L_{0,δ(T_{k})}^{2}}.

The first and last term tend to zero as k → ∞. The remaining term

sup_{t∈[−T_{k},T_{k}]} ∥L_{x#_{T_{k}}^{0} y} − L_{b}∥ ∥e^{δT_{k}}ξ_{k}∥_{L_{1,δ}^{2}}

is bounded by

sup_{t∈[−1,1]} ∥L_{x#_{T_{k}}^{0} y} − L_{b}∥ + sup_{t∈[−T_{k},−1]} ∥L_{x^{2T_{k}}} − L_{b}∥ + sup_{t∈[1,T_{k}]} ∥L_{y^{−2T_{k}}} − L_{b}∥.

All these terms tend to zero because of the exponential decay to the critical point b of the trajectories (A_{1}(t), ψ_{1}(t)) and (A_{2}(t), ψ_{2}(t)), and because of condition (5) of Definition 2.8. Thus, we have ∥L_{b} ζ_{k}e^{δT_{k}}ξ_{k}∥_{L_{0,δ}^{2}} → 0 as k → ∞.

Lemma 4.11 shows that the condition ∥L_{b} ζ_{k}e^{δT_{k}}ξ_{k}∥_{L_{0,δ}^{2}} → 0 implies the convergence ζ_{k}e^{δT_{k}}ξ_{k} → 0 in the L_{1,δ}^{2}-norm, and therefore we have

∥e^{δT_{k}}ξ_{k}∥_{L_{1}^{2}(Y ×[−T_{k},T_{k}])} → 0.

In other words, the ξ_{k} get localized at the asymptotic ends exponentially fast.

This result allows us to rephrase the convergence condition ∥L_{x#_{T_{k}}^{0} y}ξ_{k}∥ → 0 in terms of the Fredholm operators L_{x} and L_{y}:

∥L_{x}(ρ_{1−T_{k}}^{−}ξ_{k}^{−T_{k}})∥_{L_{0,δ}^{2}} ≤ ∥ρ′_{1−T_{k}}^{−}ξ_{k}^{−T_{k}}∥_{L_{0,δ}^{2}} + ∥ρ^{−}L_{x#_{T_{k}}^{0} y}ξ_{k}∥_{L_{0,δ(T_{k})}^{2}} ≤
C∥ρ′_{1}^{−}e^{δT_{k}}ξ_{k}∥_{L_{0,δ}^{2}} + ∥ρ^{−}L_{x#_{T_{k}}^{0} y}ξ_{k}∥_{L_{0,δ(T_{k})}^{2}} ≤
C∥e^{δT_{k}}ξ_{k}∥_{L_{1}^{2}(Y ×[−1,1])} + ∥L_{x#_{T_{k}}^{0} y}ξ_{k}∥_{L_{0,δ(T_{k})}^{2}} → 0,

where ρ_{1−T_{k}}^{−}(t) = ρ(t + 1 − T_{k}) and ξ_{k}^{−T_{k}}(t) = ξ_{k}(t − T_{k}).

The linearization L_{x} is a Fredholm operator. The sequence ρ_{1−T_{k}}^{−}ξ_{k}^{−T_{k}} is uniformly bounded in the L_{1,δ}^{2} norm: in fact, by the estimate (36) we have

∥ρ_{1−T_{k}}^{−}ξ_{k}^{−T_{k}}∥_{L_{1,δ}^{2}} ≤ C∥ρ_{1}^{−}e^{δT_{k}}ξ_{k}∥_{L_{1,δ}^{2}} ≤ C.

This implies the existence of an L_{1,δ}^{2}-element u in Ker(L_{x}) such that, upon passing to a subsequence, we obtain the L_{1,δ}^{2} convergence

∥ρ_{1−T_{k′}}^{−} ξ_{k′}^{−T_{k′}} − u∥_{L_{1,δ}^{2}} → 0.

This implies the convergence

∥ρ_{1}^{−}e^{δT_{k′}} ξ_{k′} − ρ^{−}e^{δT_{k′}} u^{T_{k′}} ∥_{L_{1,δ}^{2}} → 0.  (42)

Similarly we obtain an L_{1,δ}^{2}-element v in Ker(L_{y}) and a subsequence such that

∥ρ_{−1+T_{k′}}^{+} ξ_{k′}^{T_{k′}} − v∥_{L_{1,δ}^{2}} → 0.

and

∥ρ_{−1}^{+}e^{δT_{k′}} ξ_{k′} − ρ^{+}e^{δT_{k′}} v^{−T_{k′}} ∥_{L_{1,δ}^{2}} → 0.  (43)

Here we have ρ_{−1}^{+}(t) = ρ^{+}(t − 1) and ρ_{1}^{−}(t) = ρ^{−}(t + 1). ⋄

Thus, by Lemma 4.12 we have identified the eigenspace of small eigenvalues of L_{x#_{T}^{0} y}^{∗}L_{x#_{T}^{0} y} with a subspace of Ker(L_{x}) × Ker(L_{y}). Lemma 4.15 will prove the reverse inclusion. The next Lemma proves a stronger result, namely that the operator L_{x#_{T}^{0} y}^{∗}L_{x#_{T}^{0} y} is uniformly invertible, in the rescaled norm L_{1,δ(T)}^{2}, on the complement of the eigenspace of small eigenvalues. This is the essential result on the eigenvalue splitting which we are going to use in Section 6, in Lemma 6.13.

Let us introduce the following notation. Let F_{#_{T}} be the linearization of the pre-gluing map (35). Thus, F_{#_{T}} is a map

F_{#_{T}} |_{(x,y)} : Ker(L_{x}) × Ker(L_{y}) → T_{1}((e_{a}^{+})^{−1}(x_{a})) ⊂ T_{1}(B_{k,δ}^{0}(O_{a}, O_{b})). (44)

Here x_{a} is the asymptotic value as t → −∞ of x and e_{a}^{+} is the endpoint map e_{a}^{+} : S_{Γ_{ac}} → O_{a}. Since the pre-glued solution (35) involves translations, the map F_{#_{T}} in (44) has operator norm bounded by C_{T} ∼ Ce^{−δT}, namely, we have

∥F_{#_{T}} |_{(x,y)}(u, v)∥_{1,δ} ≤ Ce^{−δT} ∥(u, v)∥_{1,δ}.  (45)

We may also consider the same map acting as

F_{#_{T}} |_{(x,y)} : Ker(L_{x}) × Ker(L_{y}) → T_{1,T} ((e_{a}^{+})^{−1}(x_{a})) ⊂ T_{1,T} (B_{k,δ(T)}^{0}(O_{a}, O_{b})), (46) with T_{1,T} the virtual tangent space in the L_{1,δ(T)}^{2} norm. In this case F_{#_{T}} is bounded in the operator norm uniformly in T, given the estimate (36) for the rescaled norms.

Lemma 4.13 There exist T_{0} such that

∥L_{x#_{T}^{0} y}ξ∥_{L_{0,δ(T)}^{2}} ≥ C_{T} ∥ξ∥_{L_{1,δ}^{2}},

for all T ≥ T_{0} and for all ξ in the orthogonal complement

T^{⊥} = (F_{#_{T}} (Ker(L_{x}) × Ker(L_{y})))^{⊥}

in the space of L_{1,δ}^{2}-connections and sections. With this choice of norms, the constant C_{T} grows like Ce^{δT}. If we consider the rescaled norms L_{ℓ,δ(T)}^{2} or the original norms L_{ℓ,δ}^{2} on both the source and target space, the constant is bounded uniformly in T, given the estimate (36) for the rescaled norms.

\underline{Proof:} Suppose there is a sequence of parameters T_{k} and elements ξ_{k} in the orthogonal complement

ξ_{k} ∈ (F_{#_{T_{k}}} (Ker(L_{x}) × Ker(L_{y})))^{⊥}

in the space of L_{1,δ}^{2}-connections and sections, such that e^{δT_{k}}∥ξ_{k}∥_{L_{1,δ}^{2}} = 1 and ∥L_{x#_{T_{k}}^{0} y}ξ_{k}∥_{L_{0,δ(T_{k})}^{2}} → 0. We use the estimates of Lemma 4.12 to derive a con-

tradiction with the assumption ξ_{k} ∈ T^{⊥}. We have

1 = lim_{k} e^{2δT_{k}}∥ξ_{k}∥_{L_{1,δ}^{2}}^{2} = lim_{k}⟨ρ_{1}^{−}e^{δT_{k}}ξ_{k}, e^{δT_{k}}ξ_{k}⟩_{L_{1,δ}^{2}} + ⟨ρ_{−1}^{+}e^{δT_{k}}ξ_{k}, e^{δT_{k}}ξ_{k}⟩_{L_{1,δ}^{2}},

since the remaining term satisfies

lim_{k}⟨(1 − ρ_{1}^{−} − ρ_{−1}^{+})e^{δT_{k}}ξ_{k}, e^{δT_{k}}ξ_{k}⟩ = 0.

This follows from Lemma 4.12, since (1 − ρ_{1}^{−} − ρ_{−1}^{+}) is supported in [−2, 2], and we are considering the case where Ker(L_{b}) = 0 in L_{1,δ}^{2}.

Moreover, we know from Lemma 4.12 that, upon passing to a subsequence we have convergence as in (42) and (43). This implies that, upon passing to a subsequence, we obtain

lim_{k}⟨ρ_{1}^{−}e^{δT_{k}}ξ_{k}, e^{δT_{k}}ξ_{k}⟩_{L_{1,δ}^{2}} = lim_{k} e^{2δT_{k}}⟨ρ^{−}u^{T_{k}}, ξ_{k}⟩_{L_{1,δ}^{2}},

and correspondingly

lim_{k}⟨ρ_{−1}^{+}e^{δT_{k}}ξ_{k}, e^{δT_{k}}ξ_{k}⟩_{L_{1,δ}^{2}} = lim_{k} e^{2δT_{k}}⟨ρ^{+}v^{−T_{k}}, ξ_{k}⟩_{L_{1,δ}^{2}}.

The sum of these two terms therefore gives

lim_{k} e^{2δT_{k}}⟨F_{#_{T_{k}}} (u, v), ξ_{k}⟩_{L_{1,δ}^{2}}.

But this term is zero since we are assuming that the elements ξ_{k} are in the orthogonal complement

ξ_{k} ∈ (F_{#_{T_{k}}} (Ker(L_{x}) × Ker(L_{y})))^{⊥} .

⋄

Remark 4.14 The results of Lemma 4.12 and Lemma 4.13 can be reformulated and proved, with minor modifications, for the operator L_{x#_{T}^{0} y}^{∗} and elements

(u, v) ∈ Ker(L_{x}^{∗}) × Ker(L_{y}^{∗}).

Namely, suppose given a sequence ξ_{k} in T_{0}(S_{Γ_{ac}}), bounded uniformly in the L_{0,δ(T_{k})}^{2}-norms, such that L_{x#_{T_{k}}^{0} y}^{∗}ξ_{k} → 0 as k → ∞. Then we have convergence

∥L_{b}^{∗}ζ_{k}ξ_{k}∥_{L_{1,δ}^{2}} → 0, which implies L_{0,δ}^{2}–convergence of the elements ζ_{k}ξ_{k} to zero. Moreover, there exist elements u ∈ Coker(L_{x}) and v ∈ Coker(L_{y}) satisfying

∥ρ_{1−T_{k}}^{−}ξ_{k}^{−T_{k}} − u∥_{0,δ} → 0

and

∥ρ_{−1+T_{k}}^{+}ξ_{k}^{T_{k}} − v∥_{0,δ} → 0.

Thus, for all T ≥ T_{0}, there is an estimate

∥L_{x#_{T}^{0} y}^{∗}ξ∥_{1,δ} ≥ C∥ξ∥_{0,δ},

for all ξ in

(F_{#_{T}} (Coker(L_{x}) × Coker(L_{y})))^{⊥} ⊂ T_{0}(S_{Γ_{ac}}).

The constant C is independent of T ≥ T_{0}.

Thus, in Lemma 4.13 and Remark 4.14, we have obtained invertibility of the linearization of the pre-glued solution on the orthogonal complement of the linearization of the pre-gluing map. The norm of the inverse operator grows exponentially with the gluing parameter T. This is an effect of using the weighted Sobolev norms that are not translation invariant. If we use the rescaled norm δ(T), or the original δ–norm on both the source and target space, we obtain uniform invertibility, with the norm of the inverse operator uniformly bounded in T.

Consider the two Laplacians of the gluing construction, namely, the selfadjoint operators

H_{x#_{T}^{0} y}^{0} = L_{x#_{T}^{0} y}L_{x#_{T}^{0} y}^{∗}

and

H_{x#_{T}^{0} y}^{1} = L_{x#_{T}^{0} y}^{∗}L_{x#_{T}^{0} y}

acting on L_{0,δ(T)}^{2} (or L_{0,δ}^{2}) and on L_{1,δ(T)}^{2} (or L_{1,δ}^{2}), respectively.

Lemma 4.15 The operator H_{x#_{T}^{0} y}^{1} has at least

dim Ker(L_{x}) + dim Ker(L_{y})

independent eigenvectors with small eigenvalues μ_{T} satisfying μ_{T} → 0 as T → ∞. All these eigenvectors have fast decaying eigenvalues, namely μ_{T} satisfying μ_{T} ≤ Ce^{−δT}. The operator H_{x#_{T}^{0} y}^{0} has at least

dim Coker(L_{x}) + dim Coker(L_{y})

independent eigenvectors with small eigenvalues, namely with eigenvalues μ_{T} satisfying the property that μ_{T} → 0 as T → ∞. Again, these have fast decaying eigenvalues, namely with μ_{T} satisfying μ_{T} ≤ Ce^{−δT}.

\underline{Proof:} Let us prove the case of H_{x#_{T}^{0} y}^{1} first. Suppose given an element u in the L_{1,δ}^{2}–kernel Ker(L_{x}). Consider the element ξ_{T} = ρ^{−}u^{T}. We obtain the following estimate

∥L_{x#_{T}^{0} y}(ξ_{T})∥_{L_{0,δ(T)}^{2}} ≤ ∥L_{x}u∥_{L_{0,δ}^{2}} + ∥L_{x#_{T}^{0} y}e^{δT} ξ_{T} ∥_{L_{1,δ}^{2}(Y ×[−1,1])}.

The first term on the right vanishes, and the second term can be estimated as

∥L_{x#_{T}^{0} y}e^{δT} ξ_{T} ∥_{L_{1,δ}^{2}(Y ×[−1,1])} ≤ C∥e_{δ}u∥_{L_{1}^{2}(Y ×[T −1,T +1])},

by the effect of time translations on the norms. This last term is certainly bounded by

C · ∥ξ_{T} ∥_{L_{1,δ}^{2}},

Thus, using the relation 36, we obtain

∥L_{x#_{T}^{0} y}(ξ_{T})∥_{L_{0,δ}^{2}} ≤ Ce^{−δT} ∥ξ_{T} ∥_{L_{1,δ}^{2}}.

The same argument can be repeated starting with elements v ∈ Ker(L_{y}) and constructing ξ_{T} = ρ^{+}v^{−T}. Notice moreover that if u and w are two independent vectors in Ker(L_{x}), then the corresponding elements ξ_{T}^{u} = ρ^{−}u^{T} and ξ_{T}^{w} = ρ^{−}w^{T} are independent vectors in ApprCoker(L_{x#_{T}^{0} y}), since we have

⟨ξ_{T}^{u}, ξ_{T}^{w}⟩_{L_{1,δ(T)}^{2}} → ⟨u, w⟩_{L_{1,δ}^{2}}.

This proves the statement for H_{x#_{T}^{0} y}^{1}. The argument for H_{x#_{T}^{0} y}^{0} is analogous. ⋄

Thus we can give the following definition.

Definition 4.16 We define the approximate kernel and cokernel

ApprKer(L_{x#_{T}^{0} y}) ApprCoker(L_{x#_{T}^{0} y})

of the linearization at the pre-glued solution as the span of the eigenvectors of H_{x#_{T}^{0} y}^{1} and H_{x#_{T}^{0} y}^{0} respectively, with small eigenvalues, namely with eigenvalues μ_{T} satisfying μ_{T} → 0 as T → ∞.

Lemma 4.12 and Lemma 4.13, together with Remark 4.14, and Lemma 4.15 give the following result.

Proposition 4.17 We have isomorphisms

ApprKer(L_{x#_{T}^{0} y}) →^{≅} Ker(L_{x}) × Ker(L_{y})  (47)

and

ApprCoker(L_{x#_{T}^{0} y}) →^{≅} Coker(L_{x}) × Coker(L_{y}),  (48)

given by the projections. Moreover, we obtain constraints on the dimensions of the actual kernel and cokernel of the linearization at the pre-glued solution:

dim Ker(L_{x#_{T}^{0} y}) ≤ dim Ker(L_{x}) + dim Ker(L_{y})

and

dim Coker(L_{x#_{T}^{0} y}) ≤ dim Coker(L_{x}) + dim Coker(L_{y}).

\underline{Proof:} We have proved in Lemma 4.12 and Lemma 4.13 that

dim ApprKer(L_{x#_{T}^{0} y}) ≤ dim Ker(L_{x}) + dim Ker(L_{y}),

and in Lemma 4.15 we have proved the reverse estimate, namely that

dim ApprKer(L_{x#_{T}^{0} y}) ≥ dim Ker(L_{x}) + dim Ker(L_{y}).

This, together with the result of Lemma 4.13 shows that the linearization of the gluing map F_{#} gives the isomorphism of the Proposition. The result for the cokernels is analogous, by Remark 4.14 and Lemma 4.15. ⋄

Corollary 4.18 A direct consequence of Proposition 4.17 is the splitting of the index:

Ind(L_{x#_{T}^{0} y}) = Ind(L_{x}) + Ind(L_{y}).  (49)

This can be rephrased as

Ind_{e}(L_{x#_{T}^{0} y}) = Ind_{e}(L_{x}) + Ind_{e}(L_{y}) − 1,

in the case of gluing across a free orbit O_{b}, or as

Ind_{e}(L_{x#_{T}^{0} y}) = Ind_{e}(L_{x}) + Ind_{e}(L_{y})

in the case of the fixed point b = θ, with Ind_{e} as in Theorem 3.8, satisfying

Ind_{e}(L_{𝔸,Ψ}) = μ(O_{a}) − μ(O_{b}) + 1 − dim(G_{a}).

\underline{Proof:} The index of the linearization L_{x#_{T}^{0} y} can be computed as

Ind(L_{x#_{T}^{0} y}) = dim(ApprKer(L_{x#_{T}^{0} y})) − dim(ApprCoker(L_{x#_{T}^{0} y})).

This can be seen, for instance, by the supertrace formula for the index. Proposition 4.17 then gives the result. The statement for Ind_{e} follows from the relation

Ker(L_{x}) ≅ Ker_{e}(L_{x})/ℝ and Ker(L_{y}) ≅ Ker_{e}(L_{y})/ℝ,

with Ker_{e} the space of extended L_{1,δ}^{2} solutions as described in Theorem 3.8. ⋄

All the previous discussion makes no assumption on the cokernels of the linearizations L_{x} and L_{y}, so it applies equally to the unobstructed case analyzed in the rest of this section (the case of the moduli spaces M̂(O_{a}, O_{b}) of flow lines), and to the obstructed case of the moduli spaces M(O_{a}, O_{a′}) and M^{P} (O_{a}, O_{b}) introduced in Section 6, Theorem 6.1. The obstructed case is discussed in Lemma 6.14 and Proposition 6.17.

For the remaining of this section we assume that Coker(L_{x}) = 0 and Coker(L_{y}) = 0. When these conditions are satisfied, we say that our gluing theory is unobstructed. In this case, we obtain the following result.

Proposition 4.19 Let K be a compact set

K ⊂ M̂(O_{a}, O_{b}) ×_{O_{b}} M̂(O_{b}, O_{c}).

There exist a bound T_{0}(K) > 0 such that, for all T ≥ T_{0}(K) and for all broken trajectories

((A_{1}(t), ψ_{1}(t)), (A_{2}(t), ψ_{2}(t))) ∈ K,

the Fredholm operator L_{x#_{T}^{0} y}

L_{x#_{T}^{0} y} : T_{1}_{x#_{T}^{0} y} → T_{0}_{x#_{T}^{0} y}

is surjective. We are also assuming that Coker(L_{x}) = 0 and Coker(L_{y}) = 0. Then the composition of the pre-gluing map #_{T}^{0} with the orthogonal projection on the actual kernel Ker(L_{x#_{T}^{0} y}) gives an isomorphism

Ker(L_{x}) × Ker(L_{y}) →^{≅} Ker(L_{x#_{T}^{0} y}).  (50)

\underline{Proof:} We know that L_{x#_{T}^{0} y} is Fredholm of index μ(O_{a}) − μ(O_{c}). We also know that dim Ker_{e}(L_{x}) = μ(O_{a}) − μ(O_{b}) + dim(O_{a}) and dim Ker_{e}(L_{y}) = μ(O_{b}) − μ(O_{c}) + dim(O_{b}).

We need to know that for any pair x = (𝔸_{1}, Ψ_{1}) and y = (𝔸_{2}, Ψ_{2}) there is a bound T_{0} = T (x, y) such that L_{x#_{T}^{0} y} is surjective for T ≥ T_{0}. The compactness of K will ensure that there is a uniform such bound T (K).

Lemma 4.13 provides the crucial step in the argument: the linearization L_{x#_{T}^{0} y} has a bounded right inverse when restricted to the orthogonal complement of F_{#}(Ker(L_{x}) × Ker(L_{y})).

Now Proposition 4.19 follows form the splitting of the index 4.18, since we obtain the estimate

dim Ker(L_{x#_{T}^{0} y}) ≤ dim Ker(L_{x}) + dim Ker(L_{y})
= Ind(L_{x}) + Ind(L_{y}) = Ind(L_{x#_{T}^{0} y}) ≤ dim Ker(L_{x#_{T}^{0} y}),

hence

dim Ker(L_{x#_{T}^{0} y}) = dim Ker(L_{x}) + dim Ker(L_{y}).

⋄

In terms of the extended kernels, the isomorphism (50) gives rise to the isomorphism

Ker_{e}(L_{x#_{T}^{0} y}) ≅ Ker_{e}(L_{x}) ×_{ℝ} Ker_{e}(L_{y}).  (51)

The space on the right hand side is the tangent space to the fibered product

M(O_{a}, O_{b}) ×_{O_{b}} M(O_{b}, O_{c}).

Now we can proceed to give a proof of Theorem 4.9. \underline{Proof of Theorem 4.9:} Given the approximate solution obtained via the pregluing map #_{T}^{0} , we prove that this can be perturbed to an actual solution. Similar arguments are presented in [18], [55].

In order to deform a pre-glued approximate solution to an actual solution we need to construct a projection of pre-glued solutions onto M(O_{a}, O_{c}). Thus we need a normal bundle to M(O_{a}, O_{c}) inside B_{kδ}^{0}(O_{a}, O_{c}). We proceed according to the following construction. Let U(O_{a}, O_{c}) be the image in B^{0}(O_{a}, O_{c}) of the pre-gluing map #_{T}^{0} defined on M(O_{a}, O_{b})×_{O_{b}} M(O_{b}, O_{c}). Consider the Hilbert bundles T_{1} and T_{0} given by the L_{1,δ}^{2} and L_{0,δ}^{2} tangent bundles of A^{0}(O_{a}, O_{c}). Choose a slice

S_{Γ_{ac}} ⊂ A(O_{a}, O_{c}),

such that the pre-glued element x#_{T}^{0} y lies in a ball of radius r in A(O_{a}, O_{c}) centered at Γ_{ac}. The flow equation

f(x#_{T}^{0} y) = \{ \begin{matrix}D_{𝔸_{1}#_{T}^{0} 𝔸_{2}}(Ψ_{1}#_{T}^{0} Ψ_{2}) \\ F_{𝔸_{1}#_{T}^{0} 𝔸_{2}}^{+} − (Ψ_{1}#_{T}^{0} Ψ_{2}) · \overline{(Ψ_{1}#_{T}^{0} Ψ_{2})} − iμ − P_{(𝔸_{1}#_{T}^{0} 𝔸_{2},Ψ_{1}#_{T}^{0} Ψ_{2})} \\ G_{Γ_{ac}}^{∗}(𝔸_{1}#_{T}^{0} 𝔸_{2}, Ψ_{1}#_{T}^{0} Ψ_{2}) \end{matrix}

(52) induces a bundle map f : T_{1} → T_{0}. The linearization L_{x#_{T}^{0} y} is the fiber derivative of f.

We assume here that the linearizations L_{x} and L_{y} are surjective. Thus, consider the space

K = ⋃_{K×[T_{0},∞)} Ker_{e}(L_{x}) ×_{ℝ} Ker_{e}(L_{y}).

The image of K under the linearization F_{#} of the pre-gluing map defines a subbundle of T_{1}. We consider the space T_{x#_{T}^{0} y}^{⊥} of elements of the tangent space T_{1} at the point x#_{T}^{0} y that are orthogonal to

T_{x#_{T}^{0} y} = F_{#}(Ker_{e}(L_{x}) ×_{ℝ} Ker_{e}(L_{y})).

The space T^{⊥} gives the normal bundle of the gluing construction. In the case of the fixed point b = θ, we have

K = ∪_{K×[T_{0},∞)}Ker_{e}(L_{x}) × Ker(L_{y})

and the normal bundle of the gluing construction given by

T_{x#_{T}^{0} y}^{⊥} = (F_{#_{T}} (Ker_{e}(L_{x}) × Ker(L_{y})))^{⊥}.

Now we want to define the actual gluing map # that provides a solution of the flow equations in M(O_{a}, O_{c}). This means that we want to obtain a section σ of T_{1} such that the image under the bundle homomorphism given by the flow equation is zero in T_{0}. Moreover, we want this element

σ(x, y, T) = σ(𝔸_{1}, Ψ_{1}, 𝔸_{2}, Ψ_{2}, T)

to converge to zero sufficiently rapidly as T → ∞, so that the glued solution will converge to the broken trajectory in the limit T → ∞.

The perturbation of the approximate solution to an actual solution can be obtained as a fixed point theorem in Banach spaces, via the following contraction principle.

Remark 4.20 Suppose given a smooth map f : E → F between Banach spaces of the form

f(x) = f(0) + Df(0)x + N(x),

with Ker(Df(0)) finite dimensional, with a right inverse Df(0) ∘ G = Id_{F}, and with the nonlinear part N(x) satisfying the estimate

∥GN(x) − GN(y)∥ ≤ C(∥x∥ + ∥y∥)∥x − y∥  (53)

for some constant C > 0 and x and y in a small neighborhood B_{ϵ(C)}(0). Then, with the initial condition ∥G(f(0))∥ ≤ ϵ/2, there is a unique zero x_{0} of the map f in B_{ϵ}(0) ∩ G(F). This satisfies ∥x_{0}∥ ≤ ϵ.

The map f is given in our case by the flow equation, viewed as a bundle homomorphism T_{1} ↦ T_{0}.

Proposition 4.21 Lemma 4.13 and Proposition 4.19 imply that the linearization L_{x#_{T}^{0} y} is uniformly invertible on the orthogonal complement of the actual kernel

T_{x#_{T}^{0} y}^{⊥} = Ker_{e}(L_{x#_{T}^{0} y})^{⊥} ≅ (F_{#_{T}} (Ker_{e}(L_{x}) ×_{ℝ} Ker_{e}(L_{y})))^{⊥}.

Consider the right inverse map of L restricted to T^{⊥},

G : T_{0} → T^{⊥}.

There is a T (K) and a constant C > 0 independent of T, such that we have

∥G_{χ}ξ∥_{L_{1,δ}^{2}} ≤ C∥ξ∥_{L_{0,δ}^{2}}

for χ = (x, y, T) ∈ K × [T (K), ∞). Moreover, we can write f as a sum of a linear and a non-linear term, where the linear term is L and the nonlinear term is

N_{(A(t),ψ(t))}(α, ϕ) = (σ(ϕ, ϕ) + N q_{(A(t),ψ(t))}(α, ϕ), α · ϕ).

Here we write the perturbation q of equation (19) as sum of a linear and a nonlinear term, 2q = Dq + N q. Then the conditions of Remark 4.20 are satisfied. This provides the existence of a unique correction term

σ(𝔸_{1}, Ψ_{1}, 𝔸_{2}, Ψ_{2}, T) ∈ B_{ϵ}(0) ∩ T^{⊥}

satisfying f(σ) = 0. This element σ is smooth and it decays to zero when T is very large, as proved in Section 6, using the estimate (100). This means that the glued solution converges to the broken trajectory (x, y) for T → ∞. The gluing map is given by

(A_{1}#_{T} A_{2}, ψ_{1}#_{T} ψ_{2}) = (A_{1}#_{T}^{0} A_{2}, ψ_{1}#_{T}^{0} ψ_{2}) + σ(𝔸_{1}, Ψ_{1}, 𝔸_{2}, Ψ_{2}, T), (54)

with a rate of decay

∥σ(𝔸_{1}, Ψ_{1}, 𝔸_{2}, Ψ_{2}, T)∥_{L_{1,δ}^{2}} ≤ Ce^{−δT}  (55)

as T → ∞.

In Section 6, in Lemma 6.14 and Proposition 6.17, we shall consider a similar fixed point problem in the presence of obstructions coming form non-vanishing cokernels. The proof of Proposition 4.21 is given in Section 6, after Lemma 6.14, since it follows from the more general case discussed there. The rate of decay ∥σ(𝔸_{1}, Ψ_{1}, 𝔸_{2}, Ψ_{2}, T)∥_{1,δ} ≤ Ce^{−δT} as T → ∞ is derived in Lemma 6.14 in Section 6.2.

Remark 4.22 Notice that, if we have a nontrivial cokernel of L_{x} or L_{y}, then not only T^{⊥} is not be a bundle, but in general L_{x#_{T}^{0} y} may not be invertible on all of the orthogonal complement of the actual kernel. We still know that L_{x#_{T}^{0} y} is uniformly invertible on the orthogonal complement of the approximate kernel, but in this case the kernel is only a proper subspace of the approximate kernel. Thus there may be elements χ = (x, y, T) in K × [T_{0}, ∞) that cannot be perturbed to an actual solution. In fact, as seen in the proof of Lemma 4.13, the condition that the composition F̃_{#_{T}} of the linearization F_{#_{T}} of the pre-gluing map #_{T}^{0} with the orthogonal projection onto Ker(L_{x#_{T}^{0} y}) gives an isomorphism

Ker_{e}(L_{x}) ×_{ℝ} Ker_{e}(L_{y}) →^{≅} Ker_{e}(L_{x#_{T}^{0} y})

is equivalent to the condition that

Coker(L_{x}) × Coker(L_{y}) →^{≅} Coker(L_{x#_{T}^{0} y}).

However, in the general case, where we may have non-trivial Coker(L_{x}) and Coker(L_{y}) we expect to have the isomorphism only at the level of the approximate kernel and cokernel, whereas the actual kernel and cokernel satisfy the weaker condition

dim Ker(L_{x#_{T}^{0} y}) = dim Ker(L_{x}) + dim Ker(L_{y}) − k,

for some k ≥ 0, and

dim Coker(L_{x#_{T}^{0} y}) = dim Coker(L_{x}) + dim Coker(L_{y}) − k,

according to the computation of Lemma 4.13. This case occurs for instance in the case discussed in Section 6, Theorem 6.21 and Theorem 6.22.

To continue with the proof of Theorem 4.9, we now look at the induced map

ˆ# : K̂ × [T_{0}, ∞) → M̂(O_{a}, O_{c}).

Recall that we have an identification of the moduli spaces M̂(O_{a}, O_{b}) with the balanced moduli spaces M^{bal}(O_{a}, O_{b}) of classes in M(O_{a}, O_{b}) that satisfy the equal energy condition (29). Thus, we can define the induced gluing map ˆ# by restricting the gluing map # of (54) to the subspace

M^{bal}(O_{a}, O_{b}) ×_{O_{b}} M^{bal}(O_{b}, O_{c}) ⊂ M(O_{a}, O_{b}) ×_{O_{b}} M(O_{b}, O_{c}),

and then composing the image x#_{T} y ∈ M(O_{a}, O_{c}) with a time translation that determines an element { \atop x#_{T} y} ∈ M^{bal}(O_{a}, O_{c}). We define x ˆ#_{T} y = { \atop x#_{T} y}.

The identification M̂(O_{a}, O_{b}) ≅ M^{bal}(O_{a}, O_{b}) determines an identification of the tangent spaces

T_{x}M(O_{a}, O_{b}) ≅ T_{x}M̂(O_{a}, O_{b}) ⊕ ℝ · U_{x}  (56)

for x ∈ M^{bal}(O_{a}, O_{b}). Here we have

U_{x} = Π_{S}∇C_{ρ}(λ_{1}^{+}x(t)),

where the gauge transformation λ_{1}^{+} is as in (34) and Π_{S} is the projection on the tangent space to the slice T (S_{Γ_{ab}}). Here T denotes the L_{1,δ}^{2} tangent space.

In order to prove that ˆ# is a local diffeomorphism, it suffices to show that the linearization F̃ ˆ#_{T} is an isomorphism, and then prove injectivity of #_{T}. Recall that we have tangent spaces

T_{x}M(O_{a}, O_{b}) ≅ Ker_{e}(L_{x}) and T_{y}M(O_{a}, O_{b}) ≅ Ker_{e}(L_{y}).

We have a decomposition as in (56) with an element U_{y} defined analogously. Moreover, we have the isomorphism of Proposition 4.19, cf. Remark 4.22, which gives an isomorphism

F̃_{#_{T}} : T_{(x,y)}(M(O_{a}, O_{b}) ×_{O_{b}} M(O_{b}, O_{c})) →^{≅} T_{x#_{T} y}M(O_{a}, O_{c}), (57)

where F̃_{#_{T}} is the composition of the linearization F_{#_{T}} of the pre-gluing map with the projection onto the kernel Ker_{e}(L_{x#_{T}^{0} y}).

We can consider the subspace

T_{(x,y)}(M̂(O_{a}, O_{b}) ×_{O_{b}} M̂(O_{b}, O_{c})) ⊕ ℝ · (U_{x}, −U_{y})  (58)

inside the space

\begin{aligned}T_{(x,y)}(M(O_{a}, O_{b}) ×_{O_{b}} M(O_{b}, O_{c})) ≅ & T_{(x,y)}(M̂(O_{a}, O_{b}) ×_{O_{b}} M̂(O_{b}, O_{c})) \\ & ℝ · U_{x} ⊕ ℝ · U_{y}. \end{aligned}

Here we have x and y in M^{bal}(O_{a}, O_{b}) and M^{bal}(O_{b}, O_{c})), respectively. Claim: For all T ≥ T_{0}, sufficiently large, the restriction of the isomorphism (57) to the subspace (58) gives the desired isomorphism

F̃ ˆ#_{T} : T_{(x,y)}(M̂(O_{a}, O_{b}) ×_{O_{b}} M̂(O_{b}, O_{c})) ⊕ ℝ · (U_{x}, −U_{y}) →^{≅} T_{x ˆ#_{T} y}M̂(O_{a}, O_{c}).

\underline{Proof of Claim:} Suppose there exist a sequence T_{n} → ∞ such that there exist (ξ_{n}, η_{n}, τ_{n}) in the space (58), with the property that

F_{#_{Tn}} (ξ_{n}, η_{n}, τ_{n}) ∈ ℝ · U_{T_{n}},

where U_{T_{n}} is defined as

U_{T_{n}} = Π_{S}∇C_{ρ}(λ_{n}(x ˆ#_{T_{n}}y)(t)),

with gauge transformations λ_{n} as in (34) and the projection Π_{S} on the slice S_{Γ_{ac}}. We are using the identification

T_{x#_{Tn} y}M(O_{a}, O_{c}) ≅ T_{x#_{Tn} y}M̂(O_{a}, O_{c}) ⊕ ℝ · U_{T_{n}}.

We have

F_{#_{Tn}} (ξ_{n}, η_{n}, τ_{n}) = τ̃_{n}U_{T_{n}},

for some τ̃_{n} ∈ ℝ. We can normalize the elements (ξ_{n}, η_{n}, τ_{n}) so that τ̃_{n} = 1 for all n. By the result of Proposition 4.21, this implies that the pre-glued elements satisfy

∥F_{#_{Tn}^{0}} (ξ_{n}, η_{n}) + τ_{n}F_{#_{Tn}^{0}} (U_{x}, −U_{y}) − F_{#_{Tn}^{0}} (U_{x}, U_{y})∥_{L_{1,δ(Tn)}^{2}} → 0

as n → ∞. We are using the fact that the error term in the gluing map (54) decays to zero as T_{n} → ∞ as in Proposition 4.21 and in (100). This then implies the following convergence

∥ξ_{n} + (τ_{n} − 1)U_{x}∥_{L_{1,δ}^{2}(Y ×(−∞,T_{n}−1])} → 0

and

∥η_{n} − (1 + τ_{n})U_{y}∥_{L_{1,δ}^{2}(Y ×[+1−T_{n},∞))} → 0,

where we use the norm estimate for the map (46).

This gives a contradiction, according to the decomposition (56), which prescribes that ξ_{n} ⊥ U_{x} and η_{n} ⊥ U_{y}, cf. Proposition 2.56 of [55].

We can estimate explicitly the norm of the isomorphism F̃ ˆ#_{T}. This is bounded by the product of the norm of the linearization F_{#_{T}} at the pre-glued solution (44), the norm of the isomorphism Ker(L_{x#_{T}^{0} y}) ≅ Ker(L_{x#_{T} y}), and the norm of the isomorphism Ker(L_{x#_{T} y}) ≅ Ker(L_{x ˆ#_{T} y}) given by time translation. The first norm is bounded uniformly in T, if we use the rescaled norms L_{1,δ(T_{n})}^{2} on the tangent space of M(O_{a}, O_{c}). The second norm is bounded by a constant C_{2} = C_{2}(K) over the compact set K because of the decay of the error term in the gluing map (54), cf. (100). The third norm is bounded by Ce^{δτ(x#_{T} y)}, where τ(x#_{T} y) is the unique time shift that maps x#_{T} y to the element { \atop x#_{T} y} satisfying the equal energy condition (29). Thus, on the compact set K this norm is also bounded by a term Ce^{δ·τ(K)}. Summarizing, we have obtained the estimate ∥F̃ ˆ#_{T} ∥ ≤ C, on the operator norm of F̃ ˆ#_{T}, uniformly in T, for

F̃ ˆ#_{T} : T_{1,δ}(M(O_{a}, O_{b})) × T_{1,δ}(M(O_{b}, O_{c})) → T_{1,δ(T)}M(O_{a}, O_{c}),

or ∥F̃ ˆ#_{T} ∥ ≤ Ce^{−δT}, for

F̃ ˆ#_{T} : T_{1,δ}(M(O_{a}, O_{b})) × T_{1,δ}(M(O_{b}, O_{c})) → T_{1,δ}M(O_{a}, O_{c}).

We still have to prove the injectivity of the map #_{T} for all sufficiently large T ≥ T_{0}. Suppose there exist a sequence T_{k} → ∞ and elements (x_{1,k}, y_{1,k}) ≠ (x_{2,k}, y_{2,k}) in M^{bal}(O_{a}, O_{b}) ×_{O_{b}} M^{bal}(O_{b}, O_{c}), such that

x_{1,k} ˆ#_{T_{k}}y_{1,k} = x_{2,k} ˆ#_{T_{k}}y_{2,k}

for all k. Then, there exist a based L_{2,δ}^{2} gauge transformation λ on Y × ℝ such that we have

lim_{k→∞} ∥x_{1,k}#_{T_{k}}^{0}y_{1,k} − λ(x_{2,k}#_{T_{k}}^{0}y_{2,k})∥_{L_{1,δ}^{2}} = 0.

We are using here the fact that the time shifts agree as T_{k} → ∞. Upon passing to a subsequence, we know that there are limits x_{i} = lim_{k} x_{i,k} and y_{i} = lim_{k} y_{i,k}, for i = 1, 2, by the assumed compactness of K̂. We obtain

x_{1} ˆ#_{T_{k}}y_{1} = x_{1} ˆ#_{T_{k}}y_{2},

for all T_{k}, but for T_{k} → ∞ we have convergence of x_{i} ˆ#_{T_{k}}y_{i} to the broken trajectory (x_{i}, y_{i}). We obtain (x_{1}, y_{1}) = (x_{2}, y_{2}) = (x, y). Now consider the elements (u_{i,k}, v_{i,k}) = (x_{i,k} − x, y_{i,k} − y), for i = 1, 2. They satisfy (u_{1,k}, v_{1,k}) ≠ (u_{2,k}, v_{2,k}) and

F_{#_{T_{k}}} |_{(x,y)}(u_{1,k}, v_{1,k}) = F_{#_{T_{k}}} |_{(x,y)}(u_{2,k}, v_{2,k}),

for all k, which contradicts the fact that the map F_{#} is a local diffeomorphism, as we just proved.

Now a comment about orientations. Notice that the isomorphism of Proposition 4.19 induces an isomorphism

Λ^{max}Ker_{e}(L_{x}) ⊗ Λ^{max}Ker_{e}(L_{y}) → Λ^{max}Ker_{e}(L_{x#_{T} y}).

Thus, in the unobstructed case we are considering, where all the cokernels are trivial, we obtain that the gluing map #_{T} is compatible with the orientations. Using the decompositions (56) we obtain that the gluing map ˆ#_{T} is also compatible with the induced orientations on the moduli spaces M̂(O_{a}, O_{b}), cf. Proposition 3.6 of [55]. Notice that the orientation on

(M̂(O_{a}, O_{b}) ×_{O_{b}} M̂(O_{b}, O_{c})) × ℝ

induced, under the decomposition (56), by the product orientation on

M(O_{a}, O_{b}) ×_{O_{b}} M(O_{b}, O_{c})

agrees with the pullback of the product orientation of

M̂(O_{a}, O_{b}) × ℝ × M̂(O_{b}, O_{c})

under the map (x, y, T) → (x, T, y).

We now come to the last statement in Theorem 4.9. We prove that any sequence of trajectories in M(O_{a}, O_{c}) converging to a broken trajectory in M(O_{a}, O_{b}) ×_{O_{b}} M(O_{b}, O_{c}) lies eventually in the image of the gluing map.

This requires a preliminary statement about the endpoint maps

e_{b}^{−} : M(O_{a}, O_{b}) → O_{b} and e_{b}^{+} : M(O_{b}, O_{c}) → O_{b}.

Claim: Suppose given a sequence (x_{n}, y_{n}) in

K_{1} × K_{2} ⊂ M(O_{a}, O_{b}) × M(O_{b}, O_{c}),

with K_{1} and K_{2} compact sets in the L_{2,δ}^{2}-topology. Suppose that the endpoints e_{b}^{−}(x_{n}) and e_{b}^{+}(y_{n}) converge to the same element x_{b} on the critical orbit O_{b}, in the L_{1}^{2}-topology on B^{0}. Then there is a subsequence (x′_{n}, y′_{n}) converging to an element (x′, y′) in the fibered product M(O_{a}, O_{b}) ×_{O_{b}} M(O_{b}, O_{c}). \underline{Proof of Claim:} The compactness of K_{1} × K_{2} ensures the existence of a convergent subsequence (x′_{n}, y′_{n}) with limit (x′, y′). Moreover, we can estimate the distance

∥e_{b}^{−}(x′) − e_{b}^{+}(y′)∥_{L_{1}^{2}} ≤ ∥e_{b}^{−}(x_{n}) − e_{b}^{−}(x′)∥_{L_{1}^{2}}+
∥e_{b}^{+}(y_{n}) − e_{b}^{+}(y′)∥_{L_{1}^{2}} + ∥e_{b}^{−}(x_{n}) − e_{b}^{+}(y_{n})∥_{L_{1}^{2}}.

The last term goes to zero by hypothesis, and the first and second term on the right hand side go to zero by the continuity of the endpoint maps. Thus, we have e_{b}^{−}(x′) = e_{b}^{+}(y′) = x_{b}.

Now we return to the statement on the range of the gluing map. This is the analogue in our setting of the method of continuity used in Section 7.3 of [18], and of the arguments of Lemma 4.5.1 and Section 4.6 of [44]

Suppose given a sequence X_{k} of elements in M(O_{a}, O_{c}) which converges smoothly on compact sets (as in Theorem 4.1) to a broken trajectory (x, y) in the fibered product M(O_{a}, O_{b}) ×_{O_{b}} M(O_{b}, O_{c}).

For a sequence of gluing parameters T_{k} → ∞, let (x_{k}, y_{k}) be the projection onto the slices S_{Γ_{ab}} × S_{Γ_{bc}},

S_{Γ_{ab}} ⊂ A_{k,δ}(O_{a}, O_{b}) and S_{Γ_{bc}} ⊂ A_{k,δ}(O_{b}, O_{c}),

of the cut off elements (ρ_{1−T_{k}}^{−}X_{k}^{−T_{k}}, ρ_{−1+T_{k}}^{+}X_{k}^{T_{k}}). The elements x_{k} and y_{k} are no longer solutions of the Seiberg–Witten equations. However, we can choose the sequence T_{k} → ∞ such that the elements x_{k} and y_{k} lie respectively within neighborhoods of radius ϵ/4 of the solutions x and y. We choose ϵ so that it satisfies

ϵ ≤ min{ϵ_{ac}, ϵ_{ab}, ϵ_{bc}}.

Here we have ϵ_{ab} and ϵ_{bc} determined as in Lemma 6.14, for the contraction principle for the fixed point problem (99) in B_{2,δ}(O_{a}, O_{b}) and in B_{2,δ}(O_{b}, O_{c}), respectively. Similarly, the constant ϵ_{ac} is the constant for the contraction principle of Lemma 6.14 in B_{2,δ}(O_{a}, O_{c}). (cf. Proposition 4.21 and Remark 4.20.) By applying the result of Proposition 4.21, there are then unique elements x̃_{k} and ỹ_{k} in a ϵ/2–neighborhood of x_{k} and y_{k}, respectively, which satisfy the equations. Notice that these elements (x̃_{k}, ỹ_{k}) will in general have e_{b}^{−}(x̃_{k}) ≠ e_{b}^{+}(ỹ_{k}), hence they do not define an element in the fibered product. However, we can estimate that the distance between the endpoints goes to zero. In fact, we have

∥e_{b}^{−}(x̃_{k}) − e_{b}^{+}(ỹ_{k})∥_{L_{1}^{2}} ≤
C(dist_{L_{1}^{2}}(λ_{k}^{−}x̃_{k}(t), e_{b}^{−}(x̃_{k})) + dist_{L_{1}^{2}}(λ_{k}^{+}ỹ_{k}(t), e_{b}^{+}(ỹ_{k})))
+C(∥x̃_{k} − x∥_{L_{2,δ}^{2}} + ∥ỹ_{k} − y∥_{L_{2,δ}^{2}})
+C(dist_{L_{1}^{2}}(λ^{−}x(t), e_{b}^{−}(x)) + dist_{L_{1}^{2}}(λ^{+}y(t), e_{b}^{+}(y))).

The first two terms and last two terms on the right hand side decay exponentially like Ce^{−δ|t|} by the results of Section 3.4, Theorem 3.12. The gauge elements λ_{k}^{−}, λ_{k}^{+}, λ^{−}, and λ^{+} are as in (34), cf. Lemma 4.8. The remaining term is bounded by ϵ.

Since in the statement of Theorem 4.9 we are only interested in the codimension one boundary, we may as well assume that, upon passing to a subsequence, the sequence (x̃_{k}, ỹ_{k}) converges in the strong topology to a pair (x′, y′). By the result of the previous Claim, we know that this element (x′, y′) is in the fibered product M(O_{a}, O_{b}) ×_{O_{b}} M(O_{b}, O_{c}).

Now consider the pre-glued solutions x′#_{T_{k}}^{0}y′. By our construction, the original elements X_{k} lie within neighborhoods of radius ϵ of x′#_{T_{k}}^{0}y′ in the L_{2,δ}^{2}–norms. By Lemma 6.14, and Remark 4.20 there is a unique zero of the map f in this neighborhood, that is, a unique solution of the Seiberg–Witten equations, obtained as a small deformation of the approximate solution x′#_{T_{k}}^{0}y′. This implies the desired equality X_{k} = x′#_{T_{k}}y′.

This completes the proof of Theorem 4.9. ⋄

Before discussing the fine structure of the compactification, we can add a brief comment about the gluing result in the non-equivariant setting, as stated in Lemma 2.14. The difference in the non-equivariant setting is the presence of the extra U(1)–gluing parameter, in gluing across the reducible θ. Namely, the gluing map is of the form

# : M̂(a, θ) × M̂(θ, c) × U(1) × [T_{0}, ∞) → M̂(a, c).

The reason for the presence of the extra U(1) gluing parameter is the fact that the reducible point θ is not a smooth point in the non-equivariant moduli space, hence, in order to formulate the pre-gluing and gluing construction, it is necessary to lift the pre-gluing to the framed moduli space, in the proximity of the reducible point. More explicitly, we define the pre-gluing map as

x#{0 \atop T,u}y = \{ \begin{matrix}λ{+ \atop 1}(𝔸{2T \atop 1} , Ψ{2T \atop 1} ) & t ≤ −2 \\ λ0(A0 + ν, 0) + exp(iu(t + 2))(α1(t + 2T), ϕ1(t + 2T)) & −2 ≤ t ≤ −1 \\ λ0(A0 + ν, 0) + exp(iu)· & −1 ≤ t ≤ 1 \\ (ρ−(t)(α1(t + 2T), ϕ1(t + 2T)) + ρ+(t)(α2(t − 2T), ϕ2(t − 2T))) \\ λλ{− \atop 2}(𝔸{−2T \atop 2} , Ψ{−2T \atop 2} ) & t ≥ 1 \end{matrix}  (59)

with the extra gluing parameter λ = exp(iu) in U(1).

This pre-gluing map gives a choice of a lift of

M̂(a, θ) × M̂(θ, c) × U(1)

to M̂(O_{a}, θ)×M̂(θ, O_{c}), then the gluing construction works as in the equivariant case and composition with the projection onto the quotient by the U(1) action then provides the resulting gluing map with values in M̂(a, c).

4.3 Multiple gluing theorem and corner structure

We now generalize the result of Theorem 4.9 of the previous subsection to the case of multiple gluings of broken trajectories in boundary strata of higher codimension. The purpose is to identify the fine structure of the compactification, namely to show that the spaces M̂(O_{a}, O_{b}) of flow lines compactify to a C^{∞} manifold with corners, in the sense of [42]. We shall follow the notation M̂(O_{a}, O_{b})^{∗} to distinguish the compactification from the original space M̂(O_{a}, O_{b}).

Theorem 4.23 The compactification M̂(O_{a}, O_{b})^{∗} has the structure of a smooth manifold with corners, with codimension k boundary faces of the form

⋃_{c_{1},···c_{k}} M̂(O_{a}, O_{c_{1}})^{∗} ×_{O_{c_{1}}} M̂(O_{c_{1}}, O_{c_{2}})^{∗} × ⋯ ×_{O_{c_{k}}} M̂(O_{c_{k}}, O_{b})^{∗}.  (60)

Here the union is over all possible sequences of critical orbits O_{c_{1}}, ⋯, O_{c_{k}} with decreasing indices.

We also have the following.

Corollary 4.24 The endpoint maps e_{a}^{+} and e_{b}^{−} and their derivatives extend continuously over the boundary and on the boundary they coincide with e_{a}^{+} and e_{b}^{−} on M̂(O_{a}, O_{c_{1}}) and M̂(O_{c_{k}}, O_{b}) respectively. Thus, the maps e_{a}^{+} and e_{b}^{−} are fibrations with compact fibers in the category of smooth manifolds with corners.

\underline{Proof of Theorem 4.23:} We proceed as follows. First we prove that M̂(O_{a}, O_{b})^{∗} is a t-manifold (has a C^{∞} structure with corners) in the sense of Definition 1.6.1 of [42]. This amounts to showing that the multiple gluing maps define a compatible set of charts in the sense of Section 1.6 of [42] on open sets in M̂(O_{a}, O_{b}) near the boundary. We then show that this t-manifold has the structure of a smooth manifold with corners, in the sense of Section 1.8 of [42], by showing that the boundary faces satisfy the condition (1.8.7) of [42].

Consider broken trajectories (x_{0}, x_{1}, …, x_{k}) in a compact subset

K̂ ⊂ M̂(O_{a}, O_{c_{1}}) ×_{O_{c_{1}}} M̂(O_{c_{1}}, O_{c_{2}}) × ⋯ ×_{O_{c_{k}}} M̂(O_{c_{k}}, O_{b}),

where O_{c_{1}}, …, O_{c_{k}} are the critical points with decreasing indices

μ(O_{a}) > μ(O_{c_{1}}) > μ(O_{c_{2}}) > ⋯ > μ(O_{c_{k}}) > μ(O_{b}).

We introduce multiple gluing maps

ˆ#_{i_{1},...,i_{k}} : K̂ × [T_{0}, ∞)^{k} → M̂(O_{a}, O_{b}).

Here (T_{1}, …, T_{k}) ∈ [T_{0}, ∞)^{k} are gluing parameters, and the indices {i_{1}, …, i_{k}} are a permutation of the set {1, …, k} which specifies in which order the multiple gluing is performed as a sequence of k gluings as in Theorem 4.9. For instance, we have

ˆ#_{1,2,...,k}^{T_{1},...,T_{k}}(x_{0}, x_{1}, …, x_{k}) = (⋯ (((x_{0} ˆ#_{T_{1}}x_{1}) ˆ#^{T_{2}}x_{2}) ˆ#_{T_{3}}x_{4}) ⋯) ˆ#_{T_{k}}x_{k},

or

ˆ#_{2,1,3,...,k}^{T_{1},...,T_{k}}(x_{0}, x_{1}, …, x_{k}) = (⋯ (x_{0} ˆ#_{T_{1}}(x_{1} ˆ#^{T_{2}}x_{2})) ˆ#_{T_{3}}x_{3}) ⋯) ˆ#_{T_{k}}x_{k},

etc.

The gluing construction is non-canonical, in the sense that the identification of the normal bundle T^{⊥} in the pre-gluing construction, as in Lemma 4.13, is dependent on the order of the gluing, hence there is no obvious associativity law for multiple gluings. However, we are going to show that the maps ˆ#_{i_{1},...,i_{k}}^{T_{1},...,T_{k}}, or rather their inverses, to be consistent with the notation of (1.6.1) of [42], define a system of charts of a C^{∞} structure with corners, or t-manifold. We prove the following result, which ensures that the ˆ#_{i_{1},...,i_{k}}^{T_{1},...,T_{k}} are local diffeomorphisms as needed.

Proposition 4.25 Suppose given O_{a}, O_{c_{1}}, ⋯ O_{c_{k}} and O_{b} in M^{0} with decreasing indices. Then, given a compact set K̂ in

M̂(O_{a}, O_{c_{1}}) ×_{O_{c_{1}}} M̂(O_{c_{1}}, O_{c_{2}}) × ⋯ ×_{O_{c_{k}}} M̂(O_{c_{k}}, O_{b}),

there is a lower bound T_{0}(K) > 0 such that the gluing maps ˆ#_{i_{1},...,i_{k}}^{T_{1},...,T_{k}} define local diffeomorphisms

ˆ#_{i_{1},...,i_{k}} :  K̂ × [T_{0}(K), ∞)^{k} ⟶ M̂(O_{a}, O_{b}).

where the orientation on the left hand side is the one induced by the product orientation on

(((M̂(O_{a}, O_{c_{1}}) × ℝ × M̂(O_{c_{1}}, O_{c_{2}})) × ℝ) × ⋯ × ℝ) × M̂(O_{c_{k}}, O_{b}).

\underline{Proof:} We want to generalize the analogous statement proved in Theorem 4.9 for the case of one intermediate critical point.

First, we need to show the analogue of the error estimate (55), which shows that the glued solutions #_{i_{1},...,i_{k}}^{T_{1},...,T_{k}}(x_{0}, …, x_{k}) converge to broken trajectories in the limit when a certain subset {T_{i_{1}}, … T_{i_{m}}} of the gluing parameters satisfies T_{i_{ℓ}} → ∞.

We then need to show that a sequence of solutions in M̂(O_{a}, O_{b}) that converges to a broken trajectory (x_{0}, …, x_{k}) in the boundary

M̂(O_{a}, O_{c_{1}}) ×_{O_{c_{1}}} ⋯ ×_{O_{c_{k}}} M̂(O_{c_{k}}, O_{b})

lies eventually in the range of a gluing map #_{i_{1},...,i_{k}}^{T_{1},...,T_{k}}. We first prove that, when the gluing parameters {T_{1}, …, T_{k}} satisfy T_{i} → ∞, for all i = 1, …, k, we have an estimate

∥σ_{k}(x_{0}, …, x_{k}, T_{1}, …, T_{k})∥_{L_{1,δ}^{2}} ≤ Ce^{−δT} ,  (61)

with T = min{T_{1}, …, T_{k}}. This is the analogue of the estimate (100) for multiple gluing. Here σ_{k} is the error term

σ_{k}(x_{0}, …, x_{k}, T_{1}, …, T_{k}) = #_{i_{1},...,i_{k}}^{T_{1},...,T_{k}}(x_{0}, …, x_{k}) − #^{0}_{i_{1},...,i_{k}}^{T_{1},...,T_{k}}(x_{0}, …, x_{k}),

with #^{0}_{i_{1},...,i_{k}}^{T_{1},...,T_{k}} the pre-gluing map, namely the composite of k pre-gluing maps as in (35), in the order specified by (i_{1}, …, i_{k}).

We prove the estimate (61) for the case of the gluing map #_{1,...,k}^{T_{1},...,T_{k}}. The same procedure works with a permutation of the order of gluing. We proceed by induction on k. The case k = 1 follows from Theorem 4.9. Assume as induction hypothesis that the gluing map #_{1,...,k−1}^{T_{1},...,T_{k−1}} can be written as

#_{1,...,k−1}^{T_{1},...,T_{k−1}}(x_{0}, …, x_{k−1}) =
#^{0}_{1,...,k−1}^{T_{1},...,T_{k−1}}(x_{0}, …, x_{k−1}) + σ_{k−1}(x_{0}, …, x_{k−1}, T_{1}, …, T_{k−1}),

with #^{0}_{1,...,k−1}^{T_{1},...,T_{k−1}} the pre-gluing map (i.e. the composition of k − 1 pre-gluing maps (35), and with the error term σ_{k−1} satisfying the estimate

∥σ_{k−1}(x_{0}, …, x_{k−1}, T_{1}, …, T_{k−1})∥_{L_{1,δ}^{2}} ≤ Ce^{−δT} ,

with T = min{T_{1}, …, T_{k_{1}}}. We can then form the pre-glued solution y#_{T_{k}}^{0}x_{k}, with

y = #_{1,...,k−1}^{T_{1},...,T_{k−1}}(x_{0}, …, x_{k−1}).

By Proposition 4.21 this approximate solution can be deformed to an actual solution y#_{T_{k}}x_{k}, which by definition is the same as

y#_{T_{k}}x_{k} = #_{1,...,k}^{T_{1},...,T_{k}}(x_{0}, …, x_{k}).

We need to estimate the difference y#_{T_{k}}x_{k} − y_{0}#_{T_{k}}^{0}x_{k}, where y_{0} is the approximate solution

y_{0} = #^{0}_{1,...,k−1}^{T_{1},...,T_{k−1}}(x_{0}, …, x_{k−1}).

We have

\begin{matrix}∥y#_{T_{k}}x_{k} − y_{0}#_{T_{k}}^{0}x_{k}∥_{L_{1,δ}^{2}} ≤ ∥y#_{T_{k}}x_{k} − y#_{T_{k}}^{0}x_{k}∥_{L_{1,δ}^{2}} \\ +∥y#_{T_{k}}^{0}x_{k} − y_{0}#_{T_{k}}^{0}x_{k}∥_{L_{1,δ}^{2}}. \end{matrix}  (62)

Now we can estimate the second term on the right hand side by

∥y#_{T_{k}}^{0}x_{k} − y_{0}#_{T_{k}}^{0}x_{k}∥_{L_{1,δ}^{2}} ≤ ∥F_{#_{T_{k}}} ∥ · ∥y − y_{0}∥_{L_{1,δ}^{2}}.

We know that (44) has norm bounded by Ce^{−δT_{k}}. Moreover, the quantity ∥y − y_{0}∥ is given by

∥σ_{k−1}(x_{0}, …, x_{k−1}, T_{1}, …, T_{k−1})∥_{L_{1,δ}^{2}}.

Using the induction hypothesis, we obtain

∥y#_{T_{k}}^{0}x_{k} − y_{0}#_{T_{k}}^{0}x_{k}∥_{L_{1,δ}^{2}} ≤ Ce^{−δT} .

The first term in the right hand side of (62), on the other hand, is given by

∥y#_{T_{k}}x_{k} − y#_{T_{k}}^{0}x_{k}∥_{L_{1,δ}^{2}} = ∥σ(y, x_{k}, T_{k})∥_{L_{1,δ}^{2}}.

By the estimate (100) (cf. Proposition 4.21, Lemma 6.14, Remark 4.20, and Proposition 4.21), this satisfies the estimate

∥σ(y, x_{k}, T_{k})∥_{L_{1,δ}^{2}} ≤ Ce^{−δT_{k}}.

Thus, we have the next step of the induction, namely we have obtained

∥σ_{k}(x_{0}, …, x_{k}, T_{1}, …, T_{k})∥_{L_{1,δ}^{2}} ≤ Ce^{−δ·min{T,T_{k}}},

for

σ_{k}(x_{0}, …, x_{k}, T_{1}, …, T_{k}) = y#_{T_{k}}x_{k} − y_{0}#_{T_{k}}^{0}x_{k}.

Now consider a sequence X_{ℓ} in M(O_{a}, O_{b}) that converges smoothly on compact sets, in the sense of Theorem 4.9, to a broken trajectory (x_{0}, …, x_{k}) in the fibered product

M(O_{a}, O_{c_{1}}) ×_{O_{c_{1}}} ⋯ ×_{O_{c_{k}}} M(O_{c_{k}}, O_{b}).

We need to show that these X_{ℓ} belong eventually to the range of at least one of the multiple gluing maps #_{i_{1},···,i_{k}}. We proceed as in the analogous argument in Theorem 4.9, for a single gluing. Convergence in the sense of Theorem 4.9 implies that there exist time shifts {T_{1,ℓ}, …, T_{k,ℓ}} with T_{i,ℓ} → ∞ for i = 1, … k, and combinatorial data which describe the order in which successive limits on compact sets and reparameterizations are considered in Theorem 4.9. These data determine which gluing maps #_{i_{1},···,i_{k}} have the points X_{ℓ} in the range. These combinatorial data can be described as the set of binary trees with k + 1 leaves, one root, and k − 1 intermediate nodes. To one such tree we associate a cutoff element

(η_{0}^{T_{1,ℓ},...,T_{k,ℓ}}(X_{ℓ}), …, η_{ℓ}^{T_{1,ℓ},...,T_{k,ℓ}}(X_{ℓ}))  (63)

where the η_{i}^{T_{1,ℓ},...,T_{k,ℓ}} act as a composition of time shifts and cutoff functions, as specified by the combinatorics of the tree. For instance, the two trees of the

figure correspond respectively to the elements

(ρ_{1−T_{1,ℓ}}^{−}(ρ_{1−T_{2,ℓ}}^{−}(ρ_{1−T_{3,ℓ}}^{−}X_{ℓ}^{−T_{3,ℓ}})^{−T_{2,ℓ}})^{−T_{1,ℓ}},
ρ_{−1+T_{1,ℓ}}^{+}(ρ_{1−T_{2,ℓ}}^{−}(ρ_{1−T_{3,ℓ}}^{−}X_{ℓ}^{−T_{3,ℓ}})^{−T_{2,ℓ}})^{T_{1,ℓ}},
ρ_{−1+T_{2,ℓ}}^{+}(ρ_{1−T_{3,ℓ}}^{−}X_{ℓ}^{−T_{3,ℓ}})^{T_{2,ℓ}},
ρ_{−1+T_{3,ℓ}}^{+}X_{ℓ}^{T_{3,ℓ}})

and

(ρ_{1−T_{1,ℓ}}^{−}(ρ_{1−T_{3,ℓ}}^{−}X_{ℓ}^{−T_{3,ℓ}})^{−T_{1,ℓ}},
ρ_{1−T_{1,ℓ}}^{−}(ρ_{−1+T_{2,ℓ}}^{+}(ρ_{1−T_{3,ℓ}}^{−}X_{ℓ}^{−T_{3,ℓ}})^{T_{2,ℓ}})^{−T_{1,ℓ}},
ρ_{−1+T_{1,ℓ}}^{+}(ρ_{−1+T_{2,ℓ}}^{+}(ρ_{1−T_{3,ℓ}}^{−}X_{ℓ}^{−T_{3,ℓ}})^{T_{2,ℓ}})^{T_{1,ℓ}},
ρ_{−1+T_{3,ℓ}}^{+}X_{ℓ}^{T_{3,ℓ}}).

These correspond to cases where a sequence of solutions X_{ℓ} in, say, M(O_{a}, O_{b}) converges smoothly on compact sets, after the different reparameterizations specified above, to elements in M(O_{a}, O_{c_{1}}), M(O_{c_{1}}, O_{c_{2}}), M(O_{c_{2}}, O_{c_{3}}), and M(O_{c_{3}}, O_{b}), respectively.

We denote by (x_{1,ℓ}, …, x_{k,ℓ}) the projection onto the slices of the cutoff elements (63). The elements x_{i,ℓ} are no longer solutions of the Seiberg–Witten equations, however, by hypothesis, for ℓ large enough, they are contained in ϵ/2^{k+1}-neighborhoods of the solutions x_{i}, for i = 0, …, k. Pick ϵ satisfying

ϵ ≤ max_{0≤j≤k+1}{ϵ_{c_{j}c_{j+1}}},

where the ϵ_{c_{j}c_{j+1}} is the constant for the contraction principle, as in Lemma 6.14 in B_{2,δ}(O_{c_{j}}, O_{c_{j+1}}), and we take c_{0} = a and c_{k+1} = b. Then, proceeding as in the proof of Theorem 4.9, we find elements x̃_{i,ℓ}, for i = 0, … k that satisfy the Seiberg–Witten equations and are contained in ϵ/2^{k}-neighborhoods of the elements x_{i}. The elements (x̃_{0,ℓ}, …, x̃_{k,ℓ}) are not necessarily in the fibered product, however, by the same argument used in the proof of Theorem 4.9, we obtain convergence of the elements (x̃_{0,ℓ}, …, x̃_{k,ℓ}) to a limit (x′_{0}, …, x′_{k}) in the fibered product

M(O_{a}, O_{c_{1}}) ×_{O_{c_{1}}} ⋯ ×_{O_{c_{k}}} M(O_{c_{k}}, O_{b}).

By construction, the approximate solution

#^{0}_{i_{1},...i_{k}}^{T_{1,ℓ},...,T_{k,ℓ}}(x′_{0}, … x′_{k}),

with the order (i_{1}, … i_{k}) of gluing specified by the combinatorial data, is contained in an ϵ-neighborhood of X_{ℓ} in the L_{2,δ}^{2}-norm. By the contraction argument of Lemma 6.14 together with the first part of the proof of this Proposition, we know that we must then have

X_{ℓ} = #_{i_{1},...i_{k}}^{T_{1,ℓ},...,T_{k,ℓ}}(x′_{0}, … x′_{k}).

This completes the proof. ⋄

Thus, the changes of coordinates

ˆ#_{i_{1},...,i_{k}}^{T_{1},...,T_{k}} ∘ ( ˆ#_{j_{1},...,j_{ℓ}}^{τ_{1},...,τ_{ℓ}})^{−1}

are local diffeomorphisms between open subsets of

ℝ_{k}^{n} = ℝ^{n−k} × [T_{0}, ∞)^{k}

and

ℝ_{ℓ}^{n} = ℝ^{n−ℓ} × [T_{0}, ∞)^{ℓ},

with n = dim M̂(O_{a}, O_{b}), as prescribed in Section 1.6 of [42].

Notice, in particular, that given two multiple gluing maps #_{i_{1},...,i_{k}}^{T_{1},...,T_{k}} and #_{j_{1},...j_{k}}^{T_{1},...,T_{k}}, for two different orders of gluing (i_{1}, …, i_{k}) ≠ (j_{1}, … j_{k}), we obtain an estimate

∥#_{i_{1},...,i_{k}}^{T_{1},...,T_{k}}(x_{0}, …, x_{k}) − #_{j_{1},...j_{k}}^{T_{1},...,T_{k}}(x_{0}, …, x_{k})∥_{L_{1,δ}^{2}} ≤ Ce^{−δT} ,  (64)

for T = min{T_{1}, …, T_{k}}. The proof of this estimate is completely analogous to the argument used in the proof of Proposition 4.25. For instance, we can estimate

∥(x_{0}#_{T_{1}}x_{1})#_{T_{2}}x_{2} − x_{0}#_{T_{1}}(x_{1}#_{T_{2}}x_{2})∥ ≤

∥(x_{0}#_{T_{1}}x_{1})#_{T_{2}}x_{2} − (x_{0}#_{T_{1}}x_{1})#_{T_{2}}^{0}x_{2}∥ + ∥(x_{0}#_{T_{1}}x_{1})#_{T_{2}}^{0}x_{2} − x_{0}#_{T_{1}}^{0}x_{1}#_{T_{2}}^{0}x_{2}∥

+∥x_{0}#_{T_{1}}^{0}x_{1}#_{T_{2}}^{0}x_{2} − x_{0}#_{T_{1}}^{0}(x_{1}#_{T_{2}}x_{2})∥ + ∥x_{0}#_{T_{1}}^{0}(x_{1}#_{T_{2}}x_{2}) − x_{0}#_{T_{1}}(x_{1}#_{T_{2}}x_{2})∥

≤ Ce^{−δT_{2}} + Ce^{−δT_{1}+T_{2}} + Ce^{−δT_{1}+T_{2}} + Ce^{−δT_{1}}.

The estimate (64) implies that, when all the T_{i} go to infinity, the glued elements #_{i_{1},...,i_{k}}^{T_{1},...,T_{k}}(x_{0}, …, x_{k}) and #_{j_{1},...j_{k}}^{T_{1},...,T_{k}}(x_{0}, …, x_{k}) end up in the same coordinate patch. This may not be the case if only some of the T_{i} tend to infinity and other remain bounded.

We have shown, as a result of Proposition 4.25, and by Lemma 1.7.1 of [42], that the strata

M̂(O_{a}, O_{c_{1}})^{∗} ×_{O_{c_{1}}} × ⋯ ×_{O_{c_{k}}} M̂(O_{c_{k}}, O_{b})^{∗}

inherit the structure of smooth t-sub-manifolds of codimension k in M̂(O_{a}, O_{b})^{∗}.

The codimension one boundary strata described in Theorem 4.8, compactified to smooth t-manifolds, define the boundary hypersurfaces of M̂(O_{a}, O_{b})^{∗}, as in (1.8.3) and (1.8.4) of [42].

Let us recall that a C^{∞} manifold with corners is a t-manifold where all the boundary faces are C^{∞} embedded sub-manifolds. According to Section 1.8 of [42], to ensure that this is the case, namely that the compactification M̂(O_{a}, O_{b})^{∗} has the structure of a C^{∞} manifold with corners, one only needs to check the following fact.

Lemma 4.26 Every compactified codimension k boundary face

M̂(O_{a}, O_{c_{1}})^{∗} ×_{O_{c_{1}}} × ⋯ ×_{O_{c_{k}}} M̂(O_{c_{k}}, O_{b})^{∗}

of M̂(O_{a}, O_{b})^{∗} is a component of (precisely) one intersection of k boundary hypersurfaces

H_{i_{1}} ∩ … ∩ H_{i_{k}}.

\underline{Proof of Lemma 4.26:} All the boundary hypersurfaces are identified by Theorem 4.9. Thus, we see that the Lemma is verified by setting

H_{i_{1}}  = M̂(O_{a}, O_{c_{1}})^{∗} ×_{O_{c_{1}}} M̂(O_{c_{1}}, O_{b})^{∗}
H_{i_{2}}  = M̂(O_{a}, O_{c_{2}})^{∗} ×_{O_{c_{2}}} M̂(O_{c_{2}}, O_{b})^{∗}
⋮  ⋮
H_{i_{k−1}} = M̂(O_{a}, O_{c_{k−1}})^{∗} ×_{O_{c_{k−1}}} M̂(O_{c_{k−1}}, O_{b})^{∗}
H_{i_{k}} = M̂(O_{a}, O_{c_{k}})^{∗} ×_{O_{c_{k}}} M̂(O_{c_{k}}, O_{b})^{∗}

⋄

This completes the proof of Theorem 4.23. ⋄

Corollary 4.24 now follows, by considering the restriction of the asymptotic value maps e_{a}^{+} and e_{b}^{−} to the range of the gluing maps. By the convergence property, when a subset of the gluing parameters goes to infinity, we obtain that the asymptotic value maps restrict to the corresponding maps on the boundary strata.

5 Equivariant Homology

Let us recall briefly the construction of the de Rham model for U(1)-equivariant cohomology (and homology) on a finite dimensional manifold with a U(1) action. The main reference is [2].

Let W be the Weil algebra of the Lie algebra iℝ of U(1). This is a free commutative graded algebra in one generator θ of degree 1 and one generator Ω of degree 2, with differential δ that satisfies

δθ = Ω δΩ = 0.

Let M be a manifold with a U(1) action. Consider the complex

C^{∗} = W ⊗ Ω^{∗}(M),

with differential

d_{U(1)} = d − Ωc(T).

Here T is the vector field on M generated by the infinitesimal U(1) action and c is the unique derivation in W that satisfies

c(θ) = 1 c(Ω) = 0.

We choose a sub-complex of C^{∗} by taking the cochains on which c + c(T) and L(T) vanish, where L(T) is the Lie derivative. Let us call this sub-complex Ω_{U(1)}^{∗}(M). An alternative description of the complex Ω_{U(1)}^{∗}(M) is

Ω_{U(1)}^{∗}(M) = ℝ[Ω] ⊗ Ω_{0}^{∗}(M),

where Ω_{0}^{∗}(M) are de Rham forms that are annihilated by the Lie differentiation L(T). The cohomology H^{∗}(Ω_{U(1)}^{∗}(M), d_{U(1)}) is isomorphic to the equivariant cohomology with real coefficients,

H^{∗}(Ω_{U(1)}^{∗}(M), d_{U(1)}) ≅ H_{U(1)}^{∗}(M, ℝ).

In order to compute equivariant homology with real coefficients a de Rham complex can be constructed as in [8] by considering differential forms graded by the dimension of M minus the degree,

Ω_{∗}_{U(1)}(M) = ℝ[Ω] ⊗ Ω_{0}^{dim(M)−∗}(M),

with boundary operator ∂_{U(1)} = d − c(θ) ⊗ c(T). Austin and Braam [8] proved that this complex computes the same homology as the equivariant complex of currents introduced by Duflo and Vergne [19] and studied by Kumar and Vergne [32]. With this understood, we can consider the complex Ω_{∗}_{U(1)}(O_{a}) associated to each critical orbit O_{a} in M_{0}. This will be a copy of the polynomial algebra ℝ[Ω] for the fixed point and a complex of ℝ[Ω]-modules with de Rham forms in degree zero and one in the case of orbits that come from irreducibles.

We can give a more explicit description of the boundary operator ∂_{U(1)} in this case. Let O_{a} be a critical orbit with a free U(1) action. Then, the generators of Ω_{0}^{1−∗}(O_{a}) are a 1-form η_{a} that generates H^{1}(S^{1}), in degree zero, and a zeroform 1_{a} (the constant function equal to one on O_{a}), in degree one. Thus we have

∂_{U(1)}(Ω^{n} ⊗ 1_{a}) = 0

and

∂_{U(1)}(Ω^{n} ⊗ η_{a}) = −Ω^{n−1} ⊗ 1_{a}.

5.1 The equivariant complex

We can form the bigraded complex that computes equivariant Floer homology as in [6], defined by

C_{k}_{U(1)}(Y) = ⨁_{μ(O_{a})=i,i+j=k} Ω_{j}_{U(1)}(O_{a}),  (65)

with differentials

D_{a,b}η = \{ \begin{matrix}∂_{U(1)}η & O_{a} = O_{b} \\ (−1)^{1−r(η)}(e_{b}^{−})_{∗}(e_{a}^{+})^{∗}η & μ(O_{a}) > μ(O_{b}) \\ 0 & otherwise. \end{matrix}  (66)

Here η is an equivariant differential form on the orbit O_{a}, that is, an element of Ω_{∗}_{U(1)}(O_{a}). The number r(η) is the de Rham degree of η, that is, the maximum degree of the elements of Ω_{0}^{∗}(O_{a}) that appear in the expression of η. Recall that Ω_{∗}_{U(1)}(O_{a}) is the dual of the de Rham complex, and the forms are graded by dim(O_{a}) − ∗ = 1 − ∗.

The analogous complex that computes equivariant Floer cohomology is given by

C^{k}_{U(1)}(Y) = ⨁_{μ(O_{a})=i,i+j=k} Ω^{j}_{U(1)}(O_{a}),  (67)

with coboundaries

δ_{a,b}η = \{ \begin{matrix}d_{U(1)}η & O_{a} = O_{b} \\ (−1)^{r(η)}(e_{a}^{+})_{∗}(e_{b}^{−})^{∗}η & μ(O_{a}) > μ(O_{b}) \\ 0 & otherwise. \end{matrix}  (68)

Here η is an equivariant form, that is an element of O_{b} and r(η) is the de Rham degree of η, i.e. the maximum degree of the elements of Ω_{0}^{∗}(O_{b}) that appear in the expression of η.

The boundary map is well defined, since the endpoint maps are compatible with the boundary strata as stated in 4.24. Notice that, as pointed out in [6], for dimensional reasons the maps D_{a,b} are trivial whenever μ(O_{a}) ≥ μ(O_{b}) + 3. A more explicit description of the boundary map will be given in the following.

5.2 Equivariant Floer Homology

We can prove that the composites D^{2} and δ^{2} are zero. Moreover we can see that there is a duality at the level of forms that induces a duality between equivariant Floer homology and cohomology as in [6] and [8].

Theorem 5.1 The composite maps D^{2} and δ^{2} are zero, this means that the identities

D_{a,c}^{2} = ∑_{b} D_{a,b}D_{b,c} = 0,

δ_{a,c}^{2} = ∑_{b} δ_{b,c}δ_{a,b} = 0,

hold, with O_{b} that ranges among critical orbits satisfying μ(O_{a}) ≥ μ(O_{b}) ≥ μ(O_{c}). Moreover there is a pairing <, > of forms in C_{∗}_{U(1)} and in C^{∗}_{U(1)} that satisfies

< Dη, γ >=< η, δγ > .

\underline{Proof:} The statement is true for O_{a} = O_{c}. Given critical orbits with μ(O_{a}) > μ(O_{c}), we have the expression

D_{a,c}^{2}η = (−1)^{r(η)} ∑_{{O_{b}|μ(O_{a})>μ(O_{b})>μ(O_{c})}}(−1)^{r((e_{b}^{−})_{∗}(e_{a}^{+})^{∗}η)}

(e_{c}^{−})_{∗}(e_{b}^{+})^{∗}(e_{b}^{−})_{∗}(e_{a}^{+})^{∗}η + −(e_{c}^{−})_{∗}(e_{a}^{+})^{∗}∂_{U(1)}η + ∂_{U(1)}(e_{c}^{−})_{∗}(e_{a}^{+})^{∗}η.

This expression vanishes since the co-dimension 1 boundary of M̂(O_{a}, O_{b})^{∗} is the union of the components M̂(O_{a}, O_{b})^{∗}×_{O_{b}} M̂(O_{b}, O_{c})^{∗}, as in 4.9. Thus, if we use Stokes theorem applied to the fibration with boundary e_{c}^{−} : M̂(O_{a}, O_{c})^{∗} → O_{c}, we obtain [6] that

∂_{U(1)}(e_{c}^{−})_{∗}γ = (e_{c}^{−})_{∗}∂_{U(1)}γ − (−1)^{r(γ)−dim F} (e_{c}^{−})_{∂,∗}γ.

Here γ is an equivariant form on M̂(O_{a}, O_{c})^{∗} and the map (e_{c}^{−})_{∂,∗} is the pushforward map induced by the restriction of the bundle to the boundary of each fiber F. If we choose γ of the form γ = (e_{a}^{+})^{∗}η where η is an equivariant form on the orbit O_{a}, then the map (e_{c}^{−})_{∂,∗} can be written as

(e_{c}^{−})_{∂,∗}γ = ∑_{b}(e_{c}^{−})_{∗}(e_{b}^{+})^{∗}(e_{b}^{−})_{∗}(e_{a}^{+})^{∗}η,

for all the b that satisfy μ(O_{a}) > μ(O_{b}) > μ(O_{c}), as shown in the diagram. Notice that the sign (−1)^{r((e_{b}^{−})_{∗}(e_{a}^{+})^{∗}η)} is exactly the sign (−1)^{r((e_{a}^{+})^{∗}η)−dim F}.

\xymatrix{ &  & M̂(O_{a}, O_{c})^{∗} \ar[ddll] \ar[ddrr] \\  &  & M̂(O_{a}, O_{b})^{∗} ×_{O_{b}} M̂(O_{b}, O_{c})^{∗} \ar@{-->}[dl] \ar@{-->}[dr] \ar[u] \\ O_{a} & M̂(O_{a}, O_{b})^{∗} \ar[l] \ar[r] & O_{b} & M̂(O_{b}, O_{c})^{∗} \ar[l] \ar[r] & O_{c}}

The statement about the pairing follows from the identity

∫_{O_{b}}(e_{b}^{−})_{∗}(e_{a}^{+})^{∗}η ∧ γ = ∫_{M̂(O_{a},O_{b})^{∗}}(e_{a}^{+})^{∗}η ∧ (e_{b}^{−})^{∗}γ.

⋄

Definition 5.2 We define the equivariant Floer homology and cohomology to be

SW F_{∗}_{U(1)} = H_{∗}(C_{∗}_{U(1)}, D)

and

SW F_{U(1)}^{∗} = H^{∗}(C_{U(1)}^{∗}, δ).

We now give a more detailed description of the equivariant Floer complex, which will be useful in Section 6 and Section 7.

Proposition 5.3 Let Ω^{n} ⊗ η_{a} and Ω^{n} ⊗ 1_{a} be the generators of the equivariant Floer complex. The only possibly non-trivial coefficients of the boundary map D are those of the form

⟨Ω^{n−1} ⊗ 1_{a}, D(Ω^{n} ⊗ η_{a})⟩,

or of the form

⟨Ω^{n} ⊗ 1_{b}, D(Ω^{n} ⊗ 1_{a})⟩,
⟨Ω^{n} ⊗ η_{b}, D(Ω^{n} ⊗ η_{a})⟩,

when μ(O_{a}) − μ(O_{b}) = 1, and

⟨Ω^{n} ⊗ 1_{c}, D(Ω^{n} ⊗ η_{a})⟩,

when μ(O_{a}) − μ(O_{c}) = 2. In the case of the critical orbit θ, we obtain the boundary component

⟨Ω^{n} ⊗ 1_{c}, D(Ω^{n} ⊗ θ)⟩

when μ(θ) − μ(O_{c}) = 2, where we consider the one-dimensional space M̂(θ, O_{c}) fibering over O_{c} with zero-dimensional fiber. We also have the component

⟨Ω^{n} ⊗ θ, D(Ω^{n} ⊗ η_{a})⟩,

when μ(O_{a}) − μ(θ) = 1, where the space M̂(O_{a}, θ) is one-dimensional. In this case the coefficient is obtained by integrating the 1-form η_{a} over the 1dimensional fiber of the endpoint map e_{θ} : M̂(O_{a}, θ) → θ.

\underline{Proof:} For a given orbit O_{a} the complex of equivariant forms is given by

Ω_{∗,U(1)}(O_{a}),

with the total grading

Ω_{j,U(1)}(O_{a}) = ⨁_{2k+l=j} ℝΩ^{k} ⊗ Ω_{0}^{1−l}(O_{a}).

Here Ω is of degree 2 and l is the grading in the dual de Rham complex where forms are graded by dim(O_{a}) − ∗.

Thus we have

Ω_{0,U(1)}(O_{a}) = ℝ < 1 ⊗ η_{a} >,
Ω_{1,U(1)}(O_{a}) = ℝ < 1 ⊗ 1_{a} >,
Ω_{2,U(1)}(O_{a}) = ℝ < Ω ⊗ η_{a} >,
Ω_{3,U(1)}(O_{a}) = ℝ < Ω ⊗ 1_{a} >,

and so on, where η_{a} is the one form that generates H^{1}(S^{1}) and 1_{a} is the constant function equal to 1 on O_{a}.

The differential is given by ∂_{G} = d − Ω̃c(T), where Ω̃ is the element in the dual of the Lie algebra iℝ such that < Ω̃, Ω >= 1 under the trace pairing, and c(T) is the contraction with the vector field T generated by the infinitesimal action of U(1).

As we have already discussed at the beginning of Section 5, the differential ∂_{U(1)} of the equivariant complex Ω_{∗,U(1)}(O_{a}) acts as

D(Ω^{n} ⊗ 1_{a}) = 0

and

D(Ω^{n} ⊗ η_{a}) = −Ω^{n−1}1 ⊗ 1_{a}.

The homology of this complex (for a fixed irreducible orbit O_{a}) is therefore one copy of ℝ (corresponding to the generator 1 ⊗ η_{a}) in degree zero and zero in all other degrees. This is just the usual result that the equivariant homology of O_{a} with a free action of U(1) is isomorphic to the ordinary homology (with real coefficients) of the quotient, that is of a point.

Now let us consider the bigraded complex where several orbits are considered. The bigraded complex is of the form

⨁_{μ(O_{a})+j=∗} Ω_{j,U(1)}(O_{a}).

The equivariant boundary operator can be written explicitly in components of the form

D : \begin{aligned}Ω^{n} ⊗ 1_{a} & ↦ & −n_{ab}Ω^{n} ⊗ 1_{b} \\ Ω^{n} ⊗ η_{a} & ↦ & (n_{ab}Ω^{n} ⊗ η_{b}) ⊕ (m_{ac}Ω^{n} ⊗ 1_{c}) \\ & & ⊕(−Ω^{n−1} ⊗ 1_{a}). \end{aligned}  (69)

Here n_{ab} = #M̂(O_{a}, O_{b})^{∗}, where the relative index is μ(O_{a}) − μ(O_{b}) = 1, so that −(e_{b}^{−})_{∗}(e_{a}^{+})^{∗}1_{a} = −n_{ab}1_{b} and (e_{b}^{−})_{∗}(e_{a}^{+})^{∗}η_{a} = n_{ab}η_{b}. The coefficient m_{ac} comes from the integration of the one-form η_{a} over the one-dimensional fiber of the moduli space M(O_{a}, O_{c})^{∗} with μ(O_{a}) − μ(O_{c}) = 2, namely (e_{c}^{−})_{∗}(e_{a}^{+})^{∗}η_{a} = m_{ac}1_{c}. The components of D are represented in the following diagram.

[figure: ⋮; ⋮; ⋮; ⋯; Ω^{k} ⊗ 1_{a}; −n_{ab}; 0; Ω^{k} ⊗ 1_{b}; −n_{bc}; 0; Ω^{k} ⊗ 1_{c}; 0; ⋯; ⋯; Ω^{k} ⊗ η_{a}; n_{ab}; −1; m_{ac}; Ω^{k} ⊗ η_{b}; n_{bc}; −1; Ω^{k} ⊗ η_{c}; −1; ⋯; ⋯; Ω^{k−1} ⊗ 1_{a}; −n_{ab}; Ω^{k−1} ⊗ 1_{b}; −n_{bc}; Ω^{k−1} ⊗ 1_{c}; ⋯; ⋮; ⋮; ⋮]

Remark 5.4 In view of the result of Proposition 5.3, we notice that the full strength of Theorem 4.23 is in fact more than what is strictly necessary in order to prove that the boundary D of the equivariant Floer complex satisfies D^{2} = 0. In fact, knowing that the boundary operator only depends on the moduli spaces M̂(O_{a}, O_{b}) of dimension at most two, makes it only necessary, strictly speaking, to know the properties of the compactification for moduli spaces M̂(O_{b}, O_{c}) of dimension at most three.

Clearly, it does not make any substantial difference to derive the properties of the fine structure of the compactification M̂(O_{a}, O_{b})^{∗} in this reduced case or in the general case proved in Theorem 4.23. However, both here and in Section 6, where we analyze the case of obstructed gluing, it seems useful to identify explicitly what is the minimum requirement on the fine structure of the compactification that is needed in the arguments, cf. Remark 6.2 and Remark 6.3.

5.3 Intermezzo: the coefficient m_{ac}

We now give a topological description of the coefficient m_{ac} that appears in the component

⟨D(Ω^{n} ⊗ η_{a}), Ω^{n} ⊗ 1_{c}⟩ = m_{ac}

of the boundary operator, for μ(O_{a}) − μ(O_{c}) = 2. This coefficient is defined as the pullback and pushforward of the 1-form η_{a} along the U(1)-equivariant endpoint fibrations

e_{a}^{+} : M̂(O_{a}, O_{c})^{∗} → O_{a}

and

e_{c}^{−} : M̂(O_{a}, O_{c})^{∗} → O_{c},

that is,

m_{ac} = (e_{c}^{−})_{∗}(e_{a}^{+})^{∗}η_{a}.

The 2-dimensional U(1)-manifold M̂(O_{a}, O_{c})^{∗} fibers over the circles O_{a} and O_{c} with 1-dimensional fibers. Topologically, M̂(O_{a}, O_{c})^{∗} consists of a collection of finitely many cylinders and finitely many tori.

Lemma 5.5 The pullback 1-form (e_{a}^{+})^{∗}η_{a} defines a Cartan connection on the principal U(1)-bundle M̂(O_{a}, O_{c}) → M̂(a, c).

\underline{Proof:} We want to obtain a splitting of the exact sequence

0 → T (U(1)) → T M̂(O_{a}, O_{c}) → T M̂(O_{a}, O_{c})/T (U(1)) ≅ T M̂(a, c) → 0.

Since the endpoint map e_{a}^{+} is U(1)-equivariant, and η_{a} is the generator of H^{1}(O_{a}), the kernel Ker((e_{a}^{+})^{∗}η_{a}) defines a horizontal subspace of T M̂(O_{a}, O_{c}). ⋄

Lemma 5.6 Over each component of M̂(O_{a}, O_{c}), the pushforward

(e_{c}^{−})_{∗}(e_{a}^{+})^{∗}η_{a}

computes the winding number W (γ) of the 1-dimensional γ = (e_{c}^{−})^{−1}(x_{c}), for a point x_{c} ∈ O_{c}, around the fiber of the U(1)-fibration π : M̂(O_{a}, O_{c}) → M̂(a, c).

\underline{Proof:} Consider a connected component X̂ of M̂(a, c). For μ(O_{a}) − μ(O_{c}) = 2, the component

π^{−1}(X̂) ⊂ M̂(O_{a}, O_{c})

is a cylinder or a torus. Choosing a point x_{c} ∈ O_{c}, one can identify the fiber

(e_{c}^{−})^{−1}(x_{c}) ∩ π^{−1}(X̂)

with a lift γ of the path X̂ ⊂ M̂(a, c) to M̂(O_{a}, O_{c}). In fact, the endpoint fibration e_{c}^{−} is compatible with the boundary strata M̂(O_{a}, O_{b})×_{O_{b}} M̂(O_{b}, O_{c}).

Since the 1-form η_{a} is the generator of H^{1}(O_{a}), the pullback (e_{a}^{+})^{∗}η_{a} integrated along the path

γ = (e_{c}^{−})^{−1}(x_{c}) ∩ π^{−1}(X̂)

measures precisely the number of times this path winds around the fibers of the map π,

∫_{(e_{c}^{−})^{−1}(x_{c})∩π^{−1}(X̂)}(e_{a}^{+})^{∗}η_{a} = W (γ).

⋄

Notice that in M̂(O_{a}, O_{c}) we have a choice of a horizontal direction ℓ(x_{a}), given by the integral lines of Ker((e_{a}^{+})^{∗}η_{a}), for any initial condition x_{a} ∈ O_{a}, and a direction given by the path (e_{c}^{−})^{−1}(x_{c}) for fixed x_{c} in O_{c}. The horizontal line ℓ(x_{a}) can be identified with the fiber of the other endpoint map (e_{a}^{+})^{−1}(x_{a}). Both the lines (e_{c}^{−})^{−1}(x_{c}) and (e_{a}^{+})^{−1}(x_{a}) represent a homeomorphic lift of M̂(a, c) to M̂(O_{a}, O_{c}), however, in general, an element [𝔸, Ψ] ∈ (e_{c}^{−})^{−1}(x_{c}) has a limit at t → −∞ which is some λx_{a} in O_{a}, not equal to x_{a}, and similarly an element [𝔸, Ψ] ∈ (e_{a}^{+})^{−1}(x_{a}) has a limit at t → +∞ which is some λx_{c} in O_{c} not equal to x_{c}. Different elements in (e_{a}^{+})^{−1}(x_{a}) have different λx_{c} limits in O_{c} and similarly for the other endpoint fibration. The intersection between the two lines then consists of finitely many points. By our identification of ℓ(x_{a}) = (e_{a}^{+})^{−1}(x_{a}) with the horizontal direction, this number is equal to the winding number of the path (e_{c}^{−})^{−1}(x_{c}). This intersection number counts the solutions [𝔸, Ψ] ∈ M̂(O_{a}, O_{c}) that actually have prescribed endpoints x_{a} and x_{c} at both ends.

Now consider the U(1)-bundle M(O_{a}, O_{c}) → M(a, c) over the parameterized moduli space. Connected components X of M(a, c) are infinite cylinders or strips of the form X ≅ [0, 1] × ℝ. Consider a choice of a base point (y_{0}, t_{0}) on Y × ℝ, and a complex line ℓ_{y_{0}} in the fiber W_{y_{0}} of the spinor bundle W = S ⊗ L over y_{0} ∈ Y. We choose ℓ_{y_{0}} so that it does not contain the spinor part ψ of any irreducible critical point. Since there are only finitely many critical points we can guarantee such choice exists. We consider the line bundle

L_{ac} = M(O_{a}, O_{c}) ×_{U(1)} (W_{y_{0}}/ℓ_{y_{0}}) → M(a, c)  (70)

with a section given by

s([𝔸, Ψ]) = ([𝔸, Ψ], Ψ(y_{0}, t_{0})).  (71)

Lemma 5.7 For a generic choice of (y_{0}, t_{0}) the section s of (71) has no zeroes on the boundary strata M̂(a, b) × M̂(b, c). This determines a trivialization of L_{ac} away from a compact set in M(a, c). The line bundle L_{ac} over M(a, c), with the trivialization φ specified above, has relative Euler class satisfying

e(L_{ac}, φ) = m_{ac}.

\underline{Proof:} The section s induces a trivialization of the line bundle L_{ac} away from a compact set in M(a, c). In fact, for T → ∞, we have

s((𝔸, Ψ)^{−T}) → s((A_{a}, ψ_{a})) ≠ 0

and

s((𝔸, Ψ)^{T}) → s((A_{c}, ψ_{c})) ≠ 0.

This corresponds to trivializing the fibration M̂(O_{a}, O_{c}) × {T } with the horizontal lines ℓ(x_{a}, λ) = (e_{a}^{+})^{−1}(λx_{a}), for T ≤ −T_{0} and with the lines ℓ(x_{c}, λ) = (e_{c}^{−})^{−1}(λx_{c}) for T ≥ T_{0}. The obstruction to extending the trivialization over all of M(a, c) is then measured precisely by the winding number m_{ac}. This is therefore the relative Euler class of L_{ac}, which can also be computed as

e(L_{ac}, φ) = #s^{−1}(0).

⋄

In particular, this implies that the coefficient m_{ac} can be computed by counting zeroes of any transverse section

s : M(a, c) → L_{ac}

which is non-vanishing away from a compact set, and induces the same trivialization φ,

m_{ac} = #s^{−1}(0).

We discuss some identities satisfied by the coefficients m_{ac}.

Remark 5.8 Let a and d be two irreducible critical points with μ(a)−μ(d) = 3. Assume that all the critical points c_{i} with μ(a) > μ(c_{i}) > μ(d) are irreducible. Then we have the identity

∑_{c_{1}:μ(a)−μ(c_{1})=1} n_{a,c_{1}}m_{c_{1},d} − ∑_{c_{2}:μ(c_{2})−μ(d)=1} m_{a,c_{2}}n_{c_{2},d} = 0.

When the reducible critical point θ has index between (μ(a), μ(c)), with μ(a) = 1 and μ(d) = −2, we have the identity

∑_{c_{1}:μ(c_{1})=0} n_{a,c_{1}}m_{c_{1},d} + n_{aθ}n_{θd} − ∑_{c_{2}:μ(c_{2})=−1} m_{a,c_{2}}n_{c_{2},d} = 0.  (72)

This identity follows directly from the definition of the invariant m_{ac} as pullback and pushforward of the 1-form η_{a}, using Stokes’ theorem for fibrations with boundary on the compactified moduli space M̂(O_{a}, O_{d})^{∗}. The sign comes from the orientations in the gluing theorem. However, the identity (72) also has a topological interpretation in terms of relative Euler classes. In fact, the counting above is the counting of the boundary points of the zero set of a strata transverse section of the line bundle L_{ad} over the unparameterized moduli space M(a, d). For such section, s^{−1}(0) is a 1-dimensional manifold with boundary. The counting above corresponds to a section that has

∂s^{−1}(0) = ⋃_{c_{1}:μ(a)−μ(c_{1})=1} M̂(a, c_{1}) × (s^{−1}(0) ∩ M(c_{1}, d))

∪ ⋃_{c_{2}:μ(c_{2})−μ(d)=1}(−s^{−1}(0) ∩ M(a, c_{2})) × M̂(c_{2}, d).

The other identity can be proved by the similar methods, with an extra U(1) gluing parameter when gluing along θ, as in Lemma 2.14.

Now consider the case of the 2-dimensional moduli spaces M̂(O_{a}, θ), for μ(O_{a}) − μ(θ) = 2, and M̂(θ, O_{c}) for μ(θ) − μ(O_{c}) = 3, with θ the unique reducible (the U(1)-fixed point in M^{0}).

The moduli spaces M̂(O_{a}, θ), with μ(O_{a}) − μ(θ) = 2, do not contribute components to the boundary operator D of the Floer complex, because we have

(e_{θ})_{∗}(e_{a}^{+})^{∗}η_{a} = 0,

since the fibers of the endpoint map

e_{θ} : M̂(O_{a}, θ) → θ

are 2-dimensional.

It is still true that (e_{a}^{+})^{∗}η_{a} defines a connection on the principal U(1)-bundle over M(a, θ). Moreover, a component X̂ ⊂ M̂(a, θ) is again topologically a circle or a line segment. Given a choice of a lift γ of X̂ inside the component π^{−1}(X̂) ⊂ M̂(O_{a}, θ), we can still compute its winding number and follow the same construction given above. However, now the choice of the lift γ is no longer determined by the endpoint fibration

e_{θ} : M̂(O_{a}, θ) → θ.

Different possible choices of the lift γ can have different winding numbers W (γ).

Similarly, we can still construct the bundle L_{a,θ} over M(a, θ). However, the section (71), in this case, does not give a trivialization away from a compact set. Similarly, the fibers of the endpoint map e_{θ} do not provide a trivialization at the T → ∞ end, since they do not determine a lift of M(a, θ).

However, we claim that it is still possible to associate an invariant m_{aθ} to the moduli space M(a, θ), which can be interpreted topologically as a relative Euler class. We briefly illustrate here why this should be the case, then in Section 6.3, in Remark 6.24, we give a definition of the invariant m_{aθ} as counting of the zeroes of a particular section of L_{a,θ}, the obstruction section.

The choice of the base point (y_{0}, t_{0}) determines a trivialization of the U(1)- fibration B^{0} over the configuration space B, in a small neighborhood of each irreducible critical point. If we think of parameterized flow lines between two critical points a and c as embedded in the irreducible part B^{∗} of B, then the U(1)-fibration M(O_{a}, O_{c}) has an induced trivialization away from a compact set in M(a, c), which agrees with the trivialization φ given by the cross section (71) of the associated line bundle L_{ac}. We can choose a small neighborhood U_{θ} of the reducible critical point θ such that U_{θ}^{∗} = U_{θ} ∩ B^{∗} is 1-connected, hence there is also a unique trivialization of B^{0} over U_{θ}^{∗}. We assume that this trivialization is given by a constant cross section. Thus, there is an induced trivialization away from a compact set in M(a, θ) (where μ(O_{a}) − μ(θ) = 2) or M(θ, O_{c}) (where μ(θ) − μ(O_{c}) = 3). Using this trivialization, we can define m_{aθ} and m_{θc} to be the relative Euler class of the corresponding associated line bundles L_{aθ} and L_{θc}, respectively.

We return to the description of the invariant m_{ac} for the case of O_{a} and O_{c} free orbits of relative index two, in order to illustrate the point of view we shall adopt in Remark 6.24 in describing the invariant m_{aθ}.

In general, we can construct the relative Euler class which is a cycle defined by the zeros of the transverse section as follows. Over the unparameterized moduli space M̂(a, c), consider the line bundle

L_{ac} = M̂(O_{a}, O_{c}) ×_{U(1)} ℂ,

associated to the principal U(1)-fibration

π : M̂(O_{a}, O_{c}) → M̂(a, c).

The pullback under the quotient map M(a, c) → M̂(a, c) induces a pullback line bundle, which we still denote L_{ac}, over the parameterized moduli space M(a, c).

Lemma 5.9 Over the boundary strata

M̂(a, b) × M̂(b, c)

of M̂(a, c), the line bundle L_{ac} has the form

L_{ac} ≅ π_{1}^{∗}L_{ab} ⊗ π_{2}^{∗}L_{bc}.

\underline{Proof:} The result follows by showing that, for all critical orbits O_{p} and O_{q} with μ(O_{p}) > μ(O_{q}), the principal U(1)-bundles

π : M̂(O_{p}, O_{q}) → M̂(p, q)

are compatible with the gluing maps, namely they define a line bundle

L_{pq} = M̂(O_{p}, O_{q})^{∗} ×_{U(1)} ℂ → M̂(p, q)^{∗}

in the category of manifolds with corners. The compatibility of the U(1)- fibration with the gluing maps follows from our proof of Theorem 4.9. ⋄

We now specify a choice of a class of transverse sections of L_{ac} over M(a, c). We prescribe a choice of non-zero complex numbers s_{a} for each critical point a. For each pair of critical points a, b of relative index 1, we choose a path of non-zero complex numbers s_{ab} : ℝ → ℂ − {0} connecting s_{a} and s_{b}, which is contractible in ℂ − {0} and is constant outside a compact set in ℝ. Over the 1-dimensional M(a, b) we define nowhere vanishing sections of L_{ab} as pullbacks of the s_{ab} on M̂(a, b). We still denote these sections over M(a, b) by s_{ab}. Over the 2-dimensional M(a, c) we consider the class of all transverse sections s_{ac} of L_{ac} that satisfy

s_{ac} = π_{1}^{∗}s_{ab} ⊗ π_{2}^{∗}s_{bc}

over the product submanifolds

M̂(a, b) × M̂(b, c) × [0, ϵ) × ℝ

where [0, ϵ) is the gluing parameter, and such that, for a sufficiently large T, s_{ac} is the constant section s_{a} over

M̂(a, c) × (−∞, −T]

and the constant section s_{b} over

M̂(a, c) × [T, ∞).

Clearly, any such section s_{ac} gives the same trivialization φ of L_{ac} away from a compact set in M(a, c). We can therefore compute the relative Euler class by counting the zeros of this transverse section, that is,

e(L_{ac}, φ) = #s_{ac}^{−1}(0).

For higher dimensional moduli spaces M(a, c), we proceed analogously, by inductively defining the class of transverse sections of L_{ac} which are compatible with the boundary strata of M(a, c), that is, we require that

s_{ac} = π_{1}^{∗}s_{ab} ⊗ π_{2}^{∗}s_{bc}

over all the product submanifolds

M̂(a, b) × M̂(b, c) × [0, ϵ) × ℝ

and that s_{ac} agrees with the constant section s_{a} over

M̂(a, c) × (−∞, −T]

and with the constant section s_{b} over

M̂(a, c) × [T, ∞).

Lemma 5.10 Consider the line bundle L_{ac} over a parameterized moduli space M(a, c), where μ(a) − μ(c) = 3 and a is irreducible or where a is reducible and μ(c) = −4, with the cross section s_{ac} as above. Then the zero set s_{ac}^{−1}(0) is a codimension 2 submanifold of M(a, c) which can be compactified by adding lower dimensional boundary strata such that the codimension one boundary strata are given by

∂s_{ac}^{−1}(0) = ⋃_{μ(b)−μ(c)=1} s_{ab}^{−1}(0) × M̂(b, c)

∪ ⋃_{μ(b)−μ(c)=2} M̂(a, b) × s_{bc}^{−1}(0),

for μ(a) ≠ 1;

∂s_{ac}^{−1}(0) = ⋃_{μ(b)−μ(c)=1} s_{ab}^{−1}(0) × M̂(b, c)

∪ ⋃_{μ(a)−μ(b)=1} M̂(a, b) × s_{bc}^{−1}(0)

∪s_{ac}^{−1}(0) ∩ (M̂(a, θ) × M(θ, c) × U(1))

for μ(a) = 1 and μ(c) = −2.

\underline{Proof:} The result follows from the choice of the class of sections s_{ac}, together with the result of Lemma 5.9. ⋄

If we compare the results of Lemma 5.9 and Lemma 5.10, with Lemma 5.7, we see that our previous definitions of the invariant

m_{ac} = e(L_{ac}, φ) = #s_{ac}^{−1}(0)

are a particular case of the procedure illustrated here. In particular, the identities of Remark 72 are then the direct consequences of Lemma 5.10.

6 Topological invariance

In the present section we show that the equivariant Floer groups, defined as in Definition 5.2, are topological invariants, in the sense that they are independent of the choice of the metric and of the perturbation.

Throughout all this discussion, recall that the Floer groups depend on the choice of a Spin_{c}-structure. We always assume to work with a fixed Spin_{c}- structure, so we never mention explicitly this dependence.

The easiest case, for the proof of topological invariance, is under the assumption that b^{1}(Y) ≥ 2. In fact, in this case there are no reducible solutions, hence a cobordism argument can be used to construct a morphism of the chain complexes that gives an isomorphism of the cohomology groups. In the case when b^{1}(Y) = 1, the metric and the perturbation cannot be chosen arbitrarily. This was already noticed in [5] in the case of the invariant associated to the moduli space M (which is the Euler characteristic of the Floer homology, [38]). However, given two generic metrics, it is possible to find a path of metrics joining them and an open set of sufficiently small perturbations ρ such that the invariant does not change. The dependence which is introduced in this case is rather mild and it amounts to a choice of a homology class in H^{1}(Y; ℤ). Thus, the b^{1}(Y) ≥ 1 case will follow by exactly the same argument that we present below in our simple model case of same chamber metrics and perturbations.

The case of a homology sphere [13] is rather different. In fact in this case there is an explicit dependence on the metric ([15], [31]). The interesting fact about the equivariant Floer homology is that it is instead metric independent. This means that, in this case, though we still have a condition on the kernel of the Dirac operator (Ker(∂_{A}) trivial at the reducible solution) that breaks the space of metrics and perturbations into chambers with co-dimension one walls, it is however possible to construct a chain homomorphism between the Floer complexes that correspond to different metrics, and a chain homotopy. That is, for the equivariant Floer homology the wall crossing consists only of a global grade shift.

The main theorem of this section is the following.

Theorem 6.1 Let Y be a homology sphere or a rational homology sphere. Suppose given two metrics g_{0} and g_{1} on Y and perturbations ν_{0} and ν_{1}. Then there exist a chain homomorphism I between the equivariant Floer complexes (C_{∗}_{U(1)(g_{0},ν_{0})}, D) and (C_{∗}_{U(1)(g_{1},ν_{1})}, D), defined by considering a generic path (g_{t}, ν_{t}) and moduli spaces M(O_{a}, O_{a′}) of solutions of the flow equations on (Y × ℝ, g_{t} + dt^{2}). This chain homomorphism induces an isomorphism in homology up to an index shift given by the spectral flow of the Dirac operator ∂_{ν_{t}}^{g_{t}}. The analogous construction works for cohomology groups.

We first give a “model proof” in the much simpler case of two metrics and perturbations (g_{0}, ν_{0}) and (g_{1}, ν_{1}) that are in the same chamber. The proof in this case is obviously much simpler than the general statement of Theorem 6.1. In fact, the interesting part of the statement is only for different chambers. The reason why we present explicitly the proof of this case is because, once we know that all the SW F_{∗,U(1)}(Y, (g, ν)), for (g, ν) in the same chamber, are isomorphic, we can prove the general case just by analyzing the case of two sufficiently close metrics and perturbations (g_{0}, ν_{0}) and (g_{1}, ν_{1}) in two different chambers, across a codimension one wall. Moreover, having the model case presented explicitly, we can discuss the case of different chambers by pointing out the various points in the proof where the original argument has to be modified. We prove the general statement of Theorem 6.1 in Section 6.3. \underline{Proof, Part I: the same chamber case:} Choose a path of metrics and perturbations (g_{t}, ν_{t}) with t ∈ [0, 1] that interpolates between (g_{0}, ν_{0}) and (g_{1}, ν_{1}). Consider the manifold Y × ℝ endowed with the metric g_{t} + dt^{2} on Y × {t} extended as the product metric outside Y × [0, 1].

Consider on Y × ℝ the perturbed “gradient flow” equations with respect to the metric g_{t}. Denote {O_{a}, O_{b}, ⋯} the critical orbits for the metric g_{0} and {O_{a′}, O_{b′}, ⋯} the critical orbits for the metric g_{1}.

We adopt the convention of fixing the absolute grading of the Floer complex to be the relative grading with respect to the reducible solution μ(O_{a}) − μ(θ_{0}), and μ(O_{a′}) − μ(θ_{1}), where θ_{0} = [ν_{0}, 0] and θ_{1} = [ν_{1}, 0]. We perform a shift of grading in the complex C_{k}_{U(1)}(Y, g_{1}, ν_{1}) by setting μ(θ_{1}) = −SF (∂_{ν_{t}}^{g_{t}}), where SF (∂_{ν_{t}}^{g_{t}}) is the spectral flow of the Dirac operator along the path of reducible solutions [ν_{t}, 0]. In the same chamber case this shift is irrelevant, since the spectral flow is trivial, but it will be relevant in the general case we prove in Section 6.3.

We have moduli spaces M(O_{a}, O_{a′}) of solutions modulo gauge transformations of suitably perturbed Seiberg–Witten equations on the manifold (Y × ℝ, g_{t} + dt^{2}), as specified in (84) and (85).

We need an analogue of Proposition 2.12 which ensures that, under a generic choice of the perturbation, the moduli spaces M(O_{a}, O_{a′}) are smooth manifolds of dimension μ(O_{a}) − μ(O_{a′}) + dim(O_{a}), where μ(O_{a′}) denotes the shifted grading. In Lemma 6.5, Lemma 6.6, Corollary 6.7, and Lemma 6.8 we deal with these transversality issues.

We construct a degree zero homomorphism of the Floer complexes

I : C_{k}_{U(1)}(Y, g_{0}, ν_{0}) → C_{k}_{U(1)}(Y, g_{1}, ν_{1}).

We define the homomorphism I as

I_{a,a′}η = \{ \begin{matrix}(e_{a′}^{−})_{∗}(e_{a}^{+})^{∗}η & μ(O_{a}) − μ(O_{a′}) ≥ 0 \\ 0 & μ(O_{a}) − μ(O_{a′}) < 0, \end{matrix}  (73)

with η ∈ Ω_{k−μ(O_{a}),U(1)}(O_{a}). The maps e_{a′}^{−} and e_{a}^{+} are the end point maps for M(O_{a}, O_{a′}). Clearly, such I commutes with Ωc(T). Requiring that the expression (73) is well defined, that is, that we have a well defined push-forward map (e_{a′}^{−})_{∗}, implies checking some properties of the compactification of the moduli spaces M(O_{a}, O_{a′}). For the same chamber case, the analysis of the boundary structure of M(O_{a}, O_{a′}) follows closely the model developed in Section 4 for the compactification of the moduli spaces of flow lines. However, when adapting this argument to the general case of different chambers, this compactification will require a much more delicate analysis.

The next step is then to prove that the map I is a chain homomorphism. We want the relation I ∘ D − D ∘ I = 0 to be satisfied. This corresponds to the expression

(ID − DI)(η)_{a,b′}

= \{ \begin{matrix}(e_{b′}^{−})_{∗}(e_{a}^{+})^{∗}∂_{U(1)}η − ∂_{U(1)}(e_{b′}^{−})_{∗}(e_{a}^{+})^{∗}η \\ for O_{b} = O_{a} and μ(O_{b}) ≥ μ(O_{b′}) \\ or O_{a′} = O_{b′} and μ(O_{a}) ≥ μ(O_{b′}) \\ −(−1)^{r(η)} ∑_{\{b|μ(O_{b})<μ(O_{a})\}}(e_{b′}^{−})_{∗}(e_{b}^{+})^{∗}(e_{b}^{−})_{∗}(e_{a}^{+})^{∗}η \\ +(−1)^{r((e_{a′}^{−})_{∗}(e_{a}^{+})^{∗}η)} ∑_{\{a′|μ(O_{a′})>μ(O_{b′})\}}(e_{b′}^{−})_{∗}(e_{a′}^{+})^{∗}(e_{a′}^{−})_{∗}(e_{a}^{+})^{∗}η, \\ for μ(O_{a}) > μ(O_{b}) ≥ μ(O_{b′}) and μ(O_{a}) ≥ μ(O_{a′}) > μ(O_{b′}) \end{matrix}

(74)

where the first line in the right hand side of (74) corresponds to O_{b} = O_{a} and μ(O_{b}) ≥ μ(O_{b′}) or O_{a′} = O_{b′} and μ(O_{a}) ≥ μ(O_{b′}), and the other terms correspond to μ(O_{a}) > μ(O_{b}) ≥ μ(O_{b′}) and μ(O_{a}) ≥ μ(O_{a′}) > μ(O_{b′}).

In order to show that the right hand side is zero, we consider components of the form

\begin{matrix}⋃_{\{a′|μ(O_{a′})>μ(O_{b′})\}}(M(O_{a}, O_{a′})^{∗} ×_{O_{a′}} M̂(O_{a′}, O_{b′})^{∗}) \\ ⋃_{\{b|μ(O_{b})<μ(O_{a})\}}(−M̂(O_{a}, O_{b})^{∗} ×_{O_{b}} M(O_{b}, O_{b′})^{∗}), \end{matrix}  (75)

We want to show that these are precisely all the components of the ideal boundary that appear in the actual boundary of the compactification M(O_{a}, O_{b′})^{∗}. We need Stokes’ theorem to apply, hence we need to prove that the compactification M(O_{a}, O_{a′})^{∗} with boundary strata (75) has the structure of a smooth manifold with corners. We also need an analogue of Corollary 4.24, which shows that the endpoint maps

e_{a}^{+} : M(O_{a}, O′_{a})^{∗} → O_{a}

and

e_{a′}^{−} : M(O_{a}, O′_{a})^{∗} → O_{a′}

are fibrations compatible with the boundary strata and with compact fibers, as in Theorem 6.9. Thus, using Stokes’ theorem again, we write the expression (e_{a}^{+})^{∗}η in terms of the push-forward map on the co-dimension one boundary strata. This proves that (74) vanishes identically, as expected. Notice that the sign (−1)^{r((e_{a′}^{−})_{∗}(e_{a}^{+})^{∗}η)} is just (−1)^{r(η)}.

Now consider another path of metrics and perturbations (g̃_{t}, ν̃_{t}) that connects (g_{1}, ν_{1}) to (g_{0}, ν_{0}). We construct the corresponding homomorphism

J : C_{k}_{U(1)}(Y, g_{1}, ν_{1}) → C_{k}_{U(1)}(Y, g_{0}, ν_{0}),

as in the previous case,

J_{a′,a}η = \{ \begin{matrix}(e_{a}^{−})_{∗}(e_{a′}^{+})^{∗}η & μ(O_{a′}) − μ(O_{a}) ≥ 0 \\ 0 & μ(O_{a′}) − μ(O_{a}) < 0, \end{matrix}  (76)

with η ∈ Ω_{k−μ(O_{a′}),U(1)}(O_{a′}). The maps e_{a′}^{+} and e_{a}^{−} are the end point maps for the moduli space M(O_{a′}, O_{a}) of solutions on (Y × ℝ, g̃_{t} + dt^{2}).

The statement of the theorem now follows if we show that there is a chain homotopy H such that

id_{k} − (JI)_{k} = D_{k+1}H_{k} + H_{k−1}D_{k}.  (77)

In order to define H let us consider the manifold Y × ℝ endowed with the metric γ_{1} which is

γ_{1} = \{ \begin{matrix}g_{0} + dt^{2} & t < −2 \\ g_{t+2} + dt^{2} & t ∈ [−2, −1] \\ g_{1} + dt^{2} & t ∈ [−1, 1] \\ g̃_{2−t} + dt^{2} & t ∈ [1, 2] \\ g_{0} + dt^{2} & t > 2. \end{matrix}  (78)

Consider a path of metrics γ_{σ} with σ ∈ [0, 1] that connects γ_{0} = g_{0} + dt^{2} to γ_{1}, such that for all σ the metric γ_{σ} is the product metric g_{0} + dt^{2} outside a fixed large interval [−T, T].

Let M^{P} (O_{a}, O_{b}) be the parameterized moduli space of (A(t), ψ(t), σ), solutions of the perturbed gradient flow equations with respect to the metric γ_{σ}, modulo gauge transformations. Lemma 6.5, with minor modifications to accommodate the presence of parameters, can be employed to show that , under a generic choice of the perturbation, the moduli spaces M^{P} (O_{a}, O_{b}) are smooth manifolds of dimension μ(O_{a}) − μ(O_{b}) + 2 − dim G_{a}, cut out transversely by the equations. Denote by ẽ_{a}^{+} and ẽ_{b}^{−} the end point maps from M^{P} (O_{a}, O_{b}) to O_{a} and O_{b} respectively.

Now we can define the degree-one map H to be

H : C_{k}_{U(1)}(Y, g_{0}, ν_{0}) → C_{k+1}_{U(1)}(Y, g_{0}, ν_{0})

H_{a,b} : η → (−1)^{1−r(η)}(ẽ_{b}^{−})_{∗}(ẽ_{a}^{+})^{∗}η,  (79)

with η ∈ Ω_{k−μ(O_{a}),U(1)}(O_{a}).

Again we need to verify some properties of the compactified moduli spaces M^{P} (O_{a}, O_{b})^{∗}, which ensure that the map H is well defined and has the desired properties.

The identity (77) which proves that H is a chain homotopy can be rewritten as the following two identities:

\begin{aligned}& id_{a,a} − ∑_{a′} I_{a,a′}J_{a′,a} \\ = & ∑_{\{b|μ(O_{b})≤μ(O_{a})\}} D_{a,b}H_{b,a} + ∑_{\{b|μ(O_{b})≥μ(O_{a})\}} H_{a,b}D_{b,a} \end{aligned}  (80)

and, for a ≠ b,

\begin{aligned}& − ∑_{a′} I_{a,a′}J_{a′,b} \\ = & ∑_{\{c|μ(O_{c})≤μ(O_{a})\}} D_{a,c}H_{c,b} + ∑_{\{c|μ(O_{c})≥μ(O_{a})\}} H_{a,c}D_{c,b}. \end{aligned}  (81)

The identities (80) and (81) can be proved by applying Stokes theorem again, in the way we discussed already. To this purpose, consider the components

\begin{matrix}∂^{(1)}M^{P} (O_{a}, O_{a})^{∗} = ⋃_{a′}(M(O_{a}, O_{a′})^{∗} ×_{O_{a′}} M(O_{a′}, O_{a})^{∗}) ∪ \{−O_{a}\} \\ ⋃_{\{b|μ(O_{b})≥μ(O_{a})\}}((−1)^{dimM̂(O_{b},O_{a})}M^{P} (O_{a}, O_{b})^{∗} ×_{O_{b}} M̂(O_{b}, O_{a})^{∗}) \\ ⋃_{\{b|μ(O_{b})≤μ(O_{a})\}}((−1)^{dimM^{P} (O_{b},O_{a})}M̂(O_{a}, O_{b})^{∗} ×_{O_{b}} M^{P} (O_{b}, O_{a})^{∗}) \end{matrix} (82)

and, for a ≠ b,

\begin{matrix}∂^{(1)}M^{P} (O_{a}, O_{b})^{∗} = ⋃_{a′}(M(O_{a}, O_{a′})^{∗} ×_{O_{a′}} M(O_{a′}, O_{b})^{∗}) \\ ⋃_{\{c|μ(O_{c})≥μ(O_{a})\}}((−1)^{dimM̂(O_{c},O_{b})}M^{P} (O_{a}, O_{c})^{∗} ×_{O_{c}} M̂(O_{c}, O_{b})^{∗}) \\ ⋃_{\{c|μ(O_{c})≤μ(O_{a})\}}((−1)^{dimM^{P} (O_{c},,O_{a})}M̂(O_{a}, O_{c})^{∗} ×_{O_{c}} M^{P} (O_{c}, , O_{a})^{∗}). \end{matrix} (83)

We need to know that these components are precisely the actual co-dimension one boundary in the compactifications M^{P} (O_{a}, O_{a})^{∗} and M^{P} (O_{a}, O_{b})^{∗} respectively. If we prove that the compactification M^{P} (O_{a}, O_{b})^{∗} has the structure of a smooth manifold with corners, with the actual boundary strata in the codimension one boundary consisting precisely of the components of (82) and (83), then the argument is complete.

Again, since we are dealing only with the model case of same chamber metrics and perturbations, the necessary properties of the compactification M^{P} (O_{a}, O_{b})^{∗} follow from the analysis introduced in Section 4, as in the case of moduli spaces of flow lines. This is another crucial point that requires a much more accurate analysis in the general case of different chambers. The signs in the formulae (75), (82), and (83) denote the difference of orientation, as in Theorem 6.9 and Theorem 6.10. ⋄

In particular, we can describe more explicitly the components of the maps I and H. This identifies what are, in fact, the minimal requirements on the compactifications of the moduli spaces M(O_{a}, O_{a′}) and M^{P} (O_{a}, O_{b}) that are necessary for the argument given above to work.

Let η_{a} be the one-form that generates H^{1}(S^{1}) and 1_{a} be the constant function equal to 1 on O_{a}, so that we can write the generators of the equivariant complex in the form Ω^{n} ⊗ η_{a} and Ω^{n} ⊗ 1_{a}. We have the following explicit description of the maps I and H.

Remark 6.2 The components of the map I vanish whenever we have μ(O_{a}) − μ(O_{a′}) ≥ 2. Moreover, the only possibly non-trivial components of the chain map I are

n_{aa′} = ⟨Ω^{n} ⊗ η_{a′}, I(Ω^{n} ⊗ η_{a})⟩,

and

n_{aa′} = ⟨Ω^{n} ⊗ 1_{a′}, I(Ω^{n} ⊗ 1_{a})⟩,

when μ(O_{a}) − μ(O_{a′}) = 0, with O_{a} and O_{a′} free U(1)-orbits. In this case we need to know that the moduli space M(O_{a}, O_{a′}) has a nice compactification M(O_{a}, O_{a′})^{∗} analogous to the compactification of the moduli spaces of flow lines analyzed in Sections 4.2 and 4.3. We then have

n_{aa′} = # (M(a, a′) = M(O_{a}, O_{a′})/U(1)) .

We also have

m_{ab′} = ⟨Ω^{n} ⊗ 1_{b′}, I(Ω^{n} ⊗ η_{a})⟩,

when μ(O_{a}) − μ(O_{b′}) = 1 and with O_{a} and O_{b′} free U(1)-orbits. In this case we need to know that the compactification M(O_{a}, O_{b′})^{∗} is obtained by adding the co-dimension one boundary strata (86) and satisfies the analogue of Theorem 4.9, hence the push-forward map (e_{b′}^{−})_{∗} is well defined. Moreover, in the case of the critical points θ_{0} and θ_{1} we have

n_{a,θ_{1}} = ⟨Ω^{n} ⊗ θ_{1}, I(Ω^{n} ⊗ η_{a})⟩,

when μ(O_{a}) − μ(θ_{1}) = 0. The coefficient in this case is obtained by integrating along the 1-dimensional fibers of the map e_{θ_{1}} : M(O_{a}, θ_{1}) → θ_{1}. Thus, we need to know that the moduli space M(O_{a}, θ_{1}) is compact. This gives

n_{a,θ_{1}} = #M(a, θ_{1}).

We also have the component

n_{θ_{0},b′} = ⟨Ω^{n} ⊗ 1_{b′}, I(Ω^{n} ⊗ θ_{0})⟩,

when μ(θ_{0}) − μ(O_{b′}) = 1. Here we have a 1-dimensional space M(θ_{0}, O_{b′}) fibering over O_{b′} with zero-dimensional fiber, and again we need to know that we have M(θ_{0}, O_{b′})^{∗} = M(θ_{0}, O_{b′}) in this case, which gives

n_{θ_{0},b′} = #M(θ_{0}, b′).

Remark 6.3 The components of the map H vanish whenever we have μ(O_{a}) − μ(O_{b}) ≥ 1. Moreover, the only possibly non-trivial components of the chain homotopy H are

n_{ab}^{P} = ⟨Ω^{n} ⊗ η_{b}, H(Ω^{n} ⊗ η_{a})⟩,

and

n_{ab}^{P} = ⟨Ω^{n} ⊗ 1_{b}, H(Ω^{n} ⊗ 1_{a})⟩,

when μ(O_{a}) − μ(O_{b}) = −1, with O_{a} and O_{b} free U(1)-orbits. In this case we need to know that the one-dimensional moduli space M^{P} (O_{a}, O_{b}) is compact. This gives

n_{ab}^{P} = # (M^{P} (a, b) = M^{P} (O_{a}, O_{b})/U(1)) .

We also have

m_{ac}^{P} = ⟨Ω^{n} ⊗ 1_{c}, H((Ω^{n} ⊗ η_{a})⟩,

when μ(O_{a}) − μ(O_{c}) = 0 and with O_{a} and O_{c} free U(1)-orbits. In this case we need to know that the compactification M^{P} (O_{a}, O_{c})^{∗} of the 2-dimensional moduli space M^{P} (O_{a}, O_{c}) is obtained by adding the co-dimension one boundary strata (83) and satisfies the analogue of Theorem 4.9, hence the push-forward map (ẽ_{c}^{−})_{∗} used in the definition of H is well defined. Moreover, in the case of the critical point θ_{0}

n_{a,θ_{0}}^{P} = ⟨Ω^{n} ⊗ θ_{0}, H(Ω^{n} ⊗ η_{a})⟩,

when μ(O_{a}) − μ(θ_{0}) = −1. In this case the moduli space M^{P} (O_{a}, θ_{0}) is onedimensional. Thus, in this case, the coefficient of H is obtained by integrating along the 1-dimensional fibers of the map e_{θ_{0}} : M^{P} (O_{a}, θ_{0})^{∗} → θ_{0}. Thus, we need to know that the 1-dimensional moduli space M^{P} (O_{a}, θ_{0}) has a nice compactification M^{P} (O_{a}, θ_{0})^{∗}. This gives

n_{a,θ_{0}}^{P} = #M^{P} (a, θ_{0}).

We also have the component

n_{θ_{0},b}^{P} = ⟨Ω^{n} ⊗ 1_{b}, H(Ω^{n} ⊗ θ_{0})⟩,

when μ(θ_{0}) − μ(O_{b}) = 0. Here we have a one-dimensional space M^{P} (θ_{0}, O_{b}) fibering over O_{b} with zero-dimensional fiber, and again we need to know that M^{P} (θ_{0}, O_{b})^{∗} is a nice compactification. This gives

n_{θ_{0},b}^{P} = #M^{P} (θ_{0}, b).

This gives a fair description of the model argument in the easy case of metric and perturbations in the same chamber, and of the minimal requirements on the compactifications, in order to extend the argument to the more general case. Now we can begin to analyze the general case.

6.1 The boundary structure of M(O_{a}, O_{a′})

In this section we analyze some properties of the moduli spaces M(O_{a}, O_{a′}) that are needed in the proof of Theorem 6.1. Here we make no assumption about the metrics and perturbations (g_{0}, ν_{0}), (g_{1}, ν_{1}). We shall soon restrict our attention to the case of two sufficiently close metrics and perturbations on opposite sides of a codimension one wall, which is the only case we need to complete the proof of Theorem 6.1.

Let us first give a more precise account of the construction of the moduli spaces M(O_{a}, O_{a′}). On (Y × ℝ, g_{t} + dt^{2}) we consider the equations

D_{𝔸}Ψ + ρΨ = 0  (84)

and

F_{𝔸}^{+} = Ψ · Ψ̄ + iμ + P_{(𝔸,Ψ)}.  (85)

Here the form μ is determined by the path ν_{t} of perturbations, and the additional perturbation P satisfies conditions (2)-(5) of Definition 2.8 and the modified version of condition (1): (1’) the perturbation P_{(𝔸,Ψ)} restricted to the interval (1, ∞) can be written as P_{(𝔸,Ψ)} = ∗q_{(𝔸,Ψ)}(t) + q_{(𝔸,Ψ)}(t) ∧ dt with respect to the constant metric g_{1}, where q_{(𝔸,Ψ)}(t) satisfies

q_{(𝔸,Ψ)|_{(1,∞)}^{T}}(t) = q_{(𝔸,Ψ)}(t + T),

for any T > 0 and for any t > 1. Correspondingly, when restricted to the interval (−∞, 0) the perturbation q_{(𝔸,Ψ)}(t), defined with respect to the constant metric g_{0}, satisfies

q_{(𝔸,Ψ)|_{(−∞,0)}^{T}}(t) = q_{(𝔸,Ψ)}(t + T),

for all T < 0 and t < 0.

In the Dirac equation (84), we add the perturbation term given by a 1-form ρ on Y × ℝ that is supported on some compact set Y × [−T_{0}, T_{1}], with T_{0} > 0 and T_{1} > 1. Notice that, according to condition (1’), the equations (84) and (85) are translation invariant outside the interval [−T_{0}, T_{1}]. The moduli space M(O_{a}, O_{a′}) is defined as the set of gauge equivalence classes of solutions of (84) and (85) in B_{k,δ}^{0}(O_{a}, O_{a′}).

Now, we first proceed as in Section 4.1 and prove the existence of a compactification for M(O_{a}, O_{a′}). Theorem 6.4 (the analogue of Theorem 4.1) shows the existence of a compactification and identifies the ideal boundary with fibered products of lower dimensional moduli spaces, as in (86) and (87). This convergence argument follows very closely the argument for Theorem 4.1, whereas the gluing arguments will require a more sophisticated analysis.

Theorem 6.4 Suppose either O_{a} or O_{a′} is an irreducible critical orbit. Consider the moduli space M(O_{a}, O_{a′}), with μ(O_{a}) − μ(O_{a′}) + dim(O_{a}) ≥ 1. The codimension-one boundary of M(O_{a}, O_{a′}) is a subset of the space

⋃_{O_{b}} M̂(O_{a}, O_{b}) ×_{O_{b}} M(O_{b}, O_{a′}) ∪ ⋃_{O_{b′}} M(O_{a}, O_{b′}) ×_{O_{b′}} M̂(O_{b′}, O_{a′}). (86)

Similarly, the components of higher codimension in the ideal boundary consist of fibered products of the form

\begin{matrix}⋃_{c_{1},···,c_{k},b} M̂(O_{a}, O_{c_{1}}) ×_{O_{c_{1}}} ⋯ \\ ×_{O_{c_{k}}} M̂(O_{c_{k}}, O_{b}) ×_{O_{b}} M(O_{b}, O_{a′}) \\ ⋃_{b′,c′_{1},···,c′_{ℓ}} M(O_{a}, O_{b′}) ×_{O_{b′}} M̂(O_{b′}, O_{c′_{1}}) ×_{O_{c′_{1}}} ⋯ \\ ×_{O_{c′_{ℓ}}} M̂(O_{c′_{ℓ}}, O_{a′}). \end{matrix}  (87)

\underline{Proof:} As in the proof of Theorem 4.1, we want to show that the space is precompact. Namely, any sequence x_{i} of elements in M(O_{a}, O_{a′}) has a subsequence which either converges in norm to another solution x ∈ M(O_{a}, O_{a′}), or converges to a broken trajectory.

Given a sequence x_{i} in M(O_{a}, O_{a′}), we can follow the steps of the argument given in Theorem 4.1. Claim 1: There is a subsequence {x_{i_{k}}} that converges smoothly on compact sets to a finite energy solution y of the perturbed equations (84) and (85) on (Y × ℝ, g_{t} + dt^{2}).

The proof of Claim 1 can be divided in the two cases of a compact set contained in the complement of Y × [0, 1], where the metric is constant, and of the set Y × [0, 1] where the metric changes. The first case is a direct application of the proof given in Theorem 4.1. In the second case, we obtain convergence as in [31], Lemma 3.19. We can find a sequence of gauge transformations λ_{i} such that the forms λ_{i}(𝔸_{i} − 𝔸_{0}) are co-closed and annihilate normal vectors at the boundary. Lemma 4 of [30] then shows that the sequence λ_{i}(𝔸_{i}, Ψ_{i}) has a subsequence that converges smoothly on Y × [0, 1].

Notice that the notion of finite energy, as well as the results of Corollary 3.11 and Theorem 3.12, extend to the case of solutions on (Y × ℝ, g_{t} + dt^{2}), since the metric is constant outside the compact set Y × [0, 1]. This implies that the argument given in Theorem 4.1 proves the following claim as well. Claim 2: If the limit y is an element of M(O_{a}, O_{a′}) then the convergence x_{i} → y is strong in the L_{k,δ}^{2} norm.

Thus we need to check whether there are obstructions to the convergence in norm, that is, whether broken trajectories arise as limits.

Suppose the element y is a limit smoothly on compact sets, but is not a limit in norm. Again, since the result of Corollary 3.11 holds, the solution y defines an element in some moduli space M(O_{b}, O_{b′}).

Since the metric is translation invariant only outside the interval [0, 1], consider the restriction of the sequence x_{i} and of the limit y to the interval (1, ∞) with the constant metric g_{1}. We can choose a value α satisfying

C_{ρ}(O_{b′}) > α > C_{ρ}(O_{a′}).

We can find times T_{i} > 1 such that C_{ρ}(x_{i}(T_{i})) = α. The restriction to (1, ∞) of the reparameterized sequence x_{i}^{T_{i}} is again a sequence of solutions of the flow equations with the constant metric g_{1} on Y × (1, ∞). A subsequence converges smoothly on compact sets to a solution ỹ. This defines an element in some M(O_{c′}, O_{d′}), subject to the constraint

C_{ρ}(O_{b′}) ≥ C_{ρ}(O_{c′}) > C_{ρ}(O_{d′}) ≥ C_{ρ}(O_{a′}).

Notice that the proof of Theorem 6.4 is essentially along the same lines as the proof of the analogous result, Theorem 4.1, for the moduli spaces of flow lines M̂(O_{a}, O_{b}). If we could guarantee that the cokernels of the linearizations are always trivial, then we could also extend the results of Section 4.2 and 4.3 to this case, showing that the same properties of the compactification hold true. However, the essential new feature which appears in dealing with the moduli spaces M(O_{a}, O_{a′}) is precisely that not all the cokernels are trivial. Thus, we need a much more accurate analysis of the fine structure of the compactification.

The strategy we follow is to restrict our attention to just the cases which are needed for the proof of Theorem 6.4, and show that, in those cases, we obtain the desired properties of the compactification. The purpose of the rest of Section 6 is to deal with these issues in detail.

In order to identify where the problem of cokernels may arise, we study the transversality issue for the moduli spaces M(O_{a}, O′_{a}). The following Lemma shows that the appearance of non-trivial cokernels is confined to the case of the moduli space M(θ_{0}, θ_{1}) with μ(θ_{0}) − μ(θ_{1}) < 0.

Lemma 6.5 Suppose given (g_{0}, ν_{0}), (g_{1}, ν_{1}) and a generic path (g_{t}, ν_{t}) connecting them. Consider the manifold Y ×ℝ with the metric g_{t}+dt^{2}, constant outside the interval [0, 1]. Suppose given x, a solution of equations (84) and (85), with asymptotic values on the critical orbits O_{a} and O_{a′}. Consider the linearization L_{x} of equations (84) and (85). Under a generic choice of the metric and of the perturbations P and ρ, we have the following result. If the solution x = (𝔸, Ψ) satisfies Ψ ≠ 0, then the linearization L_{x} is surjective. If we have O_{a} = θ_{0} and O_{a′} = θ_{1} and the solution x is of the form x = (𝔸, 0), then the linearization L_{x} is surjective, provided that the relative Morse index satisfies μ(θ_{0}) − μ(θ_{1}) ≥ 0.

The result of Lemma 6.5 can be rephrased as the following transversality statement for the moduli spaces M(O_{a}, O′_{a}).

Lemma 6.6 Consider the following cases:

(a) at least one of the asymptotic critical orbits O_{a} and O_{a′} has a free U(1)- action.

(b) we have O_{a} = θ_{0} and O_{a′} = θ_{1} under the condition that μ(θ_{0}) − μ(θ_{1}) ≥ 0. Then, for a generic choice of the metric and perturbations, the moduli space

M(O_{a}, O_{a′}) is a smooth oriented manifold that is cut out transversely by the equations, of dimension μ(O_{a})−μ(O_{a′})+dim(O_{a}). In particular, in case (a) the moduli space is generically empty when the virtual dimension μ(O_{a}) − μ(O_{a′}) + dim(O_{a}) is negative.

\underline{Proof of Lemma 6.5 and 6.6:} No solution with Ψ ≡ 0 arises under the hypothesis of case (a). Thus the argument of Proposition 2.12 and Proposition 3.14, applied to the linearization

L_{(𝔸,Ψ,P,ρ)}(α, Φ) = \{ \begin{matrix}D_{𝔸}Φ + αΨ + ρΦ \\ d^{+}α − \frac{1}{2}Im(Ψ · Φ̄) + DP_{(𝔸,Ψ)}(α, Φ) \\ G_{(𝔸,Ψ)}^{∗}(α, Φ) \end{matrix}

gives the desired transversality result. In case (b), we can find solutions in M(θ_{0}, θ_{1}) with Ψ ≠ 0, as well as a solution with Ψ ≡ 0. The linearization is surjective at solutions with Ψ ≠ 0 because of the argument of Proposition 2.12 and Proposition 3.14. Consider the linearization L_{x} at a solution of the form x = (𝔸, 0). We have

L_{(𝔸,0,P,ρ)}(α, Φ) = \{ \begin{matrix}D_{𝔸}Φ + ρΦ \\ d^{+}α + DP_{(𝔸,0)}(α, Φ) \\ G_{(𝔸,0)}^{∗}(α, Φ) \end{matrix}

Consider the operator

L̂_{(𝔸,0,P,ρ)}(α, Φ, p, η) = L_{(𝔸,0,P)}(α, Φ) + ( \begin{matrix}ηΦ \\ p_{(𝔸,0,P)}(α, Φ) \end{matrix} ) ,

where we vary the perturbation of the curvature equation by an element p_{(𝔸,0,P)} of T_{P} P, and the perturbation of the Dirac equation by a compactly supported 1-form η. We follow the analogous argument of Proposition 2.12. Suppose (β, ξ) is an element orthogonal to the range of L̂_{(𝔸,0,P,ρ)}. We have terms of the form

⟨β, d^{+}α + DP_{(𝔸,0)}(α, Φ) + p_{(𝔸,0,P)}(α, Φ)⟩
+⟨ξ, D_{𝔸}Φ + η · Φ⟩ = 0.

By varying p ∈ P we force β ≡ 0. If μ(θ_{0}) − μ(θ_{1}) = 0, then the condition

⟨ξ, D_{𝔸}Φ + η · Φ⟩ = 0

implies that ξ is in the kernel of D_{𝔸−η}, but for a generic choice of η this operator has trivial kernel. If μ(θ_{0}) − μ(θ_{1}) > 0, the argument is similar: choose Φ to be a non-trivial element in Ker(D_{𝔸}). This is possible since we are assuming that μ(θ_{0}) − μ(θ_{1}) > 0, hence that dim Ker(L_{x}) > 0. We show that, by varying η we force ξ to vanish. In fact, suppose

⟨ξ, η · Φ⟩ = 0

is satisfied for all possible compactly supported 1-forms η, and for a non-trivial ξ. Then we can follow the argument of [43] 6.2.1: choose some small open set where both Φ and ξ are non-trivial and approximately constant. The isomorphism

ℝ^{4} ⊗_{ℝ} ℂ → Hom_{ℂ}(S^{+}, S^{−})

induced by Clifford multiplication implies that there exist η ∈ ℝ^{4} such that

Re(⟨ξ, η · Φ⟩) > 0

at some point in the small open set U. By a trivialization of T^{∗}(Y × R) on U the vector η can be extended to a 1-form η on U. This 1-form can be extended to a compactly supported 1-form in a neighborhood of U, so that we get

∫_{Y ×ℝ} Re(⟨ξ, η · Φ⟩) > 0,

which contradicts the initial assumption.

Notice that, since we consider the gauge group G^{0}(O_{a}, O_{a′}) of gauge transformations that decay to elements λ(±∞) in the stabilizers G_{a} and G_{a′}, a constant element g is necessarily trivial, hence the element x = (𝔸, 0) is a manifold point in B^{0}(O_{a}, O_{a′}), fixed by the U(1)-action. Thus, the operator L_{(𝔸,Ψ,P,ρ)} is surjective, for a generic choice of the perturbations P and ρ. This proves that in case (a) or case (b) the moduli space M(O_{a}, O_{a′}) is cut out transversely by the equations and of dimension equal to the virtual dimension prescribed by the index theorem. The orientation of the moduli spaces M(O_{a}, O_{a′}) is obtained as in Proposition 2.13, from an orientation of the homology groups

H_{δ}^{0}(Y × ℝ) ⊕ H_{δ}^{2+}(Y × ℝ) ⊕ H_{δ}^{1}(Y × ℝ).

⋄

Notice that, since we are dealing with the case of homology spheres, we have only finitely many critical orbits O_{a} and O_{a′}. Thus we can assume that there is a fixed compact set Y × [−T_{0}, T_{1}], with T_{0} ≥ 0 and T_{1} ≥ 1, such that transversality of all the moduli spaces of case (a) and (b) of Lemma 6.6 can be achieved by a perturbation ρ supported on Y × [−T_{0}, T_{1}], together with the perturbation P of the curvature equation. Notice that outside [−T_{0}, T_{1}] the metric is translation invariant and the equations coincide with the gradient flow equations. For simplicity of notation in the following we shall assume that [−T_{0}, T_{1}] is just the interval [0, 1].

We have the following result.

Corollary 6.7 If either O_{b} or O_{b′} is a critical orbit with a free U(1)-action, then there are no reducible flow lines in the moduli space M(O_{b}, O_{b′}), and the moduli space is generically empty if the virtual dimension is negative.

Consider instead the case O_{a} = θ_{0} and O_{a′} = θ_{1} with μ(θ_{0}) − μ(θ_{1}) < 0. In this case no solution with Ψ ≠ 0 appears in the moduli space M(θ_{0}, θ_{1}).

Both statements are consequences of Lemma 6.5. In fact, if there were any solution, then by argument of Lemma 6.6 the linearization ought to be surjective for a generic choice of the perturbations, and this is incompatible with the value of the index.

This result shows that reducible solutions in M(O_{a}, O_{a′}) may appear only in the moduli spaces M(θ_{0}, θ_{1}), with μ(θ_{0})−μ(θ_{1}) < 0, which is in fact the only case not covered by the transversality result of Lemma 6.5.

We can give a more precise description of this moduli spaces as follows.

Lemma 6.8 Consider a choice of metrics and perturbations (g_{0}, ν_{0}) and (g_{1}, ν_{1}) in different chambers, with a negative spectral flow SF (∂_{ν_{t}}^{g_{t}}) < 0. In this case, the moduli space M(θ_{0}, θ_{1}) consists of a unique reducible class x = [𝔸, 0]. In particular, M(θ_{0}, θ_{1}) is non-empty, though the virtual dimension μ(θ_{0})−μ(θ_{1}) < 0 is negative. Thus, we have

dim Coker(L_{x}) = |μ(θ_{0}) − μ(θ_{1})| = −SF (∂_{ν_{t}}^{g_{t}}).

\underline{Proof:} Lemma 6.6 shows that M(θ_{0}, θ_{1}) can only contain reducible solutions. In particular, x = [𝔸, 0] is obtained as the unique solution of the equations

d^{+}𝔸 = μ and d^{∗}𝔸 = 0.

Here d^{∗} is the operator on the four-manifold Y × ℝ, with ∗_{4} the Hodge operator with respect to the metric g_{t} + dt^{2}, and the perturbation form μ is of the form μ = dν_{t} + ∗_{3}dν_{t} ∧ dt, with the ∗_{3} operator with respect to the metric g_{t}.

This solution of the four-dimensional Seiberg-Witten equations gives an element x = [𝔸, 0] which is a fixed point of the U(1)-action on the configuration space B^{0}(θ_{0}, θ_{1}). Notice that, though the point x = [𝔸, 0] is a smooth point in B^{0}(θ_{0}, θ_{1}) (fixed by the U(1)-action), it is a singular point in M(θ_{0}, θ_{1}), due to the lack of transversality, that is, the presence of a non-trivial Cokernel. ⋄

We see, from the previous discussion, that one first problem we encounter in extending the proof of the easy case of Theorem 6.1 to the harder case is the compactification of the moduli spaces M(O_{a}, O_{a′}). In fact, consider the case of the compactification of a moduli space M(θ_{0}, O_{a′}), under the assumption that μ(θ_{0}) − μ(θ_{1}) < 0.

For instance, in the case μ(θ_{0}) − μ(θ_{1}) = −2, by the result of Theorem 6.4, we know that the product

M(θ_{0}, θ_{1}) × M̂(θ_{1}, O_{a′})  (88)

is a possible component of the ideal boundary. The counting of the virtual dimensions is correct as prescribed by Theorem 6.4 for components of codimension one in the ideal boundary. In fact, we have

virtdim(M(θ_{0}, θ_{1}) × M̂(θ_{1}, O_{a′})) = μ(θ_{0}) − μ(θ_{1}) + μ(θ_{1}) − μ(O_{a′}) − 1
= μ(θ_{0}) − μ(O_{a′}) − 1 = dim(M(θ_{0}, O_{a′})) − 1.

However, from the result of Lemma 6.8, we know that the actual dimension of M(θ_{0}, θ_{1}) is zero, since it consists of the unique solution x = (𝔸, 0), hence the actual dimension of (88) is equal to μ(θ_{1}) − μ(O_{a′}). This gives

dim(M(θ_{0}, θ_{1}) × M̂(θ_{1}, O_{a′})) = μ(θ_{1}) − μ(O_{a′}) − 1
= μ(θ_{0}) + 2 − μ(O_{a′}) − 1 = dim(M(θ_{0}, O_{a′})) + 1.

Clearly, this implies that, if an entire component of the form (88) actually appears in the compactification (that is, if they can be spliced together and deformed to actual solutions as described in Section 4.2), then the resulting compactification M(θ_{0}, O_{a′})^{∗} does not have the structure of smooth manifold with corners described in Section 4.3. This could cause problems in applying the arguments based on Stokes’ theorem in the proof of Theorem 6.1. Section 6.2 deals with this problem. The main result is stated in Theorem 6.21.

We now proceed with general properties of the compactification of the moduli spaces M(O_{a}, O_{a′}) and M^{P} (O_{a}, O_{b}) which we can derive in general, regardless of the behavior of the singular components of the ideal boundary.

We have the following gluing theorem.

Theorem 6.9 For O_{a} and O_{a′} irreducible, given a compact set

K ⊂ M̂(O_{a}, O_{b}) ×_{O_{b}} M(O_{b}, O_{a′})

or

K ⊂ M(O_{a}, O_{b′}) ×_{O_{b′}} M̂(O_{b′}, O_{a′}),

there is a lower bound T_{0}(K) > 0 and gluing maps

# : K × [T, ∞) → M(O_{a}, O_{a′}),  (89)

such that #_{T} is a smooth embedding for all T ≥ T_{0}(K). The gluing maps have the property that any sequence of solutions in M(O_{a}, O_{a′}) that converges to an element in the boundary is eventually contained in the range of the gluing map. The map # is orientation preserving with respect to the orientation given by

(M̂(O_{a}, O_{b}) × ℝ^{+}) ×_{O_{b}} M(O_{b}, O_{a′})

and

M(O_{a}, O_{b′}) ×_{O_{b′}} (M̂(O_{b′}, O_{a′}) × ℝ^{+}).

Similarly, for

K ⊂ M̂(O_{a}, θ_{0}) × M(θ_{0}, O_{a′})

or

K ⊂ M(O_{a}, θ_{1}) × M̂(θ_{1}, O_{a′}),

there is a T_{0}(K) > 0 and gluing maps

# : K × [T, ∞) → M(O_{a}, O_{a′}),  (90)

which are smooth embeddings for T ≥ T_{0}(K), and with the same properties on the range and on the orientation.

\underline{Proof:} We know from the results of Lemma 6.6 that all the moduli spaces involved are smooth manifolds, cut out transversely by the equations. In particular, for any element (x, y) in a fibered product as above, the cokernels Coker(L_{x}) and Coker(L_{y}) are trivial. Thus, we can adapt the same argument used in Section 4.2 for the gluing construction, using the result of Proposition 4.21.

Namely, we proceed as follows. We show that all broken trajectories of the form M(O_{a}, O_{b′}) ×_{O_{b}} M̂(O_{b′}, O_{a′}) are indeed contained in the boundary of M(O_{a}, O_{a′}). The argument for the components of the form M̂(O_{a}, O_{b}) ×_{O_{b}} M(O_{b}, O_{a′}) is analogous. We need to show that, given x and ŷ in M(O_{a}, O_{b′}) and M̂(O_{b′}, O_{a′}) respectively, we can form a pre-glued approximate solution and perturb it to an actual solution as in Theorem 4.9. We have temporal gauge representatives

(A_{1}(t), ψ_{1}(t)) ∈ A_{k,δ}(O_{a}, O_{b′})

and

(A_{2}(t), ψ_{2}(t)) ∈ A_{k,δ}(O_{b′}, O_{a′}).

The solution y is translation invariant, but the solution x is not, since the metric is constant only outside the interval [0, 1]. Thus we have to modify slightly the definition of the pre-gluing map. Choose slices

S_{Γ_{ab′}} ⊂ A_{k,δ}(O_{a}, O_{b′})

and

S_{Γ_{b′a′}} ⊂ A_{k,δ}(O_{b′}, O_{a′}),

determined by the elements

Γ_{ab′} ∈ A_{k,δ}((A_{a}, ψ_{a}), (A_{b′}, ψ_{b′}))

and

Γ_{b′a′} ∈ A_{k,δ}((A_{b′}, ψ_{b′}), (A_{a′}, ψ_{a′})).

We choose them in such a way that we have representatives x and y in a ball of radius r centered at Γ_{ab′} and Γ_{b′a′} respectively. Then, according to Lemma 4.8, there are gauge transformations λ_{1}^{+} and λ_{2}^{−} that conjugate x and y into W_{Γ_{ab′}}^{+} and W_{Γ_{b′a′}}^{−} respectively. For t ≥ T, there is a gauge transformation λ_{b′} ∈ G(Y) such that we can write λ_{1}^{+}x(t) = λ_{b′}(A_{b′}, ψ_{b′}) + (α_{1}(t), ϕ_{1}(t)) and λ_{2}^{−}y(t) = λ_{b′}(A_{b′}, ψ_{b′}) + (α_{2}(t), ϕ_{2}(t)), where [A_{b′}, ψ_{b′}] is a point on the orbit O_{b′} and we have

lim_{t→∞}(A_{1}(t), ψ_{1}(t)) = λ_{b′}(A_{b′}, ψ_{b′}) = lim_{t→−∞}(A_{2}(t), ψ_{2}(t)).

The (right) pre-glued approximate solution is given as follows:

x#_{T}^{0} y = \{ \begin{matrix}λ_{1}^{+}x & t ≤ T − 1 \\ λ_{b′}(A_{b′}, ψ_{b′}) + ρ_{T}^{−}(t)(α_{1}(t), ϕ_{1}(t))+ \\ ρ_{T}^{+}(t)(α_{2}(t − 2T), ϕ_{2}(t − 2T)) & T − 1 ≤ t ≤ T + 1 \\ λ_{2}^{−}y^{−2T} & t ≥ T + 1. \end{matrix} (91)

Here ρ_{T}^{±}(t) are smooth cutoff functions with bounded derivative, such that ρ_{T}^{−}(t) is equal to one for t ≤ T − 1 and to zero for t ≥ T and ρ_{T}^{+}(t) is equal to zero for t ≤ T and to one for t ≥ T + 1. Lemma 4.10 shows that #_{T}^{0} descends to a well defined pre-gluing

#_{T}^{0} : M(O_{a}, O_{b′}) ×_{O_{b′}} M̂(O_{b′}, O_{a′}) → B^{0}(O_{a}, O_{a′}).

We are assuming that O_{a} and O_{a′} have a free U(1) action, thus, according to Lemma 6.6, Corollary 6.7, and Proposition 2.10, we can ensure that, for a generic choice of the metric and perturbation we have Coker(L_{x}) = 0 and Coker(L_{y}) = 0. Lemma 4.13 and Proposition 4.19 show that we have an isomorphism at the level of the actual kernel of the linearization of the pre-glued solution:

Ker(L_{x#_{T}^{0} y}) ≅ Ker(L_{x}) × Ker(L_{y}).

This implies that we have Coker(L_{x#_{T}^{0} y}) = 0 We proceed as in Proposition 4.21 and we are able to perturb the approximate solutions to an actual solutions by the fixed point argument of Theorem 4.9.

In the case of pre-gluing broken trajectories in the components

M̂(O_{a}, O_{b}) ×_{O_{b}} M(O_{b}, O_{a′})

we need to consider the (left) pre-gluing map

x#_{−T}^{0} y = \{ \begin{matrix}λ_{1}^{+}x^{2T} & t ≤ −T − 1 \\ λ_{b}(A_{b}, ψ_{b}) + ρ_{−T}^{+} (t)(α_{2}(t), ϕ_{2}(t)+ \\ ρ_{−T}^{−} (t)(α_{1}(t + 2T), ϕ_{1}(t + 2T)) & −T − 1 ≤ t ≤ −T + 1 \\ λ_{2}^{−}y & t ≥ −T + 1. \end{matrix} (92)

Here ρ_{−T}^{±} (t) are smooth cutoff functions with bounded derivative, such that ρ_{−T}^{−} (t) is equal to one for t ≤ −T − 1 and to zero for t ≥ −T and ρ_{−T}^{+} (t) is equal to zero for t ≤ −T and to one for t ≥ −T + 1. The rest of the argument is analogous. We have

Ker(L_{x#_{T}^{0} y}) ≅ Ker(L_{x}) × Ker(L_{y})

and Coker(L_{x#_{T}^{0} y}) = 0. We then proceed as in the previous case. ⋄

We have a similar result for the moduli spaces M^{P} (O_{a}, O_{c}).

Theorem 6.10 For O_{a} and O_{c} free orbits, consider a compact set

K ⊂ M^{P} (O_{a}, O_{b}) ×_{O_{b}} M̂(O_{b}, O_{c})

or

K ⊂ M̂(O_{a}, O_{b}) ×_{O_{b}} M^{P} (O_{b}, O_{c}).

There is a bound T_{0}(K) and gluing maps

# : K × [T_{0}, ∞) → M^{P} (O_{a}, O_{c})

that are smooth embeddings, compatible with the orientation induced by the product orientation of

M^{P} (O_{a}, O_{b}) × ℝ × M̂(O_{b}, O_{c})

and

M̂(O_{a}, O_{b}) × ℝ × M^{P} (O_{b}, O_{c}).

Similarly, for a compact set

K ⊂ M(O_{a}, O_{a′}) ×_{O_{a′}} M(O_{a′}, O_{c}),

there is a T_{0}(K) and an orientation preserving gluing map

# : K ⊂ M(O_{a}, O_{a′}) ×_{O_{a′}} M(O_{a′}, O_{c}) → M^{P} (O_{a}, O_{c}) ∩ {σ = 1},  (93)

that is a smooth embedding. Since by construction the moduli space M^{P} (O_{a}, O_{c}) satisfies

M^{P} (O_{a}, O_{c}) ∩ {σ ∈ (1 − ϵ, 1]} ≅ (M^{P} (O_{a}, O_{c}) ∩ {σ = 1}) × (1 − ϵ, 1],

this provides the necessary collar structure at the boundary components

M(O_{a}, O_{a′}) ×_{O_{a′}} M(O_{a′}, O_{c}).  (94)

Moreover, any sequence of solutions in M^{P} (O_{a}, O_{c}) converging to the boundary lies eventually in the range of the gluing map.

\underline{Proof:} The proof of the first two cases is analogous to the case of flow lines analyzed in Section 4, therefore we concentrate on the case of the gluing map (93). Consider the moduli space

M_{σ=1}^{P}(O_{a}, O_{c}) = M^{P} (O_{a}, O_{c}) ∩ {σ = 1},

with the metric γ_{1} = g_{t}#g̃_{t}, as in (78). We define the metric γ_{1}(R) = g_{t}#_{R}g̃_{t}, for sufficiently large R ≥ R_{0}, as

γ_{1}(R) = \{ \begin{matrix}g_{0} + dt^{2} & t < −2 − R \\ g_{t+2} + dt^{2} & t ∈ [−2 − R, −1 − R] \\ g_{1} + dt^{2} & t ∈ [−1 − R, 1 + R] \\ g̃_{2−t} + dt^{2} & t ∈ [1 + R, 2 + R] \\ g_{0} + dt^{2} & t > 2 + R. \end{matrix}  (95)

Correspondingly, we have metrics γ_{σ}(R), for σ ∈ [0, 1]. We consider the moduli spaces M^{P} (O_{a}, O_{c}) as before, defined with respect to the metrics γ_{σ}(R). All the properties of these moduli spaces discussed so far, are independent of the choice of a fixed R. However, to describe in detail the appearance of strata of the form (94) in the compactification, and their gluing properties, we need to work with a large R. This correspond to stretching a long cylinder Y ×[−1−R, 1+R] with the constant metric g_{1} + dt^{2} inside the manifold Y × ℝ. The proof is obtained via the following argument.

Proposition 6.11 Consider a solution x = [𝔸, Ψ] in M_{σ=1}^{P}(O_{a}, O_{c}) on the manifold (Y ×ℝ, γ_{1}(R)). For every ϵ > 0, there exists R_{0}, such that, if R ≥ R_{0}, the restriction of x to Y × [−2, 2] is sufficiently close to a solution [A_{a′}, ψ_{a′}] ∈ O_{a′}, for some critical orbit O_{a′}. Moreover, given a pair of solutions (x, y) in the fibered product (94), for R sufficiently large, we can consider the preglued solution obtained by splicing together the solutions x and y cut off by functions ρ_{−} and ρ_{+} supported in Y × (−∞, −1] and Y × [1, ∞), respectively. For R sufficiently large, these approximate solutions can be deformed to actual solutions. Thus, we obtain the gluing map (93) with the desired properties.

\underline{Proof of Proposition 6.11:} the proof of the first statement follows closely the argument of Proposition 8 of [30]. The gluing argument again follows very closely the argument illustrated in Section 4.2, Theorem 4.9 in the case of the moduli spaces of flow lines. Again, the key feature is the fact that the cokernels Coker(L_{x}) and Coker(L_{y}) vanish, hence the gluing argument presents no surprise with respect to the case of gluing flow lines. ⋄

All the remaining statements of Theorem 6.10 are proved by the same techniques developed in Section 4. ⋄

The main problem now is to extend these gluing theorems to the case of strata involving a contribution from the singular moduli space M(θ_{0}, θ_{1}). The strategy we develop in the rest of Section 6.1 and in Section 6.2 is to identify precisely the contribution of these singular components (88) to the compactification of the moduli spaces M(θ_{0}, O_{a′})^{∗} and to the analogous case for M^{P} (O_{a}, O_{b})^{∗}. The main technique we employ is the obstruction bundle for the gluing construction and a direct analysis of the zero set of the canonical obstruction section. We aim at showing that all the components of the actual boundary in the compactification M(O_{a}, O_{a′})^{∗} behave according to the picture described in Section 4.3, hence the arguments of the easy case of Theorem 6.1 involving the properties of the compactifications moduli spaces M(O_{a}, O_{a′})^{∗} and M^{P} (O_{a}, O_{b})^{∗} can be extended to the case of different chambers as well.

Remark 6.12 Recall that, in the proof of Theorem 6.1, we need only consider the case of two sufficiently close metrics and perturbations (g_{0}, ν_{0}) and (g_{1}, ν_{1}) in two different chambers. Thus, we can assume that the path (g_{t}, ν_{t}) satisfies SF (∂_{ν_{t}}^{g_{t}}) = −2 and the path (g̃_{t}, ν̃_{t}) satisfies SF (∂g̃_{t} ; ν̃_{t}) = +2. Thus, the compactified moduli spaces M(O_{a′}, O_{a})^{∗} used in the definition of the map J and in the proof of DJ −JD = 0 all satisfy the properties of Section 4.2 and 4.3, by simply rephrasing the same arguments. The only problem then remains in the analysis of the moduli spaces needed in the definition of I, in the proof of ID − DI = 0 and in the construction of the chain homotopy.

6.2 Obstruction bundle and gluing theorems

The obstruction bundle technique we discuss in this Section will both complete the proof of the unobstructed gluing theorems, as stated in Proposition 4.21, and identify the precise boundary structure in the gluing with obstructions.

Let U(O_{a}, O_{b}) be the image of M(O_{a}, O_{c}) ×_{O_{c}} M(O_{c}, O_{b}) under the pregluing map #^{0}, for a large gluing parameter T >> 0. In the case of a translation invariant metric, analyzed in Section 4, the statement of the gluing theorem 4.9 is equivalent to the statement that the subset

U(O_{a}, O_{b}) ⊂ B_{k,δ}^{0}(O_{a}, O_{b})

can be deformed to M(O_{a}, O_{b}), whenever U(O_{a}, O_{b}) is obtained by pasting together solutions along O_{b} with μ(O_{a}) > μ(O_{c}) > μ(O_{b}). It is this equivalent statement which we are going to verify in the proof of Proposition 4.21 given in this Section, following Lemma 6.14.

However, if we consider the moduli spaces M(O_{a}, O_{a′}) with the metric g̃_{t} that varies along the cylinder, there might be topological obstructions that make it impossible to deform the set of pre-glued trajectories to actual solutions of (18) and (19). In other words, given a point [x#_{T}^{0} y] ∈ U(O_{a}, O_{a′}) there might be an obstruction to pushing that point onto M(O_{a}, O_{a′}). Typically obstructions may arise either because of the presence of non-trivial cokernels of the linearizations L_{x} and L_{y}, or because Sard’s theorem is not available. In the problem we are considering the obstruction originates in the presence of the non-trivial cokernel of Lemma 6.8. Similar topological obstructions to gluing solutions arise in other gauge theory problems [57], [58]. Following Taubes’ construction of the obstruction bundle, it is possible to describe the obstruction completely in terms of the vanishing of a canonical section.

We maintain the notation (52) as in the proof of the gluing theorem: for an element x#_{T}^{0} y, with x = (𝔸_{1}, Ψ_{1}) and y = (𝔸_{2}, Ψ_{2}), we write

f(x#_{T}^{0} y) = \{ \begin{matrix}D_{𝔸_{1}#_{T}^{0} 𝔸_{2}}(Ψ_{1}#_{T}^{0} Ψ_{2}) + ρ(Ψ_{1}#_{T}^{0} Ψ_{2}) \\ F_{𝔸_{1}#_{T}^{0} 𝔸_{2}}^{+} − (Ψ_{1}#_{T}^{0} Ψ_{2}) · \overline{(Ψ_{1}#_{T}^{0} Ψ_{2})} − iμ − P_{(𝔸_{1}#_{T}^{0} 𝔸_{2},Ψ_{1}#_{T}^{0} Ψ_{2})} \\ G_{Γ_{aa′}}^{∗} (𝔸_{1}#_{T}^{0} 𝔸_{2}, Ψ_{1}#_{T}^{0} Ψ_{2}), \end{matrix}

for a fixed choice of a slice S_{Γ_{aa′}} in A(O_{a}, O_{a′}), such that x#_{T}^{0} y is in a ball of radius r in A(O_{a}, O_{a′}) centered at Γ_{aa′}. If the pre-glued element x#_{T}^{0} y can be deformed to an actual solution, then there exists a small (α, Φ) ∈ Ker(L_{x#_{T}^{0} y})^{⊥} such that we have

[x#_{T}^{0} y + (α, Φ)] ∈ M(O_{a}, O_{a′}),

that is, the following equation is satisfied

f(x#_{T}^{0} y) + L_{x#_{T}^{0} y}(α, Φ) + N_{x#_{T}^{0} y}(α, Φ) = 0.  (96)

Here N denotes the nonlinear term in the equation, as in Proposition 4.21.

N_{x#_{T}^{0} y}(α, Φ) = (Φ · Φ̄ + N P_{x#_{T}^{0} y}(α, Φ), α · Φ),

where the perturbation P of equation (85) is written as sum of a linear and a non-linear term, P = DP + N P.

The presence of a non-trival approximate kernel of L_{x#_{T}^{0} y}^{∗} can generate obstructions to solving equation (96) for (α, Φ). In fact, the hypothesis that L_{x#_{T}^{0} y} has a trivial approximate cokernel is essential in the proof of the gluing theorem (see Lemma 4.13 and Proposition 4.19): the same argument cannot be extended to a case with a non-trivial cokernel.

We follow [58] and introduce open sets U(μ) of elements [x#_{T}^{0} y] ∈ U(O_{a}, O_{a′}) such that μ > 0 is not an eigenvalue of the operator L_{x#_{T}^{0} y}L_{x#_{T}^{0} y}^{∗} acting on L_{0,δ(T)}^{2} connections and sections. There are projection maps Π(μ, x#_{T}^{0} y) onto the span of the eigenvectors of L_{x#_{T}^{0} y}L_{x#_{T}^{0} y}^{∗} with eigenvalue smaller than μ. These are smooth maps of [x#_{T}^{0} y] ∈ U(μ).

We have the following result, which gives the necessary eigenvalue splitting for the operators L_{x#_{T}^{0} y}L_{x#_{T}^{0} y}^{∗} and L_{x#_{T}^{0} y}^{∗}L_{x#_{T}^{0} y}.

Lemma 6.13 There exists a μ_{0} = μ(T_{0}) > 0, such that, for all T ≥ T_{0}, all the small eigenvalues μ_{T} of L_{x#_{T}^{0} y}^{∗}L_{x#_{T}^{0} y} (or L_{x#_{T}^{0} y}L_{x#_{T}^{0} y}^{∗}), satisfy μ_{T} < μ_{0} and μ_{T} → 0, and all the other eigenvalues are bounded below by μ_{0}.

\underline{Proof:} Recall that, from Lemma 4.12 we know that the number of independent eigenvectors of L_{x#_{T}^{0} y}^{∗}L_{x#_{T}^{0} y} or L_{x#_{T}^{0} y}L_{x#_{T}^{0} y}^{∗} with small eigenvalue μ_{T} → 0 is at most

dim Ker(L_{x}) + dim Ker(L_{y})

or at most

dim Coker(L_{x}) + dim Coker(L_{y}),

respectively. Moreover, by Lemma 4.15, we know that the spans of the eigenvectors of the small eigenvalues have exactly these dimensions, cf. Proposition 4.17. So there exists a μ_{0} such that, for T ≥ T_{0}, there is exactly this number of independent eigenvectors with eigenvalues μ_{T} < μ_{0} and μ_{T} → 0. In fact, we also know more precise estimates on the rate of decay of these small eigenvalues, as specified in Lemma 4.15. It remains to be shown that such μ_{0} can be chosen so that all the remaining eigenvalues are bounded below by μ_{0}. In other words, we have to show that we cannot have other sequences of eigenvalues μ_{T} (k) satisfying μ_{T} (k) → μ_{k} > 0, as T → ∞, but with the limits μ_{k} → 0 as k → ∞. This follows from Lemma 4.13. In fact, we have proved that the operator L_{x#_{T}^{0} y}^{∗}L_{x#_{T}^{0} y} (or L_{x#_{T}^{0} y}L_{x#_{T}^{0} y}^{∗}) is uniformly invertible on the orthogonal complement of the small eigenvalues eigenspace. Notice that Lemma 4.15 and Lemma 4.13 give norms for the operator L_{x#_{T}^{0} y} (and by similar arguments for the operator L_{x#_{T}^{0} y}^{∗}) which are uniformly bounded in T, when the operators are considered as acting between Sobolev spaces with the same δ weight on source and target space (or with the same rescaled weight δ(T) on both source and target space). In other words we have

∥L_{x#_{T}^{0} y}^{∗}L_{x#_{T}^{0} y}ξ∥_{L_{1,δ}^{2}} ≥ C∥ξ∥_{L_{1,δ}^{2}},

for

ξ ∈ (F_{#_{T}} (Ker(L_{x}) × Ker(L_{y})))^{⊥}

inside

T_{1,δ}((e_{a}^{+})^{−1}(x_{a})) ⊂ T_{1,δ}(S_{Γ_{aa′}}),

and

∥L_{x#_{T}^{0} y}L_{x#_{T}^{0} y}^{∗}ξ∥_{L_{0,δ}^{2}} ≥ C∥ξ∥_{L_{0,δ}^{2}},

for

ξ ∈ (F_{#_{T}} (Coker(L_{x}) × Coker(L_{y})))^{⊥}

inside T_{0,δ}(S_{Γ_{aa′}}). With this choices of norms, the constants are all independent of T. Here x_{a} ∈ O_{a} is the asymptotic value as t → −∞ of the solution x ∈ M(O_{a}, O_{a′}), and e_{a}^{+} : S_{Γ_{aa′}} → O_{a} is the endpoint map, for the fixed choice of slices. ⋄

It is important to notice that the difference between using the rescaled δ(T)– norms or the original δ–norms in the previous Lemma is reflected in the different norm bound on the gluing map F̃_{#_{T}}, as explained in the proof of Theorem 4.9.

The kind of analysis of the behavior of small eigenvalues described in Lemma 6.13 has been used in similar context in [57], [58], with a fuller account in [47], and in [10]. A similar analysis has also been used, more recently, in [11], [22].

By Lemma 6.13, we now know that there exist T_{0} >> 0 and μ(T_{0}) > 0, such that, for T > T_{0} and 0 < μ < μ(T_{0}), we can identify the projection map Π(μ, x#_{T}^{0} y) with projection on the approximate cokernel ApprCoker(L_{x#_{T}^{0} y}), as defined in Definition 4.16.

Consider a compact set K ⊂ U(O_{a}, O_{a′}). Suppose it can be covered with one open set U(μ). If not, we can paste together the projections corresponding to the finitely many open sets that cover K, using a suitable partition of unity as in [58], pg. 193 and slightly modifying the argument that follows in [58].

Since the element (α, Φ) is in Ker(L_{x#_{T}^{0} y})^{⊥}, we have

(α, Φ) = L_{x#_{T}^{0} y}^{∗}(β, ξ).

Suppose that [x#_{T}^{0} y + (α, Φ)] is a solution, for a non-trivial element (α, Φ). For μ small enough, we can assume that (β, ξ) satisfies

(β, ξ) ∈ Ker(Π(μ, x#_{T}^{0} y)).

Then (β, ξ) solves the equations

L_{x#_{T}^{0} y}L_{x#_{T}^{0} y}^{∗}(β, ξ) + (1 − Π(μ, x#_{T}^{0} y)) (N_{x#_{T}^{0} y}L_{x#_{T}^{0} y}^{∗}(β, ξ) + f(x#_{T}^{0} y)) = 0.

(97)

and

Π(μ, x#_{T}^{0} y) (N_{x#_{T}^{0} y}L_{x#_{T}^{0} y}^{∗}(β, ξ) + f(x#_{T}^{0} y)) = 0.  (98)

Lemma 6.14 below ensures that it is always possible to find a small solution of equation (97), hence the problem of deforming an approximate solution to an actual solution depends on whether equation (98) can also be solved. The latter has a geometric interpretation as the section of an obstruction bundle, as described in the following (see [16] Section 4, [57] Section 3 and 5, [58] Section 6).

Lemma 6.14 There is an ϵ > 0 and a constant C > 0 such that, for any μ with Cϵ < μ < μ_{0}, with μ_{0} the least eigenvalue of H_{x#_{T}^{0} y}^{0} on the complement of ApprCoker(L_{x#_{T}^{0} y}), the equation (97) has a unique solution

(β, ξ) ∈ Ker(Π(μ, x#_{T}^{0} y))

with ∥(β, ξ)∥_{L_{2,δ}^{2}} ≤ ϵ, provided the error term ∥f(x#_{T}^{0} y)∥ satisfies an estimate

∥f(x#_{T}^{0} y)∥ ≤ \frac{μ}{2ϵ}.

\underline{Proof:} For μ > 0 and any (𝔸, Ψ), the operator

H_{(𝔸,Ψ)}^{0} = L_{(𝔸,Ψ)}L_{(𝔸,Ψ)}^{∗}

has a bounded inverse, when restricted to the image of (1 − Π(μ, (𝔸, Ψ)), in the space of L_{δ}^{2} connections and sections, as discussed in Lemma 4.13.

Thus, the equation (97) can be rephrased as a fixed point problem

(β, ξ) = −H_{x#_{T}^{0} y}^{0}^{−1}(1 − Π(μ, x#_{T}^{0} y))(N_{x#_{T}^{0} y}L_{x#_{T}^{0} y}^{∗}(β, ξ) + f(x#_{T}^{0} y)). (99)

We need to prove that the right hand side is a contraction. For all (β, ξ) and (β′, ξ′), we have an estimate

∥H_{x#_{T}^{0} y}^{0}^{−1}(1 − Π(μ, x#_{T}^{0} y)) (N L^{∗}(β, ξ) − N L^{∗}(β′, ξ′)) ∥_{L_{δ}^{2}} ≤

\frac{1}{μ}∥(1 − Π)(N L^{∗}(β, ξ) − N L^{∗}(β′, ξ′))∥_{L_{δ}^{2}} ≤

\frac{1}{μ}∥N L^{∗}(β, ξ) − N L^{∗}(β′, ξ′)∥_{L_{δ}^{2}}.

The quadratic form (ϕ · ϕ̄, α · ϕ) satisfies an estimate of the form

∥N(x) − N(y)∥ ≤ C(∥x∥ + ∥y∥)∥x − y∥

as required for the contraction method 4.20. The perturbation term also satisfies a similar estimate for large enough T because of the assumptions on the perturbation space P. Thus we can improve the last estimate to get

\frac{1}{μ}∥N L^{∗}(β, ξ) − N L^{∗}(β′, ξ′)∥_{L_{δ}^{2}} ≤

\frac{C̃}{μ} (∥L^{∗}(β, ξ)∥ + ∥L^{∗}(β′, ξ′)∥)∥L^{∗}((β, ξ) − (β′, ξ′))∥_{L_{1,δ}^{2}} ≤

\frac{C}{μ} (∥(β, ξ)∥ + ∥(β′, ξ′)∥)∥(β, ξ) − (β′, ξ′)∥_{L_{2,δ}^{2}}.

Thus, we have

∥H_{x#_{T}^{0} y}^{0}^{−1}(1 − Π(μ, x#_{T}^{0} y)) (N_{x#_{T}^{0} y}L_{x#_{T}^{0} y}^{∗}(β, ξ) + f(x#_{T}^{0} y)) ∥ ≤

\frac{C}{μ} ∥(β, ξ)∥ + \frac{1}{μ}∥f(x#_{T}^{0} y)∥.

Suppose given ϵ and μ satisfying

max{2Cϵ, ϵ} < μ < μ_{0},

and assume the the error term also satisfies

∥f(x#_{T}^{0} y)∥ ≤ \frac{μ}{2ϵ}.

Then the map is a contraction on the ball ∥(β, ξ)∥_{L_{δ}^{2}} ≤ ϵ. ⋄

Remark 6.15 Lemma 6.14 relies on the identification of the image of 1 − Π(μ, x#_{T}^{0} y), that is, the complement of the small eigenvalues eigenspace for the Laplacian H_{x#_{T}^{0} y}^{0}, with the space

F_{#_{T}} (Coker(L_{x}) × Coker(L_{y})).

This identification is obtained from Lemma 4.13 together with Remark 4.14. In particular, recall that, by Lemma 4.13 and Remark 4.14, the isomorphism

F_{#_{T}} : CokerL_{x} × CokerL_{y} →^{≅} Image(1 − Π(μ, x#_{T}^{0} y))

has norm bounded uniformly in T if we use weighted norms with rescaled weight δ(T) on the target space and the δ-norms on the source spaces. It has norm bounded by Ce^{−δT} if the target space is also endowed with the δ–norm.

Remark 6.16 In the case of the gluing theorem for flow lines, with approximate solutions x#_{T}^{0} y in U(O_{a}, O_{c}), obtained from elements x ∈ M̂(O_{a}, O_{b}) and y ∈ M̂(O_{b}, O_{c}), it is possible to choose the constant ϵ in Lemma 6.14 independent of T ≥ T_{0}. In fact, in this case it is possible to choose μ ≤ μ_{0}, the least eigenvalue of H_{x#_{T}^{0} y}^{0}, which in this case is independent of T, cf. the spectral decomposition of Lemma 6.13. The constant C is also independent of T ≥ T_{0}, cf. Lemma 6.13. Thus, in this case, the projection Π(μ, x#_{T}^{0} y) is trivial, hence the gluing result reduces to the following argument that proves Proposition 4.21.

A proof of Proposition 4.21 follows from Lemma 6.14 in the case the cylindrical metric on Y × ℝ, and under the assumptions that Coker(L_{x}) = 0 and Coker(L_{y}) = 0. \underline{Proof of Proposition 4.21:} Let us consider the space T_{1,δ}(S_{Γ_{ac}}), endowed with the L_{1,δ}^{2}–norm. The initial condition of Remark 4.20 is provided by the exponential decay. In fact, we have

∥f(𝔸_{1}#_{T}^{0} 𝔸_{2}, Ψ_{1}#_{T}^{0} Ψ_{2})∥_{L_{1,δ}^{2}} ≤
∥F_{#_{T}} ∥ · (∥(α_{1}, ϕ_{1})|_{Y ×[−1+T,T]}∥_{L_{1,δ}^{2}} + ∥(α_{2}, ϕ_{2})|_{Y ×[−T,−T +1]}∥_{L_{1,δ}^{2}}).

The operator norm ∥F_{#_{T}} ∥, for F_{#_{T}} acting on the L_{1,δ}^{2} spaces, is bounded by Ce^{−δT}, as in (45). The terms

∥(α_{1}, ϕ_{1})|_{Y ×[−1+T,T]}∥_{L_{1,δ}^{2}} and ∥(α_{2}, ϕ_{2})|_{Y ×[−T,−T +1]}∥_{L_{1,δ}^{2}}

are bounded by a constant, uniformly in T ≥ T_{0}, because of the exponential decay of (A_{1}(t), ψ_{1}(t)) and (A_{2}(t), ψ_{2}(t)) to the endpoints. Thus, for all T ≥ T_{0}, we have obtained an estimate

∥f(𝔸_{1}#_{T}^{0} 𝔸_{2}, Ψ_{1}#_{T}^{0} Ψ_{2})∥_{1,δ} ≤ Ce^{−δT} .  (100)

⋄

This result can be adapted to prove all the unobstructed gluing results, such as the ones stated in Theorems 6.9 and 6.10. Now we turn to the more interesting case with obstructions.

Using the result of Lemma 6.14, we can construct the obstruction bundle and the canonical section, in the case with non-trivial approximate cokernel (i.e. non-trivial small eigenvalue eigenspace of H_{x#_{T}^{0} y}^{0}).

Proposition 6.17 There is a local bundle E over U(O_{a}, O_{a′}) and a section s_{μ} such that the following property holds. A point x#_{T}^{0} y in U(O_{a}, O_{a′}) can be deformed to a solution of the flow equations (18) and (19) iff

x#_{T}^{0} y ∈ s_{μ}^{−1}(0).

\underline{Proof:} Consider the section

s_{μ}(x#_{T}^{0} y) = Π(μ, x#_{T}^{0} y) (N_{x#_{T}^{0} y}L_{x#_{T}^{0} y}^{∗}(β, ξ) + f(x#_{T}^{0} y))

of the vector bundle over U(O_{a}, O_{a′}) with fiber ApprCoker(L_{x#_{T}^{0} y}). Here (β, ξ) is the unique solution of (97).

Since (β, ξ) satisfies (97), we have

[x#_{T}^{0} y + L_{x#_{T}^{0} y}^{∗}(β, ξ)] ∈ M(O_{a}, O_{a′})

iff

s_{μ}(x#_{T}^{0} y) = 0.

⋄

In the case of moduli spaces of solutions of the Seiberg–Witten equations on Y × ℝ with the changing metric g_{t} + dt^{2}, as in the construction of the map I and the subsequent arguments, Proposition 6.17 characterizes which points of U(O_{a}, O_{a′}) can be deformed to actual solutions. We need a more explicit description of this obstruction bundle in the specific problem at hand, namely in the case of approximate solutions U(θ_{0}, O_{a′}), with the pre-gluing map

#^{0} : M(θ_{0}, θ_{1}) × M̂(θ_{1}, O_{a′}) × [T_{0}, ∞) → U(θ_{0}, O_{a′})

or U(O_{a}, θ_{1}) with the pre-gluing map

#^{0} : M̂(O_{a}, θ_{0}) × M(θ_{0}, θ_{1}) × [T_{0}, ∞) → U(O_{a}, θ_{1}).

We first need to derive the equivalent of Remark 6.16. Namely, we need a suitable choice of μ and ϵ, so that the map in the fixed point problem (99) is a contraction. This choice depends upon estimating the error term ∥f(x#_{T}^{0} y)∥. We have the following result.

Lemma 6.18 Let x = [𝔸, 0] be the unique reducible in M(θ_{0}, θ_{1}), with μ(θ_{0}) − μ(θ_{1}) = −2. Consider (right)pre-glued solutions x#_{T}^{0} y in U(θ_{0}, O_{a′}), with y ∈ M̂(θ_{1}, O_{a′}). We have an estimate on the error term ∥f(x#_{T}^{0} y)∥ of the form

∥f(x#_{T}^{0} y)∥_{L_{1,δ}^{2}} ≤ C̃,

for some constant independent of T ≥ T_{0}, satisfying C̃ < μ_{0}. Moreover, choose C̃ < μ < μ_{0}, independent of T, with C̃ as above, and with μ_{0} the lower bound on the spectrum of the Laplacian H_{x#_{T}^{0} y}^{0} acting on the complement of the space of small eigenvalues, as in Lemma 6.13. Then for max{2Cϵ, ϵ} < μ, we obtain that the map of (99) is a contraction, and the fixed point problem has a unique solution as in Lemma 6.14. The case of (right)pre-gluing maps is completely analogous.

\underline{Proof:} Recall that (𝔸, 0) is the unique solution to d^{+}𝔸 = μ and d^{∗}𝔸 = 0. Here we have the 4-dimensional d^{∗}, with ∗_{4} with respect to the metric g_{t} + dt^{2}, and μ of the form μ = dν_{t} + ∗_{3}dν_{t} ∧ dt, with ∗_{3} with respect to the metric g_{t}. We have

∥f(x#_{T}^{0} y)∥ ≤ C∥F_{#_{T}} ∥·
(∥(α_{1}, ϕ_{1})∥_{L_{1,δ}^{2}(Y ×[−T −1,−T +1])} + ∥𝔸 − 𝔸_{1}∥_{L_{1,δ}^{2}(Y ×[T −1,T +1])}).

Here 𝔸_{1} is the solution of d^{+}𝔸 = μ_{1} and d^{∗}𝔸 = 0, with μ_{1} = dν_{1} + ∗_{3}dν_{1} ∧ dt. Here we have the 4-dimensional d^{∗}, with ∗_{4} with respect to the metric g_{1} + dt^{2} and ∗_{3} with respect to the metric g_{1}. In the rest of the proof we write d^{∗_{g_{t}}} and d^{∗_{g_{1}}} and ∗_{g_{t}} and ∗_{g_{1}} for the 4-dimensional ∗-operators with respect to the two different metrics. (We hope this will not cause confusion with our previous notation ∗_{4} and ∗_{3}.) The norm ∥F_{#_{T}} ∥ is exponentially small ∥F_{#_{T}} ∥ ≤ Ce^{−δT}. The term ∥(α_{1}, ϕ_{1})∥_{L_{1,δ}^{2}(Y ×[−T −1,−T +1])} is bounded by a constant because of the exponential decay of (α_{1}, ϕ_{1}), where we write y = λ_{1}(A_{1} + ν_{1}, 0) + (α_{1}, ϕ_{1}), as t → −∞. The estimate for the remaining term is obtained as follows. We write

𝔸 − 𝔸_{1} = d^{∗_{g_{t}}} (μ_{t} − μ_{1}) + (d^{∗_{g_{t}}} − d^{∗_{g_{1}}} )μ_{1}.

Thus, we have

∥𝔸 − 𝔸_{1}∥_{L_{1}^{2}(Y ×[T −1,T +1])} ≤ ∥d^{∗_{g_{t}}} (μ_{t} − μ_{1})∥_{L_{1}^{2}(Y ×[T −1,T +1])}
+∥(d^{∗_{g_{t}}} − d^{∗_{g_{1}}} )μ_{1}∥_{L_{1}^{2}(Y ×[T −1,T +1])}.

The first term in the right hand side goes to zero for T sufficiently large, because the path ν_{t} ends at ν_{1}. Upon writing

d^{∗_{g_{t}}} − d^{∗_{g_{1}}} = ∗_{g_{t}} ∘ d ∘ (∗_{g_{t}} − ∗_{g_{1}}) + (∗_{g_{t}} − ∗_{g_{1}}) ∘ d ∘ ∗_{g_{1}}

we estimate the second term by a constant times the norm ∥ ∗_{g_{t}} − ∗_{g_{1}} ∥, using the fact that d is invertible on Ker(d^{∗}), since Y is a rational homology sphere. The difference of the ∗-operators is controlled by the difference |\sqrt{|g_{t}|} − \sqrt{|g_{1}|}|, with |g| the absolute value of the determinant of the metric. For (g_{0}, ν_{0}) and (g_{1}, ν_{1}) two sufficiently close points in two different chambers, the path (g_{t}, ν_{t}) can be chosen so that this term is small enough, so that we obtain an estimate

∥F_{#_{T}} ∥ · ∥𝔸 − 𝔸_{1}∥_{L_{1,δ}^{2}(Y ×[T −1,T +1])} ≤ C̃,

with C̃ < μ_{0}, after combining the resulting estimate

∥𝔸 − 𝔸_{1}∥_{L_{1}^{2}(Y ×[T −1,T +1])} ≤ C_{1},

which gives

∥𝔸 − 𝔸_{1}∥_{L_{1,δ}^{2}(Y ×[−T −1,−T +1])} ≤ C_{1}e^{δT}

with the estimate ∥F_{#_{T}} ∥ ≤ Ce^{−δT}.

This gives the required estimate for the error term. We then have the estimate

∥H_{x#_{T}^{0} y}^{0}^{−1}(1 − Π(μ, x#_{T}^{0} y)) (N_{x#_{T}^{0} y}L_{x#_{T}^{0} y}^{∗}(β, ξ) + f(x#_{T}^{0} y)) ∥ ≤

\frac{C}{μ} ∥(β, ξ)∥ + \frac{1}{μ}∥f(x#_{T}^{0} y)∥ ≤

\frac{C}{μ} ϵ + \frac{C̃}{μ} .

With the choice of μ and ϵ as indicated, the map is a contraction. The case of (right)pre-gluing maps is completely analogous. ⋄

For T ≥ T_{0} sufficiently large, the projection Π(μ, x#_{T}^{0} y) is identified with the projection on the approximate cokernel

ApprCoker(L_{x#_{T}^{0} y}) ≅ Coker(L_{y}) ≅ ℂ.

Thus, the construction of the obstruction bundle in [57] can be rephrased as in Theorem 4.53 of [16]. Namely, we obtain the following description of the obstruction.

Theorem 6.19 Consider the space U(θ_{0}, O_{a′}), obtained by gluing x = (𝔸, 0) in M(θ_{0}, θ_{1}) with y ∈ M(θ_{1}, O_{a′}). There exists a bundle Ẽ over M(θ_{1}, O_{a′})/U(1) with fiber

Coker(L_{x}) = ℂ,

with the following property. The set of pre-glued elements x#_{T}^{0} y in U(θ_{0}, O_{a′}) that can be deformed to an actual solution in M(θ_{0}, O_{a′}) form a codimension 2 submanifold of

M(θ_{1}, a′) ≅ M(θ_{1}, O_{a′})/U(1),

given by the zero set of a transverse section of Ẽ. The case of U(O_{a}, θ_{1}) is analogous.

\underline{Proof:} As a result of Lemma 6.18, we know that, in the case of U(θ_{0}, O_{a′}), the obstruction bundle described in Proposition 6.17 is a bundle E over M(θ_{0}, θ_{1})× M(θ_{1}, O_{a′}) with fiber ApprCoker(L_{x#_{T}^{0} y}). Since we are considering the case with μ(θ_{0}) − μ(θ_{1}) < 0, according to Lemma 6.6, the moduli space M(θ_{0}, θ_{1}) consists of a unique gauge class [𝔸, 0]. Thus we have a diffeomorphism

π : M(θ_{1}, O_{a′}) →^{≅} U(θ_{0}, O_{a′})

induced by the pre-gluing map. According to Corollary 4.17, since we can restrict our attention to the case μ(θ_{0}) − μ(θ_{1}) = −2, we have

ApprCoker(L_{x#_{T}^{0} y}) ≅ Coker(L_{x}) ≅ ℂ.

Consider the bundle Ẽ over M(θ_{1}, O_{a′}) with fiber

Coker(L_{x}) ≅ ℂ,

obtained as pullback of the obstruction bundle E via the diffeomorphism π of the base spaces. There is a section s̃ = π^{∗}s_{μ} of Ẽ that corresponds to the canonical section s_{μ} of Proposition 6.17. Recall that there is a free U(1)-action over the moduli space M(θ_{1}, O_{a′}), hence there is a smooth projection to the quotient

M(θ_{1}, O_{a′})/U(1) ≅ M(θ_{1}, a′).

The bundle Ẽ can be regarded as the pullback, under this quotient map, of a bundle Ē over M(θ_{1}, O_{a′})/U(1), with fiber

Coker(L_{x}) ≅ ℂ.

The section s̃ is the pullback of a corresponding section s̄ of Ē. We are going to proceed as follows. We show that, for a generic choice of the perturbations ρ and P of equations (84) and (85), this section s̄ is a generic section of Ē.

Consider the universal pre-glued space

Ũ(θ_{1}, O_{a′}) = {(ρ, P, x#_{T}^{0} y)}

with ρ a compactly supported form in Λ_{c}^{1}(Y × ℝ), P a perturbation of P as in Definition 2.8 (with the modified property (1’) specified at the beginning of Section 6.1). The pre-glued element x#_{T}^{0} y is obtained from x and y, solutions respectively of the equations (84), (85) and (20), (21), with the perturbations ρ and P. We can extend the bundle Ẽ to a local bundle over Ũ(θ_{1}, O_{a′}) with fiber Coker(L_{x}). We can still identify this as the pullback of a local bundle Ē on the quotient Ũ(θ_{1}, O_{a′})/U(1). There is an induced section s̄ of Ē whose pull-back agrees with the section π^{∗}s_{μ} for fixed perturbations (ρ, P). We prove that this section s̄ over Ũ(θ_{1}, O_{a′})/U(1) is transverse to the zero section, hence the restriction to a generic (ρ, P) gives a generic section over U(θ_{1}, O_{a′})/U(1).

The section s̄ is given by

s̄(ρ, P, x#_{T}^{0} y) = Π_{Coker(L_{x})} (N_{(ρ,P,x#_{T}^{0} y)}L_{(ρ,P,x#_{T}^{0} y)}^{∗}(β, ξ) + f_{(ρ,P)}(x#_{T}^{0} y)) .

Suppose given a point (ρ, P, x#_{T}^{0} y) in s̄^{−1}(0). Consider a small variation of the perturbation ρ + ϵη. The variation of the term f_{(ρ+ϵη,P)}(x#_{T}^{0} y) is given by

η · Ψ_{1}#_{T}^{0} Ψ_{2} = η · ρ_{T}^{+}Ψ_{2}^{−2T} .

Let Φ_{1} and Φ_{2} in Ker(D_{𝔸−ρ}) be the generators of the 2-dimensional space Coker(L_{x}). Consider a small open sets U_{i} where Φ_{i} and ρ_{T}^{+}Ψ_{2}^{−2T} are nonvanishing, and almost constant. There exist 1-forms η_{i} supported in small neighborhoods of the open sets U_{i} such that

⟨Φ_{i}, η_{i} · ρ_{T}^{+}Ψ_{2}^{−2T} ⟩

is non-zero on U_{i}. Thus, we obtain

∫_{Y ×ℝ}⟨Φ_{i}, η_{i} · ρ_{T}^{+}Ψ_{2}^{−2T} ⟩dv ≠ 0.

Thus, by varying the perturbation ρ alone, it is possible to achieve surjectivity of the linearization of the section s̄ onto Coker(L_{x}).

There is a free U(1)-action on the space M(θ_{1}, O_{a′}), whereas the element [x] = [𝔸, 0] in M(θ_{0}, θ_{1}) is fixed by the U(1) action. The section

s̃ = π^{∗}s_{μ} : M(θ_{1}, O_{a′}) → Ẽ

is invariant under the U(1)-action, being the pullback of s̄. We have seen that, for a generic choice of the perturbation (ρ, P), the section s̄ is a generic section. Thus, the approximate solutions in U(θ_{0}, O_{a′}) that can be glued to actual solutions in M(θ_{0}, O_{a′}) are identified with the co-dimension 2 U(1)-submanifold

s̃^{−1}(0) ⊂ M(θ_{1}, O_{a′}).

In other words, we have then proved that the bundle Ē over the moduli space M(θ_{1}, O_{a′})/U(1) is a model of the obstruction. ⋄

We have the following consequence of Theorem 6.19 and Lemma 6.18.

Proposition 6.20 For any compact set

K ⊂ M(θ_{0}, θ_{1}) × s̃^{−1}(0),

with s̃^{−1}(0) ⊂ M(θ_{1}, O_{a′}) the zeroes of the obstruction section

s̃^{−1}(0)/U(1) = s̄^{−1}(0) ⊂ M(θ_{1}, a′),

there is an orientation preserving gluing map

# : K ⊂ M(θ_{0}, θ_{1}) × s̃^{−1}(0) → M(θ_{0}, O_{a′})

that is a smooth embedding. There is a similar gluing map for any compact set

K ⊂ s̃^{−1}(0) × M(θ_{0}, θ_{1}),

with s̃^{−1}(0) ⊂ M(O_{a}, θ_{0}) the zeroes of the obstruction section

s̃^{−1}(0)/U(1) = s̄^{−1}(0) ⊂ M(a, θ_{0}).

Finally, we have the following result on the singular components in the ideal boundary of the moduli spaces M(O_{a}, O_{a′}).

Theorem 6.21 Assume that μ(a) − μ(θ_{0}) ≥ 2 and μ(θ_{1}) − μ(a′) ≥ 3, then the contributions of the singular strata

M(θ_{0}, θ_{1}) × M(θ_{1}, O_{a′})  (101)

and

M(O_{a}, θ_{0}) × M(θ_{0}, θ_{1})  (102)

to the actual boundary of the compactified moduli spaces

M(θ_{0}, O_{a′})^{∗} and M(O_{a}, θ_{1})^{∗}

are given by the terms

M(θ_{0}, θ_{1}) × ∂^{(1)} (s̃^{−1}(0) ∩ M(θ_{1}, O_{a′}))  (103)

and

∂^{(1)} (s̃^{−1}(0) ∩ M(O_{a}, θ_{0})) × M(θ_{0}, θ_{1}),  (104)

respectively, where s̃ are the obstruction sections. Thus, the compactification of the moduli spaces M(O_{a}, O_{a′}) has the structure of a smooth manifold with corners, but in addition to the strata of the form (75), we also have the strata (103) and (104). Thus, we obtain

∂^{(1)}M(θ_{0}, O_{a′})^{∗} = ⋃_{{a|μ(O_{b′})>μ(O_{a′})}}(M(θ_{0}, O_{b′})^{∗} ×_{O_{b′}} M̂(O_{b′}, O_{a′})^{∗})
⋃_{{b|μ(O_{b})<μ(θ_{0})}}(−M̂(θ_{0}, O_{b})^{∗} ×_{O_{b}} M(O_{b}, O_{b′})^{∗})
∪M(θ_{0}, θ_{1}) × ∂^{(1)} (s̃^{−1}(0) ∩ M(θ_{1}, O_{a′})^{∗})

(105)

and

∂^{(1)}M(O_{a}, θ_{1})^{∗} = ⋃_{{a′|μ(O_{a′})>μ(θ_{1})}}(M(O_{a}, O_{a′})^{∗} ×_{O_{a′}} M̂(O_{a′}, θ_{1})^{∗})
⋃_{{b|μ(O_{b})<μ(O_{a})}}(−M̂(O_{a}, O_{b})^{∗} ×_{O_{b}} M(O_{b}, θ_{1})^{∗})
∪∂^{(1)} (s̃^{−1}(0) ∩ M(O_{a}, θ_{0})) × M(θ_{0}, θ_{1}).

(106)

\underline{Proof:} By the results of Proposition 6.20, we have gluing maps

M(θ_{0}, θ_{1}) × s̄^{−1}(0) → M(θ_{0}, a′)

and

s̄^{−1}(0) × M(θ_{0}, θ_{1}) → M(a, θ_{1})

that are smooth embeddings. The dimension count then implies that these gluing maps are diffeomorphisms of s̄^{−1}(0) to a union of connected components of M(θ_{0}, a′) or M(a, θ_{1}). Similarly, we have diffeomorphisms of U(1)-manifolds between s̃^{−1}(0) and a union of connected components of M(θ_{0}, O_{a′}) or M(O_{a}, θ_{1}), induced by the gluing maps

# : M(θ_{0}, θ_{1}) × s̃^{−1}(0) → M(θ_{0}, O_{a′})  (107)

and

# : s̃^{−1}(0) × M(θ_{0}, θ_{1}) → M(O_{a}, θ_{1}).  (108)

Under the gluing maps, the image of

M^{bal}(O_{a}, θ_{0}) ∩ s̃^{−1}(0) or M^{bal}(θ_{1}, O_{a′}) ∩ s̃^{−1}(0)

is a co-dimension 1 submanifold of M(θ_{0}, O_{a′}) or M(O_{a}, θ_{1}). This submanifold actually lies in the interior of M(θ_{0}, O_{a′}) or M(O_{a}, θ_{1}), and is not part of its boundary strata. This can be seen from the fact that the gluing maps (107) and (108) provide the collar structure around these smooth codimension one embedded submanifolds. In other words, this means that any sequence of solutions in M(θ_{0}, O_{a′}) or M(O_{a}, θ_{1}) that converges to an element in the ideal boundary components (101) and (102) is in fact already convergent in the interior (top stratum) of M(θ_{0}, O_{a′}) or M(O_{a}, θ_{1}). The only contribution of (101) and (102) to the actual boundary of the compactification then comes from the boundary points of

M(O_{a}, θ_{0}) ∩ s̃^{−1}(0)

and

M(θ_{1}, O_{a′}) ∩ s̃^{−1}(0).

This gives the formulae (105) and (106). ⋄

Similarly, we can analyze the boundary structure of M^{P} (θ_{0}, O_{a}), in the presence of obstructions. We have the following result.

Theorem 6.22 For a generic choice of perturbation, the obstruction section

s̄: M(θ_{1}, a) → L_{θ_{1},a} = M(θ_{1}, O_{a}) ×_{U(1)} ℂ

defines a codimension 2 submanifold s̄^{−1}(0) ⊂ M(θ_{1}, a), which corresponds to a U(1)-submanifold of co-dimension 2, s̃^{−1}(0) ⊂ M(θ_{1}, O_{a}), such that the gluing map

# : K ⊂ M(θ_{0}, θ_{1}) × s̃^{−1}(0) → M_{σ=1}^{P}(θ_{0}, O_{a})

is a smooth embedding, for any compact set K ⊂ M(θ_{0}, θ_{1}) × s̃^{−1}(0). Thus, the co-dimension one boundary strata of the compactification M^{P} (θ_{0}, O_{a})^{∗} are given by

∂^{(1)}M^{P} (θ_{0}, O_{a})^{∗} = ⋃_{a′}(M(θ_{0}, O_{a′})^{∗} ×_{O_{a′}} M(O_{a′}, O_{b})^{∗})
⋃ M(θ_{0}, θ_{1}) × (s̃^{−1}(0) ∩ M(θ_{1}, O_{a}))
⋃_{{c|μ(O_{c})≥μ(θ_{0})}}(M^{P} (θ_{0}, O_{c})^{∗} ×_{O_{c}} M̂(O_{c}, O_{b})^{∗})
⋃_{{c|μ(O_{c})≤μ(θ_{0})}}(M̂(θ_{0}, O_{c})^{∗} ×_{O_{c}} M^{P} (O_{c}, , O_{a})^{∗}).

(109)

with the orientations given by the gluing theorem.

The proof follows from the analysis of the obstruction bundles and sections, as in the case of Proposition 6.20 and Theorem 6.21.

6.3 Proof of topological invariance

We gave an argument for the easy case of theorem 6.1, with maps I, J, and H as in (73), (76), and (79). This gives us an isomorphism between the equivariant Floer homologies for any two choices (g_{0}, ν_{0}) and (g_{1}, ν_{1}) within the same chamber. Now we prove the general case \underline{Proof of Theorem 6.1, Part II: the general case.} In Section 6.1 and Section 6.2, we analysed the boundary structure of the moduli spaces M(O_{a}, O_{a′})^{∗} and M^{P} (O_{a}, O_{b})^{∗}, in the case of two metrics and perturbations (g_{0}, ν_{0}) and (g_{1}, ν_{1}) in different chambers, connected by a path (g_{t}, ν_{t}) satisfying SF (∂_{ν_{t}}^{g_{t}}) = −2, and a path (g̃_{t}, ν̃_{t}), in the opposite direction, satisfying SF (∂g̃_{t} ; ν̃_{t}) = −SF (∂_{ν_{t}}^{g_{t}}) = 2. We assume throughout the discussion that the metrics (g_{0}, ν_{0}) and (g_{1}, ν_{1}) are “close enough”, on the two sides of the wall. We can assume, similarly, that the paths (g_{t}, ν_{t}) and (g̃_{t}, ν̃_{t}) are also close enough.

Recall that we perform a shift of grading in the complex C_{k}_{U(1)}(Y, g_{1}, ν_{1}) by setting μ(θ_{1}) = −SF (∂_{ν_{t}}^{g_{t}}), where SF (∂_{ν_{t}}^{g_{t}}) is the spectral flow of the Dirac operator along the path of reducible solutions [ν_{t}, 0].

The analysis of the obstruction bundles in Section 6.2 implies that the boundary structure of the moduli spaces M(O_{a}, O_{a′}) and M^{P} (O_{a}, O_{b}) is modified by the presence of the zeroes of the obstruction sections. This difference determines suitable correction terms for the maps I and H, so that the argument of theorem 6.1 can be adapted to this general case.

Remark 6.23 The moduli space M(θ_{1}, θ_{0}), as a U(1)-manifold, consists of a disk, containing the fixed point x_{1} = [𝔸_{1}, 0] (the unique solution of d ∗ 𝔸_{1} = 0 and d^{+}𝔸_{1} = μ̃_{t}), and with boundary a circle, given either by a component

M(θ_{1}, O_{a}) ×_{O_{a}} M̂(O_{a}, θ_{0})

with μ(O_{a}) − μ(θ_{0}) = 1, or by a component

M̂(θ_{1}, O′_{a}) ×_{O′_{a}} M(O′_{a}, θ_{0}),

with μ(O′_{a}) − μ(θ_{0}) = 0. Thus, we have the following identity that counts the boundary components of the framed monopole moduli space M(θ_{1}, θ_{0}):

∑_{a_{(1)}} n_{θ_{1}a_{(1)}}n_{a_{(1)}θ_{0}} + ∑_{a′_{(0)}} n_{θ_{1}a′_{(0)}}n_{a′_{(0)}θ_{0}} = 1,  (110)

The sum is over all O_{a_{(1)}} in M^{0}(Y, g_{0}, ν_{0}), and O_{a′_{(0)}} in M^{0}(Y, g_{1}, ν_{1}) of (shifted) index μ(O_{a_{(1)}}) − μ(θ_{0}) = 1 and μ(O_{a′_{(0)}}) − μ(θ_{0}) = 0, respectively.

In particular, notice that we can derive the observation of Remark 6.23 if we make explicit use here of the information on the local structure of the moduli space M^{0}(Y, g_{t}, ν_{t}) of critical orbits, as the metric (g_{t}, ν_{t}) approaches the wall, This is derived in Section 7.3. We know that two possibilities arise: either an irreducible orbit O_{a}, with μ(O_{a}) − μ(θ_{0}) = 1, disappears into the reducible as the metric and perturbation (g_{t}, ν_{t}) hits the wall, or an irreducible orbit O_{a′} with μ(O_{a′}) − μ(θ_{0}) = 0 arises from the reducible, as (g_{t}, ν_{t}) hits the wall. In the first case, the disk of Remark 6.23 has boundary the circle

M(θ_{1}, O_{a}) ×_{O_{a}} M̂(O_{a}, θ_{0})

and in the second case it has boundary the circle

M̂(θ_{1}, O′_{a}) ×_{O′_{a}} M(O′_{a}, θ_{0}).

The relation (110) then yields the separate identities

#M(θ_{1}, O_{a}) = n_{θ_{1}a} = 1 and #M̂(O_{a}, θ_{0}) = n_{aθ_{0}} = 1,  (111)

for O_{a} the unique orbit that hits the reducible, and

∑_{a_{(1)}≠a} n_{θ_{1}a_{(1)}}n_{a_{(1)}θ_{0}} + ∑_{a′_{(0)}} n_{θ_{1}a′_{(0)}}n_{a′_{(0)}θ_{0}} = 0,

for μ(O_{a_{(1)}}) − μ(θ_{0}) = 1 and μ(O_{a′_{(0)}}) − μ(θ_{0}) = 0. The case of O_{a′} is analogous.

The moduli space M(θ_{1}, θ_{0}), containing this separate component with a fixed point, is one of the differences with respect to the picture for metrics and perturbations within the same chamber. Another essential difference is, of course, the presence of the singular moduli space M(θ_{0}, θ_{1}). We have already seen, in the analysis of the obtructions how this moduli space plays an essential role. However, what we wish to point out here is that the basic asymmetry between the moduli spaces M(θ_{0}, θ_{1}) and M(θ_{1}, θ_{0}) is what calls for correction terms in the maps I and H, but not in the map J.

In fact, consider first the action of the map J, defined as in (76) on the generator Ω^{n} ⊗ 1_{θ_{1}} in the Floer complex for (g_{1}, ν_{1}). We have

J(Ω^{n} ⊗ 1_{θ_{1}}) = ∑_{a_{(1)}} n_{θ_{1}a_{(1)}}Ω^{n} ⊗ 1_{a_{(1)}},

with the sum over all O_{a_{(1)}} with μ(O_{a_{(1)}}) − μ(θ_{0}) = 1.

Notice that the existence of the extra disk component in the boundary of M(θ_{1}, θ_{0}), as in remark 6.23, does not affect the identity JD − DJ = 0, in fact, even though now the count of boundary terms in M(θ_{0}, θ_{1}) satisfies (110), we still do get

⟨Ω^{n} ⊗ 1_{θ_{0}}, JD − DJ(Ω^{n} ⊗ 1_{θ_{1}})⟩ = 0.

In fact, the fibrations

M(O_{a′_{(0)}}, θ_{0}) → θ_{0},
M̂(O_{a_{(1)}}, θ_{0}) → θ_{0},

have 1-dimensional fibers, and the pushforward of a zero-form is trivial. The identity JD − DJ = 0 at all the other components follows the argument given in the proof of the easy case of Theorem 6.1, without any modification.

Now, instead, consider the case of the map I acting on Ω^{n} ⊗ 1_{θ_{0}}. The first difference we notice, with respect to the model case of metrics and perturbations in the same chamber, is that the moduli space M(θ_{0}, θ_{1}) consisting of the unique point x_{0} = [𝔸_{0}, 0] defines a non-trivial “pull-back push-forward” acting on the zero form 1_{θ_{0}}. To account for this moduli space, we have to assume the existence here of an extra component of the map I connecting Ω^{n} ⊗ 1_{θ_{0}} to Ω^{n−1} ⊗ 1_{θ_{1}}, where the drop of degree in Ω accounts for the change of grading of the reducible point, so that the map I can be of degree zero. Thus, we have a new component

⟨Ω^{n−1} ⊗ 1_{θ_{1}}, I(Ω^{n} ⊗ 1_{θ_{0}})⟩ = 1.  (112)

Notice that the necessity of the additional term (112) in the map I can be made clear by looking at the following example. Example. Consider a model case where the moduli space M^{0}(g_{0}, ν_{0}) consists solely of the fixed point θ_{0} and of an irreducible orbit O_{a} in degree one. The Floer complex for (g_{0}, ν_{0}) then has generators Ω^{n} ⊗ 1_{θ_{0}} in even degrees p = 2n ≥ 0, Ω^{n}⊗η_{a} in odd degrees p = 2n+1 ≥ 1, and Ω^{n−1}⊗1_{a} in even degrees p = 2n ≥ 2. The boundary operator D has a component

Ω^{n} ⊗ η_{a} ↦ −Ω^{n−1} ⊗ 1_{a} + n_{a0}Ω^{n} ⊗ 1_{θ_{0}}.

In the wall crossing with μ(θ_{0}) − μ(θ_{1}) = −2, the orbit O_{a} disappears into the reducible, and the Floer complex for (g_{1}, ν_{1}) only has the fixed point as generator. In this case, we have the equivariant Floer homology

SW H_{p,U(1)}(Y, (g_{0}, ν_{0})) = \{ \begin{matrix}ℝ[Ω^{n−1} ⊗ 1_{a}] & p = 2n ≥ 2 \\ 0 & otherwise, \end{matrix}

with [Ω^{n−1} ⊗1_{a}] = [Ω^{n} ⊗1_{θ_{0}}] for all n ≥ 1, and [1⊗1_{θ_{0}}] = 0. The Floer complex for (g_{1}, ν_{1}) has Ω^{n} ⊗ 1_{θ_{1}} as unique generators, and no boundary components, hence the Floer homology is

SW H_{p,U(1)}(Y, (g_{1}, ν_{1})) = \{ \begin{matrix}ℝ[Ω^{n} ⊗ 1_{θ_{1}}] & p = 2n + 2 ≥ 2 \\ 0 & otherwise, \end{matrix}

after degree shift. The map I that maps Ω^{n} ⊗1_{θ_{0}} → Ω^{n−1} ⊗1_{θ_{1}} gives the desired isomorphism. Notice that such examples can in fact be realized geometrically, for instance when considering a metric of positive scalar curvature on S^{3} as g_{1}. ⋄

It becomes then clear that, if we change the map I with an extra component as in (112), to account for the moduli space M(θ_{0}, θ_{1}), we need to add further correction terms to the original map I on other generators, so that the identity ID − DI = 0 continues to be satisfied. In assigning the necessary correction terms, we need to take into account the different structure of the compactification of the moduli spaces M(O_{a}, O_{a′}), with the boundary strata (105) and (106). Studying the boundary information of M(θ_{0}, a′_{(−2)}), we know that there is a contribution of the singular gluing from the monopoles in

s^{−1}(0) ∩ M(θ_{1}, a′_{(−2)}),

which contributes to the expression

(ID − DI)(Ω^{n} ⊗ 1_{θ_{0}}),

in addition to the ordinary components n_{θ_{0}a′_{(−1)}}Ω^{n} ⊗ 1_{a′_{(−1)}}, where n_{θ_{0}a′_{(−1)}} effectively counts the monopoles from the zeros of the obstruction section over M(θ_{1}, a′_{(−1)}). This is precisely the correction term which is needed in order to obtain the identity ID − DI = 0 once we take into account the presence of the component (112) originates from the presence of the moduli space M(θ_{0}, θ_{1}) consisting of the unique point x_{0} = [𝔸_{0}, 0].

Similarly, we need to introduce correction terms to the map H so that it continues to be a chain homotopy, satisfying id − JI = DH + HD, with respect to the modified map I and with the modified structure of the compactification of M^{P} (O_{a}, O_{b}).

In order to introduce the correction terms for the maps I and H, we need a preliminary discussion on some identities derived from the counting of zeroes of the obstruction sections.

Consider the case of the 2-dimensional moduli space M(a, θ_{0}), with a an irreducible critical point of index 2 and θ_{0} the unique reducible point.

In Section 5.3, in Lemma 5.7, we described the invariant m_{ac} as the relative Euler class of the complex line bundle L_{ac} over M(a, c). Consider now the case of the 2-dimensional moduli space M(a, θ_{0}), with O_{a} a free orbit of index 2 and θ_{0} the fixed point in M^{0}(Y, g_{0}, ν_{0}). We also introduced the invariant m_{aθ_{0}} in Section 5.3, as as the relative Euler class of the associated complex line bundle over M(a, θ_{0}), according to Lemma 5.9 and Lemma 5.10. As we point out in the following Remark, we can use the obstruction section to obtain the necessary trivialization and compute the invariant m_{aθ_{0}} as relative Euler class of L_{aθ_{0}}.

Remark 6.24 We use the same choice of trivialization s_{ab} over M(a, b), for all the free orbits O_{a} and O_{b} of relative index 1, as determined by the trivialization φ of Lemma 5.7. For M(b, θ_{0}) with μ(O_{b}) − μ(θ_{0}) = 1, we set s_{b,θ_{0}} ≠ 0 to be the obstruction section

s_{b,θ_{0}} = s̄: M(b, θ_{0}) → M(O_{b}, θ_{0}) ×_{U(1)} Coker(L_{x}) ≅ L_{bθ_{0}}.

We then set s_{a,θ_{0}} over M(a, θ_{0}), with μ(O_{a})−μ(θ_{0}) ≥ 2 to be the obstruction section

s_{a,θ_{0}} = s̄: M(a, θ_{0}) → M(O_{a}, θ_{0}) ×_{U(1)} Coker(L_{x}) ≅ L_{aθ_{0}}.  (113)

This choice satisfies the requirement of the class of sections specified above. Over a moduli spaces M(a, θ_{0}), with μ(O_{a}) − μ(θ_{0}) ≥ 2, then the obstruction section

s_{a,θ_{0}} = s̄: M(a, θ_{0}) → M(O_{a}, θ_{0}) ×_{U(1)} Coker(L_{x}) ≅ L_{aθ_{0}}

is homotopic to

π_{1}^{∗}s_{ab} ⊗ π_{2}^{∗}s_{b,θ_{0}}

over all the submanifolds

M(a, b) × M(b, θ_{0})

of M(a, θ_{0}), where we identify

L_{a,θ_{0}} = π_{1}^{∗}L_{ab} ⊗ π_{2}^{∗}L_{b,θ_{0}},

with π_{1} and π_{2} the projections on the two factors in

M(a, b) × M(b, θ_{0}),

with μ(O_{a}) > μ(O_{b}) > μ(θ_{0}), and section s_{b,θ_{0}} is the obstruction section over M(b, θ_{0}), and the section s_{ab} is a transverse section of the line bundle

L_{ab} = M(O_{a}, O_{b}) ×_{U(1)} ℂ

over M(a, b) as discussed in Section 5.3.

In summary, we have obtained the following lemma which claims that the relative Euler class m_{a_{(2)}θ_{0}} and m_{θ_{1}a′_{(−1)}} can be calculated by counting the zeros of the obstruction section over M(a_{(2)}, θ_{0}) and M(θ_{1}, a′_{(−1)}) respectively.

Lemma 6.25 For μ(a_{(2)}) − μ(θ_{0}) = 2, the relative Euler class m_{a_{(2)}θ_{0}} is given by

#s_{a_{(2)}θ_{0}}^{−1}(0)

where s_{a_{(2)}θ_{0}} is the obstruction section over M(a_{(2)}, θ_{0}), similarly, for μ(θ_{1}) − μ(a′_{(−1)}) = 3, the relative Euler class m_{θ_{1}a′_{(−1)}} is given by

#s_{θ_{0}a′_{(−1)}}^{−1}(0)

where s_{θ_{0}a′_{(−1)}} is the obstruction section over M(θ_{1}, a′_{(−1)}).

As a consequence of Remark 6.24 and Lemma 6.25, we have the following identities.

Lemma 6.26 Let O_{a_{(p)}} denote a free orbit in the moduli space M^{0}(Y, g_{0}, ν_{0}), of index μ(O_{a_{(p)}}) − μ(θ_{0}) = p. Let O_{a′_{(q)}} denote a free orbit in the moduli space M^{0}(Y, g_{1}, ν_{1}), of (shifted) index μ(O_{a′_{(q)}}) − μ(θ_{0}) = q. We have the following properties.

1. Consider the obstruction bundle over M(a_{(2)}, θ_{0}). The counting of the zeroes of the obstruction, that is, of the flowlines in M(a_{(2)}, θ_{0}) glued to the singular reducible x = [𝔸^{0}, 0] in M(θ_{0}, θ_{1}), is given by

m_{a_{(2)}θ_{0}} = #(s̄^{−1}(0) ∩ M(a_{(2)}, θ_{0})).

This satisfies

∑_{a_{(1)}} m_{a_{(3)}a_{(1)}}n_{a_{(1)}θ_{0}} − ∑_{a_{(2)}} n_{a_{(3)}a_{(2)}}m_{a_{(2)}θ_{0}} = 0.  (114)

2. The counting of the zeroes of the obstruction section s̄ of the obstruction bundle over M(θ_{1}, a′_{(−1)}), which counts the gluing of the singular reducible x = [𝔸^{0}, 0] in M(θ_{0}, θ_{1}) to flowlines in M(θ_{1}, a′_{(−1)}), is given by

m_{θ_{1}a′_{(−1)}} = #(s̄^{−1}(0) ∩ M(θ_{1}, a′_{(−1)})).

This satisfies

∑_{a′_{(−1)}} m_{θ_{1}a′_{(−1)}}n_{a′_{(−1)}a′_{(−2)}} − ∑_{a′_{(0)}} n_{θ_{1}a′_{(0)}}m_{a′_{(0)}a′_{(−2)}} = 0.  (115)

\underline{Proof:} In this case, we know that, for dimensional reasons, we have

s̄^{−1}(0) ∩ M(a_{(1)}, θ_{0}) = ∅.

Thus, by the previous discussion, and the results of Lemma 5.9, Lemma 5.10, and Remark 6.24, we know that the boundary of the 1-dimensional manifold (s̄^{−1}(0) ∩ M(a_{(3)}, θ_{0})) consists of the set

⋃_{a_{(2)}} M̂(a_{(3)}, a_{(2)}) × (s̄^{−1}(0) ∩ M(a_{(2)}, θ_{0}))

∪ ⋃_{a_{(1)}}(−s_{a_{(2)}a_{(1)}}^{−1}(0) ∩ (M(a_{(2)}, a_{(1)}) × M̂(a_{(1)}, θ_{0}))).

This implies that we have the identity

∑_{a_{(1)}} m_{a_{(3)}a_{(1)}}n_{a_{(1)}θ_{0}} − ∑_{a_{(2)}} n_{a_{(3)}a_{(2)}}m_{a_{(2)}θ_{0}} = 0,

with the sign denoting the different orientation. The remaining case is analogous. ⋄

Now we define the necessary modifications to the maps I and H.

Definition 6.27 We modify the maps I, H on Ω^{n} ⊗ 1_{θ_{0}} and Ω^{n−1} ⊗ 1_{a_{(1)}} as follows:

1.

\begin{aligned}I(Ω^{n} ⊗ 1_{θ_{0}}) = & Ω^{n−1} ⊗ 1_{θ_{1}} \\ & + ∑_{a′_{(−1)}} n_{θ_{0}a′_{(−1)}}Ω^{n} ⊗ 1_{a′_{(−1)}} \\ & + ∑_{a′_{(0)}} n_{θ_{1}a′_{(0)}}Ω^{n} ⊗ η_{a′_{(0)}}; \end{aligned}  (116)

2.

\begin{aligned}I(Ω^{n−1} ⊗ 1_{a_{(1)}}) = & n_{a_{(1)}θ_{0}}Ω^{n−1} ⊗ 1_{θ_{1}} \\ & + ∑_{a′_{(1)}} n_{a_{(1)}a′_{(1)}}Ω^{n−1} ⊗ 1_{a′_{(1)}}; \end{aligned}  (117)

3.

\begin{aligned}H(Ω^{n} ⊗ 1_{θ_{0}}) = & ∑_{a_{(0)}} n_{θ_{0}a_{(0)}}^{P}Ω^{n} ⊗ 1_{a(0)} \\ & + ∑_{a_{(1)}} n_{θ_{1}a_{(1)}}Ω^{n} ⊗ η_{a_{(1)}}. \end{aligned}  (118)

In order to show that the relations ID = DI and id − JI = DH + HD are still satisfied, we only need to check explicitly all the terms that are directly affected by the presence of the correction terms.

Lemma 6.28 We have

1. ID = DI on Ω^{n} ⊗ 1_{θ_{0}} and Ω^{n−1} ⊗ 1_{a_{(1)}};

2. ID = DI on Ω^{n} ⊗ η_{a_{(3)}};

3. ID = DI on Ω^{n} ⊗ η_{a_{(1)}};

\underline{Proof:} We prove the claim by direct analysis of the boundary strata of the various moduli spaces involved. We have

ID(Ω^{n} ⊗ 1_{θ_{0}}) = ∑_{a_{(−2)}} n_{θ_{0}a_{(−2)}}n_{a_{(−2)}a′_{(−2)}}Ω^{n} ⊗ 1_{a′_{(−2)}}

and

DI(Ω^{n} ⊗ 1_{θ_{0}}) = − ∑_{a′_{(−1)}} n_{θ_{0}a′_{(−1)}}n_{a′_{(−1)}a′_{(−2)}}Ω^{n} ⊗ 1_{a′_{(−2)}}

+ ∑_{a′_{(0)}} n_{θ_{1}a′_{(0)}}m_{a′_{(0)}a′_{(−2)}}Ω^{n} ⊗ 1_{a′_{(−2)}}.

The first claim then follows, since we know that the co-dimension one boundary of M(θ_{0}, O_{a′_{(−2)}}) consists of

⋃_{a_{(−2)}} M̂(θ_{0}, O_{a_{(−2)}}) ×_{O_{a_{(−2)}}} M(O_{a_{(−2)}}, O_{a′_{(−2)}})

∪ ⋃_{a′_{(−1)}} M(θ_{0}, O_{a′_{(−1)}}) × M̂(O_{a_{(−1)}}, O_{a′_{(−2)}})

∪M(θ_{0}, θ_{1}) × (s̃^{−1}(0) ∩ (M̂(θ_{1}, O_{a′_{(0)}}) ×_{O_{a{′ \atop (0)}}} M(O_{a′_{(0)}}, O_{a′_{(−2)}})).

In fact, according to Theorem 6.21, we have boundary strata as in (105),

M(θ_{0}, θ_{1}) × ∂^{(1)} (s̃^{−1}(0) ∩ M(θ_{1}, O_{a′_{(−2)}})^{∗}) .

Here s̃ is the obstruction section. By Remark 6.24, we have

∂^{(1)} (s̃^{−1}(0) ∩ M(θ_{1}, O_{a′_{(−2)}})^{∗})
= s̃^{−1}(0) ∩ (⋃_{a′_{(0)}} M̂(θ_{1}, O_{a′_{(0)}})^{∗} ×_{O_{a{′ \atop (0)}}} M(O_{a′_{(0)}}, O_{a′_{(−2)}})^{∗})
∪s̃^{−1}(0) ∩ (⋃_{a′_{(−1)}} M̂(θ_{1}, O_{a′_{(−1)}})^{∗} ×_{O_{a{′ \atop (−1)}}} M(O_{a′_{(−1)}}, O_{a′_{(−2)}})^{∗}).

Note that the coefficient n_{θ_{0}a′_{(−1)}} effectively counts the monopoles from the zeros of the obstruction section over M(θ_{1}, a′_{(−1)}), so we are left with the remaining counting,

∑_{a′_{(0)}} n_{θ_{1}a′_{(0)}}m_{a′_{(0)}a′_{(−2)}}Ω^{n} ⊗ 1_{a′_{(−2)}},

which proves the claim.

In order to prove Claim (2), we compute

ID(Ω^{n} ⊗ 1_{a_{(1)}}) = − ∑_{a_{(0)}} n_{a_{(1)}a_{(0)}}n_{a_{(0)}a′_{(0)}}Ω^{n} ⊗ 1_{a′_{(0)}}

and

DI(Ω^{n} ⊗ 1_{a_{(1)}}) = − ∑_{a′_{(1)}} n_{a_{(1)}a′_{(1)}}n_{a′_{(1)}a′_{(0)}}Ω^{n} ⊗ 1_{a′_{(0)}} + n_{a_{(1)}θ_{0}}n_{θ_{1}a′_{(0)}}Ω^{n} ⊗ 1_{a′_{(0)}}

Then the claim follows from the boundary structure of M(a_{(1)}, a′_{(0)}). The zeroes of the obstruction section contribute the term

n_{a_{(1)}θ_{0}}n_{θ_{1}a′_{(0)}}

in the counting of the boundary points.

The argument for Claims (3) and (4) is analogous. ⋄

Now we need to check the effect of the correction terms on the identity 1 − JI = HD + HD.

Lemma 6.29 We have the identity 1 − JI = DH + HD on Ω^{n} ⊗ 1_{θ_{0}}.

\underline{Proof:} By direct calculation, we obtain

(1 − JI)(Ω^{n} ⊗ 1_{θ_{0}})
= Ω^{n} ⊗ 1_{θ_{0}} − J(Ω^{n−1} ⊗ 1_{θ_{1}} + ∑_{a′_{(−1)}} n_{θ_{0}a′_{(−1)}}Ω^{n} ⊗ 1_{a′_{(−1)}}
+ ∑_{a′_{(0)}} n_{θ_{1}a′_{(0)}}Ω^{n} ⊗ η_{a′_{(0)}})
= Ω^{n} ⊗ 1_{θ_{0}} − ∑_{a_{(1)}} n_{θ_{1}a_{(1)}}Ω^{n−1} ⊗ 1_{a_{(1)}}
−(∑_{a′_{(−1)}} n_{θ_{0}a′_{(−1)}}n_{a′_{(−1)}a_{(−1)}} + ∑_{a′_{(0)}} n_{θ_{1}a′_{(0)}}m_{a′_{(0)}a_{(−1)}})Ω^{n} ⊗ 1_{a_{(−1)}}
− ∑_{a′_{(0)}} n_{θ_{1}a′_{(0)}}n_{a′_{(0)}a_{(0)}}Ω^{n} ⊗ η_{a_{(0)}} − ∑_{a′_{(0)}} n_{θ_{1}a′_{(0)}}n_{a′_{(0)}θ_{0}}Ω^{n} ⊗ 1_{θ_{0}}

We also get

(DH + HD)((Ω^{n} ⊗ 1_{θ_{0}})
= D(∑_{a_{(0)}} n_{θ_{0}a_{(0)}}^{P}Ω^{n} ⊗ 1_{a(0)} + ∑_{a_{(1)}} n_{θ_{1}a_{(1)}}Ω^{n} ⊗ η_{a_{(1)}})
+H(∑_{a_{(−2)}} n_{θ_{0}a_{(−2)}}Ω^{n} ⊗ 1_{a_{(−2)}})
= ∑_{a_{(−1)}}(∑_{a_{(1)}} n_{θ_{1}a_{(1)}}m_{a_{(1)}a_{(−1)}} − ∑_{a_{(0)}} n_{θ_{0}a_{(0)}}^{P}n_{a_{(0)}a_{(−1)}}
− ∑_{a_{(−2)}} n_{θ_{0}a_{(−2)}}n_{a_{(−2)}a_{(−1)}}^{P})Ω^{n} ⊗ 1_{a(−1)}
+ ∑_{a_{(1)},a_{(0)}} n_{θ_{1}a_{(1)}}n_{a_{(1)}a_{(0)}}Ω^{n} ⊗ η_{a_{(0)}} − ∑_{a_{(1)}} n_{θ_{1}a_{(1)}}Ω^{n−1} ⊗ 1_{a_{(1)}}
+ ∑_{a_{(1)}} n_{θ_{1}a_{(1)}}n_{a_{(1)}θ_{0}}Ω^{n} ⊗ 1_{θ_{0}}.

Let us first check the coefficient of Ω^{n} ⊗ 1_{θ_{0}}. Equating the coefficients of Ω^{n} ⊗ 1_{θ_{0}} in the two expressions above yields the identity

∑_{a_{(1)}} n_{θ_{1}a_{(1)}}n_{a_{(1)}θ_{0}} + ∑_{a′_{(1)}} n_{θ_{1}a′_{(0)}}n_{a′_{(0)}θ_{0}} = 1.

This identity is satisfied, since it is exactly the counting of (110).

We then check the coefficient of Ω^{n} ⊗ η_{a_{(0)}}. We need to prove the identity

∑_{a_{(1)}} n_{θ_{1}a_{(1)}}n_{a_{(1)}a_{(0)}} + ∑_{a′_{(0)}} n_{θ_{1}a′_{(0)}}n_{a′_{(0)}a_{(0)}} = 0.

This follows from the boundary structure of M(θ_{1}, O_{a_{(0)}})^{∗}, given by

⋃_{a_{(1)}} M(θ_{1}, O_{a_{(1)}})^{∗} ×_{O_{a_{(1)}}} M̂(O_{a_{(1)}}, O_{a_{(0)}})^{∗}

∪ ⋃_{a′_{(0)}} M̂θ_{1}, O_{a′_{(0)}})^{∗} ×_{O_{a{′ \atop (0)}}} M(O_{a′_{(0)}}, O_{a_{(0)}})^{∗}.

In order to compare the coefficients of Ω^{n} ⊗ 1_{a(−1)}, we need to prove the identity

\begin{aligned}& ∑_{a′_{(−1)}} n_{θ_{0}a′_{(−1)}}n_{a′_{(−1)}a_{(−1)}} − ∑_{a_{(0)}} n_{θ_{0}a_{(0)}}^{P}n_{a_{(0)}a_{(−1)}} \\ − & ∑_{a_{(−2)}} n_{θ_{0}a_{(−2)}}n_{a_{(−2)}a_{(−1)}}^{P} \\ + & ∑_{a′_{(0)}} n_{θ_{1}a′_{(0)}}m_{a′_{(0)}a_{(−1)}} + ∑_{a_{(1)}} n_{θ_{1}a_{(1)}}m_{a_{(1)}a_{(−1)}} = 0 \end{aligned}  (119)

Among these terms we can isolate a contribution

∑_{a′_{(−1)}} n_{θ_{0}a′_{(−1)}}n_{a′_{(−1)}a_{(−1)}} − ∑_{a_{(0)}} n_{θ_{0}a_{(0)}}^{P}n_{a_{(0)}a_{(−1)}} − ∑_{a_{(−2)}} n_{θ_{0}a_{(−2)}}n_{a_{(−2)}a_{(−1)}}^{P},

which is the contribution of the boundary terms in M^{P} (θ_{0}, O_{a_{(−1)}})^{∗} obtained by gluing co-dimension one boundary strata along irreducible critical orbits. The term

∑_{a′_{(−1)}} n_{θ_{0}a′_{(−1)}}n_{a′_{(−1)}a_{(−1)}}

counts the contribution of the special gluing

M(θ_{0}, θ_{1}) × M(θ_{1}, a′_{(−1)}) × M(a′_{(−1)}, a_{(−1)}),

where the coefficient

n_{θ_{0}a′_{(−1)}} = #s̄^{−1}(0)

counts the zeroes of the obstruction section in M(θ_{1}, a′_{(−1)}).

According to Theorem 6.22, the counting of the remaining boundary components for M^{P} (θ_{0}, a_{(−1)}) is given by the gluing

M(θ_{0}, θ_{1}) × s^{−1}(0),

with

s^{−1}(0) ⊂ M(θ_{1}, a_{(−1)}).

Recall that, in this case, s^{−1}(0) is zero-dimensional.

We claim that the zeroes of this obstruction section are counted precisely by the expression

∑_{a′_{(0)}} n_{θ_{1}a′_{(0)}}m_{a′_{(0)}a_{(−1)}} + ∑_{a_{(1)}} n_{θ_{1}a_{(1)}}m_{a_{(1)}a_{(−1)}}.

In fact, we can describe the obstruction bundle as the line bundle

M(θ_{1}, O_{a_{(−1)}})^{∗} ×_{U(1)} Coker(L_{x}),

over M(θ_{1}, a_{(−1)})^{∗}, with x = [𝔸_{0}, 0] the singular reducible in M(θ_{0}, θ_{1}), and Coker(L_{x}) = ℂ. With the choice of trivilizations, the zeros of the obstruction section are localized in some compact set in

⋃_{a′_{(0)}} M(θ_{1}, a′_{(0)}) × M(a′_{(0)}, a_{(−1)})

∪ ⋃_{a_{(1)}} M(θ_{1}, a_{(1)}) × M(a_{(1)}, a_{(−1)}).

Thus, counting the zeros of this obstruction section, which gives the relative Euler class of the associated complex line bundle over M(θ_{1}, a_{(1)}), contributes

\begin{aligned}#s^{−1}(0) = & ∑_{a′_{(0)}} n_{θ_{1},a′_{(0)}}m_{a′_{(0)}a_{(−1)}} \\ & + ∑_{a_{(1)}} n_{θ_{1}a_{(1)}}m_{a_{(1)}a_{(−1)}}, \end{aligned}

Then the vanishing condition of (119) follows from #(∂M^{P} (θ_{0}, a_{(−1)})) = 0. Thus, we have completed the proof of the identity 1 − JI = DH + HD on the generator Ω^{n} ⊗ 1_{θ_{0}}. ⋄

We proceed to check the remaining identities.

Lemma 6.30 We have the identity 1 − JI = DH + HD on Ω^{n} ⊗ 1_{a_{(1)}}.

\underline{Proof:} We compute

(1 − JI)(Ω^{n} ⊗ 1_{a_{(1)}})

= Ω^{n} ⊗ 1_{a_{(1)}} − J(n_{a_{(1)}θ_{0}}Ω^{n−1} ⊗ 1_{θ_{1}} + ∑_{a′_{(1)}} n_{a_{(1)}a′_{(1)}}Ω^{n−1} ⊗ 1_{a′_{(1)}})
= Ω^{n} ⊗ 1_{a_{(1)}} − ∑ã_{(1)}(∑_{a′_{(1)}} n_{a_{(1)}a′_{(1)}}n_{a′_{(1)}ã_{(1)}} + n_{a_{(1)}θ_{0}}n_{θ_{1}ã_{(1)}})Ω^{n} ⊗ 1ã_{(1)}

\begin{aligned}& (DH + HD)(Ω^{n} ⊗ 1_{a_{(1)}}) \\ = & (∑_{a_{(2)}} n_{a_{(1)}a_{(2)}}^{P}n_{a_{(2)}ã_{(1)}} + ∑_{a_{(0)}} n_{a_{(1)}a_{(0)}}n_{a_{(0)}ã_{(1)}}^{P})Ω^{n} ⊗ 1ã_{(1)}. \end{aligned}

If we compare the coefficient of each term in DH + HD − 1 + JI, we see that the coefficient of Ω^{n} ⊗ 1_{a_{(1)}} is given by

∑_{a_{(2)}} n_{a_{(1)}a_{(2)}}^{P}n_{a_{(2)}a_{(1)}} + ∑_{a_{(0)}} n_{a_{(1)}a_{(0)}}n_{a_{(0)}a_{(1)}}^{P}

+ ∑_{a′_{(1)}} n_{a_{(1)}a′_{(1)}}n_{a′_{(1)}a_{(1)}} − 1 + n_{a_{(1)}θ_{0}}n_{θ_{1}a_{(1)}}.

Notice that the sum

∑_{a_{(2)}} n_{a_{(1)}a_{(2)}}^{P}n_{a_{(2)}a_{(1)}} + ∑_{a_{(0)}} n_{a_{(1)}a_{(0)}}n_{a_{(0)}a_{(1)}}^{P} + ∑_{a′_{(1)}} n_{a_{(1)}a′_{(1)}}n_{a′_{(1)}a_{(1)}} − 1

is the algebraic counting of the boundary points in M^{P} (a_{(1)}, a_{(1)})^{∗} that correspond to boundary components of the form

⋃_{a_{(2)}} M^{P} (a_{(1)}, a_{(2)}) × M̂(a_{(2)}, a_{(1)}) ∪ ⋃_{a_{(0)}} M̂(a_{(1)}, a_{(0)}) × M^{P} (a_{(0)}, a_{(1)})

∪ ⋃_{a′_{(1)}} M(a_{(1)}, a′_{(1)}) × M(a′_{(1)}, a_{(1)}) ∪ {−a}.

Similarly, for ã_{(1)} ≠ a_{(1)}, the coefficient of Ω^{n} ⊗ 1ã_{(1)} is given by

∑_{a_{(2)}} n_{a_{(1)}a_{(2)}}^{P}n_{a_{(2)}ã_{(1)}} + ∑_{a_{(0)}} n_{a_{(1)}a_{(0)}}n_{a_{(0)}ã_{(1)}}^{P} + ∑_{a′_{(1)}} n_{a_{(1)}a′_{(1)}}n_{a′_{(1)}ã_{(1)}} + n_{a_{(1)}θ_{0}}n_{θ_{1}ã_{(1)}}.

Again, notice that the sum

∑_{a_{(2)}} n_{a_{(1)}a_{(2)}}^{P}n_{a_{(2)}ã_{(1)}} + ∑_{a_{(0)}} n_{a_{(1)}a_{(0)}}n_{a_{(0)}ã_{(1)}}^{P} + ∑_{a′_{(1)}} n_{a_{(1)}a′_{(1)}}n_{a′_{(1)}ã_{(1)}}

is the algebraic counting of the boundary points of M^{P} (a_{(1)}, ã_{(1)})^{∗} which correspond to boundary components of the form

⋃_{a_{(2)}} M^{P} (a_{(1)}, a_{(2)}) × M̂(a_{(2)}, ã_{(1)}) ∪ ⋃_{a_{(0)}} M̂(a_{(1)}, a_{(0)}) × M^{P} (a_{(0)}, ã_{(1)})

∪ ⋃_{a′_{(1)}} M(a_{(1)}, a′_{(1)}) × M(a′_{(1)}, ã_{(1)}).

We only need to prove that the counting of the remaining boundary components of M^{P} (a_{(1)}, ã_{(1)}) is given by

n_{a_{(1)}θ_{0}}n_{θ_{1}ã_{(1)}},

which counts the contribution of the zeros of the obstruction section, that is, of those monopoles of M^{P} (a_{(1)}, ã_{(1)}) which are obtained by gluing the singular reducible x ∈ M(θ_{0}, θ_{1}) with

M(a_{(1)}, θ_{0}) × U(1) × M(θ_{1}, ã_{(1)}).

Recall that the non-equivariant gluing

# : s̄^{−1}(0) ⊂ M(a_{(1)}, θ_{0}) × U(1) × M(θ_{1}, ã_{(1)}) → M^{P} (a_{(1)}, ã_{(1)})

corresponds to the equivariant gluing in the framed moduli spaces

# : s̃^{−1}(0) ⊂ M(O_{a_{(1)}}, θ_{0}) × M(θ_{1}, Oã_{(1)}) → M^{P} (O_{a_{(1)}}, Oã_{(1)}).

The pull-back and push-forward map

(e− ; ã_{(1)})_{∗}(e_{a_{(1)}}^{+})^{∗}(η_{a_{(1)}})

defines the relative Euler characteristic number on the associated line bundle of

L_{a_{(1)},ã_{(1)}} = (M(O_{a_{(1)}}, θ_{0}) × M(θ_{1}, Oã_{(1)})) ×_{U(1)} ℂ.

This gives

(e− ; ã_{(1)})_{∗}(e_{a_{(1)}}^{+})^{∗}(η_{a_{(1)}}) = n_{a_{(1)}θ_{0}}n_{θ_{1}ã_{(1)}}.

Consider the obstruction bundle

(M(O_{a_{(1)}}, θ_{0}) × M(θ_{0}, θ_{1}) × M(θ_{1}, Oã_{(1)})) ×_{U(1)} Coker(L_{x})

over the space

M(a_{(1)}, θ_{0}) × M(θ_{1}, ã_{(1)}) × U(1),

with Coker(L_{x}) = ℂ, for x ∈ M(θ_{0}, θ_{1}) the singular point x = [𝔸_{0}, 0]. Counting the zeros of the obstruction section

#(s̄^{−1}(0) ∩ (M(a_{(1)}, θ_{0}) × M(θ_{1}, ã_{(1)}) × U(1)))

gives another computation of the same relative Euler class above, that is,

\begin{aligned}& #(s̄^{−1}(0) ∩ (M(a_{(1)}, θ_{0}) × M(θ_{1}, ã_{(1)}) × U(1))) \\ = & n_{a_{(1)}θ_{0}}n_{θ_{1}ã_{(1)}}. \end{aligned}

Thus, the components of the boundary of M^{P} (a_{(1)}, ã_{(1)})^{∗} which come from the gluing with the singular x ∈ M(θ_{0}, θ_{1}) contribute a term

n_{a_{(1)}θ_{0}}n_{θ_{1}ã_{(1)}}

to the counting of the boundary points. This completes the proof of the Lemma. ⋄

Finally, we have to check the following.

Lemma 6.31 We have the identity 1 − JI = DH + HD on Ω^{n} ⊗ η_{a_{(1)}}.

\underline{Proof:} Direct calculation of the terms (1−JI)(Ω^{n}⊗η_{a_{(1)}}) and (DH +HD)(Ω^{n}⊗ η_{a_{(1)}}) shows that the conditions required in order to have the same coefficients on all the generators are precisely the conditions already verified in the case of Lemma 6.30. ⋄

Notice how clearly this argument of topological invariance breaks down for the non-equivariant Floer homology. The invariance within the same chamber is still verified: in fact, no substantial changes are necessary in that first part of the proof, in order to adapt it to the case of the non-equivariant Floer homology. However, as wee see clearly from the structure of this second part of the proof, the general argument for the proof of Theorem 6.1, for metrics and perturbations in two different chambers, relies essentially on the contribution of the reducible points, in order to construct the chain map I and chain homotopy H, as discussed in Definition 6.27, Lemma 6.28, Lemma 6.29, Lemma 6.30, and Lemma 6.31. The example presented at the beginning of Section 6.3 also clarifies why the argument cannot be adapted to the non-equivariant Floer homology.

7 Wall crossing formula for the Casson-type invariant

We want to compare the equivariant Floer homology with the ordinary Floer homology in the cases where the latter is defined, i.e. when b^{1}(Y) is non-trivial [38] or when Y is a homology sphere, [13].

In the case when b^{1}(Y) is non-trivial, we expect to find that the equivariant Floer homology, which is computed by considering framed moduli spaces, is isomorphic to the ordinary Floer homology computed in the unframed space. In fact this is the analogue of the well known result for equivariant cohomology of a finite dimensional manifold, where, if the action of the group is free, then the equivariant cohomology is just the ordinary cohomology of the quotient, H_{G}^{∗}(M; ℝ) ≅ H^{∗}(M/G; ℝ) as H^{∗}(BG; ℝ)-modules.

In the case of a homology sphere, instead, we expect to find an exact sequence that connects the equivariant Floer homology with the ordinary Floer homology and an extra copy of ℝ[Ω] that corresponds to the unique reducible solution that has been removed in the computation of the non-equivariant Floer homology.

Recall that we have the explicit description of the boundary operator in the equivariant Floer complex, as analyzed in Section 5, Proposition 5.3, which gives the formula (69),

D : \begin{aligned}Ω^{n} ⊗ 1_{a} & ↦ & −n_{ab}Ω^{n} ⊗ 1_{b} \\ Ω^{n} ⊗ η_{a} & ↦ & (n_{ab}Ω^{n} ⊗ η_{b}) ⊕ (m_{ac}Ω^{n} ⊗ 1_{c}) \\ & & ⊕(−Ω^{n−1} ⊗ 1_{a}). \end{aligned}

7.1 Comparison with the non-equivariant Floer Homology

Now we can define a chain homomorphism that maps the equivariant to the non-equivariant complex.

Let us first work in the case with no reducible solution (i.e. with b^{1}(Y) > 0). In this case for each O_{a} that appears in the equivariant complex we have a generator ℝa that appears in the non-equivariant complex (coefficients in ℝ).

Now we define the chain map

i_{k} : C_{k,U(1)}(Y) → C_{k}(Y),

so that it satisfies ∂_{k}i_{k} = i_{k−1}D_{k}. Let i_{k} act on the generators as follows

i_{k} : ⨁_{μ(a)+j=k} Ω_{j,U(1)}(O_{a}) → ∑_{μ(O_{a})=k} ℝa,  (120)

i_{k}(Ω^{n} ⊗ 1_{a}) = 0,

for all values of n and μ(O_{a}),

i_{k}(1 ⊗ η_{a}) = a,

if μ(O_{a}) = k, and in all other cases

i_{k}(Ω^{n} ⊗ η_{a}) = 0.

This means that the map i_{k} kills all the generators in degree k that are not the generator of the equivariant homology of some orbit O_{a} of degree k.

With this definition it is clear that i_{∗} is a chain map. Thus it defines a sub-complex of C_{∗,U(1)}(Y) given by Q_{∗} = Ker(i_{∗}) with the restriction of the boundary operator D.

Theorem 7.1 If there are no reducible solutions (i.e. b^{1}(Y) > 0) the map i_{∗} defined in (120) induces an isomorphism in cohomology,

SW H_{∗,U(1)}(Y) ≅ SW H_{∗}(Y).

\underline{Proof:} The complexes C_{k,U(1)}, Q_{k}, and C_{k} all have a filtration by index. For C_{k,U(1)} the filtration is given by

C_{k,U(1)}(n) = ⨁_{μ(O_{a})+j=k,μ(O_{a})≤n} Ω_{j,U(1)}(O_{a}).  (121)

The complex Q_{∗} is written as

Q_{k} = ⨁_{μ(a)+j=k,j≥1} Ω_{j,U(1)}(O_{a}).  (122)

It has a filtration by index of the form

Q_{k}(n) = ⨁_{μ(a)+j=k,j≥1,μ(a)≤n} Ω_{j,U(1)}(O_{a}).  (123)

On the other hand also the non-equivariant complex has a filtration by index of the form

C_{k}(n) = ⨁_{μ(a)=k≤n} ℝa.  (124)

Thus we can look at the spectral sequences associated to the filtrations and prove that i induces a map of spectral sequences and an isomorphism of the E^{1}-terms of the spectral sequences associated to the filtration of C_{∗,U(1)} and of C_{∗}. Thus we get the resulting isomorphism of the E^{∞} terms, i.e. of the homology of C_{∗,U(1)} and of C_{∗}.

Lemma 7.2 Let E_{U(1)}, E_{Q}, and E be the spectral sequences associated to the filtration of the complexes C_{U(1)}, Q and C respectively. The chain map i_{∗} induces a map of spectral sequences. Moreover, in the case when b^{1}(Y) > 0 the map i_{∗} induces an isomorphism of the E^{1}-terms

E_{kl,U(1)}^{1} ≅ E_{kl}^{1}.

\underline{Proof:}

Consider the filtration of C_{∗,U(1)}. The E_{kl}^{0} terms of the spectral sequence are given by

E_{kl,U(1)}^{0} = C_{k+l,U(1)}(k)/C_{k+l,U(1)}(k − 1).

From (121) we get

E_{kl,U(1)}^{0} = ⨁_{μ(O_{a})=k} Ω_{l,U(1)}(O_{a}),

and the differentials E_{kl,U(1)}^{0} → E_{kl−1,U(1)}^{0} are just given by the differential of the equivariant complex on each fixed orbit O_{a}. Thus the E^{1}-term of this spectral sequence is given by

E_{kl,U(1)}^{1} = H_{k+l}(E_{k∗,U(1)}^{0}),

E_{kl,U(1)}^{1} = ⨁_{μ(O_{a})=k} H_{k+l,U(1)}(O_{a}).

But since the O_{a} are irreducible orbits, with a free U(1) action, the equivariant homology is concentrated in degree zero,

E_{k0,U(1)}^{1} = ⨁_{μ(O_{a})=k} ℝa,

E_{kl,U(1)}^{1} = 0

for l ≠ 0.

Now let us consider the filtration of the non-equivariant complex C_{∗}. From (124) we get

E_{k0}^{0} = ⨁_{μ(O_{a})=k} ℝa

and E_{kl}^{0} = 0 for l ≠ 0. Thus in this case the only terms that survive in the E^{1} is in degree l = 0 and is

E_{k0}^{1} = ⨁_{μ(O_{a})=k} ℝa.

⋄

Thus the map i_{∗} induces an isomorphism on the E^{1}-terms of the two spectral sequences, hence on the E^{∞}-terms, namely on the homology

H_{∗}(C_{k,U(1)}(Y), D_{k}) ≅^{i} H_{∗}(C_{k}(Y), ∂_{k}).

This completes the proof of theorem 7.1. ⋄

Now let us consider the case when the manifold Y is an integral homology sphere. In this case there are also terms in the equivariant complex that come from the reducible solution θ = [ν, 0]. We assume that θ has index zero, μ(θ) = 0.

Theorem 7.3 Let Y be an integral homology sphere. Then there is an exact sequence

⋯ → H_{k,U(1)}(θ) → SW H_{k,U(1)}(Y) →^{i} SW H_{k}(Y) →^{Δ} H_{k−1,U(1)}(θ) ⋯

In the equivariant complex in degree k we have an extra generator Ω^{k} ⊗ θ. The boundary maps that come from the equivariant complex associated to the degenerate orbit ν with the trivial action of U(1) are trivial: in fact the equivariant homology of a point is

H_{∗,U(1)}(θ) = ℝ[Ω] = H_{∗}(BU(1), ℝ).

However, there are non-trivial boundary maps that hit the generators Ω^{n} ⊗ θ. These can be described as follows. Suppose O_{a} is the orbit of an irreducible solution with index μ(O_{a}) = 1. Then we have a moduli space M̂(O_{a}, θ) that is 1-dimensional and that fibers over θ with a 1-dimensional fiber. Thus the pullback-pushforward map acts as

η_{a} ↦ (e_{θ}^{−})_{∗}(e_{a}^{+})^{∗}η_{a} = m_{aθ},

where m_{aθ} is the integration along the 1-dimensional fiber of the 1-form (e_{a}^{+})^{∗}η_{a}. This gives rise to a component of the boundary map of the form

1 ⊗ η_{a} ↦ m_{aθ}1 ⊗ θ.

Moreover, there is a non-trivial boundary map that comes from the moduli spaces that connect the reducible to generators with the orbits with μ(O_{a}) = −2.

Now the map i_{k} is defined as before with the additional condition that it kills the extra generator Ω^{k} ⊗ θ.

Lemma 7.4 Let Y be a homology sphere. Then the homology of the complex Q_{∗} is just the equivariant homology of the point θ,

H_{∗}(Q) ≅ H_{∗,U(1)}(θ) = ℝ[Ω].

\underline{Proof:} The complex Q_{∗} contains the extra generator Ω^{k} ⊗ θ in degree k and this generator appears in all levels of the filtration Q_{k}(n) for any n ≥ 0, since θ is of degree zero.

Thus if we look at the spectral sequence associated to the filtration of the complex Q_{∗} we find

E_{kl,Q}^{0} = Q_{k+l}(k)/Q_{k+l}(k − 1),

that is, for k > 0 and l ≥ 0,

E_{kl,Q}^{0} = ⨁_{μ(O_{b})=k−1} Ω_{l+1,U(1)}(O_{b}).

This complex is clearly acyclic because the differentials are just the equivariant differentials for each orbit and no generator survives in homology because we are counting only l + 1 ≥ 1, hence the terms 1 ⊗ η_{b} are suppressed (in fact they are not in Ker(i_{∗})). On the other hand for k = 0 we get

E_{0l,Q}^{0} = ℝ < Ω^{l} ⊗ θ >,

with trivial differentials, so that the E^{1}-terms are

E_{kl,Q}^{1} = H_{k+l,U(1)}(θ).

This means that the homology of the complex Q_{∗} is actually H_{∗,U(1)}(θ) = ℝ[Ω].

⋄ Thus if we consider the long exact sequence induced by the short sequence

0 → Q_{k} → C_{k,U(1)} →^{i} C_{k} → 0

we have

⋯ → H_{k,U(1)}(θ) → SW H_{k,U(1)}(Y) →^{i} SW H_{k}(Y) → ⋯

This proves theorem 7.3. ⋄

The connecting homomorphism Δ in the above long exact sequences is particularly interesting.

Proposition 7.5 Suppose given a representative ∑_{a} x_{a}a in SW H_{2k+1}(Y). We assume that Y is a homology sphere. Then the connecting homomorphism in the long exact sequence is

△_{k} : SW H_{2k+1} → H_{2k,U(1)}(θ) = ℝΩ^{k},

where

△_{k}(∑_{a} x_{a}a) = ∑ x_{a}m_{ac}m_{ce} ⋯ m_{α′α}n_{αθ}Ω^{k} ⊗ θ.  (125)

Here the sum is understood over all the repeated indices, that is over all critical points with indices μ(O_{a}) = 2k + 1, μ(O_{c}) = 2k − 1, μ(O_{α′}) = 3, μ(O_{α}) = 1.

\underline{Proof:} The map is defined by the standard diagram chase and by adding boundary terms in order to find a representative of the form (125), as illustrated in the following diagram. Sums over repeated indices are understood.

\xymatrix{ & ⋮ \ar[d] &  & ⋮ \ar[d] &  & ⋮ \ar[d] \\ 0 \ar[r] & Q_{2k+1} \ar[rr] \ar[dd] &  & C_{2k+1,U(1)} \ar[rr]^{i} \ar[dd] &  & C_{2k+1} \ar[rr] \ar[dd] &  & 0 \\  &  &  &  & x_{a} 1 ⊗ η_{a} \ar@{|->}[dd] &  & x_{a} a \ar@{|->}[ll] \\ 0 \ar[r] & Q_{2k} \ar[rr] \ar[dd] &  & C_{2k,U(1)} \ar[rr]^{i} \ar[dd] &  & C_{2k} \ar[rr] \ar[dd] &  & 0 \\  &  & Δ_{k}(x_{a} a) &  & x_{a}m_{ac} 1 ⊗ 1_{c} \ar@{|->}[ll] \\  & ⋮ &  & ⋮ &  & ⋮}

For a cycle ∑_{a:μ(O_{a})=2k+1} x_{a}a we have ∑_{a:μ(a)=2k+1} x_{a}n_{ab} = 0 for any b with μ(O_{b}) = 2k. The element ∑_{a} x_{a}a has a preimage ∑ x_{a}1 ⊗ η_{a} under i_{2k+1}. The image of this element under the equivariant boundary is given by

∑ x_{a}D(1 ⊗ η_{a}) = ∑ x_{a}n_{ab}1 ⊗ η_{b} + ∑ x_{a}m_{ac}1 ⊗ 1_{c},

where the first term is zero due to our assumption on ∑ x_{a}a. The element Δ_{k}(x_{a}a) is unchanged if we add to ∑ x_{a}1⊗η_{a} an element in the kernel of i_{2k+1}. Adding the element ∑_{a,c} x_{a}m_{ac}Ω ⊗ η_{c}, which is in the kernel of i, we get the following diagram

\xymatrix{ & ⋮ \ar[d] &  & ⋮ \ar[d] &  & ⋮ \ar[d] \\ 0 \ar[r] & Q_{2k+1} \ar[rr] \ar[dd] &  & C_{2k+1,U(1)} \ar[rr]^{i} \ar[dd] &  & C_{2k+1} \ar[rr] \ar[dd] &  & 0 \\  &  &  &  & \begin{matrix}x_{a} 1 ⊗ η_{a} \\ +x_{a}m_{ac} Ω ⊗ η_{c} \end{matrix} \ar@{|->}[dd] &  & x_{a} a \ar@{|->}[ll] \\ 0 \ar[r] & Q_{2k} \ar[rr] \ar[dd] &  & C_{2k,U(1)} \ar[rr]^{i} \ar[dd] &  & C_{2k} \ar[rr] \ar[dd] &  & 0 \\  &  & Δ_{k}(x_{a} a) &  & x_{a}m_{ac}m_{ce} Ω ⊗ 1_{e} \ar@{|->}[ll] \\  & ⋮ &  & ⋮ &  & ⋮}

where we have

D(x_{a}1 ⊗ η_{a} + x_{a}m_{ac}Ω ⊗ η_{c}) = x_{a}m_{ac}1 ⊗ 1_{c}+
x_{a}m_{ac}(−1 ⊗ 1_{c} + n_{cd}Ω ⊗ η_{d} + m_{ce}Ω ⊗ 1_{e}) = x_{a}m_{ac}m_{ce}Ω ⊗ 1_{e}

with sums over repeated indices. The last equality follows from the identities ∑ m_{ac}n_{cd} +n_{ab}m_{bd} = 0 and ∑ x_{a}n_{ab}m_{bd} = 0. We can iterate the procedure. In the following step we add a term ∑ x_{a}m_{ac}m_{ce}Ω^{2} ⊗η_{e} to the preimage of ∑ x_{a}a. The corresponding image under the boundary of the equivariant complex is ∑ x_{a}m_{ac}m_{ce}m_{eg}Ω^{2} ⊗ 1_{g}, where μ(O_{c}) − μ(O_{e}) = μ(O_{e}) − μ(O_{g}) = 2. The procedure can be iterated until the reducible point θ is hit. Contributions from other irreducible critical orbits O_{p} with μ(O_{p}) − μ(θ) = 0 are killed in finitely many steps, iterating the same procedure, since the complex is finitely generated and they eventually hit the lowest index critical points. Thus, the resulting image under the connecting homomorphism △_{k}(∑_{a:μa=2k+1} x_{a}a) is given by

∑ x_{a}m_{ac}m_{ce} ⋯ m_{α′α}m_{αθ}Ω^{k}.

⋄

As in the case of instanton Floer theory [60], one expects Ker(Δ_{k}) ≅ Im(i_{2k+1}) to be the part of the Floer homology where the relative invariants of four-manifolds with boundary live.

7.2 Wall-crossing formula: the algebraic picture

In this section, we will apply the equivariant Seiberg-Witten-Floer homology theory to study the dependence of the metric for the Casson-type invariant [12]

[13] of a homology sphere. In order to define the Casson-type invariant, we choose a metric whose ordinary Dirac operator has trivial kernel. The metrics whose ordinary Dirac operator has non-trivial kernel form a chamber structure as proven in Theorem 2.5 and Theorem 2.6. The usual cobordism argument can be adopted to prove that the Casson-type invariant is constant in each chamber. The aim of this section is to get a wall-crossing formula for a path of metrics and perturbations that crosses the wall.

Denote by λ_{SW} (Y, g, ν) the Casson-type invariant for the metric and perturbation (g, ν): λ_{SW} (Y, g, ν) is the Euler characteristic of the non-equivariant Seiberg-Witten-Floer homology. Recall that this Floer homology [13] is defined by removing the reducible critical point, the trivial solution θ = [ν, 0]. We have

λ_{SW} (Y, g, ν) = ∑_{k}(−1)^{k} dim SW H_{k}(Y, g, ν).  (126)

In this Section 7.2 we derive the wall crossing formula under the following assumption. Fix metrics and perturbations (g_{0}, ν_{0}), (g_{1}, ν_{1}) in two different chambers, with the property that there exists an open set of paths (g_{t}, ν_{t}) connecting (g_{0}, ν_{0}) to (g_{1}, ν_{1}) such that (g_{t}, ν_{t}) hits a co-dimension one wall only once, transversely. This situation is certainly verified if the two points (g_{0}, ν_{0}) and (g_{1}, ν_{1}) are close enough points on either side of a wall W_{1}, in the notation of Theorem 2.6, that is, a wall of metrics and perturbations satisfying Ker(∂_{ν}^{g}) = ℂ. We also assume that, along the path (g_{t}, ν_{t}) we have SF (∂_{ν_{t}}^{g_{t}}) = 2, that is, that we have μ(θ_{0}) − μ(θ_{1}) = 2 in the notation used in Section 6, which determines a global grade shift between the equivariant Floer complexes for (g_{0}, ν_{0}) and for (g_{1}, ν_{1}).

Using the topological invariance of the equivariant Seiberg-Witten-Floer homology, we obtain the following isomorphism:

SW H_{k,U(1)}(Y, g_{0}, ν_{0}) ≅ SW H_{k+2,U(1)}(Y, g_{1}, ν_{1}).

In fact, the necessary degree shift is computed as follows: if we set μ(θ_{0}) = 0, hence μ(θ_{1}) = −2, and the map I has degree zero, we obtain that the generators Ω^{n} ⊗ 1_{θ_{0}} and Ω^{n+1} ⊗ 1_{θ_{1}} have the same degree.

First of all, we express the Casson-type invariant (126) in terms of some alternating sum of the equivariant Seiberg-Witten-Floer homology groups. We consider the Casson-type invariant for metric an perturbation (g_{0}, ν_{0}). The exact sequences given in Theorem 7.3, relating the equivariant to the non-equivariant Seiberg-Witten-Floer homology, and the fact that we have H_{∗,U(1)}(θ) = ℝ[Ω] give us the following result.

Proposition 7.6 (1) For k < 0,

SW H_{k}(Y, g_{0}, ν_{0}) ≅ SW H_{k,U(1)}(Y, g_{0}, ν_{0}).

For k ≥ 0, we have the following exact sequences

0 → SW H_{2k+1,U(1)}(Y, g_{0}, ν_{0}) → SW H_{2k+1}(Y, g_{0}, ν_{0}) → ℝΩ^{k} →
→ SW H_{2k,U(1)}(Y, g_{0}, ν_{0}) → SW H_{2k}(Y, g_{0}, ν_{0}) → 0.

Thus the dimensions are related by

dim SW H_{2k} − dim SW H_{2k+1} = dim SW H_{2k,U(1)} − dim SW H_{2k+1,U(1)} − 1.

This gives the wall-crossing formula for the Casson-type invariant.

Theorem 7.7 Suppose given two metrics and perturbations (g_{0}, ν_{0}) and (g_{1}, ν_{1}) in two different chambers and a generic path (g_{t}, ν_{t}) that connects them and that crosses the wall W once. Assume that the crossing happens at a generic point of W, so that the relative Morse index with respect to the reducible solution decreases by 2 across the wall, i.e. that we have μ(θ_{0}) − μ(θ_{1}) = 2. Then the Casson-type invariant changes by

λ_{SW} (Y, g_{1}, ν_{1}) = λ_{SW} (Y, g_{0}, ν_{0}) − 1.

\underline{Proof:} We can assume that we are in one of the following two cases for the non-equivariant Floer homology group. \underline{Case 1:} There exists an integer N such that SW H_{p}(Y, g_{0}, ν_{0}) = 0 for all p ≥ 2N but SW H_{2N−1}(Y, g_{0}, ν_{0}) ≠ 0. \underline{Case 2:} here exists an integer N such that SW H_{p}(Y, g_{0}, ν_{0}) = 0 for all p ≥ 2N+1 but SW H_{2N}(Y, g_{0}, ν_{0}) ≠ 0.

For case 1, we can assume N ≥ 0, otherwise the non-equivariant Floer homology groups are isomorphic to the equivariant Floer homology groups by Proposition 7.6. Therefore, from the exact sequences in Proposition 7.6, we get

• SW H_{p,U(1)}(Y, g_{0}, ν_{0}) = \{ \begin{matrix}0 & p > 2N, p is odd \\ ℝΩ^{m} & p = 2m ≥ 2N. \end{matrix}

•

dim SW H_{2N−2}(Y, g_{0}, ν_{0}) − dim SW H_{2N−1}(Y, g_{0}, ν_{0})

= dim SW H_{2N−2,U(1)}(Y, g_{0}, ν_{0}) − dim SW H_{2N−1,U(1)}(Y, g_{0}, ν_{0}) − 1

dim SW H_{2N−4}(Y, g_{0}) − dim SW H_{2N−3}(Y, g_{0}, ν_{0})

= dim SW H_{2N−4,U(1)}(Y, g_{0}, ν_{0}) − dim SW H_{2N−3,U(1)}(Y, g_{0}, ν_{0}) − 1

⋯

dim SW H_{0}(Y, g_{0}, ν_{0}) − dim SW H_{1}(Y, g_{0}, ν_{0})

= dim SW H_{0,U(1)}(Y, g_{0}, ν_{0}) − dim SW H_{1,U(1)}(Y, g_{0}, ν_{0}) − 1.

• For k < 0, SW H_{k}(Y, g_{0}, ν_{0}) = SW H_{k,U(1)}(Y, g_{0}, ν_{0}).

Given the above information, we can calculate the Casson-type invariant for the metric and perturbation (g_{0}, ν_{0}) as,

λ_{SW} (Y, g_{0}, ν_{0}) = ∑_{k}(−1)^{k}dimSW H_{k}(Y, g_{0}, ν_{0})
= ∑_{k<N}(dimSW H_{2k,U(1)}(Y, g_{0}, ν_{0})
− dim SW H_{2k+1,U(1)}(Y, g_{0}, ν_{0})) − N.

From the isomorphism SW H_{k,U(1)}(Y, g_{1}, ν_{1}) = SW H_{k−2,U(1)}(Y, g_{0}, ν_{0}), we have

SW H_{p,U(1)}(Y, g_{1}, ν_{1}) = \{ \begin{matrix}0 & p > 2N + 2, p is odd \\ ℝΩ^{m} & p = 2m ≥ 2N + 2. \end{matrix}

If we apply the above isomorphisms to the exact sequences in Proposition 7.6, we can see that the Casson-type invariant for metric and perturbation (g_{1}, ν_{1}) can be rewritten as

λ_{SW} (Y, g_{1}, ν_{1}) = ∑_{k}(−1)^{k} dim SW H_{k}(Y, g_{1}, ν_{1})
= ∑_{k<N+1}(dim SW H_{2k,U(1)}(Y, g_{1}, ν_{1})
− dim SW H_{2k+1,U(1)}(Y, g_{1}, ν_{1})) − (N + 1)
= λ_{SW} (Y, g_{0}, ν_{0}) − 1

For case 2, similarly, we have

• SW H_{p,U(1)}(Y, g_{0}, ν_{0}) = \{ \begin{matrix}0 & p > 2N, p is odd \\ ℝΩ^{m} & p = 2m ≥ 2N + 2. \end{matrix}

•

dim SW H_{2N}(Y, g_{0}, ν_{0}) = dim SW H_{2N,U(1)}(Y, g_{0}, ν_{0}) − 1

dim SW H_{2N−2}(Y, g_{0}, ν_{0}) − dim SW H_{2N−1}(Y, g_{0}, ν_{0})

= dim SW H_{2N−2,U(1)}(Y, g_{0}, ν_{0}) − dim SW H_{2N−1,U(1)}(Y, g_{0}, ν_{0}) − 1

dim SW H_{2N−4}(Y, g_{0}, ν_{0}) − dim SW H_{2N−3}(Y, g_{0}, ν_{0})

= dim SW H_{2N−4,U(1)}(Y, g_{0}, ν_{0}) − dim SW H_{2N−3,U(1)}(Y, g_{0}, ν_{0}) − 1

⋯

dim SW H_{0}(Y, g_{0}, ν_{0}) − dim SW H_{1}(Y, g_{0}, ν_{0})

= dim SW H_{0,U(1)}(Y, g_{0}, ν_{0}) − dim SW H_{1,U(1)}(Y, g_{0}, ν_{0}) − 1

• For k < 0, SW H_{k}(Y, g_{0}, ν_{0}) = SW H_{k,U(1)}(Y, g_{0}, ν_{0}).

Then the Casson-type invariant for (g_{0}, ν_{0}) is

λ_{SW} (Y, g_{0}, ν_{0}) = ∑_{k}(−1)^{k} dim SW H_{k}(Y, g_{0}, ν_{0})
= ∑_{k<N}(dim SW H_{2k,U(1)}(Y, g_{0}, ν_{0})
− dim SW H_{2k+1,U(1)}(Y, g_{0}, ν_{0})) − N
+ dim SW H_{2N,U(1)}(Y, g_{0}, ν_{0}) − 1.

From the isomorphism SW H_{k,U(1)}(Y, g_{1}, ν_{1}) = SW H_{k−2,U(1)}(Y, g_{0}, ν_{0}), we know that

SW H_{p,U(1)}(Y, g_{1}, ν_{1}) = \{ \begin{matrix}0 & p > 2N + 2, p is odd \\ ℝΩ^{m} & p = 2m ≥ 2N + 2. \end{matrix}

If we apply the above isomorphisms to the exact sequences in Proposition 7.6 again, we can see that the Casson-type invariant for metric and perturbation (g_{1}, ν_{1}) is

λ_{SW} (Y, g_{1}, ν_{1})
= ∑_{k}(−1)^{k} dim SW H_{k}(Y, g_{1}, ν_{1})
= ∑_{k<N+1}(dim SW H_{2k,U(1)}(Y, g_{1}, ν_{1}) − dim SW H_{2k+1,U(1)}(Y, g_{1}, ν_{1}))
−(N + 1) + dim SW H_{2N+2.U(1)}(Y, g_{1}, ν_{1}) − 1
= λ_{SW} (Y, g_{0}, ν_{0}) − 1

Thus, we have proven the wall-crossing formula:

λ_{SW} (Y, g_{1}, ν_{1}) = λ_{SW} (Y, g_{0}, ν_{0}) − 1.

⋄

Now we are interested in generalizing the argument of Theorem 7.7 to the case of a path (g_{t}, ν_{t}) that crosses the wall structure W at a point which lies in a stratum W_{n} of higher codimension.

If we know that a stratum W_{n} of metrics and perturbations satisfying Ker(∂_{ν}^{g}) = ℂ^{n} is obtained as the transverse intersection of n strata W_{1}^{i_{1}} ∩ ⋯ ∩ W_{1}^{i_{n}}, where every W_{1}^{i_{k}} consists of metrics with Ker(∂_{ν}^{g}) = ℂ, then a path (g_{t}, ν_{t}) that crosses W at a point in W_{n} can be deformed to a path that crosses each W_{1}^{i_{k}} once transversely. In this case, the wall crossing formula simply follows by applying repeatedly Theorem 7.7. However, we do not really need the assumption on the structure of W near a stratum of higher codimension. In fact, it is enough to know that the complex spectral flow SF_{ℂ}(∂_{ν_{t}}^{g_{t}}) = \frac{1}{2}SF (∂_{ν_{t}}^{g_{t}}) is equal to ±n along the path (g_{t}, ν_{t}) that crosses a point on W_{n}. In that case, we can follow the same argument in the proof of Theorem 7.7, but starting with a grade shift of 2n between the equivariant Floer complexes for (g_{0}, ν_{0}) and (g_{1}, ν_{1}). We obtain the following result.

Proposition 7.8 Let (g_{0}, ν_{0}) and (g_{1}, ν_{1}) be two metrics and perturbations in two different chambers. Suppose given a path (g_{t}, ν_{t}) joining them that crosses the wall W once transversely at a point of a stratum W_{n} of codimension 2n − 1. The relative Morse index with respect to the reducible solution decreases by 2n across the wall, that is, that we have μ(θ_{0}) − μ(θ_{1}) = 2n. Then the Casson-type invariant changes by

λ_{SW} (Y, g_{1}, ν_{1}) = λ_{SW} (Y, g_{0}, ν_{0}) − n.

Since the argument of Theorem 7.7 depends only on the counting of the grade shift between the Floer complexes for (g_{0}, ν_{0}) and (g_{1}, ν_{1}) given by the spectral flow SF (∂_{ν_{t}}^{g_{t}}) , together with the proof of topological invariance of the equivariant Floer homology (up to this grade shift), we can formulate the result under these more general hypothesis.

Proposition 7.9 Let (g_{0}, ν_{0}) and (g_{1}, ν_{1}) be two metrics and perturbations in two different chambers. Suppose given a path (g_{t}, ν_{t}) joining them that crosses the wall W transversely in finitely many points. Then the Casson invariant changes by

λ_{SW} (Y, g_{1}, ν_{1}) = λ_{SW} (Y, g_{0}, ν_{0}) − SF_{ℂ}(∂_{ν_{t}}^{g_{t}}),  (127)

where SF_{ℂ}(∂_{ν_{t}}^{g_{t}}) = \frac{1}{2}SF (∂_{ν_{t}}^{g_{t}}) is the complex spectral flow of the Dirac operator along the path of reducible solutions [ν_{t}, 0].

\underline{Proof:} If the spectral flow along the path (g_{t}, ν_{t}) is given by SF (∂_{ν_{t}}^{g_{t}}), the topological invariance of the equivariant Floer homology gives

SW H_{k,U(1)}(Y, g_{0}, ν_{0}) ≅ SW H_{k+SF (∂_{ν_{t}}^{g_{t}}),U(1)}(Y, g_{1}, ν_{1}).

We can then follow the steps of the proof of Theorem 7.7 and compare the ranks of the Floer groups and the counting of the Euler characteristic. This can be done by induction on |SF (∂_{ν_{t}}^{g_{t}})|. The result is the formula (127). ⋄

The wall crossing formula in the case of J-invariant perturbations constructed by W.Chen [14] can also be derived with the same method. This gives rise to the following wall crossing result.

Corollary 7.10 Consider the invariant λ_{SW} (Y, g, f) where f is the J-invariant perturbation of [14] Prop.2.6. Given two metrics and perturbations (g_{0}, f_{0}) and (g_{1}, f_{1}) in two different chambers and a generic path (g_{t}, f_{t}) that crosses the wall once with Ker(∂_{f}^{g}) = ℍ. The relative Morse index with respect to the reducible solution decreases by 4 across the wall, namely, we have μ(θ_{0}) − μ(θ_{1}) = 4. Then the Casson-type invariant changes by

λ_{SW} (Y, g_{1}, f_{1}) = λ_{SW} (Y, g_{0}, f_{0}) − 2.

\underline{Proof:} Again the main issue is the change of grading of the equivariant Floer homology induced by the spectral flow of the Dirac operator along the path (g_{t}, f_{t}). This time, since we are using J–invariant perturbations, the Dirac operator ∂_{f}^{g} is quaternion linear, hence, for the chosen path the spectral flow satisfies SF (∂_{f_{t}}^{g_{t}}) = 4. This implies that there is a degree shift

SW H_{k,U(1)}(Y, g_{0}, f_{0}) ≅ SW H_{k+4,U(1)}(Y, g_{1}, f_{1}).

By applying the previous Proposition 7.9, we obtain the result. ⋄

7.3 Wall-crossing formula: the geometric picture

In this section we re-derive, in a more geometric way, the wall crossing formula for a homology three-sphere Y that we proved algebraically in the previous section. We analyze the local structure of the parameterized moduli space. A geometric proof of the wall-crossing formula has been also worked out by [35].

Let M^{∗}(g, ν) denote the irreducible part of the moduli space M for the metric and perturbation (g, ν). Given a family of metrics and perturbation (g_{t}, ν_{t}) with (t ∈ [−1, 1]), the moduli spaces M^{∗}(g_{−1}, ν_{−1}) and M^{∗}(g_{1}, ν_{1}) are cobordant as long as the path (g_{t}, ν_{t}) does not cross the wall, that is the codimension one subspace W in the space of metrics and perturbations

W = {(g, ν)|Ker(∂_{ν}^{g}) ≠ 0}.

Suppose the path (g_{t}, ν_{t}) crosses the wall W just once at t = 0. Generically, Ker(∂_{ν_{0}}^{g_{0}}) ≅ ℂ. We want to analyze the local structure of the parameterized moduli space

M̄ = {M(g_{t}, ν_{t}) × {t}|t ∈ [−1, 1]}

at the reducible point ϑ_{0} = (θ_{0}, 0), where θ_{0} = [ν_{0}, 0] is the class of the reducible solution of (15) with the metric and perturbation (g_{0}, ν_{0}). There is a family of reducibles ϑ_{t} in M̄. Let M̄^{∗} be the irreducible set in M̄, U be a sufficiently small neighborhood of θ_{0} in M̄, and U^{∗} be the irreducible part of U.

We construct a bundle over neighborhood of ϑ_{0} in A × [−1, 1], together with a section ς such that

U^{∗} = (ς^{−1}(0) − {(ϑ_{t}, t)})/G.

Lemma 7.11 1) The slice of the G/U(1)-action at a point (A_{0}, 0) is V_{(A_{0},0)} = Ker(d^{∗}) × Γ_{L^{2}}(S),

(2) The slice of the G/U(1)-action at a point (A, ψ) is

V_{(A,ψ)} = {(α, ϕ)|d^{∗}(α) − 2iIm⟨ϕ, ψ⟩is a constant function on Y.}

(3) For (A, ψ) close to (A_{0}, 0) there is an isomorphism

λ_{(A,ψ)} : V_{(A,ψ)} → V_{(A_{0},0)}

\underline{Proof:} Properties (1) and (2) follow by direct computation. For (3), choose (α, ϕ) in V_{(A,ψ)} and define λ_{(A,ψ)}(α, ϕ) to be

(α − 2dξ_{(α,ϕ)}, ξ_{(α,ϕ)}ψ + ϕ)

where ξ_{(α,ϕ)} is the unique solution of the following equations:

\{ \begin{matrix}2d^{∗}dξ_{(α,ϕ)} = d^{∗}α \\ _{∫Y} ξ_{(α,ϕ)}dv = 0 \end{matrix}

Direct computation shows that λ_{(A,ψ)} is an isomorphism. ⋄

The above Lemma shows that we obtain a locally trivial vector bundle V over the space of connections and sections A endowed with a U(1)-action.

Define the section ς

ς : A × [−1, 1] → V

to be

ς(A, ψ, t) = λ_{(A,ψ)}(∗_{g_{t}}(F_{A} − dν_{t}) − σ(ψ, ψ), ∂_{A}ψ).

Near θ_{0}, we know that U = ς^{−1}(0)/G. Therefore, the local structure of U^{∗} at θ_{0} is given by the Kuranishi model of ς^{−1}(0)/G at θ_{0}.

Suppose (A_{t}, ψ_{t}) is an element in U^{∗}. Consider a formal expansion at ϑ_{0} of the form

A_{t} = ν_{t} + tα_{1} + t^{2}α_{2} + ⋯,
ψ_{t} = tψ_{1} + t^{2}ψ_{2} + · · · .

The section ς is approximated by

∗_{g_{0}}d(ν_{t} + tα_{1} + t^{2}α_{2} + ⋯) − ∗_{g_{0}}dν_{t} − σ(tψ_{1} + t^{2}ψ_{2} + ⋯, tψ_{1} + t^{2}ψ_{2} + ⋯),

∂_{ν_{t}}^{g_{0}}(tψ_{1} + t^{2}ψ_{2} + ⋯) + (tα_{1} + t^{2}α_{2} + ⋯) · (tψ_{1} + t^{2}ψ_{2} + ⋯),

where we are perturbing in a neighborhood of the wall W just by changing the perturbation and fixing the metric g_{0}.

The zero set of the section therefore determines the conditions ∗dα_{1} = 0 and d^{∗}α_{1} = 0, which imply α_{1} = 0 on a homology sphere. Moreover, we have d^{∗}α_{2} = 0 and

∗dα_{2} = σ(ψ_{1}, ψ_{1}).

On the kernel of d^{∗} the operator ∗d is invertible, hence we have

α_{2} = (∗d)^{−1}σ(ψ_{1}, ψ_{1}).

The Kuranishi model near ϑ_{0} is given by a U(1)-equivariant map

S : ℝ × Ker(∂_{ν_{0}}^{g_{0}}) → CoKer(∂_{ν_{0}}^{g_{0}}),

where U(1) acts on Ker(∂_{ν_{0}}^{g_{0}}) ≅ CoKer(∂_{ν_{0}}^{g_{0}}) ≅ ℂ by the natural multiplication on ℂ.

There exists a sufficiently small δ > 0 such that, for t ∈ [−δ, δ], we have that ∂_{ν_{t}}^{g_{0}} has exactly one small eigenvalue λ(t) with eigenvector ϕ_{t} and with λ(0) = 0, that is

∂_{ν_{t}}^{g_{0}}ϕ_{t} = λ(t)ϕ_{t}.

This implies that, if λ′(0) > 0, then the spectral flow of ∂_{ν_{t}}^{g_{0}} for (t ∈ [−1, 1]) is 1 and, if λ′(0) < 0, the spectral flow of ∂_{ν_{t}}^{g_{0}} for (t ∈ [−1, 1]) is −1.

The map S is given by

S : ℝ × ℂ → ℂ,
S(t, wϕ) = Π_{Ker(∂_{ν_{0}}^{g_{0}} )}(∂_{A_{t}}wϕ).

Here we assume that ϕ is a spinor in Ker(∂_{ν_{0}}^{g_{0}}) with ∥ϕ∥ = 1, so that Ker(∂_{ν_{0}}^{g_{0}}) ≅ ℂϕ. Consider the expression

< ∂_{ν_{t}}^{g_{0}}ϕ, ϕ >= z(t).

Notice that we have z′(0) = λ′(0), in fact, we write formally λ(t) ∼ tλ′(0), ϕ_{t} ∼ ϕ + tϕ_{1} and the Dirac operator ∂_{ν_{t}}^{g_{0}} ∼ ∂_{ν_{0}}^{g_{0}} + tC, where ν_{t} ∼ ν_{0} + tν_{1} and C acts as Clifford multiplication by ν_{1}. We can write the first order term in the relation ∂_{ν_{t}}^{g_{0}}ϕ_{t} = λ(t)ϕ_{t} as

t < ∂_{ν_{0}}^{g_{0}}ϕ_{1}, ϕ > +t < Cϕ, ϕ >= tλ′(0) + λ(0).

Here the term < ∂_{ν_{0}}^{g_{0}}ϕ_{1}, ϕ >=< ϕ_{1}, ∂_{ν_{0}}^{g_{0}}ϕ > vanishes, and also λ(0) = 0. Thus, we have the relation

< Cϕ, ϕ >= λ′(0).

On the other hand, we have

< Cϕ, ϕ >= z′(0)

from the expansion of ∂_{ν_{t}}^{g_{0}}ϕ = z(t)ϕ.

Thus the map S can be rewritten as

S(t, wϕ) = z(t)wϕ + t^{2} < α_{2}ϕ, ϕ > wϕ + O(t^{3})
= wϕ (z(t) + t^{2}r^{2} < (∗d)^{−1}σ(ϕ, ϕ), σ(ϕ, ϕ) >) + O(t^{3}).

Here we use the fact that the first order term of the Dirac equation gives ∂_{ν_{0}}^{g_{0}}ψ_{1} = 0, therefore ψ_{1} = re^{iθ}ϕ and σ(ψ_{1}, ψ_{1}) = r^{2}σ(ϕ, ϕ).

The term

γ(Y, g_{0}, ν_{0}) =< (∗d)^{−1}σ(ϕ, ϕ), σ(ϕ, ϕ) >

is a constant that only depends on the manifold and on θ_{0}. An inductive argument shows that, if γ(Y, g_{0}, ν_{0}) vanishes, then all the forms α_{i} in the formal expansion of A_{t} must also vanish identically. Thus, we can assume that γ(Y, g_{0}, ν_{0}) ≠ 0.

Notice that we have

ℝ × (Ker(∂_{ν_{0}}^{g_{0}}) − {0})/U(1) = ℝ × ℝ^{+}.

The irreducible part of ς^{−1}(0)/G is tangent to {0} × ℝ^{+} as t approaches 0, as we see in the following.

The difference λ between the Casson-type invariant at t = ±δ can be evaluated by counting the number (with sign) of oriented lines in ς^{−1}(0)/G, with t ∈ [−δ, δ], that are tangent to {0} × ℝ^{+} × {0}. Here we identify U^{∗} with the set (S^{−1}(0) − {w = 0})/U(1). The sign of the wall crossing term is determined by the section S, as follows. The zero set (S^{−1}(0) − {w = 0})/U(1) is given by the condition

t = −\frac{λ′(0)}{r^{2}γ(Y, g_{0}, ν_{0})}.

Thus, we have one line in U^{∗} which is counted with the orientation determined by the sign of −γ(Y, g_{0}, ν_{0}) and the spectral flow. Suppose that we have λ′(0) > 0, then the spectral flow is SF_{ℂ}(∂_{ν_{t}}^{g_{t}}) = 1 on the path t ∈ [−1, 1]. If we have γ(Y, g_{0}, ν_{0}) > 0, then there is a unique irreducible solution, which contributes a +1 to the invariant, that flows into the reducible as t → 0, with t < 0. If we have γ(Y, g_{0}, ν_{0}) < 0, then a unique irreducible that contributes a −1 to the invariant approaches the reducible as t → 0, t > 0. This gives the wall crossing term

λ(Y, g_{1}) = λ(Y, g_{−1}) − SF_{ℂ}(∂_{ν_{t}}^{g_{t}}).

This provides a geometric interpretation of the wall-crossing formula that we derived algebraically in the previous section from the exact sequences.

Acknowledgements. We are deeply grateful to T. Mrowka for the many invaluable comments and suggestions. We thank A. Carey and M. Rothenberg for the many useful conversations. We are grateful to L. Nicolaescu for having suggested the argument of Proposition 2.13 and to R.G. Wang for having suggested the use of condition (2). In the early stages of this work useful were comments of R. Brussee, G. Matić, R. Mazzeo, and J.W. Morgan. We thank the referee for the very detailed and useful comments and for many valuable suggestions on how to improve various sections of the manuscript. Parts of this work were carried out during visits of the two authors to the Max Planck Institut für Mathematik in Bonn, and visits of the first author to the University of Adelaide. We thank these institutions for the kind hospitality and for support.

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Matilde Marcolli Department of Mathematics 2-275, MIT, Cambridge MA 02139, USA matilde@math.mit.edu

Bai-Ling Wang Department of Pure Mathematics, The University of Adelaide, Adelaide SA 5005, Australia bwang@maths.adelaide.edu.au