PU(2) MONOPOLES AND LINKS OF TOP-LEVEL SEIBERG-WITTEN MODULI SPACES

PAUL M. N. FEEHAN AND THOMAS G. LENESS

Abstract. This is the first of two articles in which we give a proof—for a broad class of four-manifolds—of Witten’s conjecture that the Donaldson and Seiberg-Witten series coincide, at least through terms of degree less than or equal to c−2, where c = − \frac{1}{4} (7χ+11σ) and χ and σ are the Euler characteristic and signature of the four-manifold. In the present article, we construct virtual normal bundles for the Seiberg-Witten strata of the moduli space of PU(2) monopoles and compute their Chern classes.

1. Introduction

1.1. Main results. The purpose of the present article and its sequel [18], is to prove that Witten’s conjecture [76] relating the Donaldson and Seiberg-Witten invariants holds in “low degrees” for a broad class of four-manifolds, using the PU(2)-monopole cobordism [64]. We shall assume throughout that X is a closed, connected, smooth four-manifold with an orientation for which b_{2}^{+}(X) > 0. We state the simplest version of our result here; more general results are given in [18, §1]. The Seiberg-Witten (SW) invariants (see [18, §4.1]) comprise a function, SW_{X} : Spin^{c}(X) → ℤ, where Spin^{c}(X) is the set of isomorphism classes of spin^{c} structures on X. For w ∈ H^{2}(X; ℤ), one defines

SW_{X}^{w}(h) = ∑_{𝔰∈Spin^{c}(X)}(−1)^{\frac{1}{2} (w^{2}+c_{1}(𝔰)·w)}SW_{X}(𝔰)e^{⟨c_{1}(𝔰),h⟩}, h ∈ H_{2}(X; ℝ), (1.1)

by analogy with the structure of the Donaldson series D_{X}^{w}(h) [44, Theorem 1.7]. There is a map c_{1} : Spin^{c}(X) → H^{2}(X; ℤ) and the image of the support of SW_{X} is the set B of SW-basic classes [76]. A four-manifold X has SW-simple type when b_{1}(X) = 0 if c_{1}(𝔰)^{2} = 2χ + 3σ for all c_{1}(𝔰) ∈ B, where χ and σ are the Euler characteristic and signature of X. We let B^{⊥} ⊂ H^{2}(X; ℤ) denote the orthogonal complement of B with respect to the intersection form Q_{X} on H^{2}(X; ℤ). Let c(X) = −\frac{1}{4}(7χ + 11σ).

Theorem 1.1. Let X be four-manifold with b_{1}(X) = 0 and odd b_{2}^{+}(X) ≥ 3. Assume X is abundant, SW-simple type, and effective. For any Λ ∈ B^{⊥} and w ∈ H^{2}(X; ℤ) for which Λ^{2} = 2 − (χ + σ) and w − Λ ≡ w_{2}(X) (mod 2), and any h ∈ H_{2}(X; ℝ), one has

D_{X}^{w}(h) ≡ 0 ≡ SW_{X}^{w}(h) (mod h^{c(X)−2}),
D_{X}^{w}(h) ≡ 2^{2−c(X)}e^{\frac{1}{2} Q_{X}(h,h)}SW_{X}^{w}(h) (mod h^{c(X)}).

(1.2)

The vanishing assertion in low degrees for the series D_{X}^{w}(h) and SW_{X}^{w}(h) in equation (1.2) was proved in [16].

We shall explain below the terminology and notation in the statement of Theorem 1.1. Witten’s conjecture [76] asserts that a four-manifold X with b_{1}(X) = 0 and odd b_{2}^{+}(X) ≥ 3 has SW-simple type if and only if it has “KM-simple type”, that is, simple type in the sense of Kronheimer and Mrowka (see Definition 1.4 in [44]), and that the SW-basic and “KM-basic” classes (see Theorem 1.7 in [44]) coincide; if X has simple type, then

Date: July 31, 2000, math.DG/yymmnnn. First version: December 8, 1997, dg-ga/9712005 (v1), §1-3.

The first author was supported in part by an NSF Mathematical Sciences Postdoctoral Fellowship under grant DMS 9306061 and by NSF grant DMS 9704174.

D_{X}^{w}(h) = 2^{2−c(X)}e^{\frac{1}{2} Q_{X}(h,h)}SW_{X}^{w}(h), h ∈ H_{2}(X; ℝ). (1.3)

The quantum field theory argument giving equation (1.3) when b_{2}^{+}(X) ≥ 3 has been extended by Moore and Witten [55] to allow b_{2}^{+}(X) ≥ 1, b_{1}(X) ≥ 0, and four-manifolds X of non-simple type. Recall that b_{2}^{+}(X) is the dimension of a maximal positive-definite linear subspace H^{2,+}(X; ℝ) for the intersection pairing Q_{X} on H^{2}(X; ℝ).

With our stated hypotheses, Theorem 1.1 therefore proves that equation (1.3) holds mod h^{c(X)} where δ = c(X). To prove the equivalence mod h^{δ} for δ > c(X), more work is required. For example, in [21], we use the gluing theory of [19], [20]—allowing “one bubble”— to prove that equation (1.3) holds mod h^{c(X)+2} under hypotheses similar to those of Theorem 1.1. If one desires a mod h^{δ} relation such as (1.2) for larger values of δ relative to c(X), one must allow “more bubbles” and the difficulty of the calculations rapidly increases: see [18, §1] for a more detailed discussion.

The concept of “abundance” was first introduced in [16]:

Definition 1.2. [16, p. 169] We say that a closed, oriented four-manifold X is abundant if the restriction of the intersection form to B^{⊥} contains a hyperbolic sublattice.

The abundance condition is merely a convenient way of formulating the weaker, but more technical condition that one can find (for example) classes Λ_{j} ∈ B^{⊥} such that Λ_{j}^{2} = 2j − (χ + σ), for j = 1, 2, 3: this is the only property of Q_{X}|B^{⊥} which we use to prove Theorem 1.1, while slightly different choices of classes Λ with even squares are used to prove the main results (Theorems 1.1 and 1.3) of [16]. While all compact, complex algebraic, simply-connected surfaces with b_{2}^{+} ≥ 3 are abundant (see Theorem A.1 and its proof in Appendix A), some examples of manifolds which are not abundant but for which one can still find classes in B^{⊥} with the desired even squares are described in [16, §2]. It remains an interesting problem to determine whether all smooth four-manifolds have this property, whether or not they are abundant.

To see that such classes Λ_{j} exist when X is abundant, note that because Q_{X}|B^{⊥} contains a hyperbolic factor there are classes e_{0}, e_{1} ∈ H^{2}(X; ℤ) which are orthogonal to the SWbasic classes and which satisfy e_{0}^{2} = 0 = e_{1}^{2} and e_{0} · e_{1} = 1. Now set t = \frac{1}{2}(χ + σ) and define Λ_{j} ∈ H^{2}(X; ℤ) by Λ = e_{0} + (j − t)e_{1}, and observe that Λ_{j}^{2} has the desired values for j = 1, 2, 3.

In the present article and its companion [18] we prove Theorem 1.1 using the moduli space M_{𝔱} of PU(2) monopoles [64] to provide a cobordism between the link of the moduli space M_{κ}^{w} of anti-self-dual connections and links of moduli spaces of Seiberg-Witten moduli spaces, M_{𝔰}, these moduli spaces being (topologically) embedded in M_{𝔱}. Let M̄_{𝔱} denote the Uhlenbeck compactification (see Theorem 2.12) of M_{𝔱} in the space of ideal PU(2) monopoles, ∪_{ℓ=0}^{∞}(M_{𝔱_{ℓ}} × Sym^{ℓ}(X)).

Definition 1.3. We say that a closed, oriented, smooth four-manifold X with b_{1}(X) ≥ 0 and b_{2}^{+}(X) ≥ 1 is effective if X satisfies Conjecture 3.1 in [16]. This conjecture asserts that for a Seiberg-Witten moduli space M_{𝔰} appearing in level ℓ ≥ 0 of M̄_{𝔱}, the pairings of products of Donaldson-type cohomology classes on the top stratum of M_{𝔱}/S^{1} with a link of (M_{𝔰} × Sym^{ℓ}(X)) ∩ M̄_{𝔱} in M̄_{𝔱} are multiples of the Seiberg-Witten invariants for M_{𝔰}. In particular, these pairings are zero when the Seiberg-Witten invariants for M_{𝔰} are trivial.

The motivation for Conjecture 3.1 of [16] and a more detailed explanation appears in [16, §3.1]; see also [23]. It is almost certainly true that this conjecture holds for all fourmanifolds, based on our work in [19], [20], [22]. We verify Conjecture 3.1 in [16] by direct calculation in the present pair of articles for Donaldson invariants defined by M_{κ}^{w} ↪ M_{𝔱} and Seiberg-Witten moduli spaces M_{𝔰} ↪ M_{𝔱}, while in [21] we verify the conjecture for Seiberg-Witten moduli spaces M_{𝔰} contained in the “first level” of the Uhlenbeck compactification M̄_{𝔱}. However, we strongly expect the conjecture to hold for Seiberg-Witten moduli spaces M_{𝔰} contained in any level of the compactification M̄_{𝔱}.

Theorem 1.1 gives a relation, mod h^{δ}, between the Donaldson and Seiberg-Witten series when δ = c(X). Now if X̃ = X#\overline{ℂℙ}^{2} is the blow-up of X, then c(X̃) = c(X) + 1. However, this does not imply that one can compute a term of higher degree in h in the Donaldson series D_{X}^{w}(h) from Theorem 1.1 merely by blowing up. To understand why, it suffices to examine the case where X also has KM-simple type. Let PD[e] ∈ H^{2}(X̃; ℤ) denote the Poincaré dual of the exceptional class e ∈ H_{2}(X̃; ℤ). Then Proposition 1.9 in [44] implies that for all h ∈ H_{2}(X̃; ℤ) ≅ ℝ[e] ⊕ H_{2}(X; ℤ),

Dw+PD[e] ; X̃(h) = −D_{X}^{w}(h) exp(−\frac{1}{2}e · e) sinh(e · h).

Thus, Dw+PD[e] ; X̃ has a zero at the origin of order one greater than that of D_{X}^{w}.

On the other hand, If Λ ∈ B^{⊥}(X), then we also have Λ ∈ B^{⊥}(X̃) and w + PD[e] − Λ ≡ w_{2}(X) + PD[e] ≡ w_{2}(X̃) (mod 2), while χ(X̃) + σ(X̃) = χ(X) + σ(X), so Λ^{2} still has the desired even square in the case of X̃. If X is SW-simple type and abundant then the same holds for X̃ (and as mentioned, it is almost certainly true that all four-manifolds are effective). Theorem 1.1 then yields, for all h ∈ H_{2}(X̃; ℤ),

Dw+PD[e] ; X̃(h) ≡ 0 ≡ SWw+PD[e] ; X̃(h) (mod h^{c(X)−1}),
Dw+PD[e] ; X̃(h) ≡ 2^{2−c(X)−1} exp (\frac{1}{2}QX̃(h, h)) SWw+PD[e] ; X̃(h) (mod h^{c(X)+1}).

One sees that the formula does not give new higher-degree terms in the Donaldson series.

As we mentioned just prior to its statement, Theorem 1.1 is the simplest formulation of the main results of the present article and its companion [18]. More general results are described in [18, §1], where we allow four-manifolds with b_{2}^{+}(X) ≥ 1 and drop the assumptions that X has SW-simple type or is abundant and relax the constraints on the existence of classes Λ ∈ B^{⊥} with prescribed squares; we also consider some limited cases where b_{1}(X) > 0. A full account of the case b_{1}(X) > 0 is of potential interest but is beyond the scope of the present pair of articles: we shall describe this case elsewhere.

1.2. An outline of the proof of Theorem 1.1. The proof of Theorem 1.1 splits into two steps. Step (i), which we carry out in this article, is to construct links of Seiberg-Witten moduli spaces in the top level of the Uhlenbeck-compactified moduli space of PU(2) monopoles, as boundaries of tubular neighborhoods in certain “thickened” or “virtual” moduli spaces of PU(2) monopoles (see Theorems 3.19 and 3.21) and then compute the Chern character and Chern classes of the vector bundles defining these tubular neighborhoods (see Theorem 3.29 and Corollary 3.30).

Step (ii), which we take up in the companion article [18], is to compute the pairings of products of cohomology classes on the moduli space of PU(2) monopoles with the links of the anti-self-dual moduli space of SO(3) connections and with links of the moduli spaces of Seiberg-Witten monopoles. These computations rely on our calculation of the Chern characters of the normal bundles of the strata of Seiberg-Witten monopoles, and a comparison of the orientations of the moduli spaces of anti-self-dual connections and Seiberg-Witten monopoles, and their links in the moduli space of PU(2) monopoles. Applying the PU(2)- monopole cobordism then yields an expression for the Donaldson invariants in terms of Seiberg-Witten invariants and hence completes the proof of Theorem 1.1.

1.3. A guide to the article. An index of notation appears just before §2. The present article is a revision of sections 1–3 of the preprint [17], while the companion article [18] is a revision of sections 4–7 of [17], which was distributed in December 1997.

Section 2 of this paper gathers together the principal gauge theory results developed in [15], [24] that we shall need here for the moduli space of PU(2) monopoles M_{𝔱}. In §2.1, we review the construction of the moduli spaces of PU(2) monopoles M_{𝔱} from [24], but now phrased in the convenient and more compact framework of “spin^{u} structures” 𝔱. In §2.2 we recall our Uhlenbeck compactness and transversality results for the moduli spaces of PU(2) monopoles from [24] and [15]. In §2.3, we recall the construction of the moduli space of Seiberg-Witten monopoles M_{𝔰} as in [43], [45], [56], but with non-standard perturbations so we can directly identify these moduli spaces with strata of reducible PU(2) monopoles. Our transversality result [15] ensures that the natural stratification of the Uhlenbeck-compactified moduli space of PU(2) monopoles is smooth. In §2.4 we compute the cohomology ring of the configuration space of spin^{c} pairs and describe the cohomology classes arising in the definition of Seiberg-Witten invariants and in our later calculation (see §3.6) of the Chern character of certain universal families of vector bundles over X parameterized by M_{𝔰}.

The construction of the links in M_{𝔱} of the stratum M_{κ}^{w} of anti-self-dual connections and of the strata M_{𝔰} of reducible (or Seiberg-Witten) PU(2) monopoles occupies §3. In §3.1 we classify the possible singularities of M_{𝔱}. A link L_{𝔱,κ}^{w} of the stratum M_{κ}^{w} of anti-self-dual connections in M_{𝔱} is constructed in §3.2 using the L^{2} norm of the spinor components of PU(2) monopoles to define the distance to the stratum M_{κ}^{w}. In §3.3 we show that the subspaces of reducible PU(2) monopoles in M_{𝔱} can be identified with Seiberg-Witten moduli spaces M_{𝔰}, as defined in §2.3. As we explain in §3.4, the elliptic deformation complex for the PU(2) monopole equations (2.32) at a reducible pair splits into a tangential deformation complex — which can be identified with the elliptic deformation complex for the Seiberg-Witten monopole equations — and a normal deformation complex. Via the Kuranishi model, these deformation complexes describe the local structure of the moduli space M_{𝔱} of PU(2) monopoles near a reducible solution. The description of a neighborhood of M_{𝔰} in M_{𝔱} is complicated by the fact that we may have both a tangential deformation complex with positive index, so dim M_{𝔰} > 0, and a normal deformation complex with negative index. (This contrasts with the simpler situation considered in [12], [29, pp. 65–69], where abstract perturbations are used in conjunction with the local Kuranishi model to describe the local structure of the moduli space M_{κ}^{w} near an isolated reducible connection when the four-manifold has b_{2}^{+}(X) = 0: in this case the normal deformation complex has positive index.) The stratum M_{𝔰} will not in general be a smooth submanifold of M_{𝔱} as reducible PU(2) monopoles cannot be shown to be regular points of the zero locus of the PU(2) monopole equations (2.32). Therefore, in §3.5, we construct an ambient finite-dimensional, smooth manifold M_{𝔱}(Ξ, 𝔰) containing an open neighborhood in M_{𝔱} of the stratum M_{𝔰} and containing M_{𝔰} as a smooth submanifold. We can then define a link L_{𝔱,𝔰} of M_{𝔰} in M_{𝔱}/S^{1} as the S^{1} quotient of the intersection with M_{𝔱}/S^{1} of the boundary of a tubular neighborhood of M_{𝔰} in the ambient manifold M_{𝔱}(Ξ, 𝔰). In order to compute the pairings of cohomology classes on M_{𝔱}^{∗,0}/S^{1} (the smooth locus or top stratum of M_{𝔱}/S^{1}) with the link L_{𝔱,𝔰} we shall need the Chern classes of the normal bundle N_{𝔱}(Ξ, 𝔰) of the stratum M_{𝔰} ↪ M_{𝔱}(Ξ, 𝔰): we accomplish this in §3.6, using the Atiyah-Singer index theorem for families, by computing the Chern character of this normal bundle (Theorem 3.29) and then, after imposing a constraint on H^{1}(X; ℤ) to simplify our calculations, its Chern classes (Corollary 3.30).

In Appendix A, we include a proof that all compact, complex algebraic, simply connected surfaces with b_{2}^{+} ≥ 3 are abundant (Theorem A.1). For minimal surfaces of general type, this fact was asserted in [16, p. 175].

Acknowledgments. The authors thank Adebisi Agboola, Ron Fintushel, Tom Mrowka, Peter Ozsváth, András Stipsicz, and Zoltán Szabó for helpful conversations during the course of our work on this article and its companion [18]. We especially thank Tom Mrowka for his many helpful comments during the course of this work, for bringing a correction to some examples in [17] to our attention, as well pointing out that our Theorem 1.4 in [17] could be more usefully rephrased and specialized to give the version stated as Theorem 1.4 in [18] (and Theorem 2.1 in [16]). We are also very grateful to András Stipsicz for his considerable help with the proof of Theorem A.1 (the proofs of the difficult cases are all due to him), as well as to Adebisi Agboola for his assistance with the number theoretical aspects of that proof. We thank the Columbia University Mathematics Department, the Institute for Advanced Study, Princeton, and the Max Planck Institute für Mathematik, Bonn for their generous support and hospitality during a series of visits while this article and its companion [18] were being prepared. Finally, we thank the anonymous referee and Simon Donaldson for their editorial suggestions and comments on the previous versions of this article.

Index of Notation

(A, Φ), [A, Φ] §2.1.5 [A, Φ, x] §2.2 A^{ad}, Â §2.1.4 A^{det} Equation (2.24) (B, Ψ), [B, Ψ] Equation (2.52) A_{L} Lemma 2.9 A_{Λ} Equation (2.24) F_{A}, F_{A}^{+} Equation (2.32) F_{B}, F_{B}^{+}, Tr(F_{B}) Equations (2.6) & (2.55) A_{𝔰} §2.1.2 A_{𝔱} Lemma 2.11 A_{κ}^{w}, B_{κ}^{w} §2.1.6 B_{𝔱} §3.2 C_{𝔰}, C_{𝔰}^{0}, C̃_{𝔰} Equations (2.54) & (2.52) Ĉ_{𝔰}, Ĉ_{𝔰}^{0} Equation (2.68) C_{𝔰}^{p,0} Lemma 2.19 C_{𝔱}, C̃_{𝔱} Equations (2.34) & (2.31) C_{𝔱}^{0}, C_{𝔱}^{∗}, C_{𝔱}^{∗,0} Equation (2.39) Cℓ(T^{∗}X), ℂℓ(T^{∗}X) §2.1.1 D_{A}, D_{B} §2.1.2 D_{A,Φ} Equation (2.48) D_{A,Φ}^{n}, D_{A,Φ}^{t} Equation (3.35) D_{B,Ψ} Equation (2.62) D §3.5 D^{n} Equation (3.42) D′, D′′ Equations (3.62) & (3.58) E, F Equation (3.54) E′, F′, E′′, F′′ Equation (3.57) E_{ℓ} Equation (2.43) F_{j}, F_{j}^{t}, F_{j}^{n} Equations (3.19) & (3.21) G_{𝔰} §2.3 Ĝ_{𝔰} Equation (2.66) G_{𝔰}^{p} Equation (2.72) G_{𝔱} Definition 2.6 G_{κ}^{w} Lemma 2.7 H_{A,Φ}^{i}, H_{B,Ψ}^{i} Lemma 3.15 Jac(X) Equation (2.74) L_{𝔱,κ}^{w} Definition 3.7 L_{𝔱,𝔰} Equation (3.47) 𝕃_{𝔰} Equation (2.69) M_{𝔰}, M_{𝔰}^{0} Equation (2.57) M_{𝔱} Equation (2.33) M_{𝔱}^{∗}, M_{𝔱}^{0}, M_{𝔱}^{∗,0} Equation (2.40) M̄_{𝔱} Equation (2.46) M_{κ}^{w}, M_{κ}^{w,∗} Equation (2.41) M_{𝔱}(Ξ, 𝔰) Definition 3.20 N_{𝔱}(Ξ, 𝔰) Equation (3.42) N_{𝔱}^{ε}(Ξ, 𝔰), ℙN_{𝔱}(Ξ, 𝔰) Definition 3.22 Ñ_{𝔱}(Ξ, 𝔰) §3.5.4 𝔖 Equations (2.32) & (2.55) Spin^{c}(X) §1.1 V, V^{±} §2.1.1 𝔙, 𝔙^{t}, 𝔙^{n} Equations (3.38) & (3.40) W, W^{±} Definition 2.1 c(X) §1.1 c_{1}(𝔰) Equation (2.3) c_{1}(𝔱), p_{1}(𝔱), w_{2}(𝔱) Equation (2.19) d_{ℝ}, d_{ℂ} Lemma 2.9 d_{a}(𝔱) Equation (2.51) d_{s}(𝔰) Equation (2.63) di ; Â Equation (3.6) d_{A,Φ}^{i} Equations (2.49) & (2.50) d_{A,Φ}^{i,t} Equations (3.26) & (3.29) d_{A,Φ}^{i,n} Equations (3.27) & (3.30) d_{B,Ψ}^{i} Equations (2.58) & (2.60)

det^{\frac{1}{2}} (V^{+}) Equation (2.9) 𝔤_{𝔱} Equation (2.18) h_{p} §2.4.1 n_{a}(𝔱) Equation (2.51) n′_{s}(𝔱, 𝔰), n′′_{s}(𝔱, 𝔰) Equation (3.72) r Lemma 2.20 r_{Ξ} Theorem 3.29 𝔰 Definition 2.1 𝔰 ⊗ L Equation (2.4) 𝔱 Definition 2.2 𝔱_{ℓ} Equation (2.44) w §2.1.6 Δ Equation (2.76) Ξ, Π_{Ξ} Lemma 3.19 γ, γ̃ Equations (3.44) & (3.51) γ_{i}, γ_{i}^{∗}, γ_{i}^{J,∗} Definition 2.21 δ_{L} Equation (3.58) η Equation (2.55) ϑ Equation (2.32) ι Equations (3.4) & (3.9) μ_{𝔰} Equation (2.71) ρ Equation (2.1) ϱ Equation (3.10) ϱ_{L} Proof of Lemma 3.11 σ, χ §1.1 τ Equation (2.32) φ Equation (3.45)

2. Preliminaries

In this section we recall the framework for gauge theory for PU(2) monopoles established in [15], [24], [23], though we introduce some notational and other simplifications here. In §2.1 we describe the PU(2) monopole equations while in §2.2 we recall our Uhlenbeck compactness and transversality results from [15], [24]. We review the usual construction of the moduli space of Seiberg-Witten monopoles in §2.3, although we use the perturbation parameter τ ∈ Ω^{0}(GL(Λ^{+})) to achieve transversality for the moduli space of Seiberg-Witten monopoles rather than employ the customary perturbation parameter η ∈ Ω^{+}(X, iℝ).

One slightly awkward issue here then concerns the possible presence of zero-section Seiberg-Witten monopoles. We recall from [56, §6.3] that these may be avoided when b_{2}^{+}(X) > 0 by choosing a generic perturbation η ∈ Ω^{+}(X, iℝ). For trivial reasons, variation of the parameter τ has no effect on the presence or absence of zero-section solutions and so, even for a generic Riemannian metric on X, they cannot be avoided in all cases. Instead, we circumvent this problem by a employing a restriction on the second Stiefel-Whitney class of an SO(3) bundle comprising part of the definition of the “spin^{u} structure” introduced below, following [25], which precludes the existence of flat, reducible connections. With the aid of the blow-up trick of [57], there is no resulting loss of generality when computing Donaldson or Seiberg-Witten invariants [27], [26].

Finally, in §2.4 we compute the cohomology ring of the configuration space of Seiberg-Witten pairs and express the Seiberg-Witten μ-classes in terms of these generators.

Although we use the generic metrics and Clifford maps of [15] to achieve transversality for the moduli space of PU(2) monopoles, we note that all of the results in this article can be obtained using the holonomy perturbations of [24].

2.1. PU(2) monopoles. Throughout this article, X denotes a closed, connected, oriented, smooth four-manifold. We begin our discussion with a brief review of the definition of spin^{c} structures and introduce the concept of a “spin^{u} structure”, which will then allow us to give a convenient definition of the PU(2)-monopole equations.

2.1.1. Hermitian Clifford modules, module derivations, and spin connections. Let V be a complex vector bundle over a Riemannian manifold (X, g). A real-linear map ρ : T^{∗}X → End(V) is a Clifford map, which is compatible with g, if it satisfies

ρ(α)ρ(α) = −g(α, α)id_{V} , α ∈ C^{∞}(T^{∗}X). (2.1)

Equation (2.1) and the universal property of Clifford algebras [49, Proposition I.1.1] imply that ρ extends to a real algebra homomorphism ρ : Cℓ(T^{∗}X) → End(V), where Cℓ(T^{∗}X) is the real Clifford algebra, and a complex algebra homomorphism, ρ : ℂℓ(T^{∗}X) → End(V), where ℂℓ(T^{∗}X) = Cℓ(T^{∗}X) ⊗_{ℝ} ℂ is the complex Clifford algebra. In particular, the pair (ρ, V) defines a complex Clifford module. Recall that Cℓ(T^{∗}X) is canonically isomorphic to Λ^{•}(T^{∗}X) as an orthogonal vector bundle (see [49, Proposition II.3.5] or [8, Proposition 3.5]). The fibers of V are irreducible modules for the fibers of ℂℓ(T^{∗}X), [49, Theorem I.5.8] if and only if V has complex rank 2^{n}, when X has dimension 2n. If X is even-dimensional, as we shall assume from now on, the bundle V admits a splitting V = V^{+} ⊕ V^{−}, where the subbundles V^{±} are the eigenspaces of ρ(vol) and are irreducible Cℓ^{0}(T^{∗}X) modules [49, p. 98–99], where Cℓ^{0}(T^{∗}X) ≅ Λ^{even}(T^{∗}X). If V is a Hermitian vector bundle, then we require that ρ(α) ∈ End(V) be skew-Hermitian for all α ∈ C^{∞}(T^{∗}X) and call (ρ, V) a Hermitian Clifford module.

A unitary connection A on V, where (ρ, V) is a Hermitian Clifford module, defines a Clifford module derivation ∇_{A} on C^{∞}(V) if

∇_{η}^{A}(ρ(α)Φ) = ρ(∇_{η}α)Φ + ρ(α)∇_{η}^{A}Φ, (2.2)

where α ∈ C^{∞}(T^{∗}X), η ∈ C^{∞}(T X), and Φ ∈ C^{∞}(V) and ∇ is an orthogonal connection on T^{∗}X. (We follow the convention of [8, Definition 3.39].) If ∇_{A} is a module derivation, it preserves the splitting V = V^{+} ⊕ V^{−} [49, Corollary II.4.12]. We shall also let ∇ denote the canonically induced orthogonal connections on Λ^{•} = Λ^{•}(T^{∗}X) and the real Clifford algebra Cℓ(T^{∗}X) [49, Proposition II.4.8].

Conversely, if we require that ∇_{A} be a Clifford module derivation, then the relation (2.2) defines an orthogonal connection ∇ on T^{∗}X [68, §6.1]. The connection A is called spin if ∇ is the Levi-Civita connection on T^{∗}X.

We assume for the remainder of the article that (X, g) is an oriented, Riemannian fourmanifold. The subbundles V^{±} may then be characterized by requiring that ρ(ω)V^{−} = 0 for all ω ∈ Ω^{+}(X, ℝ) and similarly for V^{+}.

2.1.2. Spin^{c} structures. Here we specialize to the case where the complex Clifford module has rank four.

Definition 2.1. We call 𝔰 = (ρ, W) a spin^{c} structure over an oriented Riemannian fourmanifold (X, g) if (ρ, W) is a Hermitian Clifford module and W has complex rank four.

Given a spin^{c} structure 𝔰 = (ρ, W), one defines

c_{1}(𝔰) = c_{1}(W^{+}). (2.3)

If L is a complex line bundle over X, we obtain a new spin^{c} structure on (X, g),

𝔰 ⊗ L = (ρ, W ⊗ L), (2.4)

with c_{1}(𝔰 ⊗ L) = c_{1}(𝔰) + 2c_{1}(L). If 𝔰, 𝔰′ are any two spin^{c} structures on (X, g) then one has 𝔰′ = 𝔰 ⊗ L, for a complex line bundle L on X uniquely determined by 𝔰 and 𝔰′ [68].

There is a canonical isomorphism of orthogonal vector bundles,

𝔲(W) ≅ i\underline{ℝ} ⊕ 𝔰𝔲(W).

where we abbreviate the trivial real line subbundle iℝ id_{W} ⊂ 𝔲(W) by i\underline{ℝ} = X×iℝ. The Clifford multiplication ρ defines canonical isomorphisms Λ^{±} ≅ 𝔰𝔲(W^{±}), where Λ^{±} = Λ^{±}(T^{∗}X) are the bundles of self-dual and anti-self-dual two-forms, with respect to the Riemannian metric g on T^{∗}X. A unitary connection ∇_{A} on W determines a unitary connection on det(W); conversely, a choice of orthogonal connection on 𝔰𝔲(W) and a unitary connection on det(W) uniquely determine a unitary connection on W [24, §2.1.1]. In particular, any two unitary connections on W, which are both spinorial with respect to ∇, differ by an element of Ω^{1}(X, iℝ), since the induced connection on 𝔰𝔲(W) ≅ Λ^{2} is constrained by the choice of ∇ on T^{∗}X. The Dirac operator D = ρ ∘ ∇ on C^{∞}(W) is not self-adjoint unless the connection ∇ on T^{∗}X is torsion-free—that is, ∇ is the Levi-Civita connection—as one can see from examples.

For k ≥ 2, we let A_{𝔰} denote the space of L_{k}^{2} spin connections on W. From the preceding remarks (see [68, Lemma 6.1] for details), A_{𝔰} is an affine space for the Hilbert space L_{k}^{2}(X, iΛ^{1}) of imaginary one-forms: following the convention of [43, §2(i)], the action is given by

(B, b) ↦ B + b id_{W} . (2.5)

Thus, denoting the trace on two-by-two complex matrices by Tr, one has

Tr(F_{B+b}) = Tr(F_{B}) + 2db and D_{B+b} = D_{B} + ρ(b), (2.6)

as we later use when describing the deformation complex for the Seiberg-Witten equations.

2.1.3. Spin^{u} structures. An elegant reformulation of the PU(2) monopole equations, as discussed in [64] and [24], has been described by Mrowka [59], based on his joint work with Kronheimer and, motivated by his comments, we shall give a more invariant definition of PU(2) monopoles than the one we presented in [24].

Definition 2.2. We call 𝔱 = (ρ, V) a spin^{u} structure over an oriented Riemannian fourmanifold (X, g) if (ρ, V) is a Hermitian Clifford module and V has complex rank eight.

In the case of even-dimensional manifolds X with spin structures and complex modules (of arbitrary dimension) for the real Clifford algebra, Cℓ(T^{∗}X), the following result appears as Proposition 3.35 in [8].

Lemma 2.3. Let W and V be complex Clifford modules over a Riemannian four-manifold X, of rank four and eight respectively. Then there is a rank-two complex vector bundle E over X, unique up to isomorphism, and an isomorphism of complex Clifford modules,

V ≅ W ⊗ E.

If W and V are Hermitian Clifford modules, then the bundle E can be assumed Hermitian and the isomorphism V ≅ W ⊗ E can be taken to be an isomorphism of Hermitian Clifford modules.

Proof. Let Δ be an irreducible ℂℓ(ℝ^{4}) module, recalling that any such module is unique up to equivalence [49, Theorem I.5.7], with ℂℓ(ℝ^{4}) ≅ M_{4}(ℂ) as a complex algebra acting on Δ ≅ ℂ^{4} by the standard representation. (We use M_{d}(ℂ) to denote the algebra of complex d × d matrices.) Hence, the only ℂℓ(ℝ^{4})-module endomorphisms of Δ are given by complex scalar multiplication and we have an isomorphism of complex vector spaces, ℂ ≅ End_{ℂℓ(ℝ^{4})}(Δ), given by z ↦ z id_{Δ}. Therefore, we obtain an isomorphism of complex vector spaces

ℂ^{2} ≅ Hom_{ℂℓ(ℝ^{4})}(Δ, Δ ⊕ Δ), (2.7)

where (z_{1}, z_{2}) ∈ ℂ^{2} is identified with the ℂℓ(ℝ^{4})-module homomorphism Δ → Δ ⊕ Δ given by v ↦ (z_{1}v, z_{2}v). Indeed, the map (2.7) is surjective because we can compose any homomorphism Δ → Δ ⊕ Δ with projection onto each factor and then use the fact that End_{ℂℓ(ℝ^{4})}(Δ) ≅ ℂ. Moreover, the map (2.7) is injective by construction.

Given a complex Clifford module W of rank four, define

E = Hom_{ℂℓ(T^{∗}X)}(W, V). (2.8)

For every x ∈ X and isomorphism ℂℓ(T^{∗}X)|_{x} ≅ ℂℓ(ℝ^{4}) of complex Clifford algebras, there are isomorphisms of complex Clifford modules, W |_{x} ≅ Δ and V |_{x} ≅ Δ ⊕ Δ. The isomorphism (2.7) then implies that E is a rank-two complex vector bundle over X. The map W ⊗ E → V given by Φ ⊗ M → M(Φ), where Φ ∈ W |_{x} and M ∈ E|_{x} for some x ∈ X, is a ℂℓ(ℝ^{4})-module isomorphism since it is fiberwise injective, the ranks of the two bundles agree, and it is a complex Clifford module homomorphism by construction. To see that the map W ⊗ E → V is injective, observe that if M(Φ) = 0, then either Φ = 0 or Φ ≠ 0 and M has a non-trivial kernel, in which case one can see from the explicit form of the map (2.7) that M|_{x} must then be zero.

Next we show that E is unique up to isomorphism. From equation (2.9) below, we have

c_{1}(E) = \frac{1}{2}c_{1}(V^{+}) − c_{1}(W^{+}).

From equation (2.16) below we see that the bundle 𝔰𝔲(V) determines the subbundle 𝔰𝔲(E) up to isomorphism, independently of W, and hence 𝔰𝔲(V) uniquely determines p_{1}(𝔰𝔲(E)). Then

c_{2}(E) = −\frac{1}{4}(p_{1}(𝔰𝔲(E)) − c_{1}(E)^{2}),

and E is determined up to isomorphism by c_{1}(E) and c_{2}(E). Finally, if W and V are Hermitian then E is Hermitian and the isomorphism V ≅ W ⊗ E can be taken to be Hermitian.

The presentation of V as W ⊗ E is not unique. If W is replaced by W ⊗ L, where L is a complex line bundle over X, then E may be replaced by L^{−1} ⊗ E and so one also has V ≅ (W ⊗ L) ⊗ (L^{−1} ⊗ E). By Lemma 2.3, we may always write

(ρ, V) = (ρ, W ⊗ E),

for some spin^{c} structure 𝔰 = (ρ, W) and Hermitian bundle E. Although W and E are not determined by (ρ, V), the spin^{u} structure (ρ, V) does determine a complex line bundle, which is independent of tensor-product decomposition V ≅ W ⊗ E,

(2.9) det^{\frac{1}{2}} (V^{+}) = det(W^{+}) ⊗ det(E),

noting that det(V^{+}) ≅ det(V^{−}), so det(V) ≅ det(V^{+})^{⊗2} since V = V^{+} ⊕ V^{−}.

The spin^{u} structure (ρ, V) also defines an SO(3) bundle over X. To see this, recall that there are isometries of orthogonal vector bundles,

(2.10) 𝔲(E) ≅ i\underline{ℝ} ⊕ 𝔰𝔲(E), M ↦ (Tr M)id_{E} ⊕ (M − (Tr M)id_{E}) ,

where \underline{ℝ} = ℝ id_{E} here, and

𝔤𝔩(E) ≅ i𝔲(E) ⊕ 𝔲(E), M ↦ \frac{1}{2}(M + M^{†}) ⊕ \frac{1}{2}(M − M^{†}), (2.11)

with similar isomorphisms for W. Since 𝔤𝔩(V) ≅ 𝔤𝔩(W) ⊗ 𝔤𝔩(E), the decomposition (2.11) yields an isometry,

𝔲(V) ≅ i𝔲(W) ⊗_{ℝ} 𝔲(E). (2.12)

Combining the identifications (2.10) and (2.12), we obtain isometries of orthogonal vector bundles, 𝔲(V) ≅ i\underline{ℝ} ⊕ 𝔰𝔲(V) where \underline{ℝ} = ℝ id_{V} here, and an orthogonal decomposition,

𝔰𝔲(V) ≅ 𝔰𝔲(W) ⊕ i𝔰𝔲(W) ⊗ 𝔰𝔲(E) ⊕ 𝔰𝔲(E), (2.13)

where we have identified ℝ id_{W} ⊗ 𝔰𝔲(E) with 𝔰𝔲(E) and 𝔰𝔲(W) ⊗ ℝ id_{E} with 𝔰𝔲(W).

Recall that the Clifford map ρ : T^{∗}X → End(W) defines isometries of orthogonal vector bundles ρ : Λ^{±} → 𝔰𝔲(W^{±}) and ρ : Λ^{2} → 𝔰𝔲(W) [68, §4.8]. More generally, one has:

Lemma 2.4. Let (ρ, W) be a spin^{c} structure over an oriented, Riemannian four-manifold X. Then the map ρ : Λ^{•} ⊗_{ℝ} ℂ → End(W) defined by composing the isomorphism Λ^{•} ⊗_{ℝ} ℂ ≅ ℂℓ(T^{∗}X) and the Clifford representation ℂℓ(T^{∗}X) → End(W) yields an isometric isomorphism of orthogonal vector bundles,

Λ^{1} ⊕ Λ^{2} ⊕ i(Λ^{3} ⊕ Λ^{4}) ≅ 𝔰𝔲(W). (2.14)

Proof. The isomorphism Λ^{•} ≅ Cℓ(T^{∗}X) is given explicitly by [8, Proposition 3.5]

e^{i_{1}} ∧ ⋯ ∧ e^{i_{p}} ↦ ρ(e^{i_{1}}) ⋯ ρ(e^{i_{p}}),

if {e^{1}, …, e^{4}} is a local oriented, orthonormal frame for T^{∗}X. Using this isomorphism and the fact that ρ(α)^{†} = −ρ(α) when α ∈ Ω^{1}(X, ℝ), it is easy to see that the lefthand side of (2.14) is mapped into 𝔲(W), the skew-Hermitian endomorphisms of W. The elements of ρ(Λ^{1}) and ρ(iΛ^{3}) give isomorphisms in Hom(W^{±}, W^{∓}) and their block-matrix representations in End(W^{+} ⊕ W^{−}) have zero components in End(W^{±}), so they have zero trace. As remarked earlier, the elements of ρ(Λ^{2}) are known to have zero trace. Next, ρ(vol)|_{W^{±}} = ±id_{W} by definition of W^{±} and thus ρ(iΛ^{4}) has zero trace. Hence the left-hand side of (2.14) is also mapped into 𝔰𝔩(W), the traceless endomorphisms of W.

The map ℂℓ(T^{∗}X) → End(W) is an isomorphism [8, Proposition 3.19] of Clifford modules and in particular is injective. Since the sum of the ranks of the bundles on the left-hand side of (2.14) is 15, which is equal to the rank of 𝔰𝔲(W), we see that the map (2.14) must be an isomorphism, as claimed.

Note that ρ : Λ^{•} ⊗_{ℝ} ℂ assigns ρ(1) = id_{V}. For convenience, we define

Λ^{♯} = Λ^{1} ⊕ Λ^{2} ⊕ i(Λ^{3} ⊕ Λ^{4}). (2.15)

From the decomposition (2.13), we see that the Clifford map ρ : Λ^{♯} → 𝔰𝔲(V) embeds Λ^{♯} as a subbundle of 𝔰𝔲(V) and Λ^{±} ≅ 𝔰𝔲(W^{±}) as subbundles of 𝔰𝔲(V^{±}) via ω ↦ ρ(ω). In particular, the orthogonal decomposition (2.13) is equivalent to

𝔰𝔲(V) ≅ ρ(Λ^{♯}) ⊕ iρ(Λ^{♯}) ⊗ 𝔰𝔲(E) ⊕ 𝔰𝔲(E). (2.16)

and, upon restriction to 𝔰𝔲(V^{±}),

𝔰𝔲(V^{±}) ≅ ρ(Λ^{±}) ⊕ iρ(Λ^{±}) ⊗ 𝔰𝔲(E) ⊕ 𝔰𝔲(E). (2.17)

Plainly, the decompositions (2.16) and (2.17) are independent of the tensor-product decomposition V ≅ W ⊗ E, and so the spin^{u} structure 𝔱 determines an SO(3) bundle,

𝔤_{𝔱} = 𝔰𝔲(E). (2.18)

Indeed, the subbundle 𝔤_{𝔱} ⊂ 𝔰𝔲(V) can be characterized invariantly as span of the sections ξ ∈ C^{∞}(𝔰𝔲(V)) for which [ρ(ω), ξ] = 0 for all ω ∈ C^{∞}(Λ^{•}).

The spin^{u} structure 𝔱 thus defines the characteristic classes,

c_{1}(𝔱) = c_{1}(det^{\frac{1}{2}} (V^{+})), p_{1}(𝔱) = p_{1}(𝔤_{𝔱}), and w_{2}(𝔱) = w_{2}(𝔤_{𝔱}). (2.19)

These classes obey the constraints that w_{2}(𝔱) ∈ H^{2}(X; ℤ/2ℤ) has an integral lift and

\begin{aligned}c_{1}(𝔱) − w_{2}(𝔱) & ≡ w_{2}(X) (mod 2), \\ p_{1}(𝔱) & ≡ w_{2}(𝔱)^{2} (mod 4), \end{aligned} (2.20)

recalling that w_{2}(𝔰𝔲(E)) ≡ c_{1}(E) (mod 2) and c_{1}(W^{+}) ≡ w_{2}(X) (mod 2). Note that w_{2}(𝔱) is not necessarily equal to c_{1}(𝔱) (mod 2).

Conversely, given a triple (Λ, w, p), where Λ, w ∈ H^{2}(X; ℤ) obey Λ−w ≡ w_{2}(X) (mod 2) and p ∈ ℤ obeys p ≡ w^{2} (mod 4), there is a spin^{u} structure 𝔱 = (ρ, V) with c_{1}(𝔱) = Λ, w_{2}(𝔱) ≡ w (mod 2), and p(𝔱) = p. Indeed, it suffices to choose a spin^{c} structure (ρ, W) with c_{1}(W^{+}) = Λ − w and a U(2) bundle E with c_{1}(E) = w and c_{2}(E) = −\frac{1}{4}(p − w^{2}); one then sets V = W ⊗ E.

2.1.4. Spin^{u} structures and spin connections. Suppose we are given a spin^{u} structure (ρ, V). If A is a unitary connection on V, it induces an orthogonal connection on 𝔲(V) ≅ i\underline{ℝ}⊕𝔰𝔲(V) and 𝔰𝔲(V), denoted A^{ad}, by setting

∇_{η}^{A}(MΦ) = (∇_{η}^{A^{ad}}M)Φ + M∇_{η}^{A}Φ, (2.21)

where M ∈ C^{∞}(𝔰𝔲(V)), η ∈ C^{∞}(T X), and Φ ∈ C^{∞}(V); more succinctly, we have

∇_{η}^{A^{ad}}M = [∇_{η}^{A}, M]. (2.22)

Suppose A defines a Clifford module derivation ∇_{A}. The bundle 𝔰𝔲(V) is a real Clifford module and since ∇_{A} is a Clifford module derivation on C^{∞}(V), it induces a Clifford module derivation on C^{∞}(𝔰𝔲(V)), denoted ∇_{A^{ad}}, via equations (2.2) and (2.21), so that

∇_{η}^{A^{ad}}(ρ(α)M) = ρ(∇_{η}α)M + ρ(α)∇_{η}^{A^{ad}}M, (2.23)

where α ∈ C^{∞}(T^{∗}X), η ∈ C^{∞}(T X), and M ∈ C^{∞}(𝔰𝔲(V)). Hence, if A is a spin connection on V then A^{ad} is a spin connection on 𝔰𝔲(V), inducing the Levi-Civita connection ∇ on T^{∗}X and hence on Λ^{2}.

Lemma 2.5. Let (ρ, V) be a spin^{u} structure over a Riemannian four-manifold (X, g) and let A be a spin connection on V. Then the induced orthogonal connection A^{ad} on 𝔰𝔲(V) is spin and preserves the three subbundles in the orthogonal decomposition (2.16) for 𝔰𝔲(V), inducing the Levi-Civita connection ∇ on the subbundle ρ(Λ^{♯}), an SO(3) connection Â on the subbundle 𝔰𝔲(E), and the tensor-product connection ∇⊗Â on the subbundle iρ(Λ^{♯})⊗𝔰𝔲(E).

Proof. The connection A^{ad} is spin by the remarks preceding the statement of the lemma. From equations (2.23) and (2.22) one can see that

∇_{η}^{A^{ad}}(ρ(ω)id_{V}) = ρ(∇_{η}ω)id_{V} ,

for all η ∈ C^{∞}(T^{∗}X) and ω ∈ C^{∞}(Λ^{♯}), so A^{ad} preserves the subspace ρ(Λ^{♯}) ⊂ 𝔰𝔲(V), inducing the Levi-Civita connection ∇.

Recall that sections ξ of 𝔰𝔲(E) ⊂ 𝔰𝔲(V) can be characterized as sections of 𝔰𝔲(V) having zero commutator with all ρ(ω), for ω ∈ C^{∞}(Λ^{♯}). Thus, for any such ξ, ω, and η ∈ C^{∞}(T^{∗}X), we have [ξ, ρ(ω)] = 0 and

∇_{η}^{A^{ad}}[ξ, ρ(ω)] = [∇_{η}^{A^{ad}}ξ, ρ(ω)] + [ξ, ∇_{η}^{A^{ad}}ρ(ω)]
= [∇_{η}^{A^{ad}}ξ, ρ(ω)] + [ξ, ρ(∇_{η}ω)] = [∇_{η}^{A^{ad}}ξ, ρ(ω)].

Hence, [∇_{η}^{A^{ad}}ξ, ρ(ω)] = 0 for all ω and therefore A^{ad} preserves the subspace 𝔰𝔲(E) ⊂ 𝔰𝔲(V), on which it induces an orthogonal connection Â.

Sections of the subbundle iρ(Λ^{♯}) ⊗ 𝔰𝔲(E) are linear combinations of sections of the form iρ(ω)ξ, where ω ∈ C^{∞}(Λ^{♯}) and ξ ∈ C^{∞}(𝔰𝔲(E)). Since ∇_{A^{ad}} is a Clifford module derivation and induces the connections ∇ and Â on the subbundles ρ(Λ^{♯}) and 𝔰𝔲(E), respectively, we have

∇_{η}^{A^{ad}}(ρ(ω)ξ) = ρ(∇_{η}ω)ξ + ρ(ω)∇Â ; ηξ,

so the orthogonal connection induced by A on iρ(Λ^{♯}) ⊗ 𝔰𝔲(E) is given by ∇ ⊗ Â.

We shall fix, once and for all, a smooth, unitary connection A_{Λ} on the square-root determinant line bundle, det^{\frac{1}{2}} (V^{+}), and henceforth require that our unitary connections A on V = V^{+} ⊕ V^{−} induce the resulting unitary connection on det(V^{+}),

(2.24) A^{det} = 2A_{Λ} on det(V^{+}), where we write A^{det} for the connection on det(V^{+}) induced by A|_{V^{+}}. If a unitary connection A on V induces a connection A^{det} = 2A_{Λ} on det(V^{+}), then it necessarily induces the connection A_{Λ} on det^{\frac{1}{2}} (V^{+}).

2.1.5. PU(2) monopoles. For k ≥ 2, we let A_{𝔱} denote the space of L_{k}^{2} spin connections on V. From the preceding subsection, A_{𝔱} is an affine space for the Hilbert space L_{k}^{2}(Λ^{1} ⊗ 𝔤_{𝔱}),

(A, a) ↦ A + a, (2.25)

via the inclusion (2.13) given by 𝔰𝔲(E) ⊂ 𝔰𝔲(V), a ↦ id ⊗ a. This descends to an action on the affine space of SO(3) connections on 𝔤_{𝔱},

(Â, a) ↦ Â + ad(a), (2.26)

with ad(a) ∈ L_{k}^{2}(Λ^{1} ⊗ 𝔰𝔬(𝔤_{𝔱})). We have

ad^{−1}(F{ \atop A+a}) = ad^{−1}(FÂ) + dÂa + a ∧ a and D_{A+a} = D_{A} + ρ(a), (2.27)

as we later use when describing the deformation complex for the PU(2)-monopole equations; note that the map ad : 𝔤_{𝔱} → 𝔰𝔬(𝔤_{𝔱}) is an isomorphism.

Recall from the proof of Lemma 2.3 that the fibers of V are isomorphic to Δ ⊕ Δ as complex Clifford algebra modules, where Δ is the unique (up to equivalence) irreducible ℂℓ(ℝ^{4}) module. From equation (2.7) that there is an isomorphism ℂ^{2} ≅ Hom_{ℂℓ(ℝ^{4})}(Δ, Δ ⊕ Δ), identifying (z_{1}, z_{2}) with the homomorphism Δ → Δ ⊕ Δ, v ↦ (z_{1}v, z_{2}v). Hence, there is an isomorphism of complex algebras,

(2.28) m : M_{2}(ℂ) ≅ End_{ℂℓ(ℝ^{4})}(Δ ⊕ Δ), M ↦ id_{Δ} ⊗ M,

where M_{2}(ℂ) is the space of complex 2 × 2 matrices. If u ∈ End_{ℂℓ(ℝ^{4})}(Δ ⊕ Δ), we define the Clifford determinant of u by

det_{ℂℓ}(u) = det(m^{−1}(u)), (2.29)

where det(m^{−1}(u)) is the usual complex determinant of m^{−1}(u) as a 2 × 2 complex matrix. Since the Clifford determinant (2.29) is invariant under conjugation by Clifford automorphisms of Δ ⊕ Δ and complex automorphisms of ℂ^{2}, we can therefore define the Clifford determinant, det_{ℂℓ}(u), of a complex Clifford algebra endomorphism u of the bundle V by taking the corresponding definition on the fibers of V.

Definition 2.6. We say that u is a spin^{u} automorphism of V if it is an L_{k+1}^{2} unitary, complex Clifford algebra automorphism of V with Clifford-determinant one. We let G_{𝔱} denote the Hilbert Lie group of spin^{u} automorphisms of V.

Let G_{κ}^{w} be the group of L_{k+1}^{2} unitary, determinant-one automorphisms of E, with Lie algebra L_{k+1}^{2}(𝔰𝔲(E)), where w = c_{1}(E) and κ = c_{2}(E) − \frac{1}{4}c_{1}(E)^{2}. While at first glance it might seem more natural to relax the requirement that u ∈ G_{𝔱} have Clifford-determinant one to that of complex-determinant one, our refinement gives us a useful identification of G_{𝔱} with G_{κ}^{w}:

Lemma 2.7. Suppose (ρ, V) is a spin^{u} structure with V = W ⊗E, for some spin^{c} structure (ρ, W) and rank-two Hermitian bundle E. Then the following map is an isomorphism of Hilbert Lie groups:

m : G_{κ}^{w} ≅ G_{𝔱}, u ↦ id_{W} ⊗ u. (2.30)

Proof. Plainly, the map is an injective homomorphism, so it remains to show that it is surjective. Suppose v ∈ G_{𝔱}. For any x ∈ X, the remarks preceding Definition 2.6 imply that we may write v|_{x} = id_{W_{x}} ⊗ u_{x}, where u_{x} ∈ End_{ℂ}(E_{x}) and so we have v = id_{W} ⊗ u for some u ∈ End_{ℂ}(E). But det(u) = det_{ℂℓ}(v) = 1 and as id_{W} ⊗ u^{†}u = v^{†}v = id_{V}, we must have u^{†}u = id_{E} and thus u ∈ G_{κ}^{w}, as desired.

Consequently, the Hilbert Lie group G_{𝔱} has Lie algebra L_{k+1}^{2}(𝔤_{𝔱}) ⊂ L_{k+1}^{2}(𝔰𝔲(V)). Note that if u is a spin^{u} automorphism of V, then Ad (u) preserves three factors in the orthogonal decomposition (2.17) of 𝔰𝔲(V); it acts as the identity on the subbundle ρ(Λ^{2}), as an orthogonal gauge transformation û on 𝔰𝔲(E), and as id⊗û on the subbundle iρ(Λ^{2})⊗𝔰𝔲(E).

For an L_{k}^{2} section Φ of V^{+}, we let Φ^{∗} denote its pointwise Hermitian dual and let (Φ⊗Φ^{∗})_{00} be the component of Φ ⊗ Φ^{∗} ∈ i𝔲(V^{+}) which lies in the factor 𝔰𝔲(W^{+}) ⊗ 𝔰𝔲(E) of the decomposition (2.13) of i𝔲(V^{+}) ≅ \underline{ℝ} ⊕ i𝔰𝔲(V^{+}) (with V^{+} in place of V). The Clifford multiplication ρ defines an isomorphism ρ : Λ^{+} → 𝔰𝔲(W^{+}) and thus an isomorphism ρ = ρ ⊗ id_{𝔰𝔲(E)} of Λ^{+} ⊗ 𝔰𝔲(E) with 𝔰𝔲(W^{+}) ⊗ 𝔰𝔲(E).

The pre-configuration space of pairs on V is given by

C̃_{𝔱} = A_{𝔱} × L_{k}^{2}(V^{+}), (2.31)

with tangent spaces L_{k}^{2}(𝔤_{𝔱}) ⊕ L_{k}^{2}(V^{+}). We call a pair (A, Φ) ∈ C̃_{𝔱} a PU(2) monopole if

𝔖(A, Φ) = (\begin{matrix}ad^{−1}(F + ; Â) − τρ^{−1}(Φ ⊗ Φ^{∗})_{00} \\ D_{A}Φ + ρ(ϑ)Φ \end{matrix}) = 0, (2.32)

where F + ; Â ∈ L_{k−1}^{2}(Λ^{+} ⊗ 𝔰𝔬(𝔤_{𝔱})) is the self-dual component of the curvature FÂ of Â while D_{A} = ρ ∘ ∇_{A} : L_{k}^{2}(V^{+}) → L_{k−1}^{2}(V^{−}) is the Dirac operator, τ ∈ L_{k+1}^{2}(X, GL(Λ^{+})) and ϑ ∈ L_{k+1}^{2}(Λ^{1} ⊗ ℂ) are perturbation parameters. We let

(2.33) M_{𝔱} = {[A, Φ] ∈ C_{𝔱} : (A, Φ) satisfies (2.32)},

be the moduli space of solutions to (2.32) cut out of the configuration space,

C_{𝔱} = C̃_{𝔱}/G_{𝔱}, (2.34)

where u ∈ G_{𝔱} acts by u(A, Φ) = (u_{∗}A, uΦ). The linearization of the map G_{𝔱} → C̃_{𝔱}, u ↦ u(A, Φ), at id_{V} ∈ G_{𝔱}, is given by

ζ ↦ −d_{A,Φ}^{0}ζ = (−dÂζ, ζΦ), (2.35)

with L^{2}-adjoint d_{A,Φ}^{0,∗}, where ζ ∈ L_{k+1}^{2}(𝔤_{𝔱}).

The circle S^{1} defines a family of unitary gauge transformations on V acting by scalar multiplication, so that

S^{1} × C̃_{𝔱} → C̃_{𝔱}, (e^{iθ}, (A, Φ)) ↦ (A, e^{iθ}Φ). (2.36)

Because scalar multiplication by S^{1} commutes with G_{𝔱}, the action (2.36) descends to an action on C_{𝔱} and on M_{𝔱}.

Remark 2.8. Note that we break here with our former convention [24], [23] of considering M_{𝔱} and C_{𝔱} as quotients by S^{1} ×_{{±1}} G_{𝔱}, rather than G_{𝔱} as defined here.

We call a spin connection A on V reducible if it splits as a direct sum of connections on V = W ⊕ W′, where 𝔰 = (ρ, W) and 𝔰′ = (ρ, W′) are spin^{c} structures, and call A irreducible otherwise. We write 𝔱 = 𝔰 ⊕ 𝔰′ and A = B ⊕ B, for the induced spin connections B on W and B′ on W′.

We call a connection Â on an SO(3) bundle F reducible if it splits as a direct sum d_{ℝ}⊕A_{L} on F = i\underline{ℝ} ⊕ L, where A_{L} is a unitary connection on a complex line bundle L over X and d_{ℝ} is the product connection on i\underline{ℝ}, and call Â irreducible otherwise. The following lemma summarizes the relationship between these notions of reducibility:

Lemma 2.9. Let (ρ, V) be a spin^{u} structure on X. Then a spin connection A on V is reducible with respect to a splitting V = W ⊕W′, where (ρ, W) and (ρ, W′) are spin^{c} structures, if and only if the induced SO(3) connection Â on 𝔤_{𝔱} is reducible with respect to a splitting 𝔤_{𝔱} = i\underline{ℝ}⊕L, where L is a complex line bundle over X such that W′ = W ⊗L. If A = B ⊕B′ is reducible, then there is a unitary connection A_{L} on L such that

B′ = B ⊗ A_{L} and Â = d_{ℝ} ⊕ A_{L}. (2.37)

If A^{det} = 2A_{Λ} on det(V^{+}), then

A_{L} = A_{Λ} ⊗ (B^{det})^{∗}. (2.38)

Remark 2.10. We write B^{det} for the connection on det(W^{+}) induced by B|_{W^{+}}, where B is a spin connection on W = W^{+} ⊕ W^{−}.

Proof. Suppose that A = B ⊕ B′ is a reducible spin connection on V with respect to the splitting V = W ⊕ W′. There is a unique complex line bundle L over X such that W′ = W ⊗ L. If A_{L} is a unitary connection on L, then both B′ and B ⊗ A_{L} are spin connections on W′. But any two spin connections on W′ differ by an element of Ω^{1}(X, iℝ), via the action (2.5), and thus we can write B′ = B ⊗ A_{L} for some unitary connection A_{L} on L.

The unitary connection Ã = d_{ℂ} ⊕ A_{L} on E = \underline{ℂ} ⊕ L induces the SO(3) connection d_{ℝ} ⊕ A_{L} on 𝔰𝔲(E) ≅ i\underline{ℝ} ⊕ L. To see this, we pass to a local trivialization of E and view Ã as a connection matrix one-form,

Ã = (\begin{matrix}0 & 0 \\ 0 & A_{L} \end{matrix}) ∈ Ω^{1}(𝔲(E)).

The matrix Ã acts on a section ξ of 𝔰𝔲(E) via the adjoint representation,

Â · ξ = ad(Ã)ξ = [Ã, ξ] = (\begin{matrix}0 & −Ā_{L}z̄ \\ A_{L}z & 0 \end{matrix}) , where ξ = (\begin{matrix}ν & −z̄ \\ z & −ν \end{matrix}) ∈ C^{∞}(𝔰𝔲(E)).

Thus, Â · (ν, z) = (0, A_{L}z), with respect to the isomorphism 𝔰𝔲(E) ≅ i\underline{ℝ} ⊕ L of Lemma 3.10 and so, as a connection, we see that Â = d_{ℝ} ⊕ A_{L}.

Therefore, the spin connection A = B⊗(d_{ℂ}⊕A_{L}) on W ⊗E induces the SO(3) connection d_{ℝ} ⊕ A_{L} on 𝔰𝔲(E) ≅ i\underline{ℝ} ⊕ L via the decomposition (2.13),

𝔰𝔲(V) ≅ 𝔰𝔲(E) ⊕ 𝔰𝔲(W) ⊕ 𝔰𝔲(W) ⊗ 𝔰𝔲(E).

The connections B on W and d_{ℂ}⊕A_{L} on E induce the connection B^{det} ⊗A_{L} on det(W^{+})⊗ det(E) = det^{\frac{1}{2}} (V^{+}). By convention, our spin connections A on V induce the connection 2A_{Λ} on det(V^{+}), and thus A_{Λ} = B^{det} ⊗ A_{L}.

Conversely, if Â = d_{ℝ} ⊕ A_{L} is an SO(3) connection which is reducible with respect to the splitting 𝔰𝔲(E) = i\underline{ℝ} ⊕ L, then Â lifts to a U(2) connection d_{ℂ}⊕ A_{L} on E which is reducible with respect to the splitting E = \underline{ℂ} ⊕ L, and lifts to a spin connection A = B ⊗ (d_{ℂ}⊕ A_{L}) = B ⊕B′ on V which is reducible with respect to the splitting W ⊗(\underline{ℂ}⊕L) = W ⊕W ⊗L.

We say that a pair (A, Φ) ∈ C̃_{𝔱} is irreducible (respectively, reducible) if the connection A is irreducible (respectively, reducible). We let C_{𝔱}^{∗} ⊂ C_{𝔱} be the open subspace of gaugeequivalence classes of irreducible pairs. If Φ ≡ 0 on X, we call (A, Φ) a zero-section pair. We let C_{𝔱}^{0} ⊂ C_{𝔱} be the open subspace of gauge-equivalence classes of non-zero-section pairs and recall that

C_{𝔱}^{∗,0} = C_{𝔱}^{∗} ∩ C_{𝔱}^{0} (2.39)

is a Hausdorff, Hilbert manifold [24, Proposition 2.8] represented by pairs with trivial stabilizer in G_{𝔱}. Let

M_{𝔱}^{∗,0} = M_{𝔱} ∩ C_{𝔱}^{∗,0} (2.40)

be the open subspace of the moduli space M_{𝔱} represented by irreducible, non-zero-section PU(2) monopoles; the subspaces M_{𝔱}^{∗} and M_{𝔱}^{0} are defined analogously.

2.1.6. Spaces of SO(3) connections. Given a class in H^{2}(X; ℤ/2ℤ) with an integral lift w ∈ H^{2}(X; ℤ) and an integer p obeying p^{2} ≡ w^{2} (mod 4), set κ = −\frac{1}{4}p, and choose any Hermitian rank-two bundle E with c_{1}(E) = w, so w_{2}(𝔰𝔲(E)) = w (mod 2), and p_{1}(𝔰𝔲(E)) = −4κ. Let A_{κ}^{w}(X) denote the affine space of L_{k}^{2} orthogonal connections on 𝔰𝔲(E) and let B_{κ}^{w}(X) = A_{κ}^{w}(X)/G_{κ}^{w} be the quotient, where G_{κ}^{w} acts on 𝔰𝔲(E) via the adjoint action.

Let A_{κ}^{w,∗}(X) and B_{κ}^{w,∗}(X) be the subspace of irreducible SO(3) connections and its quotient. The moduli space of anti-self-dual connections on 𝔰𝔲(E) is then defined by

M_{κ}^{w}(X) = {[Â] ∈ B_{κ}^{w}(X) : F + ; Â = 0}, (2.41)

with M_{κ}^{w,∗} = M_{κ}^{w}∩B_{κ}^{w,∗}. We follow the convention of [44] in taking the quotient by the group of determinant-one, unitary automorphisms of E rather than the group of determinant-one, orthogonal automorphisms of 𝔰𝔲(E).

Lemma 2.5 implies that a spin connection A on V = W ⊗ E determines a unique orthogonal connection Â on 𝔰𝔲(E) and, conversely, that every orthogonal connection on 𝔰𝔲(E) is in the image of the map A ↦ Â. Hence, one can easily translate between the conventions employed in the present article and those of [24], [15], [23]:

Lemma 2.11. Let A_{Λ} be the fixed unitary connection on the complex line bundle det^{\frac{1}{2}} (V^{+}) and let A_{𝔱} denote the corresponding space of spin connections on V, as described in §2.1.4. For each SO(3) connection Â on 𝔰𝔲(E), let A denote the corresponding spin connection on V. With respect to the action of G_{κ}^{w} on A_{κ}^{w} and of G_{𝔱} on A_{𝔱}, and the identification G_{κ}^{w} ≅ G_{𝔱}, the following map is a gauge-equivariant bijection:

A_{κ}^{w} ≅ A_{𝔱}, Â ↦ A. (2.42)

2.2. Uhlenbeck compactness and transversality for PU(2) monopoles. We briefly recall our Uhlenbeck compactness and transversality results [15], [24] for the moduli space of PU(2) monopoles with the perturbations discussed in the preceding section. The moduli space of PU(2) monopoles is non-compact but has an Uhlenbeck compactification analogous to the compactification M̄_{κ}^{w} of the moduli space of anti-self-dual connections on an SO(3) bundle [14, §4.4].

Given a non-negative integer ℓ and a spin^{u} structure 𝔱 = (ρ, V), Lemma 2.3 allows us to write V = W ⊗ E for some choice of spin^{c} structure (ρ, W) and corresponding Hermitian, rank-two bundle E. Let E_{ℓ} → X be the Hermitian, rank-two bundle with

c_{1}(E_{ℓ}) = c_{1}(E) and c_{2}(E_{ℓ}) = c_{2}(E) − ℓ. (2.43)

We then define a spin^{u} structure 𝔱_{ℓ} = (ρ, V_{ℓ}) on X by setting

V_{ℓ} = W ⊗ E_{ℓ}, (2.44)

and observe that

c_{1}(𝔱_{ℓ}) = c_{1}(𝔱), p_{1}(𝔱_{ℓ}) = p_{1}(𝔱) + 4ℓ, and w_{2}(𝔱_{ℓ}) = w_{2}(𝔱). (2.45)

We say that a sequence of points [A_{α}, Φ_{α}] in C_{𝔱} converges to a point [A, Φ, x] in C_{𝔱_{ℓ}}×Sym^{ℓ}(X) if the following hold:

• There is a sequence of L_{k+1,loc}^{2} spin^{u} bundle isomorphisms u_{α} : V |_{X\x} → V_{ℓ}|_{X\x} such that the sequence of monopoles u_{α}(A_{α}, Φ_{α}) converges to (A, Φ) in L_{k,loc}^{2} over X \ x, and

• The sequence of measures |F_{A_{α}}|^{2} converges in the weak-* topology on measures to |F_{A}|^{2} + 8π^{2} ∑_{x∈x} δ(x).

There is a natural extension of this definition of convergence of points in C_{𝔱} to one for sequences in the space of “ideal pairs”, ⊔_{ℓ=0}^{∞}(C_{𝔱_{ℓ}} × Sym^{ℓ}(X)), and this serves to define the “Uhlenbeck topology” on the space of ideal pairs. We define the topological space

(2.46)  M̄_{𝔱} = Closure(M_{𝔱}) ⊂ ⨆_{ℓ=0}^{∞}(M_{𝔱_{ℓ}} × Sym^{ℓ}(X)),

where the closure is taken with respect to the Uhlenbeck topology on the (second countable, Hausdorff) space of ideal PU(2) monopoles, ⊔_{ℓ=0}^{∞}(M_{𝔱_{ℓ}} ×Sym^{ℓ}(X)). We call the intersection of M̄_{𝔱} with M_{𝔱_{ℓ}} × Sym^{ℓ}(X) a lower-level of the compactification M̄_{𝔱} if ℓ > 0 and call M_{𝔱} the top or highest level.

Theorem 2.12. [24] Let X be a closed, oriented, Riemannian, smooth four-manifold with spin^{u} structure 𝔱. Then there is a positive integer N, depending at most on the curvature of the fixed unitary connection on det(V^{+}) together with p_{1}(𝔱), such that the topological space M̄_{𝔱} is second countable, Hausdorff, compact, and given by the closure of M_{𝔱} in the space of ideal PU(2) monopoles, ⊔_{ℓ=0}^{N}(M_{𝔱_{ℓ}} × Sym^{ℓ}(X)), with respect to the Uhlenbeck topology.

Theorem 2.12 is a special case of the more general result proved in [24] for the moduli space of solutions to the PU(2) monopole equations in the presence of holonomy perturbations. The existence of an Uhlenbeck compactification for the moduli space of solutions to the unperturbed PU(2) monopole equations (2.32) was announced by Pidstrigatch [63] and an argument was outlined in [64]. A similar argument for the equations (2.32) (without perturbations) was outlined by Okonek and Teleman in [61]. An independent proof of Uhlenbeck compactness for (2.32) and other perturbations of these equations is also given in [73].

We recall from [24, Equation (2.37)] that the elliptic deformation complex for the moduli space M_{𝔱} is given by

L_{k+1}^{2}(𝔤_{𝔱}) ⟶^{d_{A,Φ}^{0}} \begin{matrix}L_{k}^{2}(Λ^{1} ⊗ 𝔤_{𝔱}) \\ ⊕ \\ L_{k}^{2}(V^{+}) \end{matrix} ⟶^{d_{A,Φ}^{1}} \begin{matrix}L_{k−1}^{2}(Λ^{+} ⊗ 𝔤_{𝔱}) \\ ⊕ \\ L_{k−1}^{2}(V^{−}) \end{matrix} (2.47)

with elliptic deformation operator

D_{A,Φ} = d_{A,Φ}^{0,∗} + d_{A,Φ}^{1} (2.48) and cohomology H_{A,Φ}^{•}. Here, d_{A,Φ}^{1} is the linearization at the pair (A, Φ) of the gaugeequivariant map 𝔖 defined by the equations (2.32), so

d_{A,Φ}^{1}(a, ϕ) = (D𝔖)_{A,Φ}(a, ϕ) = (\begin{matrix}d+ ; Âa − τρ^{−1}(ϕ ⊗ Φ^{∗} + Φ ⊗ ϕ^{∗})_{00} \\ (D_{A} + ρ(ϑ))ϕ + ρ(a)Φ \end{matrix}) , (2.49)

while −d_{A,Φ}^{0} is the differential of the map G_{𝔱} → C̃_{𝔱}, u ↦ u(A, Φ) = (A − (d_{A}u)u^{−1}, uΦ), so

d_{A,Φ}^{0}ζ = (d_{A}ζ, ζΦ). (2.50)

The space H_{A,Φ}^{0} = Ker d_{A,Φ}^{0} is the Lie algebra of the stabilizer in G_{𝔱} of a pair (A, Φ) and H_{A,Φ}^{1} is the Zariski tangent space to M_{𝔱} at a point [A, Φ]. If H_{A,Φ}^{2} = 0, then [A, Φ] is a regular point of the zero locus of the PU(2) monopole equations (2.32) on C_{𝔱}.

We now turn to the question of transversality. Let D_{A,∇} and D_{A} be the Dirac operators on V^{+} defined, respectively, by unitary connections A on V which are spin with respect to an SO(4) connection ∇ and spin, in the usual sense, with respect to the Levi-Civita connection on T^{∗}X: the two operators differ by an element ρ(ϑ′) ∈ Hom(V^{+}, V^{−}) [15, Lemma 3.1], where ϑ′ ∈ Ω^{1}(X, ℂ). Even though a unitary connection on V will not necessarily induce a torsion-free connection on T^{∗}X for generic pairs (g, ρ) of Riemannian metrics and compatible Clifford maps, we can assume that the Dirac operator D_{A} in (2.32) is defined using the Levi-Civita connection for the metric g by absorbing the difference term ρ(ϑ′) into the perturbation term ρ(ϑ). Given any fixed pair (g_{0}, ρ_{0}) satisfying (2.1) and automorphism f ∈ C^{∞}(GL(T^{∗}X)), then (g, ρ) = (f^{∗}g_{0}, f^{∗}ρ_{0}) is again a compatible pair; the pair (g, ρ) is generic if f is generic.

Theorem 2.13. [15] Let X be a closed, oriented, smooth four-manifold with spin^{u} structure 𝔱. Then for a generic, C^{∞} pair (g, ρ) satisfying (2.1) and generic, C^{∞} parameters (τ, ϑ), the moduli space M_{𝔱}^{∗,0} = M_{𝔱}^{∗,0}(g, ρ, τ, ϑ) of PU(2) monopoles is a smooth manifold of the expected dimension,

dim M_{𝔱}^{∗,0} = d_{a} + 2n_{a},

where

\begin{aligned}d_{a}(𝔱) & = −2p_{1}(𝔱) − \frac{3}{2}(χ + σ), \\ n_{a}(𝔱) & = \frac{1}{4}(p_{1}(𝔱) + c_{1}(𝔱)^{2} − σ). \end{aligned} (2.51)

In equation (2.51), the quantity d_{a}(𝔱) is the expected dimension of the moduli space of anti-self-dual connections on 𝔤_{𝔱}, while n_{a}(𝔱) is the complex index of the Dirac operator on C^{∞}(V^{+}). Theorem 2.13 was proved independently, using a somewhat different method, by A. Teleman [73].

2.3. Moduli spaces of Seiberg-Witten monopoles. We recall the definition of the space of Seiberg-Witten monopoles. Let 𝔰 = (ρ, W) be a spin^{c} structure on X and let

C̃_{𝔰} = A_{𝔰} × L_{k}^{2}(W^{+}) (2.52)

be the pre-configuration space of pairs (B, Ψ), with tangent space L_{k}^{2}(X, iℝ) ⊕ L_{k}^{2}(W^{+}) due to the affine structure (2.5). We call a unitary automorphism of W a spin^{c} automorphism if it is a complex Clifford module endomorphism of W. Thus, spin^{c} automorphisms induce the identity on the factor 𝔰𝔲(W) ≅ Λ^{2}(T^{∗}X) of the orthogonal decomposition 𝔲(W) = i\underline{ℝ} ⊕ 𝔰𝔲(W). We let G_{𝔰} denote the group of L_{k+1}^{2} spin^{c} automorphisms of W and observe that there is an isomorphism [56, §§4.3–4.5]

L_{k+1}^{2}(X, S^{1}) → G_{𝔰}, s ↦ s · id_{W} .

Hence, the group G_{𝔰} has Lie algebra L_{k+1}^{2}(X, iℝ) via the identification

L_{k+1}^{2}(X, iℝ) ≅ T_{id}G_{𝔰}, f ↦ f · id_{W} ,

and acts smoothly by pushforward on C̃_{𝔰},

(s, (B, Ψ)) ↦ (s_{∗}B, sΨ) = (B − s^{−1}ds, sΨ). (2.53)

We thus obtain a configuration space

C_{𝔰} = C̃_{𝔰}/G_{𝔰} (2.54)

and let C̃_{𝔰}^{0} ⊂ C̃_{𝔰} denote the open subspace of pairs (B, Ψ) with Ψ ≢ 0, where G_{𝔰} acts freely on C̃_{𝔰}^{0} with quotient C_{𝔰}^{0}, a smooth Hilbert manifold.

We call a pair (B, Ψ) ∈ C̃_{𝔰} a Seiberg-Witten monopole if

𝔖(B, Ψ) = (\begin{matrix}Tr(F_{B}^{+}) − τρ^{−1}(Ψ ⊗ Ψ^{∗})_{0} − η \\ D_{B}Ψ + ρ(ϑ)Ψ, \end{matrix}) = 0, (2.55)

where F_{B}^{+} ∈ L_{k−1}^{2}(Λ^{+} ⊗ 𝔲(W^{+})) is the self-dual component of the curvature F_{B} of B and Tr(F_{B}^{+}) ∈ L_{k−1}^{2}(Λ^{+} ⊗ 𝔰𝔲(W^{+})) is the trace-free part, D_{B} = ρ ∘ ∇_{B} : C^{∞}(W^{+}) → C^{∞}(W^{−}) is the Dirac operator defined by the spin connection B, the perturbation terms τ and ϑ are as defined in our version of the PU(2) monopole equations (2.32), and where η ∈ C^{∞}(iΛ^{+}) is an additional perturbation (see Remark 2.14). The quadratic term Ψ ⊗ Ψ^{∗} lies in C^{∞}(i𝔲(W^{+})) and (Ψ ⊗ Ψ^{∗})_{0} denotes the traceless component lying in C^{∞}(i𝔰𝔲(W^{+})), so ρ^{−1}(Ψ ⊗ Ψ^{∗})_{0} ∈ C^{∞}(iΛ^{+}).

Remark 2.14. We note that in the usual presentation of the Seiberg-Witten equations [43], [60], one takes τ = id_{Λ^{+}} and ϑ = 0, while η is a generic perturbation. However, we shall see in Lemma 3.12 that in order to identify solutions to the Seiberg-Witten equations (2.55) with reducible solutions to the PU(2) monopole equations (2.32), we need to employ the perturbations given in equation (2.55) and choose

η = F_{A_{Λ}}^{+}, (2.56)

where A_{Λ} is the fixed unitary connection on the line bundle det^{\frac{1}{2}} (V^{+}) with Chern class denoted by c_{1}(𝔱) = Λ ∈ H^{2}(X; ℤ) and represented by the real two-form (1/2πi)F_{A_{Λ}}. In particular, this choice of η is not generic in the sense of [56, Proposition 6.3.1 & Corollary 6.3.2] if c_{1}(𝔰) − Λ ∈ H^{2}(X; ℤ) is a torsion class. If c_{1}(𝔰) − Λ is not torsion, then this class is represented by (1/2πi)(Tr(F_{B}) − F_{A_{Λ}}) and, if b_{2}^{+}(X) > 0 and the metric g is generic, then there are no zero-section solutions to the Seiberg-Witten equations (2.55) by [56, Proposition 6.3.1].

The moduli space of Seiberg-Witten monopoles is defined by

\begin{aligned}M_{𝔰} & = \{(B, Ψ) ∈ C̃_{𝔰} : 𝔖(B, Ψ) = 0\}/G_{𝔰}. \\ M_{𝔰}^{0} & = M_{𝔰} ∩ C_{𝔰}^{0}. \end{aligned} (2.57)

The usual Seiberg-Witten moduli space—obtained from our definition with τ = id_{Λ^{+}} and ϑ = 0—is compact [43, Corollary 3], [56, Proposition 6.4.1] and, for generic η, the open subspace M_{𝔰}^{0} is a smooth manifold of the expected dimension [43, Lemma 5], [56, Proposition 6.2.2]. In the subsections below we describe transversality and compactness properties for the space M_{𝔰}, as defined here.

2.3.1. The deformation complex. Given (B, Ψ) ∈ C̃_{𝔰}, the smooth map G_{𝔰} → C̃_{𝔰} defined by (2.53) has differential at id_{W},

f ↦ −d_{B,Ψ}^{0}f = (−df, fΨ), f ∈ L_{k+1}^{2}(X, iℝ). (2.58)

For all (b, ψ) ∈ L_{k}^{2}(iΛ^{1}) ⊕ L_{k}^{2}(W^{+}) and f ∈ L_{k+1}^{2}(X, iℝ) we have

((df, −fΨ), (b, ψ))_{L^{2}} = (df, b)_{L^{2}} + (−fΨ, ψ)_{L^{2}}
= (f, d^{∗}b)_{L^{2}} − (f, \overline{⟨Ψ, ψ⟩})_{L^{2}}
= (f, d^{∗}b)_{L^{2}} − (f, ⟨ψ, Ψ⟩)_{L^{2}}
= (f, d^{∗}b)_{L^{2}} − (f, i Im⟨ψ, Ψ⟩)_{L^{2}} ,

noting that Ω^{0}(X, iℝ) has a real L^{2} inner product, and thus (compare [56, Lemma 4.5.5])

d_{B,Ψ}^{0,∗}(b, ψ) = d^{∗}b − i Im⟨ψ, Ψ⟩. (2.59)

The Seiberg-Witten equations (2.55) define a G_{𝔰}-equivariant map 𝔖 : C̃_{𝔰} → L_{k−1}^{2}(iΛ^{+}) × L_{k−1}^{2}(W^{−}) with differential at the point (B, Ψ) given by

d_{B,Ψ}^{1}(b, ψ) = (D𝔖)_{B,Ψ}(b, ψ) = (\begin{matrix}2d^{+}b − τρ^{−1}(ψ ⊗ Ψ^{∗} + Ψ ⊗ ψ^{∗})_{0} \\ D_{B}ψ + ρ(ϑ)ψ + ρ(b)Ψ \end{matrix}) . (2.60)

Since 𝔖(s(B, Ψ)) = (𝔖_{1}(B, Ψ), s𝔖_{2}(B, Ψ)) = s·𝔖(B, Ψ), the differential of the composition G_{𝔰} → L_{k}^{2}(iΛ^{+}) × L_{k}^{2}(W^{−}) is given by

f ↦ d_{B,Ψ}^{1} ∘ d_{B,Ψ}^{0}f = (0, f𝔖_{2}(B, Ψ)), f ∈ L_{k+1}^{2}(X, iℝ),

and so d_{B,Ψ}^{1} ∘ d_{B,Ψ}^{0} = 0 if and only if 𝔖_{2}(B, Ψ) = 0. In particular, if 𝔖(B, Ψ) = 0 we have an elliptic deformation complex

L_{k+1}^{2}(X, iℝ) ⟶^{d_{B,Ψ}^{0}} \begin{matrix}L_{k}^{2}(iΛ^{1}) \\ ⊕ \\ L_{k}^{2}(W^{+}) \end{matrix} ⟶^{d_{B,Ψ}^{1}} \begin{matrix}L_{k−1}^{2}(iΛ^{+}) \\ ⊕ \\ L_{k−1}^{2}(W^{−}) \end{matrix} (2.61)

with cohomology H_{B,Ψ}^{•}. We write

D_{B,Ψ} = d_{B,Ψ}^{0,∗} + d_{B,Ψ}^{1} (2.62)

for the rolled-up deformation operator of the complex (2.61).

The deformation complex (2.61) is the same, up to slight differences in the zeroth order terms of the differential d_{B,Ψ}^{1}, as the deformation complex [56, §4.6] for the usual Seiberg-Witten equations and so has the same index. Therefore, provided the zero locus M_{𝔰}^{0} of the section 𝔖 is regular we have

d_{s}(𝔰) = dim M_{𝔰}^{0} = \frac{1}{4} (c_{1}(𝔰)^{2} − 2χ − 3σ) . (2.63)

With the deformation complex in place, we now turn to the issues of compactness and transversality.

2.3.2. Compactness and transversality. We first observe that the standard arguments [43], [56], [76] establishing that the Seiberg-Witten equations as stated in those references define a compact moduli space carry over to show that the moduli space M_{𝔰} defined by (2.55) is compact; the only slight change is the requirement that τ be C^{0}-close to the identity on Λ^{+}:

Proposition 2.15. Let 𝔰 = (ρ, W) be a spin^{c} structure on a closed, oriented, Riemannian four-manifold, (X, g), parameters η ∈ Ω^{0}(iΛ^{+}) and ϑ ∈ Ω^{1}(X, ℂ), and perturbation τ ∈ Ω^{0}(GL(Λ^{+})) such that ∥τ − id_{Λ^{+}}∥_{C^{0}} < \frac{1}{64}. Then the moduli space M_{𝔰}(g, η, τ, ϑ) is compact.

Our next task is to establish an transversality result for M_{𝔰}^{0}(g, η, τ, ϑ) with generic parameter τ analogous to the one in [43] where η is generic.

Proposition 2.16. Let 𝔰 = (ρ, W) be a spin^{c} structure on a closed, oriented, Riemannian four-manifold, (X, g). Then for any fixed parameters η ∈ Ω^{0}(iΛ^{+}) and ϑ ∈ Ω^{1}(X, ℂ) there is a first-category subset of Ω^{0}(GL(Λ^{+})) such that for all C^{∞} parameters τ in the complement of this subset, the moduli space M_{𝔰}^{0}(g, τ, η, ϑ) is a smooth manifold of the expected dimension.

Proof. We closely follow the method described in [24, §5.1 & §5.2]. Our Banach space of parameters is given by

P = C^{r}(GL(Λ^{+})).

We define an extended G_{𝔰}-equivariant map,

\underline{𝔖} = (\underline{𝔖}_{1}, \underline{𝔖}_{2}) : P × C̃_{𝔰} → L_{k−1}^{2}(iΛ^{+}) × L_{k−1}^{2}(W^{−}),

by setting

\underline{𝔖}(τ, B, Ψ) = (\begin{matrix}Tr(F_{B}^{+}) − τρ^{−1}(Ψ ⊗ Ψ^{∗})_{0} − η \\ D_{B}Ψ + ρ(ϑ)Ψ \end{matrix}) .

The parametrized moduli space 𝔐_{𝔰} is then (\underline{𝔖})^{−1}(0)/G_{𝔰} ⊂ C_{𝔰} and 𝔐_{𝔰}^{0} = 𝔐_{𝔰} ∩ (P × C_{𝔰}^{0}). The map \underline{𝔖} has differential at the point (τ, [B, Ψ]) ∈ 𝔐_{𝔰}^{0} given by

D\underline{𝔖}(δτ, b, ψ) = (\begin{matrix}2d^{+}b − τρ^{−1}(ψ ⊗ Ψ^{∗} + ψ ⊗ Ψ^{∗})_{0} + (δτ)τρ^{−1}(Ψ ⊗ Ψ^{∗})_{0} \\ (D_{B} + ρ(ϑ))ψ + ρ(b)ψ \end{matrix}) ,

where (b, ψ) ∈ L_{k}^{2}(iΛ^{1}) ⊕ L_{k}^{2}(W^{+}). Suppose

(c, φ) ∈ L_{k−1}^{2}(iΛ^{+}) ⊕ L_{k−1}^{2}(W^{−})

is L^{2} orthogonal to the image of D\underline{𝔖}. We may assume without loss of generality that (τ, B, Ψ) is a C^{r} representative of the point (τ, [B, Ψ]) (see, for example, [24, Proposition 3.7]) and so, as (D\underline{𝔖})(D\underline{𝔖})^{∗}(c, φ) = 0, elliptic regularity implies that (c, φ) is C^{r}. Then

0 = (D\underline{𝔖}(δτ, 0, 0), (c, φ))_{L^{2}} = ((δτ)τ(Ψ ⊗ Ψ^{∗})_{0}, c)_{L^{2}}, (2.64)

for all δτ ∈ C^{r}(𝔤𝔩(Λ^{+})), which yields the pointwise identity

⟨(δτ)_{x}τ_{x}(Ψ_{x} ⊗ Ψ_{x}^{∗})_{0}, c_{x}⟩_{x} = 0, x ∈ X. (2.65)

If c_{x} ≠ 0 for some x ∈ X (and thus c ≠ 0 on a non-empty open neighborhood in X), then (2.65) implies that Ψ = 0 on the non-empty open subset {c ≠ 0} ⊂ X. Aronszajn’s theorem then implies that Ψ ≡ 0 on X, contradicting our assumption that (τ, B, Ψ) is a point in 𝔐_{𝔰}^{0}. Hence, we must have c ≡ 0 on X, so (2.64) now yields

0 = (D\underline{𝔖}(0, b, 0), (0, φ))_{L^{2}} = (ρ(b)Ψ, φ)_{L^{2}},

for all b ∈ L_{k}^{2}(iΛ^{1}). Since Ψ ≢ 0, this implies that φ ≡ 0, just as in the proof of [43, Lemma 5]. Hence, Coker(D\underline{𝔖})_{τ,[B,Ψ]} = 0 at each point (τ, [B, Ψ]) in (\underline{𝔖})^{−1}(0) and so the parametrized moduli space (\underline{𝔖})^{−1}(0) = 𝔐_{𝔰}^{0} is a smooth Banach submanifold of P × C_{𝔰}^{0}. Therefore, the Sard-Smale theorem, in the form of [14, Proposition 4.3.11] (see the proof of Corollary 5.3 in [24]), implies that there is a first-category subset of C^{r}(GL(Λ^{+})) such that for all C^{r} parameters τ in the complement of this subset, the zero locus M_{𝔰}^{0}(τ) is a regular submanifold of C_{𝔰}^{0}. Lastly, we can constrain the parameter τ to be C^{∞} (and not just C^{r}) by the argument used in [24, §5.1.2]—see [68, §8.4].)

The following lemma relates the Seiberg-Witten moduli spaces defined with the perturbations of [43], [56], [76] to those used in this article.

Lemma 2.17. Let (X, g) be a closed, oriented, Riemannian four-manifold with b_{2}^{+}(X) > 0 and spin^{c} structure 𝔰.

1. Suppose that the moduli spaces M_{𝔰}^{0}(g, id_{Λ^{+}}, η′, 0) and M_{𝔰}^{0}(g, τ, η, ϑ) are regular zero loci of the Seiberg-Witten maps \underline{𝔖} in (2.55) for these perturbation parameters. If b_{+}^{2}(X) > 1, then these moduli spaces are related by an oriented, smooth cobordism.

2. Suppose that M_{𝔰}(g, id_{Λ^{+}}, η′, 0) contains no zero-section solutions. Then M_{𝔰}(g, τ, η, ϑ) contains no zero-section solutions for any triple (τ, ϑ, η) which is C^{r}-close enough to (id_{Λ^{+}}, 0, η′).

Proof. If P denotes the Banach space of C^{r} perturbation parameters (τ, η, ϑ), then a generic, smooth path in P joining (id_{Λ^{+}}, η′, 0) to (τ, η, ϑ) induces a smooth, oriented cobordism in P × C_{𝔰}^{0} joining M_{𝔰}^{0}(g, id_{Λ^{+}}, η′, 0) to M_{𝔰}^{0}(g, τ, η, ϑ). This proves Assertion (1).

If the compact space M_{𝔰}(g, id_{Λ^{+}}, η′, 0) is contained in the open subset C_{𝔰}^{0} ⊂ C_{𝔰}, then the same holds for M_{𝔰}(g, τ, η, ϑ), that is, the latter subspace also contains no zero-section pairs, for any triple (τ, ϑ, η) which is C^{r}-close enough to (id_{Λ^{+}}, 0, η′). This proves Assertion (2).

2.4. Cohomology ring of the configuration space of spin^{c} pairs. We next compute the cohomology ring of the configuration space C_{𝔰}^{0} and describe the cohomology classes used to define Seiberg-Witten invariants and arising in the calculation (see §3.6) of the Chern classes of certain vector bundles over M_{𝔰}. For this purpose we may make use of the fact since U(1) is Abelian, we have a global slice theorem for C_{𝔰} modulo the harmonic gauge transformations. Computations of the cohomology ring of C_{𝔰}^{0} have already appeared in, for example [68, Chapter 7] and [62, Lemma 5]. The version we present here is convenient for the computation of the Chern class of the universal line bundle in Lemma 2.24.

2.4.1. Harmonic gauge transformations. A gauge transformation s ∈ Map(X, S^{1}) is harmonic if s^{−1}ds ∈ L_{k}^{2}(iΛ) satisfies d^{∗}(s^{−1}ds) = 0 (we always have d(s^{−1}ds) = 0). Every component of Map(X, S^{1}) contains a harmonic representative which is unique up to multiplication by a constant element of S^{1}. The harmonic elements of Map(X, S^{1}) form a subgroup,

Ĝ_{𝔰} = {s ∈ Map(X, S^{1}) : d^{∗}(s^{−1}ds) = 0} ≅ S^{1} × H^{1}(X, iℤ), (2.66)

where, given a point p ∈ X, the isomorphism is defined by s ↦ (s(p), s^{−1}ds). Conversely, we have a map

h_{p} : H^{1}(X; iℤ) → Map(X, S^{1})

defined by setting h_{p}(β) equal to the unique harmonic gauge transformation satisfying h_{p}(β)(p) = 1 (see, for example, [68, §5.4]). The group S^{1} × H^{1}(X; iℤ) acts on C̃_{𝔰} by

((e^{iθ}, β), (B, Ψ)) ↦ (B − β id_{W} , e^{iθ}h_{p}(β)Ψ), (2.67)

where (e^{iθ}, β) ∈ S^{1} × H^{1}(X; iℤ) and (B, Ψ) ∈ C̃_{𝔰}. We then have the following global “slice result” for C_{𝔰}:

Lemma 2.18. Let B_{0} be an L_{k}^{2} spin connection on W. Then the inclusion

Ĉ_{𝔰} = (B_{0} + (Ker d^{∗})id_{W}) × L_{k}^{2}(W^{+}) ↪ C̃_{𝔰}, (2.68) where Ker d^{∗} ⊂ L_{k}^{2}(iΛ^{1}), is equivariant with respect to the action (2.67) of S^{1}×H^{1}(X; iℤ) ⊂ G_{𝔰} and descends to a homeomorphism

(B_{0} + (Ker d^{∗})id_{W}) ×_{S^{1}×H^{1}(X;iℤ)} L_{k}^{2}(W^{+}) → C_{𝔰},

and a diffeomorphism on Ĉ_{𝔰}^{0}, the complement in Ĉ_{𝔰} of the space of zero-section pairs.

Proof. Any L_{k}^{2} spin connection on W can be written as B = B_{0} + b id_{W}, where b ∈ L_{k}^{2}(iΛ^{1}). An element s of the gauge group G_{𝔰} acts on this representation of B by sending b to b−s^{−1}ds. The argument in [68, §7.2] (also see [14, pp. 54–55]) implies that there is a solution of the equation d^{∗}(b − s^{−1}ds) = 0, which is unique up to a harmonic gauge transformation, so the map on quotients is surjective. If (B_{j}, Ψ_{j}), j = 1, 2, are pairs in C̃_{𝔰} such that d^{∗}(B_{j} −B_{0}) = 0 and s ∈ G_{𝔰} satisfies s(B_{1}, Ψ_{1}) = (B_{2}, Ψ_{2}), that is

(B_{1} − s^{−1}ds id_{W} , sΨ_{1}) = (B_{2}, Ψ_{2}),

then d^{∗}(B_{2} − B_{1}) = 0 implies d^{∗}(s^{−1}ds) = 0, so s is harmonic and the map on quotients is injective.

2.4.2. Cohomology ring of the configuration space. We now use the reduction of the gauge group from G_{𝔰}to Ĝ_{𝔰} to compute the integral cohomology ring of C_{𝔰}^{0}. We begin by defining the universal complex line bundle over C_{𝔰}^{0} × X,

𝕃_{𝔰} = C̃_{𝔰}^{0} ×_{G_{𝔰}} \underline{ℂ} ≅ Ĉ_{𝔰}^{0} ×Ĝ_{𝔰} \underline{ℂ}. (2.69)

where \underline{ℂ} = X × ℂ and the action of G_{𝔰} is given for s ∈ G_{𝔰}, x ∈ X and z ∈ ℂ by

(s, (B, Ψ), (x, z)) → (s_{∗}(B, Ψ), (x, s(x)^{−1}z)) . (2.70)

As in [68, §7.4], we define cohomology classes on C_{𝔰}^{0} (the “Seiberg-Witten μ-classes”) by

μ_{𝔰} : H_{•}(X; ℝ) → H^{2−•}(C_{𝔰}^{0}; ℝ), μ_{𝔰}(α) = c_{1}(𝕃_{𝔰})/α, (2.71)

where α is either the positive generator x ∈ H_{0}(X; ℤ) or a class γ ∈ H_{1}(X; ℝ). Let G_{𝔰}^{p} be the subgroup of maps s ∈ G_{𝔰} with s(p) = 1 and note that

G_{𝔰} = G_{𝔰}^{p} × S^{1}. (2.72)

The following alternative characterization of μ_{𝔰}(x) in appears in much of the literature on Seiberg-Witten invariants.

Lemma 2.19. Let C_{𝔰}^{p,0} = C̃_{𝔰}^{0}/G_{𝔰}^{p} be the configuration space of framed pairs. Then

c_{1}(C_{𝔰}^{p,0} ×_{S^{1}} ℂ) = μ_{𝔰}(x). (2.73)

Proof. The class μ_{𝔰}(x) is the first Chern class of 𝕃_{𝔰} restricted to C_{𝔰}^{0} × {p}. Equation (2.73) then follows from the splitting G_{𝔰} = G_{𝔰}^{p} × S^{1}.

Using the isomorphism of universal line bundles in (2.69), we shall now give another description of the cohomology classes μ_{𝔰}(x) and μ_{𝔰}(γ), and show they generate the cohomology ring H^{∗}(C_{𝔰}^{0}; ℤ). We define

Jac(X) = H^{1}(X; iℝ)/H^{1}(X; iℤ). (2.74)

We then have:

Lemma 2.20. Let 𝔰 be a spin^{c} structure on a smooth, closed, oriented four-manifold X. Then, there is an S^{1}-equivariant retraction r : C_{𝔰}^{p,0} → Jac(X) × (L_{k}^{2}(W^{+}) − {0}) where S^{1} acts trivially on Jac(X) and by complex multiplication on L_{k}^{2}(W^{+}).

Proof. From the Hodge decomposition,

L_{k}^{2}(iΛ^{1}) = H^{1}(X; iℝ) ⊕ Ran d ⊕ Ran d^{∗},

where H^{1}(X; iℝ) denotes the harmonic, imaginary-valued one-forms on X, we see that

Ker d^{∗} = (Ran d)^{⊥} = H^{1}(X; iℝ) ⊕ Ran d^{∗}.

Thus, we can write the pre-configuration space Ĉ_{𝔰}^{0} as

Ĉ_{𝔰}^{0} = (B_{0} + H^{1}(X; iℝ) ⊕ Ran d^{∗}) × (L_{k}^{2}(W^{+}) − {0}) .

As usual, the factor S^{1} of the harmonic gauge group Ĝ_{𝔰} = S^{1} × H^{1}(X; iℤ) acts trivially on B_{0} + H^{1}(X; iℝ) ⊕ Ran d^{∗} and by complex multiplication on L_{k}^{2}(W^{+}) − {0}. An element β ∈ H^{1}(X; iℤ) ≅ H^{1}(X; iℤ) acts on an element (B, Ψ) = (B_{0} + b′ + b′′, Ψ) of Ĉ_{𝔰}^{0} (where b′ ∈ H^{1}(X; iℝ) and b′′ ∈ Ran d^{∗}) by

(β, (B_{0} + b′ + b′′, Ψ)) ↦ (B_{0} + (b′ + β) + b′′, h_{p}(β)Ψ).

Thus, replacing b′′ ∈ Ran d^{∗} with tb′′, 0 ≤ t ≤ 1, defines a Ĝ_{𝔰}-equivariant retraction of Ĉ_{𝔰}^{0} onto

(B_{0} + H^{1}(X; iℝ)) × (L_{k}^{2}(W^{+}) − {0}) .

This yields a deformation retraction of Ĉ_{𝔰}^{0} onto the projectivization of the complex vector bundle

(B_{0} + H^{1}(X; iℝ)) ×_{H^{1}(X;iℤ)} L_{k}^{2}(W^{+}) → Jac(X). (2.75)

By Kuiper’s theorem (see Corollary 3.18) this vector bundle is trivial, which completes the proof.

Definition 2.21. Let {γ_{i}} be a basis for H_{1}(X; ℤ)/ Tor and let {γ_{i}^{∗}} be the dual basis for H^{1}(X; ℤ), so ⟨γ_{i}^{∗}, γ_{j}⟩ = δ_{ij}. Then \{\sqrt{−1}γ_{j}^{∗}\} generates π_{1}(Jac(X)). We will write these elements of π_{1}(Jac(X)) as γ_{j}^{J} to avoid confusion. Let γ_{j}^{J,∗} ∈ H^{1}(Jac(X); ℤ) = Hom(π_{1}(Jac(X)), ℤ) be defined by ⟨γ_{i}^{J,∗}, γ_{j}^{J}⟩ = δ_{ij}. Given such a basis {γ_{i}} for H_{1}(X; ℤ)/ Tor, we call {γ_{i}^{J}} and {γ_{i}^{J,∗}} the related generators and basis for π_{1}(Jac(X)) and H^{1}(Jac(X); ℤ) respectively.

Note that H^{1}(X; ℤ) and H^{1}(Jac(X); ℤ) are free ℤ-modules by the universal coefficient theorem [70, Theorem 5.5.3].

Corollary 2.22. Continue the notation and assumptions of Lemma 2.20. Let r : C_{𝔰}^{0} → Jac(X)×(L_{k}^{2}(W^{+}) − {0}) /S^{1} be the retraction given by Lemma 2.20. Then the cohomology ring of C_{𝔰}^{0} is generated by {r^{∗}h, r^{∗}γ_{i}^{J,∗}}, where {γ_{1}^{J,∗}, …, γ_{b_{1}}^{J,∗}} is a basis for H^{1}(Jac(X); ℤ) and h the first Chern class of the S^{1} bundle given by

(B_{0} + H^{1}(X; iℝ)) ×_{H^{1}(X;iℤ)} (L_{k}^{2}(W^{+}) − {0})
↓
(B_{0} + H^{1}(X; iℝ)) ×_{H^{1}(X;iℤ)} (L_{k}^{2}(W^{+}) − {0}) /S^{1}.

2.4.3. Seiberg-Witten μ-classes and generators of the cohomology ring of the configuration space. Finally, we show that the Seiberg-Witten μ-classes can be expressed in terms of the classes r^{∗}h and r^{∗}γ_{i}^{∗} defined in Corollary 2.22. For this purpose, we use the complex line bundle,

Δ = H^{1}(X; iℝ) × X ×_{H^{1}(X;iℤ),h_{p}} ℂ → Jac(X) × X, (2.76)

whose first Chern class we now compute.

Lemma 2.23. Continue the notation of Definition 2.21. Then c_{1}(Δ) = ∑_{i=1}^{b_{1}(X)} γ_{i}^{J,∗} × γ_{i}^{∗}.

Proof. Since H^{1}(X; ℤ) and H^{1}(Jac(X); ℤ) are free, we can write c_{1}(Δ) = c_{0,2} + c_{1,1} + c_{2,0} with respect to the decomposition given by the Kunneth formula:

H^{2}(Jac(X) × X; ℤ) ≅ ⨁_{i=0}^{2} H^{i}(Jac(X); ℤ) ⊗ H^{2−i}(X; ℤ),
c_{i,2−i} ∈ H^{i}(Jac(X); ℤ) ⊗ H^{2−i}(X; ℤ).

For [B] ∈ Jac(X) and p ∈ X, the restrictions of Δ to {[B]} × X and to Jac(X) × {p} are trivial so c_{1}(Δ) = c_{1,1}. We now calculate the first Chern class of Δ restricted to the tori γ_{i}^{J} × γ_{j}. Let γ̃_{i}^{J}(t) be the path in H^{1}(X; iℝ), starting at zero and covering γ_{i}^{J}. Let s : X → S^{1} be an element of Ĝ_{𝔰} satisfying s(p) = 1 and s^{−1}ds = γ̃_{i}^{J}(1). Thus, the degree of the map s : γ_{j} → S^{1} is δ_{ij}. The restriction of the line bundle Δ to γ_{i}^{J} × γ_{j} is given by taking the trivial bundle over I × γ_{j} and making the identification

(0, q, z) ∼ (1, q, s^{−1}(q)(z)), where q ∈ γ_{j}, z ∈ ℂ.

Since the degree of the map s : γ_{j} → S^{1} is δ_{ij}, we see

c_{1}(Δ|_{γ_{i}^{J} ×γ_{j}}) = δ_{ij}γ_{i}^{J,∗} × γ_{j}^{∗}.

This completes the proof.

We now describe the relation between the cohomology classes of Corollary 2.22 and the μ_{𝔰}-map.

Lemma 2.24. Let {γ_{i}} be a basis for H_{1}(X; ℤ)/ Tor, let {γ_{i}^{∗}} be the related basis for H^{1}(Jac(X); ℤ), and let x ∈ H_{0}(X; ℤ) be a generator. Let r_{J} : C_{𝔰}^{0} → Jac(X) be the composition of the retraction r defined in Lemma 2.20 with the projection to Jac(X). Then

c_{1}(𝕃_{𝔰}) = h × 1 + (r_{J} × id_{X})^{∗}c_{1}(Δ), (2.77)

and thus

μ_{𝔰}(x) = r^{∗}h, μ_{𝔰}(γ_{i}) = r_{J}^{∗}γ_{i}^{J,∗}.

Proof. Because r is a retraction, it suffices to compute the restriction of the universal line bundle 𝕃_{𝔰} to the image of the retraction where 𝕃_{𝔰} is given by

H^{1}(X; iℝ) × (L_{k}^{2}(W^{+}) − \{0\}) ×_{S^{1}×H^{1}(X;iℤ)} \underline{ℂ}. (2.78)

The bundle

H^{1}(X; iℝ) ×_{H^{1}(X;iℤ)} (L_{k}^{2}(W^{+}) − {0}) ×_{S^{1}} ℂ ; ↓ ; H^{1}(X; iℝ) ×_{H^{1}(X;iℤ)} (L_{k}^{2}(W^{+}) − {0}) /S^{1} (2.79)

is the restriction of the framed configuration space, C_{𝔰}^{p,0} ×_{S^{1}} ℂ to the image of r and thus has first Chern class h. Then the restriction of 𝕃_{𝔰} in equation (2.78) is isomorphic to the tensor product of the bundle (2.79) with the pullback of the bundle Δ by the map r_{J} ×id_{X}. Thus c_{1}(𝕃_{𝔰}) = h × 1 + (r_{J} × id_{X})^{∗}c_{1}(Δ). The computations of μ_{𝔰}(β) then follow from the definition of the map μ_{𝔰} in equation (2.71) and the computation of c_{1}(Δ) in Lemma 2.23.

3. Singularities and their links

As we shall see in this section, the moduli space M_{𝔱} of solutions to the perturbed PU(2) monopole equations (2.32) is a smoothly stratified space. According to Theorem 2.13, the subspace M_{𝔱}^{∗,0} of solutions to (2.32) which are neither zero-section nor reducible is a smooth manifold of the expected dimension and this comprises the top stratum of M_{𝔱}. In §3.1 we classify the PU(2) monopoles where S^{1} acts trivially, noting that these are given by the zero-section or reducible solutions and comprise the singular points of M_{𝔱}. We exclude the possibility of PU(2) monopoles with flat associated SO(3) connections by a suitable choice of w_{2}(𝔱). We then analyze these lower strata of solutions in M_{𝔱} which are either zero-section or reducible or both and show that they are identified with the moduli space M_{κ}^{w} of anti-self-dual connections on the associated SO(3) bundle 𝔤_{𝔱} and moduli spaces of Seiberg-Witten monopoles, M_{𝔰} (see §3.3). We describe the link in M_{𝔱}^{∗,0} of the stratum M_{κ}^{w} in §3.2 and the links in M_{𝔱}^{∗,0} of the strata M_{𝔰} in §3.4 and §3.5. For M_{κ}^{w}, it will suffice to describe the normal cone (in the sense of [35, p. 41]) at a generic point in M_{κ}^{w}, while in the case of the link of a stratum M_{𝔰}, a global description is required. Finally, in §3.6 we compute the Chern character of the normal bundle of M_{𝔰} with respect to a finite-dimensional, open, S^{1}-equivariant smooth manifold containing a neighborhood of the image of M_{𝔰} in M_{𝔱}.

3.1. Classification of fixed points under the circle action. In this section we classify the fixed points in M_{𝔱} under the circle action induced by scalar multiplication on V.

For pairs (A, Φ) in C̃_{𝔱}, it is useful to distinguish between two kinds of circle action. First, S^{1} can act on V by scalar multiplication:

S^{1} × V → V, (e^{iθ}, Φ) ↦ e^{iθ}Φ. (3.1)

Second, if V = W ⊕ W ⊗ L is a direct sum of Clifford modules, then S^{1} can act by scalar multiplication on the factor W ⊗ L and the identity on W:

S^{1} × V → V, (e^{iθ}, Ψ ⊕ Ψ′) ↦ Ψ ⊕ e^{iθ}Ψ′, (3.2)

where Ψ ∈ C^{∞}(W) and Ψ′ ∈ C^{∞}(W ⊗ L). With respect to the splitting V = W ⊕ W ⊗ L, the two actions are related by

(\begin{matrix}1 & 0 \\ 0 & e^{i2θ} \end{matrix}) = e^{iθ}u, where u = (\begin{matrix}e^{−iθ} & 0 \\ 0 & e^{iθ} \end{matrix}) ∈ G_{𝔱},

and so, when we pass to the induced circle actions on the quotient C_{𝔱} = C̃_{𝔱}/G_{𝔱}, we obtain the same space C_{𝔱}/S^{1} for both circle actions.

If we write V = W ⊗ E, where E = \underline{ℂ} ⊕ L, the circle action (3.2) on V is equivalent to the one induced by scalar multiplication on the complex line bundle L and the trivial action on \underline{ℂ}. Hence, the induced circle action (3.3) on 𝔰𝔲(E) ⊂ 𝔰𝔲(V) with respect to the decomposition (2.13) takes the form

S^{1} × 𝔰𝔲(E) → 𝔰𝔲(E), (e^{iθ}, (ν, z)) ↦ (ν, e^{iθ}z), (3.3) with respect to the isomorphism 𝔰𝔲(E) ≅ i\underline{ℝ} ⊕ L in Lemma 3.10: the circle acts as the identity on i\underline{ℝ} and by scalar multiplication on the complex line bundle L. On the other hand, the circle action (3.1) on V induces the trivial action on both 𝔰𝔲(V) and 𝔰𝔲(E).

Proposition 3.1. Let 𝔱 = (ρ, V) be a spin^{u} structure over a closed, oriented, smooth fourmanifold X, with b_{2}^{+}(X) ≥ 1 and generic Riemannian metric. If Â is a non-flat connection on 𝔤_{𝔱}, then [A, Φ] ∈ M_{𝔱} is a fixed point with respect to the S^{1} action on M_{𝔱} if and only if one of the following hold:

1. The connection Â is anti-self-dual, irreducible, and Φ ≡ 0. The pair (A, Φ) is a fixed point with respect to the circle action (3.1).

2. The pair (A, Φ) is reducible with respect to a splitting V = W ⊕ W ⊗ L but Φ ≢ 0,

(A, Φ) = (B ⊕ B ⊗ A_{L}, Ψ ⊕ 0).

The pair (A, Φ) is a fixed point with respect to the circle action (3.2). The connection Â on 𝔤_{𝔱} is reducible with respect to the splitting 𝔤_{𝔱} ≅ i\underline{ℝ} ⊕ L, with Â = d_{ℝ} ⊕ A_{L}.

Proof. Suppose [A, Φ] ∈ M_{𝔱} is a fixed point of the S^{1} action on M_{𝔱}. Consequently, there is an element e^{iθ} ≠ ±1 such that e^{iθ}[A, Φ] = [A, Φ] and hence a gauge transformation u ∈ G_{𝔱} such that

(A, e^{iθ}Φ) = u(A, Φ) = (u(A), uΦ).

Thus, u is in the stabilizer of A, and hence that of Â, and e^{iθ}Φ = uΦ.

(1) If Â has the trivial stabilizer {±id_{V} } in G_{𝔱}, then e^{iθ}Φ = ±Φ and so Φ ≡ 0 because e^{iθ} ≠ ±1. The connection Â is irreducible [29, Theorem 10.8] since it is an SO(3) connection with minimal stabilizer in G_{𝔱}. The curvature equation in (2.32) implies that F + ; Â = 0, so Â is anti-self-dual. The pair (A, Φ) is fixed by the S^{1} action (3.1).

(2) If Â has non-trivial stabilizer in G_{𝔱}, then Â is reducible with respect to a splitting 𝔤_{𝔱} ≅ i\underline{ℝ} ⊕ L [29, Theorem 10.8], for some complex line bundle L, and takes the form Â = d_{ℝ} ⊕ A_{L}. The connection Â has stabilizer SO(2) ≅ S^{1} acting on L by complex multiplication and trivially on i\underline{ℝ}. Lemma 2.9 implies that A is reducible with respect to the splitting V = W ⊕ W ⊗ L, taking the form A = B ⊕ B ⊗ A_{L}.

If u = ±id_{V}, then Φ ≡ 0 just as in case (1) and so Â would be a reducible, anti-self-dual SO(3) connection on 𝔤_{𝔱}. But b_{2}^{+}(X) > 0 and the Riemannian metric g on X is generic, so this possibility is excluded by Corollary 4.3.15 in [14] (the four-manifold X does not need to be simply connected).

Hence, we must have u ≠ ±id_{V} and Φ ≢ 0. Because Ad (u) stabilizes Â, it induces an action on L via complex multiplication by some e^{−i2μ} ≠ 1 and the trivial action on i\underline{ℝ}, as in (3.3). Hence, with respect to the splitting V = W ⊕ W ⊗ L, the gauge transformation u takes the form

u = (\begin{matrix}e^{iμ} & 0 \\ 0 & e^{−iμ} \end{matrix}) .

Because e^{−iθ}uΦ = Φ and writing Φ = Ψ ⊕ Ψ′ with respect to the splitting V = W ⊕ W ⊗ L, the section Φ is fixed by

e^{−iθ}u = (\begin{matrix}e^{i(μ−θ)} & 0 \\ 0 & e^{−i(μ+θ)} \end{matrix}) .

If Ψ ≢ 0, then μ = θ (mod 2π) and we must have Ψ′ ≡ 0; conversely, if Ψ′ ≢ 0, then μ = −θ (mod 2π) and we must have Ψ ≡ 0. The two cases differ only by how the factors in the splitting of V are labeled, so we can assume that Ψ′ ≡ 0 and Ψ ∈ C^{∞}(W^{+}). In particular, the pair (A, Φ) is fixed by the S^{1} action (3.2).

Our proof that Φ is a section of W^{+} when Â is a non-flat reducible connection could be replaced by an appeal to Lemma 5.22 in [24], but the argument here seems more direct.

Proposition 3.1 only classifies the fixed points [A, Φ] ∈ M_{𝔱} of the circle action under the assumption that Â is not flat. However, the following lemma gives a simple condition which guarantees that there will be no pairs in M_{𝔱} with flat associated SO(3) connections. Because it relies only on a choice of integral class w (mod 2), the lemma applies simultaneously to all spin^{u} structures 𝔱 = (ρ, V) with a fixed w_{2}(𝔱) ≡ w (mod 2) and all oriented, orthogonal 3-plane bundles F with w_{2}(F) ≡ w (mod 2), and thus simultaneously to all levels of the Uhlenbeck compactifications M̄_{κ}^{w} and M̄_{𝔱}.

Lemma 3.2. [57, p. 226] Let X be a closed, oriented four-manifold. Then

• If w ∈ H^{2}(X; ℤ) and e ∈ H_{2}(X; ℤ) is a spherical class such that ⟨w, e⟩ ≢ 0 (mod 2), then no oriented, orthogonal 3-plane bundle F over X with w_{2}(F) ≡ w (mod 2) admits a flat connection.

• In particular, if w ∈ H^{2}(X; ℤ) and X̃ = X#\overline{ℂℙ}^{2} is the blow-up, with exceptional class e = [\overline{ℂℙ}^{1}] ∈ H_{2}(X; ℤ) and Poincaré dual e^{∗} ∈ H^{2}(X; ℤ), then no SO(3) bundle F over X̃ with w_{2}(F) = w + e^{∗} (mod 2)admits a flat connection.

Therefore, if X has a spherical class e ∈ H_{2}(X; ℤ) such that ⟨w, e⟩ ≠ 0 (mod 2), then Lemma 3.2 implies that there are no pairs in M_{𝔱} with flat SO(3) connections when w_{2}(𝔱) ≡ w (mod 2). In this situation, Proposition 3.1 implies that the only fixed points of the circle action on M_{𝔱} are given by either points [A, Φ] with A irreducible and Φ ≡ 0 or A reducible and Φ ≢ 0.

Suppose w ∈ H^{2}(X; ℤ) is any integral class and that 𝔱 = (ρ, V) is a spin^{u} structure over X̃ with w_{2}(𝔱) ≡ w + e^{∗} (mod 2). Then the SO(3) bundle 𝔤_{𝔱} over X̃ obeys the Morgan-Mrowka criterion of Lemma 3.2. Because the Donaldson and Seiberg-Witten invariants of X̃ determine and are determined by those of X, (see [26], [27], [57]), no information about these invariants is lost by passing to the blow-up in this way.

In Lemma 3.13 we show that the moduli space of Seiberg-Witten monopoles, M_{𝔰}^{0}, can be identified with the subspace of PU(2) monopoles in M_{𝔱}^{0} which are reducible with respect to the splitting 𝔱 = 𝔰 ⊕ 𝔰 ⊗ L, while zero-section points in M_{𝔰} are mapped to zero-section reducibles in M_{𝔱} (though not necessarily injectively). If w_{2}(𝔱) obeys the Morgan-Mrowka criterion then Proposition 3.1 implies that reducible PU(2) monopoles cannot be zerosection pairs, leading to the following

Corollary 3.3. If M_{𝔰} ↪ M_{𝔱} is a Seiberg-Witten moduli subspace and w_{2}(𝔱) obeys the Morgan-Mrowka criterion of Lemma 3.2, then M_{𝔰} contains no zero-section solutions.

3.2. The link of the stratum of anti-self-dual PU(2) monopoles. In this section we describe the Kuranishi model for a neighborhood of a zero-section solution in the PU(2) monopole moduli space and define a link of the stratum M_{κ}^{w} in M_{𝔱}. The differential D𝔖 will not be surjective at all the zero-section monopoles (A, 0), but we will show that the cokernel of (D𝔖)_{A,0}, namely H_{A,0}^{2}, can be identified with the cokernel of the perturbed Dirac operator, D_{A} + ρ(ϑ).

If we are given a decomposition, V = W ⊗ E, Lemma 2.7 provides a canonical identification of automorphism groups, G_{κ}^{w} ≅ G_{𝔱}, and with respect to this identification and choice of fixed connection A_{Λ} on det^{\frac{1}{2}} (V^{+}), Lemma 2.11 provides a canonical gauge-equivariant isomorphism

A_{κ}^{w} ≅ A_{𝔱}, Â ↦ A,

where the space A_{κ}^{w} of SO(3) connections on 𝔰𝔲(E) ≅ 𝔤_{𝔱} was defined in §2.1.6, with p_{1}(𝔤_{𝔱}) = −4κ and w_{2}(𝔤_{𝔱}) ≡ w (mod 2).

Let B_{𝔱} = A_{𝔱}/G_{𝔱} be the quotient of the space of L_{k}^{2} spin connections on V. By the preceding discussion there is a canonical homeomorphism,

B_{κ}^{w} ≅ B_{𝔱}, [Â] ↦ [A],

restricting to diffeomorphisms on smooth strata. There are a canonical smooth embedding A_{𝔱} ↪ C̃_{𝔱} and a “smoothly stratified embedding” B_{𝔱} ↪ C_{𝔱}—that is, a topological embedding restricting to smooth embeddings on smooth strata—given by A ↦ (A, 0) and [A] ↦ [A, 0]. Combining these identifications and embeddings, we obtain a gauge-equivariant smooth embedding and a smoothly stratified embedding,

ι : A_{κ}^{w} ↪ C̃_{𝔱}, Â ↦ (A, 0) and ι : B_{κ}^{w} ↪ C_{𝔱}, [Â] ↦ [A, 0]. (3.4)

Hence, we can identify the image of the induced smoothly stratified embedding

ι : M_{κ}^{w}(X) ↪ C_{𝔱}, [Â] ↦ [A, 0]. (3.5)

with the subspace of M_{𝔱} given by the zero-section solutions to the PU(2) monopole equations (2.32). Therefore, we shall refer to pairs or connections representing points in M_{κ}^{w} as zero-section PU(2) monopoles or anti-self-dual connections, depending on the context.

For a generic Riemannian metric on X, the moduli space M_{κ}^{w,∗} of irreducible anti-self-dual connections is a smooth manifold [14, Corollary 4.3.18], [29] of the expected dimension d_{a}(𝔱) given in equation (2.51).

At a zero-section PU(2) monopole (A, 0), the elliptic deformation complex (2.47) splits into a direct sum of complexes:

\begin{matrix}C^{∞}(𝔤_{𝔱}) & ⟶^{dÂ} & C^{∞}(Λ^{1} ⊗ 𝔤_{𝔱}) & ⟶^{d+ ; Â} & C^{∞}(Λ^{+} ⊗ 𝔤_{𝔱}), \\ C^{∞}(V^{+}) & ⟶^{D_{A}+ρ(ϑ)} & C^{∞}(V^{−}). \end{matrix} (3.6)

The first complex in (3.6) is simply the elliptic deformation complex for the moduli space M_{κ}^{w} of anti-self-dual connections, with cohomology H• ; Â. The following lemma is then a clear consequence of the preceding decomposition:

Lemma 3.4. If (A, 0) is a zero-section PU(2) monopole then there are canonical isomorphisms,

H_{A,0}^{0} ≅ H0 ; Â,
H_{A,0}^{1} ≅ H1 ; Â ⊕ Ker(D_{A} + ρ(ϑ)),
H_{A,0}^{2} ≅ H2 ; Â ⊕ Coker(D_{A} + ρ(ϑ)).

If the manifold X is not simply connected, there exist anti-self-dual connections which, while not globally reducible, become reducible when restricted to certain open sets in X: this complicates the construction of the restriction maps used in the definition of geometric representatives on moduli spaces of anti-self-dual connections in [44] and for PU(2) monopoles here. Therefore, we recall some of the technical points from [44, pp. 586–588] which we shall need to address these additional complications when X is not simply connected. A connection Â on an SO(3) bundle F over X is called a twisted reducible if it preserves a splitting F = λ ⊕ N, where λ is a non-trivial real line bundle and N is an O(2) bundle. The curvature FÂ of a twisted reducible connection Â has rank one even though Â is not reducible. When restricted to open sets in X over which the real line bundle λ is trivial, the connection Â becomes reducible. Let H^{i}(X; λ) denote the cohomology group of dimension b^{i}(λ) with coefficients in the local system given by λ. The bundle λ ⊗ λ is trivial and so the cup product gives a pairing H^{2}(X; λ) ⊗ H^{2}(X; λ) → H^{4}(X; ℝ) ≅ ℝ. We define b^{+}(λ) and b^{−}(λ) to be the maximum dimensions of subspaces of H^{2}(X; λ) on which this pairing is respectively positive or negative definite. If F admits a reduction λ ⊕ N, then the corresponding subspace of twisted reducibles in M_{κ}^{w} (where p_{1}(F) = −4κ and w_{2}(F) ≡ w (mod 2)) defined by this splitting is a torus of dimension b^{1}(λ). Whether or not such connections exist in M_{κ}^{w} for a generic metric depends on b^{+}(λ), as described in the following lemma.

Lemma 3.5. [44, Lemma 2.4 & Corollary 2.5] Let X be a smooth four-manifold, λ a nontrivial real line bundle on X, and F an oriented, orthogonal three-plane bundle for which F admits a reduction F = λ ⊕ N. Suppose p_{1}(F) = −4κ ≠ 0 and w_{2}(F) ≡ w (mod 2), where w ∈ H^{2}(X; ℤ).

1. If b^{+}(λ) = 0, then b^{1}(λ) = −(b^{+} − b^{1} + 1) and M_{κ}^{w} contains twisted reducibles corresponding to F = λ ⊕ N for all Riemannian metrics on X. If, in addition, b^{1}(λ) ≥ 0, then −2p_{1}(F) − 3(1 − b^{1} + b^{+}) > b^{1}(λ) and if the Riemannian metric on X is generic, each anti-self-dual connection Â, which is a twisted reducible with respect to F = λ⊕N,

has H2 ; Â = 0.

2. If b^{+}(λ) ≥ 1, then for generic Riemannian metrics on X, the space M_{κ}^{w} contains no twisted reducibles for the splitting F = λ ⊕ N.

Therefore, when b_{2}^{+}(X) > 0, the only twisted reducibles appearing in M_{κ}^{w} have codimension at least −2p_{1}(𝔤_{𝔱}) − 2(b_{2}^{+} − b_{1} + 1) and are smooth points. The Kuranishi lemma, [14, Proposition 4.2.19] or [29, Lemma 4.7], gives the following description of a neighborhood of a zero-section monopole in M_{𝔱}.

Corollary 3.6. Let 𝔱 be a spin^{u} structure on a closed, oriented, smooth four-manifold X with b_{2}^{+}(X) > 0 and generic Riemannian metric. Let [A, 0] be a point in the image of M_{κ}^{w} ↪ M_{𝔱}, so H0 ; Â = 0 = H2 ; Â for [Â] ∈ M_{κ}^{w}. Then there are

• An open, S^{1}-invariant neighborhood O_{A} of the origin in TÂM_{κ}^{w} ⊕ Ker(D_{A} + ρ(ϑ)) together with a smooth, S^{1}-equivariant embedding

γ_{A} : O_{A} ⊂ TÂM_{κ}^{w} ⊕ Ker(D_{A} + ρ(ϑ)) ↪ C̃_{𝔱},

with γ_{A}(0, 0) = (A, 0) and M_{𝔱} ∩ γ_{A}(O_{A}) an open neighborhood of [A, 0] in M_{𝔱}, and • A smooth, S^{1}-equivariant map

φ_{A} : O_{A} ⊂ TÂM_{κ}^{w} ⊕ Ker(D_{A} + ρ(ϑ)) → Coker(D_{A} + ρ(ϑ))

such that γ_{A} restricts to an S^{1}-equivariant, smoothly-stratified diffeomorphism from φ_{A}^{−1}(0) ∩ O_{A} onto M_{𝔱} ∩ γ_{A}(O_{A}).

If n_{a} ≤ 0, then at a generic point [A, 0] ∈ M_{𝔱} where Ker(D_{A} + ρ(ϑ)) = {0}, (assuming the map from M_{κ}^{w} to the space of Fredholm operators of index n_{a} is transverse to the “jumping lines strata” as described in [41]) the Kuranishi model in Lemma 3.6 shows that a neighborhood of [A, 0] ∈ M_{𝔱} contains no elements of M_{𝔱}^{0}. Thus, such points in the antiself-dual moduli space are isolated from the subspace M_{𝔱}^{0} of non-zero-section points. For this reason, we will restrict our attention to the cases where n_{a} > 0.

Although we can only describe a neighborhood of the anti-self-dual connections locally, because of the problem of spectral flow, we can still introduce a global, codimension-one subspace of the compactification M̄_{𝔱} which will serve as a link. This space might not have a fundamental class because it is not known to have locally finite topology near the lower strata of the Uhlenbeck compactification M̄_{𝔱}. However, we shall see that the local Kuranishi model in Corollary 3.6 will suffice to define intersections under some additional assumptions.

Definition 3.7. The link of M̄_{κ}^{w} in M̄_{𝔱} is given by

L_{𝔱,κ}^{w,ε} = {[A, Φ, x] ∈ M̄_{𝔱}/S^{1} : ∥Φ∥_{L^{2}}^{2} = ε}.

We write L_{𝔱,κ}^{w} when the positive constant ε is understood.

Lemma 3.8. For generic ε > 0, the link L_{𝔱,κ}^{w,ε} is closed under the S^{1} action, is a smoothly stratified, closed subspace of M̄_{𝔱}, and has codimension one in every stratum of M̄_{𝔱} which it intersects.

Proof. That L_{𝔱,κ}^{w,ε} is closed under the S^{1} action follows directly from the definitions. It is clear that the function

ℓ : M̄_{𝔱} → ℝ, [A, Φ, x] ↦ ∥Φ∥_{L^{2}}^{2},

is smooth on each stratum of M̄_{𝔱}: we claim it is continuous on M̄_{𝔱}. Let [A_{α}, Φ_{α}] be a sequence of points in M̄_{𝔱} which converge to [A_{∞}, Φ_{∞}, x]. We may assume, by an appropriate choice of a sequence of L_{k+1}^{2} spin^{u} gauge transformations of V, that the sequence {Φ_{α}} converges to Φ_{∞} in C^{∞} on X − B(x, r), where we define B(x, r) = ∪_{x∈x}B(x, r). Therefore,

|∥Φ_{α}∥_{L^{2}(X)}^{2} − ∥Φ_{∞}∥_{L^{2}(X)}^{2}| ≤ ∥Φ_{α} − Φ_{∞}∥_{L^{2}(X−B(x,r))}^{2} + ∥Φ_{α} − Φ_{∞}∥_{L^{2}(B(x,r))}^{2}
≤ ∥Φ_{α} − Φ_{∞}∥_{L^{2}(X−B(x,r))}^{2} + Cr^{4},

where the second inequality follows from the universal a priori C^{0} bound for the sequence Φ_{α} and Φ_{∞} given by [24, Lemma 4.4]. Thus,

lim sup_{α→∞} |∥Φ_{α}∥_{L^{2}(X)}^{2} − ∥Φ_{∞}∥_{L^{2}(X)}^{2}| ≤ Cr^{4}, for all r > 0,

and so

lim_{α→∞} ∥Φ_{α}∥_{L^{2}(X)}^{2} = ∥Φ_{∞}∥_{L^{2}(X)}^{2},

as desired. A generic ε > 0 is a regular value for the function ℓ on each smooth stratum of M̄_{𝔱}. For such an ε, the preimage ℓ^{−1}(ε) is a smooth submanifold of each stratum and because the function ℓ : M̄_{𝔱} → ℝ is continuous, these smooth submanifolds fit together to form a smoothly stratified subspace of M̄_{𝔱} (see Remark 3.3 in [18]).

Remark 3.9. In defining a link, it might seem more natural to work with the image of the ε-sphere in the normal bundle given by Ker(D_{A} + ρ(ϑ)), at least on the image of M_{κ}^{w} ↪ M_{𝔱} where the cokernel of the Dirac operator vanishes. This definition would have the disadvantage of not being a global object because of the jumping-line problem (that is, spectral flow). However, it can be shown that the two functions defined on the open set O of the Kuranishi model of [A, 0] in Corollary 3.6, one given by the L^{2} norm of the element of Ker(D_{A} + ρ(ϑ)), the other defined by ℓ ∘ γ are C^{1} close as ε goes to zero, as they differ by the difference of γ and the identity. As we shall see in §3.4.2 in [18] it is sufficient that these two links are cobordant.

3.3. Reducible PU(2) monopoles and their identification with Seiberg-Witten monopoles. In this section we show that the subspaces of reducible PU(2) monopoles in M_{𝔱} can be identified with the moduli spaces of Seiberg-Witten monopoles defined in §2.3.

3.3.1. Decomposing PU(2) bundles. We now describe the canonical isomorphisms of associated bundles induced from a splitting E = L_{1} ⊕ L_{2} of a Hermitian two-plane bundle E; the lemmas below are of course elementary and we just state them in order to make our conventions clear.

Lemma 3.10. If E is a Hermitian two-plane bundle and L_{1}, L_{2} are Hermitian line bundles over a manifold X such that E = L_{1} ⊕ L_{2}, then there is a canonical isometry of SO(3) bundles, 𝔰𝔲(E) ≅ i\underline{ℝ} ⊕ (L_{2} ⊗ L_{1}^{∗}), given by

(\begin{matrix}ν & −z̄ \\ z & −ν \end{matrix}) ⟼ (ν, z). (3.7)

Proof. With respect to the decomposition E = L_{1} ⊕ L_{2}, any element M ∈ 𝔤𝔩(E) takes the form

M = (\begin{matrix}M_{11} & M_{12} \\ M_{21} & M_{22} \end{matrix}) ,

where M_{jk} ∈ L_{j} ⊗ L_{k}^{∗}, for 1 ≤ j, k ≤ 2. Thus, M_{11}, M_{22} ∈ ℂ, after identifying L_{1} ⊗ L_{1}^{∗} ≅ L_{2} ⊗ L_{2}^{∗} ≅ ℂ. If M ∈ 𝔰𝔲(E), then M_{11} ∈ iℝ and M_{12} ∈ L_{1} ⊗ L_{2}^{∗} with M_{22} = −M_{11} and M_{21} = −M̄_{12}, so any element of 𝔰𝔲(E) takes the shape

M = (\begin{matrix}ν & −z̄ \\ z & −ν \end{matrix}) , ν ∈ ℂ, z ∈ L_{2} ⊗ L_{1}^{∗},

and the desired isomorphism 𝔰𝔲(E) ≅ i\underline{ℝ} ⊕ (L_{2} ⊗ L_{1}^{∗}) is given by M ↦ (ν, z).

We recall from [24, §2.4] that the induced fiber inner product on 𝔤𝔩(E) is defined by ⟨M, M′⟩ = \frac{1}{2} tr(M′M^{†}). Thus, if M′, M ∈ C^{∞}(𝔰𝔲(E)) correspond to (ν′, z′), (ν, z) ∈ C^{∞}(iΛ^{0})) ⊕ C^{∞}(L), respectively, we see that

\begin{aligned}⟨M′, M⟩ & = \frac{1}{2} tr(M′M^{†}) = \frac{1}{2} tr (\begin{matrix}ν′ & −z̄′ \\ z′ & −ν′ \end{matrix}) (\begin{matrix}ν̄ & −z̄ \\ z & −ν̄ \end{matrix}) \\ & = ν′ν̄ + \frac{1}{2}(z′z̄ + z̄′z) = ν′ν̄ + Re⟨z′, z⟩_{ℂ}, \end{aligned} (3.8)

and so the isomorphism of SO(3) bundles is an isometry.

3.3.2. The identification of reducible PU(2) monopoles with Seiberg-Witten monopoles. Next, we explain how reducible pairs in the moduli space of PU(2) monopoles may be identified with Seiberg-Witten monopoles.

Recall that C_{𝔱} is the quotient space of pairs whose spin^{u} connections induce the fixed unitary connection 2A_{Λ} on the complex line bundle det(V^{+}).

Lemma 3.11. Let 𝔰 = (ρ, W) be a spin^{c} structure over an oriented, Riemannian fourmanifold X and let 𝔱 = (ρ, V) be a spin^{u} structure with V = W ⊕ W ⊗ L. Then

ι : C̃_{𝔰} ↪ C̃_{𝔱}, (B, Ψ) ↦ (B ⊕ B ⊗ A_{L}, Ψ ⊕ 0), (3.9)

is a smooth embedding, where A_{L} = A_{Λ} ⊗ (B^{det})^{∗}, and is gauge-equivariant with respect to

ϱ : G_{𝔰} ↪ G_{𝔱}, s ↦ s id_{W} ⊕ s^{−1} id_{W ⊗L}, (3.10)

so that

ι(s(B, Ψ)) = ϱ(s)ι(B, Ψ).

The image of the map (3.9) contains all pairs in C̃_{𝔱} fixed by the action (3.2) of S^{1} on V. The induced map,

ι : C_{𝔰} ↪ C_{𝔱}, [B, Ψ] ↦ [B ⊕ B ⊗ A_{L}, Ψ ⊕ 0], (3.11) is continuous and, when restricted to C_{𝔰}^{0}, a topological embedding. If w_{2}(𝔱) ≠ 0, then (3.11) is a topological embedding of C_{𝔰}. If w_{2}(𝔱) = 0 then the map (3.11) takes zero-section points in C_{𝔰} to zero-section reducibles in C_{𝔱}, although this identification of zero-section points need not be injective if b_{1}(X) > 0.

Proof. The map ι : C̃_{𝔰} → C̃_{𝔱} is clearly a C^{∞} embedding. Furthermore,

ϱ(s)ι(B, Ψ) = (s_{∗}B ⊕ (s^{−1})_{∗}(B ⊗ A_{L}), sΨ ⊕ 0).

Since A_{L} = A_{Λ} ⊗ (B^{det})^{∗}, we see that s ∈ G_{𝔰} acts on A_{L} as (s^{−2})_{∗}A_{L} (see also §3.4.2) and so (s^{−1})_{∗}(B ⊗ A_{L}) = s_{∗}B ⊗ (s^{−2})_{∗}A_{L}. Thus, ϱ(s)ι(B, Ψ) = ι(s_{∗}B, sΨ), as desired.

Next, we characterize the image of the map ι : C̃_{𝔰} → C̃_{𝔱}. Suppose (A, Φ) ∈ C̃_{𝔱} is fixed by the S^{1} action (3.2) on V: this action descends to the action (3.3) on 𝔤_{𝔱} = i\underline{ℝ} ⊕ L, which fixes the induced connection Â. As in the proof of Assertion (2) of Proposition 3.1, the connection Â must then be reducible with respect to this splitting, taking the form Â = d_{ℝ} ⊕ A_{L} on 𝔤_{𝔱} and A = B ⊕ B ⊗ A_{L} on V. Thus, (A, Φ) is in the image of ι.

If U ⊂ C_{𝔰} is an open subset, so it is easy to see that ι(U) is open in ι(C_{𝔰}) with respect to the subspace topology induced by C_{𝔱}.

We now show ι : C_{𝔰}^{0} → C_{𝔱} is injective and that ι : C_{𝔰} → C_{𝔱} is injective when w_{2}(𝔱) ≢ 0. Suppose (B, Ψ) and (B′, Ψ′) are pairs in C̃_{𝔰}^{0} and that u ∈ G_{𝔱} satisfies uι(B, Ψ) = ι(B′, Ψ′), and thus [ι(B, Ψ)] = [ι(B′, Ψ′)] in C_{𝔱}. With respect to the decomposition V = W ⊗ E and splitting E = \underline{ℂ} ⊕ L, we can consider u to be a gauge transformation of E = \underline{ℂ} ⊕ L via the isomorphism G_{κ}^{w} ≅ G_{𝔱} in Lemma 2.7 (and acting as the identity on W) and write

u = (\begin{matrix}s & v \\ −v̄ & s̄ \end{matrix}) , where s ∈ Map(X, ℂ) and v ∈ C^{∞}(Hom(L, ℂ)),

noting that u^{−1} = u^{†} and det(u) = 1. As in the proof of Lemma 2.9, we may write ι(B) = B ⊗ A_{E} and ι(B′) = B ⊗ A′_{E}, where A_{E} and A′_{E} = u(A_{E}) are U(2) connections on E which are reducible with respect to the splitting E = \underline{ℂ} ⊕ L. Let ϱ_{L} : S^{1} → End(E) denote the action of S^{1} on E by the trivial action on \underline{ℂ} and scalar multiplication on L. Then ϱ_{L}(S^{1}) fixes u(A_{E}) and thus for all real θ

u^{−1}ϱ_{L}(e^{iθ})u ∈ Stab_{U(2)}(A_{E}) = S^{1} × S^{1},

where Stab_{U(2)}(A_{E}) is the stabilizer of A_{E} in the space of unitary automorphisms of E. Consequently, for every real θ there are real constants μ and ν such that

(\begin{matrix}1 & 0 \\ 0 & e^{iθ} \end{matrix}) (\begin{matrix}s & v \\ −v̄ & s̄ \end{matrix}) = (\begin{matrix}s & v \\ −v̄ & s̄ \end{matrix}) (\begin{matrix}e^{iμ} & 0 \\ 0 & e^{iν} \end{matrix}) ,

and we can assume e^{iθ} ≠ 1. Simplifying, this becomes

(\begin{matrix}s & v \\ −e^{iθ}v̄ & e^{iθ}s̄ \end{matrix}) = (\begin{matrix}e^{iμ}s & e^{iν}v \\ −e^{iμ}v̄ & e^{iν}s̄ \end{matrix}) .

If s ≢ 0, then we must have e^{iμ} = 1 and thus v = 0 because e^{iθ}v̄ = v̄. Since det(u) = |s|^{2} +|v|^{2} = 1, we see that s ∈ Map(X, S^{1}) ≅ G_{𝔰}. Hence, u = ϱ(s) and [B, Ψ] = [B′, Ψ′] ∈ C_{𝔰} because s(B, Ψ) = (B′, Ψ′).

It remains to consider the case s ≡ 0, for which we must then have vv̄ = 1. First suppose Ψ ≢ 0 and observe that

u(Ψ ⊕ 0) = sΨ ⊕ −v̄Ψ = Ψ′ ⊕ 0,

so v̄Ψ ≡ 0 on X and hence Ψ ≡ 0, contradicting our hypothesis that [B, Ψ] ∈ C_{𝔰}^{0}. Therefore the case s ≡ 0 cannot occur in this situation. Thus, ι : C_{𝔰}^{0} → C_{𝔱} is injective.

Otherwise, if s ≡ 0, suppose w_{2}(𝔱) ≠ 0 and observe that the automorphism u ∈ Aut(\underline{ℂ}⊕L) induces an isomorphism L ≅ \underline{ℂ}. But c_{1}(E) = c_{1}(L) and c_{1}(E) ≡ w_{2}(𝔱) (mod 2): by the hypothesis in the final statement of the lemma, w_{2}(𝔱) ≠ 0, so this contradicts L ≅ \underline{ℂ} and therefore the case s ≡ 0 cannot occur in this situation either. Thus, if w_{2}(𝔱) ≠ 0, the map ι : C_{𝔰} → C_{𝔱} is injective.

The following lemma highlights the key property of the PU(2) monopole equations:

Lemma 3.12. Let (ρ, V) be a spin^{u} structure on X. Suppose (A, Φ) = (B ⊕B ⊗A_{L}, Ψ⊕0) is a reducible pair with respect to a splitting V = W ⊕ W ⊗ L, where B is a spin connection on W, A_{L} is a unitary connection on a complex line bundle L, and Ψ is a section of W^{+}. Then (A, Φ) solves the PU(2) monopole equations (2.32) if and only if (B, Ψ) solves the Seiberg-Witten equations (2.55) with perturbation η = F_{A_{Λ}}^{+}.

Proof. Suppose that (A, Φ) solves the PU(2) monopole equations (2.32). With respect to the splitting 𝔤_{𝔱} = i\underline{ℝ} ⊕ L, Lemma 2.9 implies that Â = d_{ℝ} ⊕ A_{L}, where d_{ℝ} is the product connection on i\underline{ℝ} and A_{L} = A_{Λ} ⊗(B^{det})^{∗} is a unitary connection on the complex line bundle L = det^{\frac{1}{2}} (V^{+}) ⊗ det(W^{+})^{∗}. Let E = \underline{ℂ} ⊕ L and observe that Ã = d_{ℂ} ⊕ A_{L} is a reducible unitary connection on E which is a lift of Â on 𝔰𝔲(E). Then

F + ; Ã = (\begin{matrix}0 & 0 \\ 0 & F_{A_{L}}^{+} \end{matrix}) = F_{A_{L}}^{+} ⊗ (\begin{matrix}0 & 0 \\ 0 & 1 \end{matrix}) ∈ C^{∞}(Λ^{+} ⊗ 𝔲(E)),

and therefore, since F + ; Â ∈ C^{∞}(Λ^{+} ⊗ 𝔰𝔬(𝔰𝔲(E))), we have

ad^{−1}(F + ; Â) = (F + ; Ã)_{0} = −F_{A_{L}}^{+} ⊗ (\begin{matrix}\frac{1}{2} & 0 \\ 0 & −\frac{1}{2} \end{matrix}) ∈ C^{∞}(Λ^{+} ⊗ 𝔰𝔲(E)). (3.12)

If Φ = Ψ ⊕ 0, where Ψ ∈ Ω^{0}(W^{+}) and writing 𝔲(V^{+}) = 𝔲(W^{+}) ⊗ 𝔲(E), we have

Φ ⊗ Φ^{∗} = (\begin{matrix}Ψ ⊗ Ψ^{∗} & 0 \\ 0 & 0 \end{matrix}) = (Ψ ⊗ Ψ^{∗}) ⊗ (\begin{matrix}1 & 0 \\ 0 & 0 \end{matrix}) ∈ C^{∞}(𝔲(W^{+}) ⊗ 𝔲(E)).

Hence, projecting to 𝔰𝔲(W^{+}) ⊗ 𝔰𝔲(E), we see that

(Φ ⊗ Φ^{∗})_{00} = (Ψ ⊗ Ψ^{∗})_{0} ⊗ (\begin{matrix}\frac{1}{2} & 0 \\ 0 & −\frac{1}{2} \end{matrix}) ∈ C^{∞}(𝔰𝔲(W^{+}) ⊗ 𝔰𝔲(E)). (3.13)

Furthermore, with respect to the splitting V^{+} = W^{+} ⊕ W^{+} ⊗ L, we clearly have

(D_{A} + ρ(ϑ))Φ = ((D_{B} + ρ(ϑ))Ψ, 0). (3.14)

Then, combining equations (2.32), (3.12), (3.13), and (3.14), and noting that

F_{A_{L}} = F_{A_{Λ}} − F_{B^{det}} = F_{A_{Λ}} − Tr(F_{B}),

shows that the pair (B, Ψ) solves

\begin{aligned}Tr(F_{B}^{+}) − τρ^{−1}(Ψ ⊗ Ψ^{∗})_{0} − F_{A_{Λ}}^{+} & = 0, \\ (D_{B} + ρ(ϑ))Ψ & = 0. \end{aligned} (3.15)

Comparing (3.15) with the Seiberg-Witten equations (2.55) concludes the proof.

Lemma 3.13. Let X be a closed, oriented, Riemannian four-manifold with spin^{u} structure 𝔱 having a splitting 𝔱 = 𝔰 ⊕ 𝔰 ⊗ L. If w_{2}(𝔱) ≠ 0 then the map (3.11) of C_{𝔰} into C_{𝔱} restricts to a topological embedding

ι : M_{𝔰} ↪ C_{𝔱}, (3.16) whose image is the subspace of M_{𝔱} represented by pairs (A, Φ) which are reducible with respect to the splitting V = W ⊕ W ⊗ L, with Φ ∈ L_{k}^{2}(W^{+}) and 𝔱 = (ρ, V) and 𝔰 = (ρ, W). If w_{2}(𝔱) = 0 then the map (3.11) takes zero-section points in M_{𝔰} to zero-section reducibles in M_{𝔱}, although this identification of zero-section points need not be injective when b_{1}(X) > 0.

Proof. Given Lemma 3.11, we need only characterize the image of ι. If (A, Φ) represents a point in M_{𝔱} and is reducible with respect to the splitting V = W ⊕W ⊗L, with Φ ∈ L_{k}^{2}(W^{+}), then Lemma 3.11 implies that (A, Φ) = ι(B, Ψ), for some pair (B, Ψ) ∈ C̃_{𝔰}. Then Lemma 3.12 implies that (B, Ψ) satisfies the Seiberg-Witten equations (2.55) since (A, Φ) satisfies the PU(2) monopole equations (2.32).

3.4. The link of a stratum of reducible monopoles: local structure. The construction of the link in M_{𝔱} of the stratum M_{𝔰} ↪ M_{𝔱} of reducible PU(2) monopoles occupies this and the next subsection. Although for generic perturbations the locus M_{𝔰} of reducible solutions in M_{𝔱} defined by a reduction 𝔱 = 𝔰⊕𝔰⊗L will be a smooth manifold, the linearization of the map 𝔖 defined by the PU(2) monopole equations (2.32) need not be surjective at a reducible solution and so the points of M_{𝔰} might not be regular points of M_{𝔱}. Moreover, unlike the case of the link of M_{κ}^{w}, a local model of the link does not suffice as only one of our cohomology classes extend over M_{𝔰} (and that one vanishes in many cases), so we cannot use geometric representatives to cut down to a generic point in M_{𝔰}.

We begin with a definition of link of a stratum in smoothly stratified space, essentially following Mather [52] and Goresky-MacPherson [35]. We need only consider the relatively simple case of a stratified space with two strata since the lower strata in

M_{𝔱} ≅ M_{𝔱}^{∗,0} ∪ M_{κ}^{w} ∪ ∪_{𝔰}M_{𝔰} (3.17)

do not intersect when M_{𝔱} contains no reducible, zero-section solutions. (The finite union above over 𝔰 is over the subset of all spin^{c} structures for which M_{𝔰} is non-empty and for which there is a splitting 𝔱 = 𝔰 ⊕ 𝔰′.)

Definition 3.14. Let Z be a closed subset of a smooth, Riemannian manifold M, and suppose that Z = Z_{0} ∪ Z_{1}, where Z_{0} and Z_{1} are locally closed, smooth submanifolds of M and Z_{1} ⊂ Z̄_{0}. (That is, Z is a smoothly stratified space with two strata in the sense of [58, Chapter 11].) Let N_{Z_{1}} be the normal bundle of Z_{1} ⊂ M and let O′ ⊂ N_{Z_{1}} be an open neighborhood of the zero section Z_{1} ⊂ N_{Z_{1}} such that there is a diffeomorphism γ, commuting with the zero section of N_{Z_{1}} (so γ|_{Z_{1}} = id_{Z_{1}}), from O′ onto an open neighborhood γ(O′) of Z_{1} ⊂ M. Let O ⋐ O′ be an open neighborhood of the zero section Z_{1} ⊂ N_{Z_{1}}, where Ō = O ∪ ∂O ⊂ O′ is a smooth manifold-with-boundary. Then L_{Z_{1}} = Z_{0} ∩ γ(∂O) is a link of Z_{1} in Z_{0}.

In the preceding definition, Z_{0} will be the intersection of M_{𝔱}^{∗,0} with a neighborhood of the image of M_{𝔰} and Z_{1} will be identified with M_{𝔰}. Although the ambient manifold M in Definition 3.14 it is not required to be finite-dimensional, we shall impose this constraint as we need to define a fundamental class for the sphere bundle of N_{Z_{1}}. Thus, we will need to define a finite-dimensional submanifold of C_{𝔱} containing—as smooth submanifolds— the image of M_{𝔰} and an open neighborhood in M_{𝔱} of the image of M_{𝔰}. We fulfill these requirements in the next subsection by using a globalized, stabilized version of the local Kuranishi model [14, §4.2.4], [47] defined by the deformation complex of the PU(2) monopole equations (2.32). For the remainder of this subsection, we describe the splitting of this deformation complex, at a reducible PU(2) monopole, into deformation complexes which are “tangential” and “normal” to the stratum M_{𝔰}.

3.4.1. Decomposing the PU(2) elliptic deformation sequence at reducible pairs. In this and the next two sub-subsections we describe how the elliptic deformation complex for M_{𝔱} at a point [A, Φ] in the image of M_{𝔰} ↪ M_{𝔱} can be split into normal and tangential components, where 𝔰 = (ρ, W) and 𝔱 = (ρ, V) with V = W ⊕ W ⊗ L as in Lemma 3.11.

Let (A, Φ) = ι(B, Ψ) be a pair in C̃_{𝔱}, although not necessarily a solution to the PU(2) monopole equations. We begin by considering the deformation sequence (2.47), namely

C_{∞}(F_{0}) ⟶^{d_{A,Φ}^{0}} C_{∞}(F_{1}) ⟶^{d_{A,Φ}^{1}} C_{∞}(F_{2}), (3.18)

at a point (A, Φ) = (B ⊕ B ⊗ A_{L}, Ψ ⊕ 0), where A_{L} = A_{Λ} ⊗ (B^{det})^{∗} and the vector bundles F_{j}, j = 0, 1, 2, are defined by

\begin{aligned}F_{0} & = 𝔤_{𝔱}, \\ F_{1} & = Λ^{1} ⊗ 𝔤_{𝔱} ⊕ V^{+}, \\ F_{2} & = Λ^{+} ⊗ 𝔤_{𝔱} ⊕ V^{−}. \end{aligned} (3.19)

We shall also use L_{k+1−j}^{2}(F_{j}), the Hilbert spaces of L_{k+1−j}^{2} sections for j = 0, 1, 2, when applying these sequences. (The sequence is a complex if and only 𝔖(A, Φ) = 0.) It will be convenient to define vector bundle splittings,

F_{j} ≅ F_{j}^{t} ⊕ F_{j}^{n}, j = 0, 1, 2, (3.20)

using the canonical isomorphism 𝔤_{𝔱} ≅ i\underline{ℝ} ⊕ L of Lemma 3.10:

F_{0}^{t} = iΛ^{0}  and F_{0}^{n} = Λ^{0} ⊗_{ℝ} L = L,
F_{1}^{t} = iΛ^{1} ⊕ W^{+} and F_{1}^{n} = Λ^{1} ⊗_{ℝ} L ⊕ W^{+} ⊗ L,
F_{2}^{t} = iΛ^{+} ⊕ W^{−} and F_{2}^{n} = Λ^{+} ⊗_{ℝ} L ⊕ W^{−} ⊗ L.

(3.21)

Conversely, the decompositions (3.20) yield inclusions which we write as

ι : F_{j}^{t} ↪ F_{j} and ι : F_{j}^{n} ↪ F_{j}, j = 0, 1, 2. (3.22)

The motivation for the splitting is due to the fact that the component C^{∞}(F_{1}^{t}) will contain vectors tangent to M_{𝔰} while C^{∞}(F_{1}^{n}) will contain those vectors normal to M_{𝔰}. We note in passing that when the above vector bundles are given their natural fiber inner products, the splittings F_{j} = F_{j}^{t} ⊕ F_{j}^{n} define isomorphisms of Riemannian vector bundles when the bundles F_{j}^{t} ⊕ F_{j}^{n} are given their direct sum fiber inner products.

3.4.2. Decomposing the group actions on the deformation sequence bundles. The embedding G_{𝔰} ↪ G_{𝔱} defined in (3.10) and two S^{1} actions on V = W ⊕ W ⊗ L of interest to us, namely (3.1) and (3.2), induce corresponding group actions on the bundles F_{j}^{t}, F_{j}^{n} in (3.21), arising in the decomposition (3.34) of the PU(2)-monopole deformation sequence (3.18). For ease of later reference, we record the induced group actions here on F_{j}^{t} and F_{j}^{n}.

Writing V = W ⊗ E and E = \underline{ℂ} ⊕ L, we defined an embedding (3.10) of gauge groups,

ϱ : G_{𝔰} ↪ G_{𝔱}, s ↦ id_{W} ⊗ (\begin{matrix}s & 0 \\ 0 & s^{−1} \end{matrix}) .

Then the homomorphism Ad : Aut(E) → Aut(𝔰𝔲(E)), u ↦ u( ·)u^{−1}, induces an action of ϱ(s) on 𝔰𝔲(E) ≅ i\underline{ℝ} ⊕ L via the isomorphism ξ ↦ (ν, z) in (3.7),

Ad (ϱ(s))ξ = (\begin{matrix}s & 0 \\ 0 & s^{−1} \end{matrix}) (\begin{matrix}ν & −z̄ \\ z & ν \end{matrix}) (\begin{matrix}s^{−1} & 0 \\ 0 & s \end{matrix}) = (\begin{matrix}ν & −s^{2}z̄ \\ s^{−2}z & ν \end{matrix}) ,

where ν ∈ C^{∞}(X, iℝ) and z ∈ C^{∞}(X, ℂ). Hence, the induced action of ϱ(s) on 𝔰𝔲(E) ≅ i\underline{ℝ} ⊕ L—via the composition of the embedding ϱ : G_{𝔰} ↪ G_{𝔱} given by (3.10), the map Ad : Aut(V) → Aut(𝔰𝔲(V)), and the projection 𝔰𝔲(V) → 𝔰𝔲(E) = 𝔤_{𝔱} given by (2.13) —is trivial on the factor i\underline{ℝ} and acts as scalar multiplication by s^{−2} on the complex line bundle L:

Ad ∘ ϱ : G_{𝔰} × (i\underline{ℝ} ⊕ L) → i\underline{ℝ} ⊕ L, (s, (ν, z)) ↦ (ν, s^{−2}z). (3.23)

Hence, the induced action of s ∈ G_{𝔰} is trivial on the bundles F_{j}^{t} while on F_{j}^{n} it acts as s^{−2} on the factors Λ^{j} ⊗_{ℝ} L and as s^{−1} on the factors W^{±} ⊗ L.

The action (3.1) of S^{1} on V = W ⊕ W ⊗ L by scalar multiplication induces the trivial action on 𝔤_{𝔱} = 𝔰𝔲(E). Hence, S^{1} acts trivially on the factors iΛ^{j} and Λ^{j} ⊗_{ℝ} L of F_{j}^{t}, F_{j}^{n}, whereas it acts scalar multiplication on the factors W^{±} and W^{±} ⊗ L.

Finally, we consider the action (3.2) of S^{1} on V = W ⊕ W ⊗ L by the identity on the factor W and as scalar multiplication on W ⊗ L. The induced action of S^{1} is trivial on the factors F_{j}^{t} while it acts by scalar multiplication on the factors F_{j}^{n}.

3.4.3. Decomposing the differentials in the deformation sequence at reducible pairs. We now describe the splitting of the differentials d_{A,Φ}^{j} in the sequence, when (A, Φ) = ι(B, Ψ) is a reducible pair in C̃_{𝔱}.

Recall from equation (2.47) that

d_{A,Φ}^{1}(a, ϕ) = (D𝔖)_{A,Φ}(a, ϕ) = (\begin{matrix}d+ ; Âa − τρ^{−1}(Φ ⊗ ϕ^{∗} + ϕ ⊗ Φ^{∗})_{00} \\ (D_{A} + ρ(ϑ))ϕ + ρ(a)Φ \end{matrix}) . (3.24)

If Φ = Ψ ⊕ 0 and ϕ = ψ ⊕ ψ′, the quadratic term takes the form

Φ ⊗ ϕ^{∗} + ϕ ⊗ Φ^{∗} = (\begin{matrix}Ψ ⊗ ψ^{∗} + ψ ⊗ Ψ^{∗} & Ψ ⊗ ψ^{′,∗} \\ ψ′ ⊗ Ψ^{∗} & 0 \end{matrix}) ,

and so

τρ^{−1}(Φ ⊗ ϕ^{∗} + ϕ ⊗ Φ^{∗})_{00}

= (\begin{matrix}\frac{1}{2}τρ^{−1}(Ψ ⊗ ψ^{∗} + ψ ⊗ Ψ^{∗})_{0} & τρ^{−1}(Ψ ⊗ ψ^{′,∗})_{0} \\ τρ^{−1}(ψ′ ⊗ Ψ^{∗})_{0} & −\frac{1}{2}τρ^{−1}(Ψ ⊗ ψ^{∗} + ψ ⊗ Ψ^{∗})_{0} \end{matrix}) ,

where the diagonal term \frac{1}{2}τρ^{−1}(Ψ ⊗ ψ^{∗} + ψ ⊗ Ψ^{∗})_{0} is in C^{∞}(iΛ^{+}) while the off-diagonal term τρ^{−1}(ψ′ ⊗ Ψ^{∗})_{0} is in C^{∞}(Λ^{+} ⊗ L). Indeed, we have

(ψ′ ⊗ Ψ^{∗})_{0} ∈ C^{∞}(𝔰𝔩(W^{+}) ⊗_{ℂ} L) = C^{∞}(𝔰𝔲(W^{+}) ⊗_{ℝ} L),

since 𝔰𝔩(W^{+}) = 𝔰𝔲(W^{+}) ⊗_{ℝ} ℂ. The complex-linear Clifford map restricts to a real-linear isomorphism ρ : Λ^{+} → 𝔰𝔲(W^{+}), so we have ρ^{−1}(ψ′ ⊗ Ψ^{∗})_{0} ∈ C^{∞}(Λ^{+} ⊗ L).

The Dirac-operator term D_{A} + ρ(ϑ) + ρ(a) : C^{∞}(V^{+}) → C^{∞}(V^{−}) splits as

(D_{A} + ρ(ϑ))ϕ + ρ(a)Φ = (\begin{matrix}(D_{B} + ρ(ϑ))ψ + ρ(α)Ψ \\ (D_{B⊗A_{L}} + ρ(ϑ))ψ′ + ρ(β)Ψ \end{matrix}) ,

noting that

a = (\begin{matrix}α & −β̄ \\ β & −α \end{matrix}) , with α ∈ C^{∞}(iΛ^{1}) and β ∈ C^{∞}(Λ^{1} ⊗ L).

By Lemma 2.9, Â = d_{ℝ} ⊕ A_{L}. The term d+ ; Âa ∈ C^{∞}(Λ^{+} ⊗ 𝔤_{𝔱}) then splits as

d+ ; Âa = (\begin{matrix}d^{+}α & −d_{A_{L}^{∗}}^{+}β̄ \\ d_{A_{L}}^{+}β & −d^{+}α \end{matrix}) , (3.25)

where the diagonal term d^{+}α is in C^{∞}(iΛ^{+}), and off-diagonal term d_{A_{L}}^{+}β is in C^{∞}(Λ^{+} ⊗L).

With respect to these identifications, the linear operator d_{ι(B,Ψ)}^{1} : C^{∞}(F_{1}) → C^{∞}(F_{2}) diagonalizes to give

d_{ι(B,Ψ)}^{1} = (\begin{matrix}d_{ι(B,Ψ)}^{1,t} & 0 \\ 0 & d_{ι(B,Ψ)}^{1,n} \end{matrix}) : C^{∞}(F_{1}^{t}) ⊕ C^{∞}(F_{1}^{n}) → C^{∞}(F_{2}^{t}) ⊕ C^{∞}(F_{2}^{n}),

where d_{ι(B,Ψ)}^{1,t} : C^{∞}(F_{1}^{t}) → C^{∞}(F_{2}^{t}) and d_{ι(B,Ψ)}^{1,n} : C^{∞}(F_{1}^{n}) → C^{∞}(F_{2}^{n}) are the “tangential” and “normal” components of d_{ι(B,Ψ)}^{1}. More explicitly, the operator

d_{ι(B,Ψ)}^{1} : \begin{matrix}C^{∞}(iΛ^{1}) ⊕ C^{∞}(W^{+}), \\ ⊕ \\ C^{∞}(Λ^{1} ⊗ L) ⊕ C^{∞}(W^{+} ⊗ L) \end{matrix} ⟶ \begin{matrix}C^{∞}(iΛ^{+}) ⊕ C^{∞}(W^{−}), \\ ⊕ \\ C^{∞}(Λ^{+} ⊗ L) ⊕ C^{∞}(W^{−} ⊗ L) \end{matrix}

diagonalizes and the tangential component of d_{ι(B,Ψ)}^{1} is given by

d_{ι(B,Ψ)}^{1,t} : C^{∞}(iΛ^{1}) ⊕ C^{∞}(W^{+}) ⟶ C^{∞}(iΛ^{+}) ⊕ C^{∞}(W^{−}),

where we define

d_{ι(B,Ψ)}^{1,t}(α, ψ) = (\begin{matrix}d^{+}α − \frac{1}{2}τρ^{−1}(Ψ ⊗ ψ^{∗} + ψ ⊗ Ψ^{∗})_{0} \\ (D_{B} + ρ(ϑ))ψ + ρ(α)Ψ \end{matrix}) , (3.26)

Note that d_{ι(B,Ψ)}^{1,t} matches the Seiberg-Witten differential d_{B,Ψ}^{1} in equation (2.60), aside from a factor of 2 in the d^{+} component: the scaling factor has no significance since we are only interested in the kernels and cokernels of these operators. The normal component of d_{ι(B,Ψ)}^{1} is given by

d_{ι(B,Ψ)}^{1,n} : C^{∞}(Λ^{1} ⊗ L) ⊕ C^{∞}(W^{+} ⊗ L) ⟶ C^{∞}(Λ^{+} ⊗ L) ⊕ C^{∞}(W^{−} ⊗ L),

where we define

d_{ι(B,Ψ)}^{1,n}(β, ψ′) = (\begin{matrix}d_{A_{L}}^{+}β − τρ^{−1}(ψ′ ⊗ Ψ^{∗})_{0} \\ (D_{B⊗A_{L}} + ρ(ϑ))ψ′ + ρ(β)Ψ \end{matrix}) . (3.27)

This completes our description of the diagonalization of the differential d_{ι(B,Ψ)}^{1}.

It remains to discuss the rather simpler diagonalization of the differential d_{ι(B,Ψ)}^{0} with respect to the splitting of C^{∞}(F_{0}) and C^{∞}(F_{1}) into normal and tangential components. From Lemma 3.10 we see that elements ζ = ζ_{0} of C^{∞}(𝔤_{𝔱}), recalling that L_{k+1}^{2}(𝔤_{𝔱}) = T_{id}G_{𝔱}, may be uniquely written as

ζ = (\begin{matrix}f & −κ̄ \\ κ & −f \end{matrix}) , (3.28)

where f ∈ C^{∞}(X, iℝ) is an imaginary-valued function and κ ∈ C^{∞}(L). Thus,

ζ = f ⊕ κ ∈ C^{∞}(F_{0}^{t}) ⊕ C^{∞}(F_{0}^{n}).

Recall from equation (2.50) that the differential d_{A,Φ}^{0} is defined by

d_{A,Φ}^{0}ζ = (dÂζ, −ζΦ) ∈ C^{∞}(Λ^{1} ⊗ 𝔤_{𝔱}) ⊕ C^{∞}(V^{+}), ζ ∈ C^{∞}(Λ^{1} ⊗ 𝔤_{𝔱}).

Note that

ζΦ = (\begin{matrix}f & −κ̄ \\ κ & −f \end{matrix}) · (\begin{matrix}Ψ \\ 0 \end{matrix}) = (\begin{matrix}fΨ \\ κΨ \end{matrix}) ,
dÂζ = (\begin{matrix}df & −d_{A_{L}^{∗}} κ̄ \\ d_{A_{L}}κ & −df \end{matrix}) .

So, at the pair (A, Φ) = ι(B, Ψ) = (B ⊕ B ⊗ A_{L}, Ψ ⊕ 0) we have

d_{ι(B,Ψ)}^{0}ζ = ((\begin{matrix}df & −d_{A_{L}^{∗}} κ̄ \\ d_{A_{L}}κ & −df \end{matrix}) , (\begin{matrix}−fΨ \\ −κΨ \end{matrix})) .

With respect to these identifications, the linear operator d_{ι(B,Ψ)}^{0} : C^{∞}(F_{0}) → C^{∞}(F_{1}) diagonalizes to give

d_{ι(B,Ψ)}^{0} = (\begin{matrix}d_{ι(B,Ψ)}^{0,t} & 0 \\ 0 & d_{ι(B,Ψ)}^{0,n} \end{matrix}) : C^{∞}(F_{0}^{t}) ⊕ C^{∞}(F_{0}^{n}) ⟶ C^{∞}(F_{1}^{t}) ⊕ C^{∞}(F_{1}^{n}),

where d_{ι(B,Ψ)}^{0,t} : C^{∞}(F_{0}^{t}) → C^{∞}(F_{1}^{t}) and d_{ι(B,Ψ)}^{0,n} : C^{∞}(F_{0}^{n}) → C^{∞}(F_{1}^{n}) are the “tangential” and “normal” components of d_{ι(B,Ψ)}^{0}. More explicitly, the operator

d_{ι(B,Ψ)}^{0} : \begin{matrix}C^{∞}(iΛ^{0})) \\ ⊕ \\ C^{∞}(L) \end{matrix} ⟶ \begin{matrix}C^{∞}(iΛ^{1}) ⊕ C^{∞}(W^{+}) \\ ⊕ \\ C^{∞}(Λ^{1} ⊗ L) ⊕ C^{∞}(W^{+} ⊗ L) \end{matrix}

diagonalizes and the tangential component of d_{ι(B,Ψ)}^{0} is given by

d_{ι(B,Ψ)}^{0,t} : C^{∞}(iΛ^{0}) ⟶ C^{∞}(iΛ^{1}) ⊕ C^{∞}(W^{+}), (3.29) where we define

d_{ι(B,Ψ)}^{0,t}f = (df, −fΨ).

Note that d_{ι(B,Ψ)}^{0,t} matches the gauge group differential d_{B,Ψ}^{0} in equation (2.58). The normal component of d_{ι(B,Ψ)}^{0} is given by

d_{ι(B,Ψ)}^{0,n} : C^{∞}(L) ⟶ C^{∞}(Λ^{1} ⊗ L) ⊕ C^{∞}(W^{+} ⊗ L), (3.30) where we define

d_{ι(B,Ψ)}^{0,n}κ = (d_{A_{L}}κ, −κΨ).

This completes our description of the diagonalization of the differential d_{ι(B,Ψ)}^{0}. Recall from [24, Equation (2.38)] that the L^{2} adjoint of d_{A,Φ}^{0} is given by

d_{A,Φ}^{0,∗}(a, ϕ) = d∗ ; Âa − (·Φ)^{∗}ϕ, (a, ϕ) ∈ C^{∞}(Λ^{1} ⊗ 𝔤_{𝔱}) ⊕ C^{∞}(V^{+}). (3.31)

The L^{2} adjoint d_{ι(B,Ψ)}^{0,∗} : C^{∞}(F_{1}) → C^{∞}(F_{0}) then diagonalizes to give

d_{ι(B,Ψ)}^{0,∗} = (\begin{matrix}d_{ι(B,Ψ)}^{0,t,∗} & 0 \\ 0 & d_{ι(B,Ψ)}^{0,n,∗} \end{matrix}) : C^{∞}(F_{1}^{t}) ⊕ C^{∞}(F_{1}^{n}) → C^{∞}(F_{0}^{t}) ⊕ C^{∞}(F_{0}^{n}),

where d_{ι(B,Ψ)}^{0,t,∗} : C^{∞}(F_{1}^{t}) → C^{∞}(F_{0}^{t}) is given explicitly by

d_{ι(B,Ψ)}^{0,t,∗} : C^{∞}(iΛ^{1}) ⊕ C^{∞}(W^{+}) → C^{∞}(iΛ^{0}),

with d_{ι(B,Ψ)}^{0,t,∗}(α, ψ) = d^{∗}α − Im⟨Ψ, ψ⟩, (3.32)

and d_{ι(B,Ψ)}^{0,n,∗} : C^{∞}(F_{1}^{n}) → C^{∞}(F_{0}^{n}) is explicitly given by

d_{ι(B,Ψ)}^{0,n,∗} : C^{∞}(Λ^{1} ⊗ L) ⊕ C^{∞}(W^{+} ⊗ L) → C^{∞}(L),

with

d_{ι(B,Ψ)}^{0,n,∗}(β, ψ′) = d_{A_{L}}^{∗}β + ψ′ ⊗ Ψ^{∗}. (3.33)

Of course, these L^{2} adjoints are defined in the usual way by

(d^{∗}α − Im⟨Ψ, ψ⟩, f)_{L^{2}} = (α, df)_{L^{2}} − (ψ, fΨ)_{L^{2}},
(d_{A_{L}}^{∗}β + ψ′ ⊗ Ψ^{∗}, κ)_{L^{2}} = (β, d_{A_{L}}κ)_{L^{2}} + (ψ′, κΨ)_{L^{2}},

for all f ∈ C^{∞}(iΛ^{0}) and κ ∈ C^{∞}(L), where

(fΨ, ψ)_{L^{2}} = (f, \overline{⟨ψ, Ψ⟩})_{L^{2}} = (f, ⟨Ψ, ψ⟩)_{L^{2}},
(κΨ, ψ′)_{L^{2}} = (κ, ψ′ ⊗ Ψ^{∗})_{L^{2}}.

This completes our description of the diagonalization of the L^{2} adjoint d_{ι(B,Ψ)}^{0,∗}.

The above discussion implies that the deformation sequence (3.18) splits when (A, Φ) = ι(B, Ψ):

\begin{matrix}C^{∞}(F_{0}^{t}) & ⟶^{d_{ι(B,Ψ)}^{0,t}} & C^{∞}(F_{1}^{t}) & ⟶^{d_{ι(B,Ψ)}^{1,t}} & C^{∞}(F_{2}^{t}), \\ C^{∞}(F_{0}^{n}) & ⟶^{d_{ι(B,Ψ)}^{0,n}} & C^{∞}(F_{1}^{n}) & ⟶^{d_{ι(B,Ψ)}^{1,n}} & C^{∞}(F_{2}^{n}). \end{matrix} (3.34)

These sequences form complexes if and only if 𝔖(A, Φ) = 0, in which case they give the tangential and normal deformation complexes for the PU(2) monopole equations at a reducible solution, so it is convenient to consider the rolled-up sequences

C^{∞}(F_{1}^{t}) ⟶^{D_{ι(B,Ψ)}^{t}} C^{∞}(F_{0}^{t}) ⊕ C^{∞}(F_{2}^{t}),

C_{∞}(F n 1) ⟶^{D_{ι(B,Ψ)}^{n}} C_{∞}(F n 0) ⊕ C_{∞}(F n 2),

comprising the tangential and normal components of the rolled-up deformation sequence

D_{ι(B,Ψ)} : C^{∞}(F_{1}) → C^{∞}(F_{0}) ⊕ C^{∞}(F_{2}).

for the PU(2) monopole equations. The tangential and normal components of the deformation operator D_{ι(B,Ψ)} = d_{ι(B,Ψ)}^{0,∗} + d_{ι(B,Ψ)}^{1} are given by

D_{ι(B,Ψ)}^{t} = d_{ι(B,Ψ)}^{0,t,∗} + d_{ι(B,Ψ)}^{1,t} and D_{ι(B,Ψ)}^{n} = d_{ι(B,Ψ)}^{0,n,∗} + d_{ι(B,Ψ)}^{1,n},

so that we have a decomposition

D_{ι(B,Ψ)} = D_{ι(B,Ψ)}^{t} ⊕ D_{ι(B,Ψ)}^{n}. (3.35)

More explicitly, we see from (3.26) and (3.32) that the tangential component

D_{ι(B,Ψ)}^{t} : C^{∞}(iΛ^{1}) ⊕ C^{∞}(W^{+}) ⟶ C^{∞}(iΛ^{0}) ⊕ C^{∞}(iΛ^{+}) ⊕ C^{∞}(W^{−}) (3.36)

of D_{ι(B,Ψ)} is given by

D_{ι(B,Ψ)}^{t}(α, ψ) = (\begin{matrix}d^{∗}α − Im⟨Ψ, ψ⟩ \\ d^{+}α − \frac{1}{2}τρ^{−1}(Ψ ⊗ ψ^{∗} + ψ ⊗ Ψ^{∗})_{0} \\ (D_{B} + ρ(ϑ))ψ + ρ(α)Ψ \end{matrix}) ,

while from (3.27) and (3.33) the normal component

D_{ι(B,Ψ)}^{n} : C^{∞}(Λ^{1} ⊗ L) ⊕ C^{∞}(W^{+} ⊗ L) ⟶ C^{∞}(L) ⊕ C^{∞}(Λ^{+} ⊗ L) ⊕ C^{∞}(W^{−} ⊗ L) (3.37) of D_{ι(B,Ψ)} is given by

D_{ι(B,Ψ)}^{n}(β, ψ′) = (\begin{matrix}d_{A_{L}}^{∗}β + ψ′ ⊗ Ψ^{∗} \\ d_{A_{L}}^{+}β − τρ^{−1}(ψ′ ⊗ Ψ^{∗})_{0} \\ (D_{B⊗A_{L}} + ρ(ϑ))ψ′ + ρ(β)Ψ \end{matrix}) .

Note the operators d_{ι(B,Ψ)}^{0,n} and d_{ι(B,Ψ)}^{1,n}, and thus D_{ι(B,Ψ)}^{n}, are complex linear when the vector bundles F_{j}^{n}, j = 0, 1, 2, are given the natural complex structures induced by the S^{1} action on L.

If 𝔖(A, Φ) = 0, the two sequences in (3.34) define cohomology groups H_{ι(B,Ψ)}^{•,t} and H_{ι(B,Ψ)}^{•,n}, respectively, which we may compare with the cohomology groups H_{ι(B,Ψ)}^{•} and H_{B,Ψ}^{•} of the PU(2) monopole deformation complex (2.47) at the pair ι(B, Ψ) ∈ M̃_{𝔱} and the cohomology groups H_{B,Ψ}^{•} of the Seiberg-Witten deformation complex (2.61) at the pair (B, Ψ) ∈ M̃_{𝔰}:

Lemma 3.15. Continue the above notation and require that ι(B, Ψ) be a (reducible) PU(2) monopole. For the elliptic deformation complex (2.47) defined by the PU(2) monopole equations (2.32) and the automorphism group G_{𝔱}, we have the following canonical isomorphisms of cohomology groups,

H_{ι(B,Ψ)}^{•,t} ≅ H_{B,Ψ}^{•},

and hence,

H_{ι(B,Ψ)}^{•} ≅ H_{B,Ψ}^{•} ⊕ H_{ι(B,Ψ)}^{•,n}.

Moreover, if Ψ ≢ 0 then H_{B,Ψ}^{0} = 0, while if H_{B,Ψ}^{2} = 0 then H_{ι(B,Ψ)}^{2} ≅ H_{ι(B,Ψ)}^{2,n}.

Proof. The isomorphisms identifying the cohomology of the tangential deformation complex with that of the Seiberg-Witten complex follow immediately from a comparison of these two complexes.

If Ψ ≢ 0 then H_{B,Ψ}^{0} = 0 and (3.30) implies H_{ι(B,Ψ)}^{0,n} = 0 and so H_{ι(B,Ψ)}^{0} = 0. Moreover, if

H_{B,Ψ}^{2} = 0, then H_{ι(B,Ψ)}^{2} = H_{ι(B,Ψ)}^{2,n}.

Transversality for the Seiberg-Witten equations (Proposition 2.16) implies H_{B,Ψ}^{2} = 0.

3.5. The link of a stratum of reducible monopoles: global structure. Our task in this subsection is to construct an ambient finite-dimensional, smooth submanifold M_{𝔱}(Ξ, 𝔰) ⊂ C_{𝔱} containing M_{𝔰} as a smooth submanifold, as required by the definition of the link in Definition 3.14. Recall that 𝔖 is the section of the S^{1}-equivariant, infinite-rank “obstruction bundle”

𝔙 = C̃_{𝔱} ×_{G_{𝔱}} L_{k−1}^{2}(F_{2}) → C_{𝔱} (3.38)

defined by the PU(2) monopole equations (2.32) and this section need not vanish transversely along M_{𝔰}.

To motivate the construction of the ambient manifold M_{𝔱}(Ξ, 𝔰) given in this subsection, suppose temporarily that the cokernel of D_{A,Φ} has constant rank as [A, Φ] varies in the image of M_{𝔰} ↪ M_{𝔱} (that is, no spectral flow occurs). Then we obtain a finite-rank, S^{1}-equivariant vector bundle Coker D over M_{𝔰}, with fibers Coker D_{A,Φ}. Let 2ν be the least positive eigenvalue of the Laplacian Δ_{A,Φ} = D_{A,Φ}D_{A,Φ}^{∗} as [A, Φ] varies along the image of the compact manifold M_{𝔰} and let Π_{A,Φ;ν} denote the L^{2} orthogonal projection from L_{k−1}^{2}(F_{2}) onto the subspace spanned by the eigenvectors of Δ_{A,Φ} with eigenvalue less than ν. The vector bundle Coker D over ι(M_{𝔰}) ⊂ C_{𝔱} then extends to a vector bundle Ξ_{ν} = Ker Π_{ν}^{⊥}Δ = Coker Π_{ν}^{⊥}D, where Π_{ν}^{⊥} = id − Π_{ν}, of the same rank over an open neighborhood of ι(M_{𝔰}) ⊂ C_{𝔱}. Arguing as in Lemma 3.16, one can see that the space M_{𝔰} would be a smooth submanifold of C_{𝔱}. Then both M_{𝔰} and an open neighborhood in M_{𝔱}^{∗,0} of ι(M_{𝔰}) would be smooth submanifolds of the S^{1}-invariant “thickened” moduli space,

M_{𝔱}(Ξ_{ν}, 𝔰) = (Π_{ν}^{⊥}𝔖)^{−1}(0) ⊂ C_{𝔱},

a finite-dimensional, smooth, S^{1}-invariant manifold which serves as the ambient, finitedimensional, smooth manifold “M” of Definition 3.14. Ambient manifolds of this form, defined by spectral projections as above, have been used by Donaldson [13], Friedman-Morgan [31], and Taubes [71], [72] to describe neighborhoods of the stratum {[Θ]} × Sym^{κ}(X) containing the trivial connection Θ in the Uhlenbeck compactification of the moduli space M_{κ}^{w} of anti-self-dual connections when w = 0.

In practice, one cannot guarantee that Coker D will either vanish or even have constant rank due to spectral flow, so we must resort to a more general construction of an S^{1}- equivariant vector bundle Ξ over an open neighborhood of the image of M_{𝔰} in C_{𝔱} which “spans” Coker D along M_{𝔰}. The method we employ is an extension of one used by Atiyah and Singer to construct the index bundle or determinant-line bundle of a family of elliptic operators [6, pp. 122–127], [14, §5.1.3 & §5.2.1]. The Atiyah-Singer method has also been exploited by Furuta [32], T-J. Li & Liu [51], J. Li & Tian [50], and Ruan [65], [66], [67] to construct certain “global Kuranishi models” (or “virtual” or “thickened” moduli spaces) parameterizing spaces of Seiberg-Witten monopoles or pseudo-holomorphic curves; related ideas are contained in [11], [69]. The principal difference between our construction and those of Li-Liu, Li-Tian, or Ruan is that the vector bundle replacing Coker D is defined over an open neighborhood in the original configuration space C_{𝔱} rather than on an artificial, augmented configuration space, such as ℂ^{r} × C_{𝔱}. In this sense, our construction is closer to that of [13], [31], [71], [72] and corresponds to the alternative construction of index bundles discussed in [4, pp. 153–166], [9, §1.7]; we find this second stabilization technique more convenient when constructing links of lower-level strata of reducibles via gluing in [19], [20], [21].

3.5.1. Smooth embeddings. We recall that our abstract definition of a link of the lower stratum of a two-stratum space (see Definition 3.14) requires an ambient smooth manifold containing the strata. As a first step in the construction of a finite-dimensional ambient manifold, we use the decompositions of §3.4 to show that the topological embedding M_{𝔰} ↪ C_{𝔱}^{0} of Lemma 3.13 is smooth, so M_{𝔰} is a smooth submanifold of C_{𝔱}^{0}.

Lemma 3.16. Let 𝔱 be a spin^{u} structure on a closed, oriented, smooth four-manifold X, with reduction 𝔱 = 𝔰⊕𝔰⊗L. Let ι : C_{𝔰}^{0} ↪ C_{𝔱}^{0} be the map in Lemma 3.13. Then the following hold:

• The map ι : C_{𝔰}^{0} → C_{𝔱}^{0} is a smooth immersion, and

• The map ι : M_{𝔰}^{0} ↪ C_{𝔱}^{0} is a smooth embedding, so M_{𝔰}^{0} is a submanifold of C_{𝔱}^{0}.

Proof. Let [B, Ψ] be a point in C_{𝔰}^{0}: by the slice theorem for C_{𝔰}^{0}, the restriction of the projection map C̃_{𝔰}^{0} → C_{𝔰}^{0} = C̃_{𝔰}^{0}/G_{𝔰} to a small enough open neighborhood of (B, Ψ) in the slice (B, Ψ) + Ker d_{B,Ψ}^{∗,0} gives a smooth parameterization of an open neighborhood of [B, Ψ] in C_{𝔰}^{0}. Similarly, by the slice theorem [24, Proposition 2.8] for C_{𝔱}^{0}, the restriction of the projection map C̃_{𝔱}^{0} → C_{𝔱}^{0} = C̃_{𝔱}^{0}/G_{𝔱} to a small enough open neighborhood of ι(B, Ψ) in the slice ι(B, Ψ) + Ker d_{ι(B,Ψ)}^{∗,0} gives a smooth parameterization of an open neighborhood of

[ι(B, Ψ)] in C_{𝔱}^{0}. Comparing the operators d_{B,Ψ}^{0,∗} in equation (2.59) and d_{ι(B,Ψ)}^{0,t,∗} in equation (3.32), we see that the differential of the smooth embedding

ι : C̃_{𝔰}^{0} → C̃_{𝔱}^{0}, (B, Ψ) ↦ (B ⊕ B ⊗ A_{L}, Ψ ⊕ 0),

of Lemma 3.11 restricts to an isomorphism

Dι : Ker d_{B,Ψ}^{∗,0} ≅ Ker d_{ι(B,Ψ)}^{∗,0,t} ⊂ Ker d_{A,Φ}^{0,t,∗} ⊕ Ker d_{A,Φ}^{0,n,∗} = Ker d_{B,Ψ}^{∗,0}.

Hence, the differential Dι : T_{B,Ψ}C_{𝔰}^{0} → T_{ι(B,Ψ)}C_{𝔱}^{0} is injective and the induced maps ι : C_{𝔰}^{0} → C_{𝔱}^{0} and ι : M_{𝔰}^{0} → C_{𝔱}^{0} are smooth immersions. Since the map ι : M_{𝔰}^{0} → C_{𝔱}^{0} is a topological embedding according to Lemma 3.13, and is a smooth immersion, the map ι : M_{𝔰}^{0} → C_{𝔱}^{0} is a smooth embedding.

3.5.2. Construction of the stabilized cokernel bundle. Again, we assume that there are no zero-section reducible monopoles in M_{𝔰}. For any representative (A, Φ) of a point in the image of ι : M_{𝔰} ↪ M_{𝔱}, we have Coker(D𝔖)_{A,Φ} = Coker d_{A,Φ}^{1,n} ⊂ L_{k−1}^{2}(F_{2}^{n}), since (D𝔖)_{A,Φ} = d_{A,Φ}^{1} (by definition) and Coker d_{A,Φ}^{1,t} = 0. The splitting of the last term of the elliptic deformation complex (3.18) at a reducible PU(2) monopole into tangential and normal deformation components (3.34) yields a splitting of the corresponding complexes of Hilbert bundles,

𝔙 = C̃_{𝔱}^{0} ×_{G_{𝔱}} L_{k−1}^{2}(F_{2}) ≅ C̃_{𝔱}^{0} ×_{G_{𝔱}} L_{k−1}^{2}(F_{2}^{t} ⊕ F_{2}^{n}), (3.39)

when restricted to ι(M_{𝔰}). This splitting can be seen by using a reduction of the structure group G_{𝔱} to G_{𝔰}:

\begin{aligned}𝔙|_{ι(M_{𝔰})} & ≅ ι(M̃_{𝔰}) ×_{G_{𝔰}} L_{k−1}^{2}(F_{2}) ≅ 𝔙^{t} ⊕ 𝔙^{n}, \\ where 𝔙^{t} & = ι(M̃_{𝔰}) ×_{G_{𝔰}} L_{k−1}^{2}(F_{2}^{t}), \\ 𝔙^{n} & = ι(M̃_{𝔰}) ×_{G_{𝔰}} L_{k−1}^{2}(F_{2}^{n}) \end{aligned} (3.40)

The action (3.2) of S^{1} on V = W ⊕ W ⊗ L induces the trivial action of S^{1} on the bundle 𝔙^{t} and the standard action by complex multiplication on 𝔙^{n}; see the action (3.3) of S^{1} on 𝔤_{𝔱} ≅ i\underline{ℝ} ⊕ L induced by the action of S^{1} on V.

Note that although the splitting (3.40) is not well-defined away from the stratum ι(M_{𝔰}), since the full gauge group G_{𝔱} does not preserve the splitting of the fiber, L_{k−1}^{2}(F_{2}) ≅ L_{k−1}^{2}(F_{2}^{t}) ⊕ L_{k−1}^{2}(F_{2}^{n}), the circle action on L_{k−1}^{2}(F_{2}) induced by the circle action (3.2) on V does preserve this splitting, with the circle acting trivially on L_{k−1}^{2}(F_{2}^{t}) and by complex multiplication on L_{k−1}^{2}(F_{2}^{n}). Hence, the vector bundle (3.39) is S^{1} equivariant with respect to the circle action on the fiber L_{k−1}^{2}(F_{2}^{t} ⊕ F_{2}^{n}) induced by the circle action (3.2) on V.

We use the preceding observation below to define a finite-rank, real subbundle Ξ ⊂ 𝔙 with an almost complex structure when restricted to ι(M_{𝔰}). (If the requirement that Ξ → ι(M_{𝔰}) has an almost complex structure were dropped then we would not require the splitting into tangential and normal components in order to construct Ξ.) We shall need the following well-known consequence of Kuiper’s result that the unitary group of a Hilbert space is contractible [46, Theorem 3] (see also [9, §I.7, p. 67] or [49, p. 208]):

Theorem 3.17. [46, p. 29] Let M be a compact topological space or a space which has the homotopy type of a CW complex. Then every vector bundle over M, with fiber an infinite-dimensional, real or complex separable Hilbert space ℌ and structure group GL(ℌ), is trivial.

We have the following version of Kuiper’s result in the smooth category:

Corollary 3.18. Let M be a smooth manifold which is compact or has the homotopy type of a CW complex. Then every C^{∞} vector bundle over M, with fiber an infinite-dimensional, real or complex separable Hilbert space ℌ and structure group GL(ℌ), has a global C^{∞} trivialization.

Proof. If 𝔚 is a vector bundle over M with fiber a Hilbert space ℌ, then Theorem 3.17 yields a C^{0} trivialization τ : 𝔚 → M × ℌ, that is, a C^{0} section τ of the C^{∞} Hilbert bundle Hom(𝔚, M × ℌ) over M which gives a linear isomorphism on each fiber. Now suppose that τ_{∞} is a C^{∞} section of the bundle Hom(𝔚, M × ℌ). If τ_{∞} is chosen so that ∥τ (p) − τ_{∞}(p)∥ is sufficiently small for each p ∈ M, then τ_{∞} is an isomorphism on each fiber and gives the desired global, C^{∞} trivialization.

Note that if we are given a vector bundle 𝔚 over a space M with fiber a separable Hilbert space ℌ and structure group G ⊂ U(ℌ), then we have an isomorphism of vector bundles 𝔚 ≅ Fr(𝔚) ×_{U(ℌ)} ℌ, where Fr(𝔚) is the principal U(ℌ) bundle of unitary frames for 𝔚. Recall that a quasi-subbundle V′ → M of a smooth vector bundle π : V → M over a manifold M is a closed subspace V′ ⊂ V such that π|_{V′} is still surjective and each fiber of π|_{V′} is a linear subspace [40, Definition 1.2].

Theorem 3.19. Assume that the moduli space M_{𝔰} contains no zero-section pairs. Then there is an open neighborhood U of the subspace ι(M_{𝔰}) in C_{𝔱}, which does not contain any zero-section pairs or other reducibles, and a finite-rank, smooth, trivial, vector subbundle Ξ → U of 𝔙 → U, which is S^{1} equivariant with respect to the circle action (3.2), such that the following hold:

1. The restriction of Ξ to ι(M_{𝔰}) is a complex vector bundle.

2. The smooth bundle map Π_{Ξ^{⊥}} : 𝔙 → Ξ^{⊥} defined by the fiberwise L^{2}-orthogonal projection Π_{Ξ^{⊥}} onto the subbundle Ξ^{⊥} → U restricts to a surjective fiber map Π_{Ξ^{⊥}} : Ran(D𝔖)_{A,Φ} → Ξ_{A,Φ}^{⊥} for any point [A, Φ] ∈ U.

3. If (A, Φ) is an L_{ℓ}^{2} representative of a point in U, for some integer k ≤ ℓ ≤ ∞, then the fiber Ξ_{A,Φ} is contained in L_{ℓ}^{2}(F_{2}) = L_{ℓ}^{2}(Λ^{+} ⊗ 𝔤_{𝔱}) ⊕ L_{ℓ}^{2}(V^{−}).

Proof. By Corollary 3.18, there is a smooth trivialization of 𝔙^{n} given by a smooth isomorphism of complex Hilbert bundles,

τ : 𝔙^{n} ≅ ι(M_{𝔰}) × L^{2}(F_{2}^{n}),

as the space G_{𝔰} (of L_{k+1}^{2} gauge transformations) is a subgroup of the unitary group of the Hilbert space L^{2}(F_{2}^{n}). We have an equality of L^{2}-orthogonal complements,

(Ran D𝔖_{A,Φ})^{⊥} = (Ran d_{A,Φ}^{1,n})^{⊥} ⊂ L^{2}(F_{2}^{n}),

for every (A, Φ) ∈ ι(M̃_{𝔰}), by Lemma 3.15 and the fact that Ran d_{A,Φ}^{1,t} = L^{2}(F_{2}^{t}) by Proposition 2.16, for generic parameters τ. Thus, the family of operators {d_{A,Φ}^{1,n} : (A, Φ) ∈ ι(M̃_{𝔰})} defines quasi-subbundles Ran d^{1,n} and (Ran d^{1,n})^{⊥} of 𝔙^{n} → ι(M_{𝔰}) with fibers

Ran d^{1,n}|_{[A,Φ]} = {(A′, Φ′, Ran d_{A′,Φ′}^{1,n}) : (A′, Φ′) ∈ [A, Φ]}/G_{𝔱}.

For each point [A, Φ] ∈ ι(M_{𝔰}), we define

V_{[A,Φ]}^{⊥} = τ (Ran d^{1,n}|_{[A,Φ]}) ⊂ L^{2}(F_{2}^{n}).

Observe that there is an open neighborhood U_{[A,Φ]} ⊂ ι(M_{𝔰}) with the property that for all [A′, Φ′] ∈ U_{[A,Φ]}, the map

τ (Ran d^{1,n}|_{[A′,Φ′]}) → V_{[A,Φ]}^{⊥}

defined by L^{2}-orthogonal projection onto V_{[A,Φ]}^{⊥} is surjective. Since ι(M_{𝔰}) is compact, it has a finite subcover U_{[A_{α},Φ_{α}]} of such neighborhoods. If V = ⊕_{α}V_{[A_{α},Φ_{α}]}, then V is a finite-dimensional, complex subspace of L^{2}(F_{2}^{n}). If we define Ξ′′ → ι(M_{𝔰}) by setting

Ξ′′ = τ^{−1} (ι(M_{𝔰}) × V) ,

then Ξ will be a complex, finite-dimensional, trivial subbundle of 𝔙^{n}|_{ι(M_{𝔰})} such that fiberwise L^{2}-orthogonal projection onto the subbundle Ξ^{⊥} ⊂ 𝔙^{n}|_{ι(M_{𝔰})} is surjective when restricted to the quasi-subbundle Ran d^{1,n}.

We now extend the bundle Ξ′′ → ι(M_{𝔰}) to a subbundle of 𝔙 over an open neighborhood U of ι(M_{𝔰}) in C_{𝔱}^{0}, which does not contain any other reducible or zero-section pairs, and which is S^{1} equivariant with respect to the circle action (3.2). The space M_{𝔰} is a smooth submanifold of the Riemannian manifold C_{𝔱}^{0} by Lemma 3.16 and so it has an S^{1}-equivariant normal bundle π : 𝔑 → ι(M_{𝔰}) and an S^{1}-invariant tubular neighborhood given by an S^{1}-equivariant diffeomorphism

g : O ⊂ 𝔑 → U ⊂ C_{𝔱}^{0},

from an open, S^{1}-invariant neighborhood O of the zero section of 𝔑 onto an open, S^{1}- invariant neighborhood of ι(M_{𝔰}) in C_{𝔱}^{0} (see, for example, [10, p. 306]). The bundle projection π and the diffeomorphism g define an S^{1}-equivariant, C^{∞} retraction,

r = π ∘ g^{−1} : U ⊂ C_{𝔱}^{0} → ι(M_{𝔰}).

The complex vector bundle Ξ′′ → ι(M_{𝔰}) then extends to a vector bundle r^{∗}Ξ′′ → U, a subbundle of r^{∗}(𝔙|_{ι(M_{𝔰})}) → U, which are both S^{1} equivariant with respect to the circle action (3.2). Since the map r is an S^{1}-equivariant, C^{∞} retraction, there is an S^{1}-equivariant, C^{∞} isomorphism [37, Theorem 4.1.5], [39],

f : 𝔙|_{U} → r^{∗}(𝔙|_{ι(M_{𝔰})}).

We obtain a C^{∞} subbundle of the vector bundle 𝔙|_{U} by setting

Ξ′ = f^{−1}(r^{∗}Ξ′′) → U,

both S^{1} equivariant with respect to the action (3.2). Because Ξ′′ → ι(M_{𝔰}) is isomorphic to ι(M_{𝔰}) × ℂ^{r_{Ξ}} as a complex vector bundle, for some r_{Ξ} ∈ ℕ, we obtain an isomorphism of S^{1}-equivariant vector bundles,

Ξ′ ≅ U × ℂ^{r_{Ξ}},

since the maps f and r are S^{1} equivariant; the circle acts non-trivially on U, except along the stratum ι(M_{𝔰}), and acts by complex multiplication on ℂ^{r_{Ξ}}.

By construction, the fiberwise L^{2}-orthogonal projection 𝔙|_{A,Φ} → (Ξ′_{A,Φ})^{⊥} restricts to a surjective map Ran d_{A,Φ}^{1} → (Ξ′_{A,Φ})^{⊥} for any pair (A, Φ) representing a point in U, after shrinking U if necessary.

Given an L_{k}^{2} pair (A, Φ) representing a point in U ⊂ C_{𝔱}^{0}, our construction yields a subspace

Ξ′_{A,Φ} ⊂ L^{2}(F_{2}) = L^{2}(Λ^{+} ⊗ 𝔤_{𝔱}) ⊕ L^{2}(V^{−}).

If ℓ ≥ k is an integer, it does not necessarily follow that Ξ′_{A,Φ} is contained in the subspace of L_{ℓ−1}^{2} pairs when (A, Φ) is an L_{ℓ}^{2} pair. However, for any t > 0 the heat operator

exp(−t(1 + d_{A,Φ}^{1}d_{A,Φ}^{1,∗})) : L^{2}(Λ^{+} ⊗ 𝔤_{𝔱}) ⊕ L^{2}(V^{−}) → L_{ℓ−1}^{2}(Λ^{+} ⊗ 𝔤_{𝔱}) ⊕ L_{ℓ−1}^{2}(V^{−})

is a bounded, G_{𝔱}-equivariant, S^{1}-equivariant, linear map and we can define

Ξ_{A,Φ} = exp(−t(1 + d_{A,Φ}^{1}d_{A,Φ}^{1,∗}))Ξ′_{A,Φ}.

For small enough t = t(U), the approximation properties of the heat kernel (see [24, Lemma A.1] for a similar application), ensure that

• Ξ → U is a trivial, S^{1}-equivariant vector bundle, with the same rank as Ξ′, and

• L^{2}-orthogonal projection 𝔙|_{A,Φ} → Ξ_{A,Φ}^{⊥} restricts to a surjective map Ran d_{A,Φ}^{1} → Ξ_{A,Φ}^{⊥}

for any pair (A, Φ) representing a point in U. We let Π_{Ξ} : 𝔙 → Ξ be the C^{∞} bundle map defined by fiberwise L^{2}-orthogonal projection. This completes the proof.

3.5.3. Definition of the thickened moduli space and the link. With Theorem 3.19 in place, we can finally construct the required ambient, finite-dimensional, smooth manifold of Definition 3.14 and the link of the singular stratum of reducibles.

Definition 3.20. Assume M_{𝔰} contains no zero-section pairs. Let Ξ be a finite-rank, smooth, S^{1}-equivariant vector bundle over an open neighborhood of ι(M_{𝔰}) in C_{𝔱} such that, as in Theorem 3.19, L^{2}-orthogonal projection gives a surjective map of quasi vector bundles Ran D𝔖 → Ξ^{⊥} over M_{𝔰}. We say that Ξ is a stabilizing bundle for Ran D𝔖 and call

M_{𝔱}(Ξ, 𝔰) = (Π_{Ξ^{⊥}}𝔖)^{−1}(0) ⊂ C_{𝔱} (3.41)

the thickened moduli space defined by Ξ. If (A, Φ) ∈ C̃_{𝔱} represents [A, Φ] ∈ C_{𝔱}, we write Ξ|_{(A,Φ)} for the subspace of L_{k−1}^{2}(F_{2}) representing the fiber of Ξ over [A, Φ]. We then define

N_{𝔱}(Ξ, 𝔰) = Ker(Π_{Ξ^{⊥}}D^{n}), (3.42)

a complex, finite-rank, smooth vector bundle over M_{𝔰} with fibers

N_{𝔱}(Ξ, 𝔰)|_{[A,Φ]} ≅ Ker (d_{A,Φ}^{0,n,∗} + Π_{Ξ^{⊥}}d_{A,Φ}^{1,n}) ,

noting that D_{A,Φ}^{n} = d_{A,Φ}^{0,n,∗} + d_{A,Φ}^{1,n} for [A, Φ] ∈ ι(M_{𝔰}).

Theorem 3.21. Suppose that the spin^{u} structure 𝔱 admits a reduction 𝔱 = 𝔰⊕𝔰⊗L. Assume M_{𝔰} contains no zero-section pairs. Then the following hold:

1. There is an S^{1}-invariant, open neighborhood U of ι(M_{𝔰}) in C_{𝔱} such that the zero locus U ∩ M_{𝔱}(Ξ, 𝔰) is regular and so a manifold of dimension dim M_{𝔱} + rank_{ℝ} Ξ.

2. The space M_{𝔰} is a smooth, S^{1}-invariant submanifold of M_{𝔱}(Ξ, 𝔰).

3. The bundle N_{𝔱}(Ξ, 𝔰) is a normal bundle for the submanifold ι : M_{𝔰} ↪ M_{𝔱}(Ξ, 𝔰) and the tubular map is equivariant with respect to the circle action on N_{𝔱}(Ξ, 𝔰) given by the trivial action on the base M_{𝔰} and complex multiplication on the fibers, and the circle action on M_{𝔱}(Ξ, 𝔰) induced from the S^{1} action (3.2).

4. The restriction of the section 𝔖 to M_{𝔱}(Ξ, 𝔰) takes values in Ξ and vanishes transversely on M_{𝔱}(Ξ, 𝔰) − ι(M_{𝔰}).

Proof. The space M_{𝔱}(Ξ, 𝔰) is the zero locus of the section 𝔗 = Π_{Ξ^{⊥}}𝔖 of a vector subbundle Ξ^{⊥} → U ⊂ C_{𝔱} of 𝔙|_{U} constructed in Theorem 3.19, for some open neighborhood U of ι(M_{𝔰}) in C_{𝔱}. For any point [A, Φ] ∈ ι(M_{𝔰}) ⊂ 𝔖^{−1}(0), we have

(D𝔗)_{A,Φ} = Π_{Ξ^{⊥}}(D𝔖)_{A,Φ}.

(The differential of Π_{Ξ} does not appear here since 𝔖[A, Φ] = 0.) According to Definition 3.20, the L^{2}-orthogonal projection Π_{Ξ^{⊥}} gives a surjective map

Π_{Ξ^{⊥}} : Ran(D𝔖)_{A,Φ} → Ξ_{A,Φ}^{⊥}

and thus (D𝔗)_{A,Φ} is surjective at all points [A, Φ] in the image of M_{𝔰}. Surjectivity is an open condition, so we may assume that D𝔗 is surjective on the open neighborhood U of ι(M_{𝔰}) ⊂ C_{𝔱}, after shrinking U if necessary. The zero locus of 𝔗 in this open set is regular and thus a smooth submanifold of C_{𝔱}, which gives Assertion (1).

By Lemma 3.16 the space M_{𝔰} is a smooth submanifold of C_{𝔱} and as its image is contained in M_{𝔱}(Ξ, 𝔰), it is also a smooth submanifold of M_{𝔱}(Ξ, 𝔰) and Assertion (2) follows, as the zero locus of an S^{1}-equivariant section is S^{1} invariant.

We observe that N_{𝔱}(Ξ, 𝔰) is the normal bundle of M_{𝔰} in M_{𝔱}(Ξ, 𝔰), since

T_{A,Φ}M_{𝔱}(Ξ, 𝔰) = Ker d_{A,Φ}^{0,∗} ∩ Ker Π_{Ξ^{⊥}}(D𝔖)_{A,Φ}
= (Ker d_{A,Φ}^{0,t,∗} ⊕ Ker d_{A,Φ}^{0,n,∗}) ∩ (Ker d_{A,Φ}^{1,t} ⊕ Ker Π_{Ξ^{⊥}}d_{A,Φ}^{1,n})
= Ker D_{A,Φ}^{t} ⊕ Ker Π_{Ξ^{⊥}}D_{A,Φ}^{n}
= T_{A,Φ}ι(M_{𝔰}) ⊕ N_{𝔱}(Ξ, 𝔰)|_{A,Φ}.

(3.43)

In the second equality above, we make use of the fact that the fibers of the vector bundle Ξ → ι(M_{𝔰}) are contained in L_{k−1}^{2}(F_{2}^{n}) and so (by definition) Π_{Ξ} = 0 on L_{k−1}^{2}(F_{0}^{t}) ⊕ L_{k−1}^{2}(F_{2}^{t}) ⊃ Ran D_{A,Φ}^{t}. Also, the splitting (3.43) of tangent spaces corresponds to the splitting of Hilbert spaces L_{k}^{2}(F_{1}) ≅ L_{k}^{2}(F_{1}^{t}) ⊕ L_{k}^{2}(F_{1}^{n}), defined by the subspaces (3.21). Hence, the isomorphism (3.43) is equivariant with respect to the circle action (3.2) on the subspace T_{A,Φ}M_{𝔱}(Ξ, 𝔰) of L_{k}^{2}(F_{1}), the trivial action on the subspace T_{A,Φ}ι(M_{𝔰}) of L_{k}^{2}(F_{1}^{t}), and complex multiplication on the subspace N_{𝔱}(Ξ, 𝔰)|_{A,Φ} of L_{k}^{2}(F_{1}^{n}). This proves Assertion (3).

Because M_{𝔱}(Ξ, 𝔰) is given by U ∩ (Π_{Ξ^{⊥}}𝔖)^{−1}(0) = U ∩ 𝔖^{−1}(Ξ), we see that 𝔖 takes values in Ξ on M_{𝔱}(Ξ, 𝔰) and so Π_{Ξ^{⊥}}𝔖 = 𝔖 on M_{𝔱}(Ξ, 𝔰). According to Theorem 2.13, our transversality result for the section 𝔖 of 𝔙|_{U} = Ξ^{⊥} ⊕ Ξ defined by the PU(2) monopole equations (2.32), the section 𝔖 vanishes transversely on U − ι(M_{𝔰}) with zero locus (U ∩ M_{𝔱}) − ι(M_{𝔰}). This implies that for each [A, Φ] in (U − ι(M_{𝔰})) ∩ 𝔖^{−1}(0), the differential

(D𝔖)_{A,Φ} : T_{[A,Φ]}C_{𝔱}^{∗,0} → 𝔙|_{[A,Φ]}

is surjective. But we have the identifications

T_{[A,Φ]}M_{𝔱}(Ξ, 𝔰) = Ker(D(Π_{Ξ^{⊥}}𝔖))_{[A,Φ]}
= Ker(Π_{Ξ^{⊥}}D𝔖)_{[A,Φ]} (since 𝔖(A, Φ) = 0)
= (D𝔖)_{[A,Φ]}^{−1}(Ξ|_{[A,Φ]}),

and so one has a surjective differential

(D𝔖)_{[A,Φ]} : T_{[A,Φ]}M_{𝔱}(Ξ, 𝔰) → Ξ|_{[A,Φ]},

for [A, Φ] ∈ (U − ι(M_{𝔰})) ∩ 𝔖^{−1}(0). Thus, the sections Π_{Ξ^{⊥}}𝔖 and 𝔖—which are equal on M_{𝔱}(Ξ, 𝔰)—vanish transversely when restricted to M_{𝔱}(Ξ, 𝔰), proving Assertion (4).

The equivariant tubular neighborhood theorem (see [10, p. 306] for the finite-dimensional, G-equivariant case and [48] for the infinite-dimensional case) provides an embedding

γ : O ↪ C_{𝔱}, (3.44)

mapping an open, S^{1}-invariant neighborhood O of the zero-section M_{𝔰} ⊂ N_{𝔱}(Ξ, 𝔰) onto an open neighborhood of the submanifold ι : M_{𝔰} ↪ M_{𝔱}(Ξ, 𝔰) which covers the embedding ι; the map (3.44) is S^{1}-equivariant with respect to scalar multiplication on the fibers of N_{𝔱}(Ξ, 𝔰) and the circle action induced by (3.2) on C_{𝔱}.

The map γ then descends to a homeomorphism, and a diffeomorphism on smooth strata, from the zero locus φ^{−1}(0)/S^{1} in N_{𝔱}(Ξ, 𝔰)/S^{1} onto an open neighborhood of ι(M_{𝔰}) in the actual moduli space, M_{𝔱}, where we define

φ = Π_{Ξ}𝔖 ∘ γ, (3.45)

to be an S^{1}-equivariant obstruction section over O ⊂ N_{𝔱}(Ξ, 𝔰) of the S^{1} equivariant obstruction bundle

γ^{∗}Ξ → N_{𝔱}(Ξ, 𝔰). (3.46)

This descends to a vector bundle

(γ^{∗}Ξ)/S^{1} → N_{𝔱}(Ξ, 𝔰)/S^{1}

on the complement of the zero section, M_{𝔰} ⊂ N_{𝔱}(Ξ, 𝔰)/S^{1}, whose Euler class may be computed from

(γ^{∗}Ξ)/S^{1} ≅ (π_{N}^{∗}Ξ)/S^{1} → N_{𝔱}(Ξ, 𝔰)/S^{1},

where π_{N} : N_{𝔱}(Ξ, 𝔰) → M_{𝔰} is the projection.

We can now construct the link of M_{𝔰} in M_{𝔱}, following Definition 3.14.

Definition 3.22. Assume M_{𝔰} contains no zero-section pairs. Let N_{𝔱}^{ε}(Ξ, 𝔰) ⊂ N_{𝔱}(Ξ, 𝔰) be the sphere bundle of fiber vectors of length ε and set

ℙN_{𝔱}(Ξ, 𝔰) = N_{𝔱}^{ε}(Ξ, 𝔰)/S^{1}.

The link of the stratum ι(M_{𝔰}) ⊂ M_{𝔱} of reducible PU(2) monopoles is defined by

L_{𝔱,𝔰} = γ (φ^{−1}(0) ∩ ℙN_{𝔱}(Ξ, 𝔰)) ⊂ M_{𝔱}^{∗,0}/S^{1}. (3.47)

The orientation for L_{𝔱,𝔰} is defined by the orientation on M_{𝔰} and in turn from the homology orientation Ω (see [56, §6.6]) and the complex structure on the fibers of N_{𝔱}(Ξ, 𝔰) given by the S^{1} action. We treat the quotient ℙN_{𝔱}(Ξ, 𝔰) as the complex projectivization of a complex vector bundle; we use the complex orientation on the obstruction bundle γ^{∗}Ξ in (3.46).

The following lemma shows that the link L_{𝔱,𝔰} can be represented homologically.

Lemma 3.23. Assume M_{𝔰} contains no zero-section pairs. Then for generic ε, the section φ of the obstruction bundle γ^{∗}Ξ in (3.46) vanishes transversely on N_{𝔱}^{ε}(Ξ, 𝔰).

Proof. From the final statement of Theorem 3.21, we see that φ vanishes transversely on M_{𝔱}(Ξ, 𝔰) − ι(M_{𝔰}) and so φ cuts out the zero locus, M_{𝔱} − ι(M_{𝔰}), as a regular submanifold of M_{𝔱}(Ξ, 𝔰) − ι(M_{𝔰}). Then for generic values of ε, the zero locus will intersect N_{𝔱}^{ε}(Ξ, 𝔰) transversely.

Lemma 3.23 implies that

[L_{𝔱,𝔰}] = e ((γ^{∗}Ξ)/S^{1}) ∩ [ℙN_{𝔱}(Ξ, 𝔰)] (3.48)

is the homology class of the link in Definition 3.22.

3.5.4. Group actions and lifts of the normal bundle embedding to the pre-configuration space. In [18] we shall need a lift of the S^{1}-equivariant diffeomorphism γ from a neighborhood O ⊂ N_{𝔱}(Ξ, 𝔰) of the zero-section M_{𝔰} onto on open neighborhood of ι(M_{𝔰}) in the thickened moduli space M_{𝔱}(Ξ, 𝔰) ⊂ C_{𝔱},

γ̃ : Õ ⊂ Ñ_{𝔱}(Ξ, 𝔰) → M̃_{𝔱}(Ξ, 𝔰), (3.49)

where Õ ⊂ Ñ_{𝔱}(Ξ, 𝔰) is an S^{1} and G_{𝔰} invariant open neighborhood of M̃_{𝔰} and M̃_{𝔱}(Ξ, 𝔰) = π^{−1}(M_{𝔱}(Ξ, 𝔰)). It is convenient to describe the construction here. As usual, we need only consider the case where M_{𝔰} contains no zero-section pairs.

To see what should be the “correct” equivariance properties of the lift (3.49), we first consider the obvious extension ι : N_{𝔱}(Ξ, 𝔰) → C_{𝔱} of the embedding ι : M_{𝔰} ↪ C_{𝔱}, since γ approximates this extension on a small open neighborhood of the zero section, M_{𝔰}. Let M̃_{𝔰} = π^{−1}(M_{𝔰}) ⊂ C̃_{𝔰} be the preimage of M_{𝔰} ⊂ C_{𝔰} under the projection π : C̃_{𝔰} → C̃_{𝔰}/G_{𝔰}, and let

Ñ_{𝔱}(Ξ, 𝔰) = π^{∗}N_{𝔱}(Ξ, 𝔰) → M̃_{𝔰}

be the G_{𝔰}-equivariant pullback bundle, so

Ñ_{𝔱}(Ξ, 𝔰) ⊂ M̃_{𝔰} × F_{1}^{n},

where F_{1}^{n} ⊂ F_{1} ≅ F_{1}^{t} ⊕ F_{1}^{n} is the complex Hilbert subspace in (3.21). The bundle Ñ_{𝔱}(Ξ, 𝔰) is complex, with trivial circle action on M̃_{𝔰} and action by complex multiplication on F_{1}^{n}.

The S^{1}-equivariant map ι : N_{𝔰}(Ξ, 𝔰) → C_{𝔱}, [B, Ψ, η] ↦ [ι(B, Ψ) + η], is covered by

ι : Ñ_{𝔱}(Ξ, 𝔰) → C̃_{𝔱}, (B, Ψ, η) ↦ ι(B, Ψ) + η. (3.50)

This map is equivariant with respect to the embedding ϱ : G_{𝔰} ↪ G_{𝔱} in definition (3.10), for the following domain and range G_{𝔰} actions:

• The action of s ∈ G_{𝔰} on Ñ_{𝔱}(Ξ, 𝔰) given by the usual gauge group action on (B, Ψ) ∈ M̃_{𝔰} and the action of G_{𝔰} on η = (β, ψ′) ∈ F_{1}^{n} induced by the isomorphism F_{1} ≅ F_{1}^{t} ⊕ F_{1}^{n} and the embedding ϱ : G_{𝔰} ↪ G_{𝔱}, as described in §3.4.2, so

(s, (B, Ψ, β, ψ′)) ↦ (s(B, Ψ), s^{−2}β, s^{−1}ψ′).

• The action of ϱ(s) ∈ G_{𝔱} on C̃_{𝔱} and F_{1} in definition (3.10), so

(s, ι(B, Ψ) + η) ↦ ϱ(s)(ι(B, Ψ) + η).

The fact that ϱ(s)ι(B, Ψ) = ι(s(B, Ψ)) was noted in Lemma 3.11, while we see that ι(s^{−2}β, s^{−1}ψ′) = ϱ(s)ι(β, ψ′) by the remarks following equation (3.23) in §3.4.2.

The map (3.50) is also S^{1} equivariant for the following domain and range circle actions:

• The trivial S^{1} action on M̃_{𝔰} and action by complex multiplication on F_{1}^{n},

(e^{iθ}, (B, Ψ, η)) ↦ (B, Ψ, e^{iθ}η).

• The S^{1} action (3.2) on C̃_{𝔱}, induced by the trivial action on the factor W of V = W ⊕ W ⊗ L and complex multiplication on W ⊗ L, so on the image (3.50) one has

(e^{iθ}, ι(B, Ψ) + η) ↦ ι(B, Ψ) + e^{iθ}η,

recalling that the points ι(B, Ψ) are fixed by the action (3.2). We now turn to the construction of the map (3.49), which we shall require to have the

same G_{𝔰} and S^{1} equivariance properties as the map (3.50). If [B, Ψ] ∈ M_{𝔰}, then Proposition 2.8 in [24] yields an open neighborhood Ũ_{ι(B,Ψ)} of ι(B, Ψ) in the slice ι(B, Ψ) + Ker d_{ι(B,Ψ)}^{0,∗}, such that projection onto π(Ũ_{ι(B,Ψ)}) = U_{ι(B,Ψ)} gives a local parameterization for an open neighborhood of [ι(B, Ψ)] in M_{𝔱}(Ξ, 𝔰); the pair ι(B, Ψ) has trivial stabilizer in G_{𝔱} since Ψ ≢ 0. Because M_{𝔰} is compact we can assume (by shrinking O if necessary) that

γ[B, Ψ, η] ∈ U_{[ι(B,Ψ)]},

for all [B, Ψ, η] ∈ O ⊂ N_{𝔱}(Ξ, 𝔰). Hence,

γ[B, Ψ, η] = [ι(B, Ψ) + γ_{0}(B, Ψ, η)]

for a slice element γ_{0}(B, Ψ, η) ∈ Ker d_{ι(B,Ψ)}^{0,∗} uniquely determined by ι(B, Ψ, η), and we can therefore set

γ̃(B, Ψ, η) = ι(B, Ψ) + γ_{0}(B, Ψ, η), (B, Ψ, η) ∈ Õ ⊂ Ñ_{𝔱}(Ξ, 𝔰), (3.51)

where Õ is the preimage of O = Õ/G_{𝔰}. Note that γ_{0}(B, Ψ, η) ≈ η. If s ∈ G_{𝔰}, then the preceding equation yields

ϱ(s)γ̃(B, Ψ, η) = ι(s(B, Ψ)) + ϱ(s)γ_{0}(B, Ψ, η), (3.52)

and we again have ϱ(s)γ_{0}(B, Ψ, η) ∈ Ker d_{ι(s(B,Ψ))}^{0,∗}, by the G_{𝔱}-equivariance of the slice condition and the fact that ϱ(s)ι(B, Ψ) = ι(s(B, Ψ)). On the other hand,

γ̃(s(B, Ψ, η)) = ι(s(B, Ψ)) + γ_{0}(s(B, Ψ, η)), (3.53)

while γ[s(B, Ψ, η)] = γ[B, Ψ, η] ∈ O = Õ/G_{𝔰}. Hence, comparing equations (3.52) and (3.53), we see that we must have

γ̃(s(B, Ψ, η)) = ϱ(s)γ̃(B, Ψ, η).

Therefore, the map γ̃ has the same G_{𝔰}-equivariance properties as the map ι : Ñ_{𝔱}(Ξ, 𝔰) → C̃_{𝔱} in (3.50); a very similar argument shows that it has the same S^{1}-equivariance properties.

3.6. The link of a stratum of reducible monopoles: Chern character of the normal bundle. To compute intersection pairings on the space ℙN_{𝔱}(Ξ, 𝔰), we need to know the Chern classes of N_{𝔱}(Ξ, 𝔰). The Chern character of the vector bundle N_{𝔱}(Ξ, 𝔰) → M_{𝔰} is computed by observing that, as elements of K(M_{𝔰}), we have [N_{𝔱}(Ξ, 𝔰)] = Index D^{n} + [Ξ], where D^{n} is the normal component (3.37) of the family of deformation operators D parameterized by M_{𝔰}. For convenience in this section, we shall often omit explicit mention of the embedding map ι : M_{𝔰} ↪ M_{𝔱}(Ξ, 𝔰) and write the bundles ι^{∗}N_{𝔱}(Ξ, 𝔰) and ι^{∗}(Index D^{n}) over M_{𝔰} simply as N_{𝔱}(Ξ, 𝔰) and Index D^{n}, respectively. We then use the Atiyah-Singer index theorem for families to express ch(N_{𝔱}(Ξ, 𝔰)) in terms of the cohomology classes on H^{•}(C_{𝔰}^{0}; ℝ) described in §2.4.2.

From (3.37) we obtain a family of elliptic differential operators

C^{∞}(E) ⟶^{D^{n}} C^{∞}(F)
↓  ↓
M_{𝔰} × X ⟶^{id} M_{𝔰} × X

where C^{∞}( ·) denotes the space of smooth sections of the families of finite-rank vector bundles E, F over M_{𝔰} × X defined by

\begin{aligned}E & = M̃_{𝔰} ×_{G_{𝔰}} F_{1}^{n} \\ & = M̃_{𝔰} ×_{G_{𝔰}} ((Λ^{1} ⊗ L) ⊕ W^{+} ⊗ L) , \\ F & = M̃_{𝔰} ×_{G_{𝔰}} (F_{0}^{n} ⊕ F_{2}^{n}) \\ & = M̃_{𝔰} ×_{G_{𝔰}} ((Λ^{0} ⊕ Λ^{+}) ⊗ L ⊕ W^{−} ⊗ L) . \end{aligned} (3.54)

Recall from the paragraph following (3.23) that an element s ∈ G_{𝔰} acts on the bundles listed in (3.54) as multiplication by s^{−2} on the factors Λ^{j} ⊗ L and as multiplication by s^{−1} on the factors W^{±} ⊗ L.

Recall that N_{𝔱}(Ξ, 𝔰) = Ker Π_{Ξ^{⊥}}D^{n} by definition (3.42). On the other hand, by equation (3.37), we see that

Coker Π_{Ξ^{⊥}}D_{ι(B,Ψ)}^{n} = Coker d_{ι(B,Ψ)}^{0,n,∗} ⊕ Coker Π_{Ξ^{⊥}}d_{ι(B,Ψ)}^{1,n}.

For any [B, Ψ], we have Coker d_{ι(B,Ψ)}^{0,∗} = 0, as the stabilizer of ι(B, Ψ) is trivial in G_{𝔱}, so

Coker d_{ι(B,Ψ)}^{0,n,∗} = 0, while Lemma 3.15 and Proposition 2.16 implies that Coker d_{ι(B,Ψ)}^{1,t} = 0. Hence,

Coker Π_{Ξ^{⊥}}d_{ι(B,Ψ)}^{1,t} = 0

and

Coker Π_{Ξ^{⊥}}D_{ι(B,Ψ)}^{n} = Coker Π_{Ξ^{⊥}}d_{ι(B,Ψ)}^{1,n} ⊕ Coker Π_{Ξ^{⊥}}d_{ι(B,Ψ)}^{1,t}
= Coker Π_{Ξ^{⊥}}d_{ι(B,Ψ)}^{1} = Coker Π_{Ξ^{⊥}}(D𝔖)_{ι(B,Ψ)}.

Because L^{2}-orthogonal projection from Ran(D𝔖)_{ι(B,Ψ)} surjects onto Ξ_{ι(B,Ψ)}^{⊥} for all points [B, Ψ] ∈ M_{𝔰}, we obtain the identity

Coker Π_{Ξ^{⊥}}D^{n} = Coker Π_{Ξ^{⊥}} ≅ Ξ.

The subbundle Ξ of 𝔙 → M_{𝔰} is trivial by the construction of Theorem 3.19. From the stabilization construction of [9, §I.7.B] and the preceding remarks we see that, as elements of the K-theory group K(M_{𝔰}),

Index D^{n} = [Ker Π_{Ξ^{⊥}}D^{n}] − [Coker Π_{Ξ^{⊥}}D^{n}]
= [N_{𝔱}(Ξ, 𝔰)] − [Ξ]
= [N_{𝔱}(Ξ, 𝔰)] − [M_{𝔰} × ℂ^{r_{Ξ}}],

(3.55)

where r_{Ξ} = rank_{ℂ} Ξ. To compute ch(Index D^{n}) we shall apply the Atiyah-Singer index theorem for families of Dirac operators.

Proposition 3.24. [6], [14, Theorem 5.1.16] Let X be a closed, oriented, smooth fourmanifold with spin^{c} structure (ρ, W). Suppose E → T × X is a locally trivial family of complex vector bundles over X, parameterized by a compact space T, with a connection A_{t} on the bundle E_{t} = E|_{{t}×X} for all t ∈ T. Then the Chern character of the index bundle of the family of Dirac operators parameterized by T,

C^{∞}(E ⊗ W^{+}) ⟶^{D} C^{∞}(E ⊗ W^{−})
↓  ↓
T × X  ⟶^{id}  T × X

given by D_{A_{t}} : C^{∞}(E_{t} ⊗ W^{+}) → C^{∞}(E_{t} ⊗ W^{−}), t ∈ T, is

ch(Index(D, E, W)) = ch(E) ch(Index(D, W))
= ch(E)e^{\frac{1}{2} c_{1}(W^{+})}Â(X)/[X],

where D : C^{∞}(W^{+}) → C^{∞}(W^{−}) is the Dirac operator defined by the given spin^{c} structure.

The index of the family of operators in Proposition 3.24 defines a group homomorphism [14, p. 184],

Index(D, · ) : K(T × X) → K(T),

by taking the element [E] of K(T × X) to Index(D, E, W) = [Ker D] − [Coker D] in K(T), a virtual vector bundle over the parameter space T. We now compute ch(N_{𝔱}(Ξ, 𝔰)) in the following steps:

• Identify D^{n} with the sum of a pair of families of Dirac operators, D′ and D′′ .

• Compute ch(N_{𝔱}(Ξ, 𝔰)), using Proposition 3.24 to compute the Chern characters of the

index bundles of these families of Dirac operators. The first step is accomplished in part by introducing the following families of operators,

\begin{matrix}C^{∞}(E′) & ⟶^{δ} & C^{∞}(F′) \\ ↓ & & ↓ \\ M_{𝔰} × X & ⟶^{id} & M_{𝔰} × X \end{matrix}  and  \begin{matrix}C^{∞}(E′′) & ⟶^{D′′} & C^{∞}(F′′) \\ ↓ & & ↓ \\ M_{𝔰} × X & ⟶^{id} & M_{𝔰} × X \end{matrix} (3.56)

where E = E′ ⊕ E′′ and F = F′ ⊕ F′′, and we have defined

E′ = M̃_{𝔰} ×_{G_{𝔰}} (Λ^{1} ⊗ L)  and E′′ = M̃_{𝔰} ×_{G_{𝔰}} (W^{+} ⊗ L),
F′ = M̃_{𝔰} ×_{G_{𝔰}} ((Λ^{0} ⊕ Λ^{+}) ⊗ L) and F′′ = M̃_{𝔰} ×_{G_{𝔰}} (W^{−} ⊗ L). (3.57)

Explicitly, if (B, Ψ) is a pair representing a point [B, Ψ] in M_{𝔰}, so the spin connection in the pair ι(B, Ψ) ∈ M̃_{𝔱} is given by B ⊕ B ⊗ A_{L} and A_{L} = A_{Λ} ⊗ (B^{det})^{∗} is the induced unitary connection on the subbundle L of 𝔤_{𝔱} ≅ i\underline{ℝ} ⊕ L, the operators on the fibers defined by these families are then given by

\begin{aligned}δ_{A_{L}} ≡ d_{A_{L}}^{∗} + d_{A_{L}}^{+} & : C^{∞}(Λ^{1} ⊗ L) → C^{∞}((Λ^{0} ⊕ Λ^{+}) ⊗ L), \\ D′′_{B⊗A_{L}} & : C^{∞}(W^{+} ⊗ L) → C^{∞}(W^{−} ⊗ L), \end{aligned} (3.58)

where D′′_{B⊗A_{L}} is the Dirac operator. We use the preceding families to rewrite Index D^{n} in terms of index bundles whose Chern characters are more readily computable:

Lemma 3.25. Continue the above notation. Then, as elements of K(M_{𝔰}),

Index D^{n} = Index δ + Index D′′.

Proof. As an element of K(M_{𝔰}), the index bundle Index D^{n} depends only on the homotopy class of the leading symbol [49, Theorem III.8.6]. Thus, the index bundle of the family (B, Ψ) ↦ D_{B,Ψ}^{n} is equivalent to that of the family (B, Ψ) ↦ D_{B,0}^{n}, where D_{B,0}^{n} = δ_{A_{L}} + D′′_{B⊗A_{L}}.

We now identify the family of operators δ in (3.56) with a family of Dirac operators. (Identifications of this type were used in [5] to compute the index of the elliptic deformation complex for the anti-self-dual equation.) The vector bundle End(W) ≅ W ⊗W^{∗} is a Clifford module by Clifford multiplication on the factor W in the tensor product and so we obtain a Dirac operator,

D : C^{∞}(W^{±} ⊗ W^{∗}) → C^{∞}(W^{∓} ⊗ W^{∗}), (3.59)

on End(W) [49, p. 122-123]. Under the identification Λ^{•} ⊗_{ℝ} ℂ with End(W) given by Clifford multiplication, the operator d^{∗} + d on Λ^{•} is identified with the Dirac operator (3.59) (see [49, Theorem II.5.12]). Restricting the domain of Clifford multiplication and tensoring with the line bundle L gives isomorphisms

Λ^{1} ⊗_{ℝ} L ≅ Hom(W^{+}, W^{−}) ⊗_{ℂ} L,
(Λ^{0} ⊕ Λ^{+}) ⊗_{ℝ} L ≅ End(W^{+}) ⊗_{ℂ} L, (3.60)

and therefore isomorphisms

E′ ≅ E′_{−} and F′ ≅ E′_{+},

where the bundles E′_{±} are defined by

E′_{−} = M̃_{𝔰} ×_{G_{𝔰}} (Hom(W^{+}, W^{−}) ⊗ L),
E′_{+} = M̃_{𝔰} ×_{G_{𝔰}} (End(W^{+}) ⊗ L).

(3.61)

The operator d_{A_{L}}^{∗} + d_{A_{L}}^{+} is the restriction of d_{A_{L}}^{∗} + d_{A_{L}} to Λ^{1} ⊗_{ℝ} L composed with the projection from (Λ^{0} ⊕ Λ^{2}) ⊗_{ℝ} L to (Λ^{0} ⊕ Λ^{+}) ⊗_{ℝ} L. Thus, the isomorphisms (3.60) identify the family of operators δ in (3.56) with the family of Dirac operators:

\begin{matrix}C^{∞}(E′_{−}) & ⟶^{D′} & C^{∞}(E′_{+}) \\ ↓ & & ↓ \\ M_{𝔰} × X & ⟶^{id} & M_{𝔰} × X \end{matrix} (3.62)

Explicitly, if (B, Ψ) is a pair representing a point [B, Ψ] in M_{𝔰}, the Dirac operator on the fiber given by this family,

D′_{B⊗A_{L}} : C^{∞}(Hom(W^{+}, W^{−}) ⊗ L) → C^{∞}(End(W^{+}) ⊗ L), (3.63)

is defined by the spin^{c} connection on End(W) induced by the spin^{c} connection B on W and the unitary connection A_{L} = A_{Λ} ⊗ (B^{det})^{∗} on L. The preceding identification yields:

Lemma 3.26. Continue the above notation. Then, as elements of K(M_{𝔰}),

Index δ = Index D′.

We now begin the second step, which is to compute the Chern characters of the index bundles of the families of Dirac operators in (3.56) and (3.62). The following technical lemma helps to identify the universal bundles E′_{±}, E′′, and F′′.

Lemma 3.27. If Q_{i} → M, i = 1, 2, are S^{1} bundles over a manifold M and L_{i} = Q_{i} ×_{S^{1}} ℂ, i = 1, 2 are the associated complex line bundles, and k ∈ ℤ, then the following hold:

1. If V → M is a complex vector bundle, and e^{iθ} ∈ S^{1} acts on the fiber product Q_{1} ×_{M} V by e^{iθ} · (q_{1}, v) = (e^{iθ}q_{1}, e^{ikθ}v), then (Q_{1} ×_{M} V)/S^{1} ≅ L_{1}^{−k} ⊗ V.

2. If e^{iθ} ∈ S^{1} acts on the fiber product Q_{1} ×_{M} Q_{2} by e^{iθ} ·(q_{1}, q_{2}) = (e^{iθ}q_{1}, e^{ikθ}q_{2}), then the first Chern class of the S^{1}-bundle (Q_{1} ×_{M} Q_{2})/S^{1} → M is c_{1}(Q_{2}) − kc_{1}(Q_{1}), where the action of S^{1} on (Q_{1} ×_{M} Q_{2})/S^{1} is induced by the S^{1} action on Q_{2} of weight one.

Proof. The associated line bundles L_{i} = Q_{i} ×_{S^{1}} ℂ are given by the quotients of Q_{i} × ℂ by the relation (q_{i}, z) ∼ (e^{−iθ}q_{i}, e^{iθ}z), for (q_{i}, z) ∈ Q_{i}×ℂ and e^{iθ} ∈ S^{1}; under the same relation, Q_{1} ×_{S^{1}} S^{1} = Q_{1}. A tensor product of a complex line bundle L with a complex vector bundle V is given by the quotient of the fiber product L ×_{M} V by the relation (z, v) ∼ (w^{−1}z, wv), for z ∈ L, v ∈ V, and w ∈ ℂ^{∗}. Hence,

L_{1}^{−k} ⊗ V = {([q_{1}, z], v) ∈ L_{1} ×_{M} V : ([q_{1}, z], v) ∼ ([q_{1}, wz], w^{k}v), w ∈ ℂ^{∗}}
= {([q_{1}, e^{iμ}], v) ∈ Q_{1} ×_{M} V : ([q_{1}, e^{iμ}], v) ∼ ([q_{1}, e^{iθ}e^{iμ}], e^{ikθ}v), e^{iθ} ∈ S^{1}}

= {(p_{1}, v) ∈ Q_{1} ×_{M} V : (p_{1}, v) ∼ (e^{iθ}p_{1}, e^{ikθ}v), e^{iθ} ∈ S^{1}} (where p_{1} = [q_{1}, e^{iμ}])

= (Q_{1} ×_{M} V)/S^{1},

where in the second line above we can assume without loss that z ≠ 0. This proves Assertion (1) and Assertion (2) follows trivially from this.

Since it will not cause confusion, we will write 𝕃_{𝔰} for the restriction of the universal bundle of equation (2.69) to M_{𝔰} × X,

𝕃_{𝔰} = M̃_{𝔰} ×_{G_{𝔰}} \underline{ℂ},

and define

𝕃_{𝔰,x} = 𝕃_{𝔰}|_{M_{𝔰}×{x}}. (3.64)

By construction, c_{1}(𝕃_{𝔰,x}) = c_{1}(𝕃_{𝔰})/x = μ_{𝔰}(x) where x ∈ H_{0}(X; ℤ) is a generator. Let π_{M} : M_{𝔰} × X → M_{𝔰} and π_{X} : M_{𝔰} × X → X be the projections.

Lemma 3.28. Assume M_{𝔰} contains no zero-section pairs. If r^{∗}Δ → M_{𝔰}×X is the restriction of the pullback by the retraction r of Lemma 2.20 of the line bundle (2.76) then, using Hom(W^{+}, W^{−}) ≅ W^{−} ⊗ W^{+,∗} and End(W^{+}) ≅ W^{+} ⊗ W^{+,∗}, there are isomorphisms,

\begin{matrix}E′_{−} ≅ E′ ⊗ π_{X}^{∗}W^{−}, & E′_{+} ≅ E′ ⊗ π_{X}^{∗}W^{+}, \\ E′′ ≅ E′′ ⊗ π_{X}^{∗}W^{+}, & F′′ ≅ E′′ ⊗ π_{X}^{∗}W^{−}, \end{matrix}

where we have defined

\begin{aligned}E′ & = (π_{M}^{∗}𝕃_{𝔰,x} ⊗ r^{∗}Δ)^{⊗2} ⊗ π_{X}^{∗}(W^{+,∗} ⊗ L), \\ E′′ & = π_{M}^{∗}𝕃_{𝔰,x} ⊗ r^{∗}Δ ⊗ π_{X}^{∗}L. \end{aligned} (3.65)

Proof. There are isomorphisms,

E′_{±} ≅ (M̃_{𝔰} ×_{G_{𝔰}} L) ⊗ Hom(W^{+}, W^{±}), (3.66)

where, from equation (3.23), the action of s ∈ G_{𝔰} on M̃_{𝔰} × L is given by (B, Ψ, z) ↦ (s(B, Ψ), s^{−2}z). Let M̃_{𝔰} ×_{G_{𝔰}} (X × S^{1}) be the unit sphere bundle of 𝕃_{𝔰}. If e^{iμ} ∈ S^{1} acts on

(M̃_{𝔰} ×_{G_{𝔰}} (X × S^{1})) ×_{M_{𝔰}×X} π_{X}^{∗}L

by [(B, Ψ), x, e^{iθ}, z] ↦ [(B, Ψ), x, e^{iμ}e^{iθ}, e^{−i2μ}z]—where (B, Ψ) ∈ M̃_{𝔰}, x ∈ X, e^{iθ} ∈ S^{1} and z ∈ π_{X}^{∗}L—then we obtain an isomorphism of complex line bundles

\begin{matrix}((M̃_{𝔰} ×_{G_{𝔰}} (X × S^{1})) ×_{M_{𝔰}×X} π_{X}^{∗}L) /S^{1} & → M̃_{𝔰} ×_{G_{𝔰}} L, \\ [(B, Ψ), x, e^{iθ}, z] & ↦ [(B, Ψ), e^{2iθ}z]. \end{matrix} (3.67)

Lemma 3.27 then implies that

M̃_{𝔰} ×_{G_{𝔰}} L ≅ 𝕃_{𝔰}^{⊗2} ⊗ π_{X}^{∗}L (3.68) and equation (3.68) gives

E′_{±} ≅ 𝕃_{𝔰}^{⊗2} ⊗ π_{X}^{∗}(L ⊗ W^{+,∗} ⊗ W^{±}). (3.69)

The desired expression for E′_{±} then follows from the isomorphism 𝕃_{𝔰} ≅ 𝕃_{𝔰,x} ⊗ r^{∗}Δ given by Lemma 2.24.

In the bundles

E′′ ≅ M̃_{𝔰} ×_{G_{𝔰}} (W^{+} ⊗ L) and F′′ ≅ M̃_{𝔰} ×_{G_{𝔰}} (W^{−} ⊗ L),

an element s ∈ G_{𝔰} acts on M̃_{𝔰} × W^{±} ⊗ L by ((B, Ψ), Ψ′) ↦ (s(B, Ψ), s^{−1}Ψ′), where Ψ′ ∈ C^{∞}(W^{±} ⊗ L), as noted in the remark following equation (3.23). Lemma 3.27 and the argument yielding equation (3.68) then imply that there are isomorphisms of complex vector bundles

E′′ ≅ 𝕃_{𝔰} ⊗ π_{X}^{∗}(W^{+} ⊗ L) and F′′ ≅ 𝕃_{𝔰} ⊗ π_{X}^{∗}(W^{−} ⊗ L). (3.70)

The isomorphisms in the conclusion of the lemma now follow from equation (3.70) and the isomorphism 𝕃_{𝔰} ≅ π_{M}^{∗}𝕃_{𝔰,x} ⊗ r^{∗}Δ implied by equation (2.77).

Given Lemma 3.28, the index bundles of the families of Dirac operators in (3.56) and (3.62) now take the shape:

\begin{matrix}C^{∞}(E′ ⊗ W^{−}) & ⟶^{D′} & C^{∞}(E′ ⊗ W^{+}) \\ ↓ & & ↓ \\ M_{𝔰} × X & ⟶^{id} & M_{𝔰} × X \end{matrix} and \begin{matrix}C^{∞}(E′′ ⊗ W^{+}) & ⟶^{D′′} & C^{∞}(E′′ ⊗ W^{−}) \\ ↓ & & ↓ \\ M_{𝔰} × X & ⟶^{id} & M_{𝔰} × X \end{matrix}

We abuse notation slightly by continuing to denote the families of Dirac operators on these isomorphic bundles by D′ and D′′.

The final step in the computation of the Chern character of N_{𝔱}(Ξ, 𝔰) is to compute the Chern characters of these families.

Given the decomposition (3.37) of D_{ι(B,Ψ)}^{n}, it will be convenient to define

(3.71) Index_{ℂ} D_{ι(B,Ψ)}^{n} = n_{s} = n′_{s} + n′′_{s}, where n′_{s} = Index_{ℂ}(d_{A_{L}}^{∗} + d_{A_{L}}^{+}), n′′_{s} = Index_{ℂ} D′′_{B⊗A_{L}}.

Viewing E = \underline{ℂ} ⊕ L and 𝔰𝔲(E) = i\underline{ℝ} ⊕ L, we can compute the complex index of d_{A_{L}}^{∗} + d_{A_{L}}^{+} from the real index of d∗ ; Â +d+ ; Â on C^{∞}(Λ^{1} ⊗𝔰𝔲(E)) (for example, see [14, Equation (4.2.22)]) and the fact (Lemma 2.9) that d∗ ; Â +d+ ; Â decomposes as the direct sum of d^{∗} +d^{+} on C^{∞}(iΛ^{1}) and d_{A_{L}}^{∗} + d_{A_{L}}^{+} on C^{∞}(Λ^{1} ⊗ L); the complex index of the spin^{c} Dirac operator is given in [56, p. 47]. Thus,

\begin{aligned}n′_{s}(𝔱, 𝔰) & = − (c_{1}(𝔰) − c_{1}(𝔱))^{2} − \frac{1}{2}(χ + σ), \\ n′′_{s}(𝔱, 𝔰) & = \frac{1}{8}((2c_{1}(𝔱) − c_{1}(𝔰))^{2} − σ). \end{aligned} (3.72)

We can now state and prove one of the main results of our article:

Theorem 3.29. Let 𝔱 be a spin^{u} structure over a closed, oriented, Riemannian, smooth four-manifold X. Assume 𝔱 admits a splitting 𝔱 = 𝔰 ⊕ 𝔰 ⊗ L, where L is a complex line bundle. Suppose that there are no zero-section pairs in M_{𝔰}. Let μ_{𝔰} be the Seiberg-Witten μ-map, as in definition (2.71). Let x ∈ H_{0}(X; ℤ) be the positive generator and for {γ_{i}} a basis for H_{1}(X; ℤ)/ Tor, let {γ_{i}^{∗}} be the dual basis for H^{1}(X; ℤ) introduced in Definition 2.21. Let r_{Ξ} = rank_{ℂ} Ξ denote the rank of the stabilizing bundle. Then the Chern character of the normal bundle N_{𝔱}(Ξ, 𝔰) of the stratum M_{𝔰} ↪ M_{𝔱}(Ξ, 𝔰) is

\begin{aligned}ch(N_{𝔱}(Ξ, 𝔰)) & = r_{Ξ} + n′′_{s}e^{μ_{𝔰}(x)} + n′_{s}e^{2μ_{𝔰}(x)} \\ & − 8 ∑_{i<j}⟨γ_{i}^{∗}γ_{j}^{∗}(c_{1}(𝔱) − c_{1}(𝔰)), [X]⟩e^{2μ_{𝔰}(x)}μ_{𝔰}(γ_{i}γ_{j}) \\ & + \frac{1}{2} ∑_{i<j}⟨γ_{i}^{∗}γ_{j}^{∗}(2c_{1}(𝔱) − c_{1}(𝔰)), [X]⟩e^{μ_{𝔰}(x)}μ_{𝔰}(γ_{i}γ_{j}) \\ & + (e^{μ_{𝔰}(x)} − 32e^{2μ_{𝔰}(x)}) ∑_{i<j<k<ℓ}⟨γ_{i}^{∗}γ_{j}^{∗}γ_{k}^{∗}γ_{ℓ}^{∗}, [X]⟩μ_{𝔰}(γ_{i}γ_{j}γ_{k}γ_{ℓ}). \end{aligned} (3.73)

where μ_{𝔰}(γ_{i}γ_{j}) = μ_{𝔰}(γ_{i})μ_{𝔰}(γ_{j}) and similarly for μ_{𝔰}(γ_{i}γ_{j}γ_{k}γ_{ℓ}).

Proof. For convenience in the proof, we write μ = μ_{𝔰}(x) and γ_{i}^{J,∗} = μ_{𝔰}(γ_{i}), as in Lemma 2.24. The K-theory identification (3.55) of N_{𝔱}(Ξ, 𝔰), Lemmas 3.25 and 3.26, and the homomorphism property of the Chern character (see, for example, [49, Proposition III.11.16]) imply that

ch(N_{𝔱}(Ξ, 𝔰)) = ch(Index D^{n}) + ch(Ξ)
= ch(Index D′) + ch(Index D′′) + r_{Ξ}.

(3.74)

Recall that Index(D^{∗}, E, W) denotes the index bundle of the family of operators obtained, as in Proposition 3.24, by twisting the Dirac operator D^{∗} : C^{∞}(W^{−}) → C^{∞}(W^{+}) by a family of connections on the bundle E. Note that ch(Index(D^{∗}, W)) = − ch(Index(D, W)), where D^{∗} : C^{∞}(W^{−}) → C^{∞}(W^{+}). We now use Lemma 3.28 and Proposition 3.24 to partly compute the Chern characters of the index bundle Index D′. Bundles over X will be considered to be bundles over M_{𝔰} × X; the pullback π_{X}^{∗} will be omitted.

ch( Index D′)= ch(Index(D^{∗}, E′, W)) = ch(E′) ch(Index(D^{∗}, W))
= − ch ((π_{M}^{∗}𝕃_{𝔰,x} ⊗ r^{∗}Δ)^{⊗2} ⊗ W^{+,∗} ⊗ L) ch(Index(D, W))
= −π_{M}^{∗} ch(𝕃_{𝔰,x})^{2}r^{∗} ch(Δ)^{2} ch(W^{+,∗})e^{c_{1}(L)}e^{\frac{1}{2} c_{1}(W^{+})}Â(X)/[X].

(3.75)

Similarly, we use Lemma 3.28 and Proposition 3.24 to partly compute the Chern character of the index bundle Index D′′:

ch(Index D′′) = ch(Index(D, E′′, W)) = ch(E′′) ch(Index(D, W))
= ch(π_{M}^{∗}𝕃_{𝔰,x} ⊗ r^{∗}Δ ⊗ L)e^{\frac{1}{2} c_{1}(W^{+})}Â(X)/[X]
= π_{M}^{∗} ch(𝕃_{𝔰,x})r^{∗} ch(Δ)e^{c_{1}(L)}e^{\frac{1}{2} c_{1}(W^{+})}Â(X)/[X].

(3.76)

In the first lines of (3.75) and (3.76) we simply rewrite the index bundles using the notation of Proposition 3.24.

For the calculation of ch(Index D′′) in (3.76), we compute ch(Δ) using the expression for r^{∗}c_{1}(Δ) in Lemmas 2.23 and 2.24:

r^{∗} ch(Δ) = ∏_{i=1}^{b_{1}(X)} e^{γ_{i}^{J,∗}×γ_{i}^{∗}} = ∏_{i=1}^{b_{1}(X)} (1 + γ_{i}^{J,∗} × γ_{i}^{∗}) .

Because c_{1}(𝕃_{𝔰,x}) = μ as noted before Lemma 3.28, we see π_{M}^{∗} ch(𝕃_{𝔰,x}^{∗}) = π_{M}^{∗}e^{μ}. We shall write μ for π_{M}^{∗}μ. Applying this to (3.76) and noting that c_{1}(L) + \frac{1}{2}c_{1}(W^{+}) = c_{1}(𝔱) − \frac{1}{2}c_{1}(𝔰), we obtain

ch(Index D′′)

= e^{μ} ( ∏_{i=1}^{b_{1}(X)} e^{γ_{i}^{J,∗}×γ_{i}^{∗}} ) e^{c_{1}(𝔱)− \frac{1}{2}c1(𝔰)} (1 − \frac{1}{24}p_{1}(X)) /[X]

= e^{μ} (1 + ∑_{i<j}(γ_{i}^{J,∗}γ_{j}^{J,∗}) × (γ_{i}^{∗}γ_{j}^{∗}) + ∑_{i<j<k<ℓ}(γ_{i}^{J,∗}γ_{j}^{J,∗}γ_{k}^{J,∗}γ_{ℓ}^{J,∗}) × (γ_{i}^{∗}γ_{j}^{∗}γ_{k}^{∗}γ_{ℓ}^{∗}))

× (1 + \frac{1}{2}(2c_{1}(𝔱) − c_{1}(𝔰)) + \frac{1}{8}(2c_{1}(𝔱) − c_{1}(𝔰))^{2}) (1 − \frac{1}{24}p_{1}(X)) /[X],

and therefore, using σ = \frac{1}{3}⟨p_{1}(X), [X]⟩ (see [7, Theorem I.3.1])

ch(Index D′′) = e^{μ}(\frac{1}{8}(2c_{1}(𝔱) − c_{1}(𝔰))^{2} − \frac{1}{8}σ

+ \frac{1}{2} ∑_{i<j}(γ_{i}^{J,∗}γ_{j}^{J,∗})⟨γ_{i}^{∗}γ_{j}^{∗}(2c_{1}(𝔱) − c_{1}(𝔰)), [X]⟩

+ ∑_{i<j<k<ℓ}(γ_{i}^{J,∗}γ_{j}^{J,∗}γ_{k}^{J,∗}γ_{ℓ}^{J,∗})⟨(γ_{i}^{∗}γ_{j}^{∗}γ_{k}^{∗}γ_{ℓ}^{∗}), [X]⟩),

(3.77)

completing the calculation of ch(Index D′′).

We now complete the calculation of ch(Index δ) in (3.75). To compute ch(W^{+,∗}) we observe that c_{1}(W^{+,∗}) = −c_{1}(W^{+}) while c_{2}(W^{+,∗}) = c_{2}(W^{+}) and so, using this and the isomorphism 𝔰𝔲(W^{+}) ≅ Λ^{+}, we have

ch(W^{+,∗}) = 2 + c_{1}(W^{+,∗}) + \frac{1}{2} (c_{1}(W^{+,∗})^{2} − 2c_{2}(W^{+,∗})
= 2 − c_{1}(W^{+}) + \frac{1}{2}c_{1}(W^{+})^{2} + \frac{1}{4} (p_{1}(𝔰𝔲(W^{+})) − c_{1}(W^{+})^{2})
= 2 − c_{1}(W^{+}) + \frac{1}{4} (p_{1}(Λ^{+}) + c_{1}(W^{+})^{2}) .

Applying this to the terms involving W^{+} in (3.75) and writing c_{1}(W^{+}) = c_{1}(𝔰) yields

− ch(W^{+,∗})e^{\frac{1}{2} c_{1}(W^{+})}Â(X)/[X]
= − (2 − c_{1}(𝔰) + \frac{1}{4}(c_{1}(L)^{2} + p_{1}(Λ^{+}))) (1 + \frac{1}{2}c_{1}(𝔰) + \frac{1}{8}c_{1}(L)^{2}) (1 − \frac{1}{24}p_{1}(X)) /[X]
= −(2 + \frac{1}{4}p_{1}(Λ^{+})) (1 − \frac{1}{24}p_{1}(X)) /[X].

Hence, using the preceding expression in (3.75) and c_{1}(L) = c_{1}(𝔱) − c_{1}(𝔰), we see that

ch(Index δ)

= −π_{M}^{∗} ch(𝕃_{𝔰,x})^{2}r^{∗} ch(Δ)^{2} ch(L)(2 + \frac{1}{4}p_{1}(Λ^{+})) (1 − \frac{1}{24}p_{1}(X)) /[X]

= −e^{2μ} ( ∏_{i=1}^{b_{1}(X)} (1 + 2γ_{i}^{J,∗} × γ_{i}^{∗})) (1 + (c_{1}(𝔱) − c_{1}(𝔰)) + \frac{1}{2}(c_{1}(𝔱) − c_{1}(𝔰))^{2})

× (2 + \frac{1}{4}p_{1}(Λ^{+})) (1 − \frac{1}{24}p_{1}(X)) /[X].

Simplifying the preceding expression for ch(Index δ) and recalling that p_{1}(Λ^{+}) = 2χ + 3σ (from [38, Satz 1.5]), yields

ch(Index δ)

= −e^{2μ}(1 + 4 ∑_{i<j}(γ_{i}^{J,∗}γ_{j}^{J,∗}) × (γ_{i}^{∗}γ_{j}^{∗}) + 16 ∑_{i<j<k<ℓ}(γ_{i}^{J,∗}γ_{j}^{J,∗}γ_{k}^{J,∗}γ_{ℓ}^{J,∗}) × (γ_{i}^{∗}γ_{j}^{∗}γ_{k}^{∗}γ_{ℓ}^{∗}))

× (1 + (c_{1}(𝔱) − c_{1}(𝔰)) + \frac{1}{2}(c_{1}(𝔱) − c_{1}(𝔰))^{2}) (2 + \frac{1}{4}p_{1}(Λ^{+})) (1 − \frac{1}{24}p_{1}(X)) /[X],

and thus,

\begin{aligned}ch(Index δ) & = e^{2μ}(−(c_{1}(𝔱) − c_{1}(𝔰))^{2} − \frac{1}{2}σ − \frac{1}{2}χ \\ & − 8 ∑_{i<j}(γ_{i}^{J,∗}γ_{j}^{J,∗})⟨(γ_{i}^{∗}γ_{j}^{∗})(c_{1}(𝔱) − c_{1}(𝔰)), [X]⟩ \\ & − 32 ∑_{i<j<k<ℓ}(γ_{i}^{J,∗}γ_{j}^{J,∗}γ_{k}^{J,∗}γ_{ℓ}^{J,∗})⟨γ_{i}^{∗}γ_{j}^{∗}γ_{k}^{∗}γ_{ℓ}^{∗}, [X]⟩). \end{aligned} (3.78)

The desired expression for ch(N_{𝔱}(Ξ, 𝔰)) follows from (3.72), (3.74), (3.77), and (3.78).

Finally, we calculate the total Chern class c(N_{𝔱}(Ξ, 𝔰)), as an element of rational cohomology, under a simplifying assumption.

Corollary 3.30. Continue the hypotheses of Theorem 3.29 and assume that α ⌣ α′ = 0, for every α, α′ ∈ H^{1}(X; ℤ). Then, as elements of H^{•}(M_{𝔰}; ℝ),

c(N_{𝔱}(Ξ, 𝔰)) = (1 + 2μ_{𝔰}(x))^{n′_{s}} (1 + μ_{𝔰}(x))^{n′′_{s}} .

Proof. For convenience, we write μ = μ_{𝔰}(x). The Chern character determines the Chern polynomial as an element of rational cohomology (see the formula in [53, Problem 16-A] or [3, pp. 156–157]), so ch(N_{𝔱}(Ξ, 𝔰)) determines

c_{t}(N_{𝔱}(Ξ, 𝔰)) = ∑_{i=0}^{r_{Ξ}} c_{i}(N_{𝔱}(Ξ, 𝔰))t^{i} = ∏_{j=1}^{r_{Ξ}}(1 + α_{i}t),

where the α_{i} are the Chern roots of N_{𝔱}(Ξ, 𝔰). Theorem 3.29 implies that, with the constraint on H^{1}(X; ℤ),

ch(N_{𝔱}(Ξ, 𝔰)) = ∑_{j=1}^{r_{Ξ}} exp(α_{j}) = r_{Ξ} + n′_{s}e^{2μ} + n′′_{s}e^{μ}.

Suppose n′_{s} ≥ 0 and n′′_{s} ≥ 0. If ∏_{j=1}^{r_{Ξ}}(1 + α_{i}t) = (1 + 2μt)^{n′_{s}}(1 + μt)^{n′′_{s}}, then α_{j} = +2μ for 1 ≤ j ≤ n′_{s} and α_{j} = μ for n′_{s} < j ≤ n′_{s} + n′′_{s}, while α_{j} = 0 for n′_{s} + n′′_{s} < j ≤ r_{Ξ}. Then one can easily see that ch(N_{𝔱}(Ξ, 𝔰)) is equal to the Chern character associated to the Chern polynomial (1 + 2μt)^{n′_{s}}(1 + μt)^{n′′_{s}}. The case where either n′_{s} or n′′_{s} is negative follows from the observation that if G ∈ K(M_{𝔰}), then ch(−G) = ch(G) while c(−G) = c(G)^{−1}.

Appendix A. Abundant four-manifolds

Our goal in this section is to prove the

Theorem A.1. Every compact, complex algebraic, simply connected surface with b_{2}^{+} ≥ 3 is abundant.

In [16, p. 175] we asserted without further explanation that simply connected, minimal, complex algebraic surfaces of general type were abundant. On the other hand, some of the fake K3-surfaces of [33] fail to be abundant. If log transforms are performed on tori in three distinct nuclei then the intersection form on B^{⊥} is a degenerate form with threedimensional radical and having an −E_{8} ⊕ −E_{8} summand [16, p. 175]. We apply Theorem A.1 to produce classes Λ ∈ H^{2}(X; ℤ) which are orthogonal to the SW-basic classes, with square equal to prescribed even integers. As far as we can tell, one can always find such classes Λ for compact four-manifolds with b_{2}^{+} ≥ 3, even if non-abundant: it is interesting problem to determine if indeed this is true.

We are extremely grateful to András Stipsicz for describing a proof of Theorem A.1 in the case where X is a minimal surface of general type with odd intersection form: see Lemmas A.4 and A.6, as well as the ideas for the case of odd minimal surfaces of general type in §A.2—these are all due to him. The argument for Lemma A.4 relies on the “odd four-square theorem” (Lemma A.5), for whose proof we are indebted to A. Agboola [1].

For the remainder of this section, unless further restrictions are mentioned, we suppose X is a compact, connected, smooth four-manifold with an orientation for which b_{2}^{+}(X) > 0.

A.1. Four-manifolds whose basic classes are multiples of a fixed class. We consider the cases of even and odd intersection forms separately, beginning with the even case:

Lemma A.2. Suppose (L, Q) is an indefinite, integral, unimodular lattice and that κ ∈ L. If Q is even and |σ(Q)| ≤ rank(Q) − 4 then κ^{⊥}, the Q-orthogonal complement of κ in L, contains a hyperbolic sublattice.

Proof. We may assume without loss that κ is primitive because a sublattice H is orthogonal to κ = dκ′ if and only if it is orthogonal to κ′, where d ∈ ℤ and κ′ is primitive.

From the classification of indefinite, integral, unimodular forms (for example, see Theorem 1.2.21 in [34]) we have

(L, Q) ≅ \frac{1}{8}σ(Q)E_{8} ⊕ \frac{1}{2}(rank(Q) − |σ(Q)|)H. (A.1)

By hypothesis, rank(Q)−|σ(Q)| ≥ 4, so if R = H ⊕H then (R, Q|_{R}) is a sublattice of (L, Q). Let κ_{R} denote the component of κ in R. Because R is even, we have Q(κ_{R}, κ_{R}) = 2h for some h ∈ ℤ. Let e_{1}, e_{2} and f_{1}, f_{2} be bases for the two hyperbolic sublattices of R, so Q(e_{1}, e_{2}) = Q(f_{1}, f_{2}) = 1 and all other pairings of these four vectors vanish. Define v = e_{1} + he_{2}, so v^{2} = 2h and v is primitive. The hyperbolic sublattice F = ℤf_{1} + ℤf_{2} ⊂ R is orthogonal to v. According to Theorem 1 in [75], the orthogonal group of (R, Q|_{R}) acts transitively on primitive vectors of a given square. Hence, there is an automorphism A of R such that Q(Ax, Ay) = Q(x, y) for all x, y ∈ R and Av = κ_{R}. Because F is contained in v^{⊥}, then A(F) ⊂ κ_{R}^{⊥}. Hence, the hyperbolic sublattice A(F) is contained in κ^{⊥}.

Corollary A.3. Let X be a simply-connected spin four-manifold with b_{2}^{+}(X) ≥ 3. If the SW-basic classes of X are multiples of a class K ∈ H^{2}(X; ℤ), then X is abundant.

Proof. We consider the intersection form Q to be a form on H^{2}(X; ℤ). By Rochlin’s theorem (see Theorem 1.2.29 in [34]), we know that σ(X) ≡ 0 (mod 16) and so the classification (A.1) of forms yields (H^{2}(X; ℤ), Q) ≅ 2kE_{8} ⊕ lH with k, l ∈ ℤ. A result of Furuta [34, Theorem 1.2.31] then implies that l ≥ 2|k|+1, where k = \frac{1}{16}σ(X) and l = \frac{1}{2}(b_{2}(X)−σ(X)). If σ(X) = 0 and b_{2}^{+}(X) ≥ 3, then b_{2}(X) ≥ 4 and so 0 ≤ rank(Q) − 4; if σ(X) ≠ 0, then k ≠ 0 and Furuta’s theorem implies l ≥ 2 and so again |σ(Q)| ≤ rank(Q) − 4. Therefore, K^{⊥} contains a hyperbolic sublattice by Lemma A.2.

We turn to the more complicated case, where the intersection form is odd.

Lemma A.4. Suppose (L, Q) is an indefinite, integral, unimodular lattice. If Q is odd with b_{2}^{+} ≥ 5 and b_{2}^{−} ≥ 3, and κ ∈ L is characteristic, then κ^{⊥} contains a hyperbolic sublattice.

Proof. We may assume without loss that κ is primitive for, if not, write κ = dκ′ where d ∈ ℤ and κ′ ∈ L is primitive. Then Q(κ, x) = dQ(κ′, x) ≡ Q(x, x) (mod 2) for all x ∈ L since κ is characteristic. If d were even we would have Q(x, x) ≡ 0 (mod 2) for all x, contradicting our hypothesis that Q is an odd form. Hence, d is odd and dQ(κ′, x) ≡ Q(κ′, x) ≡ Q(x, x) (mod 2) for all x and thus κ′ is also characteristic. Then a sublattice H is orthogonal to κ if and only if it is orthogonal to κ′.

The classification of indefinite, integral, unimodular forms shows that, because Q is odd, we can find a basis {e_{i}, f_{j}} for L for which e_{i}^{2} = 1, f_{i}^{2} = −1, and

(L, Q) ≅ ( ⊕_{i=1}^{b_{2}^{+}} ℤe_{i}) ⊕ ( ⊕_{j=1}^{b_{2}^{−}} ℤf_{j}) .

For odd integers a_{1}, a_{2}, a_{3}, a_{4}, c_{1} yet to be determined, choose

ℓ = a_{1}e_{1} + a_{2}e_{2} + a_{3}e_{3} + a_{4}e_{4} + ∑_{i=5}^{b_{2}^{+}} e_{i} + c_{1}f_{1} + ∑_{j=2}^{b_{2}^{−}} f_{j} ∈ L.

The element ℓ is primitive since at least one basis coefficient is equal to one. By hypothesis, b_{2}^{+} ≥ 5 and b_{2}^{−} ≥ 3, so we can define

H = ℤ(e_{5} + f_{2}) + ℤ(e_{5} + f_{3})

and observe that (H, Q|_{H}) is hyperbolic and orthogonal to ℓ. Then

Q(ℓ, ℓ) = a_{1}^{2} + a_{2}^{2} + a_{3}^{2} + a_{4}^{2} + (b_{2}^{+} − 4) − c_{1}^{2} − (b_{2}^{−} − 1)
= a_{1}^{2} + a_{2}^{2} + a_{3}^{2} + a_{4}^{2} + σ(Q) − c_{1}^{2} − 3,

and so

Q(ℓ, ℓ) − σ(Q) + c_{1}^{2} + 3 = a_{1}^{2} + a_{2}^{2} + a_{3}^{2} + a_{4}^{2}. (A.2)

Since the coefficients of ℓ are odd, we have Q(ℓ, e_{i}) ≡ Q(ℓ, f_{j}) ≡ 1 (mod 2) for all i, j; thus ℓ is characteristic and we have ℓ^{2} ≡ σ(Q) (mod 8). As c_{1} is odd, so we can write c_{1} = 2u + 1 for some u ∈ ℤ. The left-hand side of the preceding equation therefore yields

Q(ℓ, ℓ) − σ(Q) + c_{1}^{2} + 3 ≡ (2u + 1)^{2} + 3 ≡ 4 + 4u(u + 1) ≡ 4 (mod 8).

Now any positive integer which is congruent to 4 (mod 8) can be written as the sum of four odd squares (see Lemma A.5). So we select any odd integer c_{1} for which Q(κ, κ) − σ(Q) + c_{1}^{2} + 3 > 0 and then choose odd integers a_{1}, a_{2}, a_{3}, a_{4} so that

a_{1}^{2} + a_{2}^{2} + a_{3}^{2} + a_{4}^{2} = Q(κ, κ) − σ(Q) + c_{1}^{2} + 3. (A.3)

Therefore equations (A.2) and (A.3) give

Q(ℓ, ℓ) = Q(κ, κ).

A result of Wall, (see [34, Proposition 1.2.18]) implies that the orthogonal group of (L, Q) acts transitively on the primitive, characteristic elements with a given square. Hence, we can find an orthogonal automorphism A of (L, Q) with Aℓ = κ. Then A(H) is a hyperbolic sublattice of L which is orthogonal to κ.

Lagrange’s theorem tells us that every positive integer k is the sum of four integral squares [36, Theorem 20.5]. While the following refinement of this result must surely be well known, we cannot find references to it in standard texts on elementary number theory, so we include a proof here which was generously supplied to us by Adebisi Agboola [1]. András Stipsicz has pointed out to us that Lemma A.5 was used by Dieter Kotschick in [42] to show that every finitely presentable group is the fundamental group of a closed, almost complex four-manifold.

Lemma A.5. A positive integer is the sum of four odd squares if and only if it is congruent to 4 modulo 8.

Proof. Let k be a positive integer. The set of squares modulo 8 is 0, 1, 4. Hence, if k is the sum of four odd squares, then we must have k ≡ 4 (mod 8).

Conversely, suppose that k ≡ 4 (mod 8). By Lagrange’s theorem we can write

k = w^{2} + x^{2} + y^{2} + z^{2}.

Since k is even, we must have one of the following possibilities (up to rearranging the terms on the right-hand side):

(a) w, x are even, and y, z are odd.

(b) w, x, y, z are all even.

(c) w, x, y, z are all odd (as desired). Now since k ≡ 4 (mod 8), it is not hard to check that case (a) cannot happen. In fact, it is easy to see that if case (a) is true, then k ≡ 2 or 6 (mod 8).

To see that case (c) can occur, write k = 8m + 4, where m is a non-negative integer, so

k = 1^{2} + (8m + 3).

Legendre’s theorem tells us that a positive integer is the sum of three integral squares if and only if it cannot be written in the form 4^{a}(8b + 7), for some a, b (see [36, §20.10] for the statement and [2] for a proof). Now 1 is odd, and since 8m + 3 cannot be of the form 4^{a}(8b + 7), we can express 8m + 3 as a sum of three squares. So we can write

k = 1^{2} + x^{2} + y^{2} + z^{2}.

Since 1 is odd, and we have ruled out case (a) above, it follows that x, y, z are all odd.

It remains to consider an “unstable range”, where b_{2}^{+}(Q) and b_{2}^{−}(Q) are small. Define χ(Q) = 2 + b_{2}(Q) and note that 2χ(Q) + 3σ(Q) = 4 + 5b_{2}^{+}(Q) − b_{2}^{−}(Q).

Lemma A.6. Suppose (L, Q) is an integral, indefinite, unimodular lattice and that κ ∈ L. Assume Q is odd and κ is characteristic. If one of the following hold, then κ^{⊥} contains a hyperbolic sublattice:

(a) b_{2}^{+} = 3 and b_{2}^{−} ≥ 5,

(b) b_{2}^{+} = 3 and 2 ≤ b_{2}^{−} ≤ 4, and κ has square 2χ(Q) + 3σ(Q).

Proof. Interchanging the role of b_{2}^{+} and b_{2}^{−} in the proof of Lemma A.4 takes care of case (a). Thus we need only consider case (b).

Continuing the notation of the proof of Lemma A.4, we choose

ℓ = 3e_{1} + 3e_{2} + e_{3} + ∑_{j=1}^{b_{2}^{−}} f_{j} ∈ L.

Plainly, ℓ is characteristic, primitive, and is orthogonal to the hyperbolic sublattice H = ℤ(e_{3} + f_{1}) + ℤ(e_{3} + f_{2}) of (L, Q), while (as 2χ(Q) + 3σ(Q) = 19 − b_{2}^{−})

Q(ℓ, ℓ) = 19 − b_{2}^{−} = Q(κ, κ).

Since Q is odd, we cannot have b_{2}^{−} = 3 (which would give Q(κ, κ) = 16) so b_{2}^{−} = 2 or b_{2}^{−} = 4, which gives Q(κ, κ) = 17 or 15, respectively, neither of which is divisible by d^{2}, d ∈ ℤ, unless d = ±1. Thus κ is primitive.

Just as in the proof of Lemma A.4, Wall’s theorem implies that we can find an orthogonal automorphism A of (L, Q) with Aℓ = κ, since κ, ℓ are both primitive, characteristic, and have equal square. Then A(H) is a hyperbolic sublattice of κ^{⊥}. This takes care of case (b) and completes the proof of the lemma.

Lemmas A.4 and A.6 thus yield:

Corollary A.7. Let X be a simply-connected four-manifold having odd intersection form Q_{X}, with b_{2}^{+}(X) ≥ 5 and b_{2}^{−}(X) ≥ 3 or b_{2}^{+}(X) = 3 and b_{2}^{−}(X) ≥ 2. Suppose that the SWbasic classes of X are integer multiples of a class K ∈ H^{2}(X; ℤ), where K is characteristic. Then X is abundant.

Combining Corollaries A.3 and A.7 yields:

Proposition A.8. Let X be a simply-connected four-manifold. Suppose the SW-basic classes of X are multiples of a class K ∈ H^{2}(X; ℤ), where K is characteristic. If any one of the following hold, then X is abundant:

(a) Q_{X} is even with b_{2}^{+}(X) ≥ 3,

(b) Q_{X} is odd with b_{2}^{+}(X) ≥ 5 and b_{2}^{−}(X) ≥ 3,

(c) Q_{X} is odd with b_{2}^{+}(X) = 3 and b_{2}^{−}(X) ≥ 5,

(d) Q_{X} is odd with b_{2}^{+}(X) = 3 and 2 ≤ b_{2}^{−}(X) ≤ 4, and K^{2} = 2χ(X) + 3σ(X).

A.2. Compact, complex algebraic, simply connected surfaces. We now combine the results of the preceding subsection to prove the principal result of this appendix:

Proof of Theorem A.1. Let X be a compact, complex algebraic, simply connected surface with b_{2}^{+}(X) ≥ 3. Suppose X is minimal. The Enriques-Kodaira classification then implies that X is one of the following (see [34, §3.4]):

(a) A K3 surface,

(b) An elliptic surface,

(c) A surface of general type.

The cases where X is diffeomorphic to ℂℙ^{2}, ℂℙ^{2}#\overline{ℂℙ}^{2}, or ℂℙ^{1} × ℂℙ^{1} (when X has Kodaira dimension −∞, see [34, p. 88 & Theorem 3.4.13]) are eliminated by our requirement that b_{2}^{+}(X) ≥ 3.

If X is elliptic, then it is diffeomorphic to E(n)_{p,q}, for some n, p, q ∈ ℕ, p ≤ q, gcd(p, q) = 1 (see Theorems 3.4.12 and 3.4.13 in[34]); if X is a K3 surface, then it is diffeomorphic to the surface E(2) (see Theorem 3.4.9 in [34]), which is included in this family as E(n)_{1,1} = E(n) (see [34, p. 83] for the construction of this family).

Let f be the homology class of a regular fiber of E(n)_{p,q} and observe that f_{pq} = \frac{1}{pq} f is a primitive, integral homology class. According to [28] (see [34, Theorem 3.3.6]), the SWbasic classes of E(n)_{p,q} are multiples of the Poincaré dual PD[f_{pq}]. Thus, X is abundant by Proposition A.8 (cases (a), (b), and (c)) and the observation that b_{2}^{+}(E(n)_{p,q}) = 2n − 1 and b_{2}^{−}(E(n)_{p,q}) = 10n − 1 (see [34, Lemma 3.3.4]), so b_{2}^{+}(X) ≥ 3 and b_{2}^{−}(X) ≥ 19 for n ≥ 2.

If X is a minimal algebraic surface of general type, then its SW-basic classes are ±K_{X} by [30], where K_{X} is the canonical class. If Q_{X} is even, then X is abundant by Proposition A.8. If Q_{X} is odd, the Bogomolov-Miyaoka-Yau inequality, c_{1}^{2}(X) ≤ 3c_{2}(X) (see [7, Theorem VII.1.1 (iii)]), implies that

3σ ≤ χ,

since c_{1}^{2}(X) = K_{X}^{2} = 2χ + 3σ and c_{2}(X) = χ. Thus

b_{2}^{+}(X) ≤ 2b_{2}^{−}(X) + 1. (A.4)

Equality in (A.4), or c_{1}^{2}(X) = 3c_{2}(X), holds only if the universal covering space of X is the closed unit ball in ℂ^{2} by [34, Theorem 7.2.24] (or see [7, Corollary I.15.5] and the discussion in [7, p. 230] or [54, Theorem 4]). Since X is simply-connected and closed by hypothesis, we must have

b_{2}^{+}(X) < 2b_{2}^{−}(X) + 1.

If b_{2}^{+}(X) = 3, then the preceding inequality yields b_{2}^{−}(X) ≥ 2 while if b_{2}^{+}(X) ≥ 5, it yields b_{2}^{−}(X) ≥ 3. Hence, for Q_{X} odd, X is again abundant by Proposition A.8.

This proves the theorem for minimal surfaces. If X is an abundant, smooth four-manifold, there is a hyperbolic sublattice H ⊂ H^{2}(X; ℤ) such that K is orthogonal to H if K is an SW-basic class. The blow-up X#\overline{ℂℙ}^{2} is also abundant, since we may view H as a hyperbolic sublattice of H^{2}(X#\overline{ℂℙ}^{2}; ℤ), all SW-basic classes of X#\overline{ℂℙ}^{2} have the form K ± PD[e] (see [26]), and such classes are again orthogonal to H. Hence, if X is abundant, all its blow-ups are abundant too. This completes the proof of the theorem.

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Department of Mathematics, Ohio State University, Columbus, OH 43210

   E-mail address: feehan@math.ohio-state.edu
   URL: http://www.math.ohio-state.edu/∼feehan/

Current address: Max Planck Institut für Mathematik, Vivatsgasse 7, Bonn, D-53111, Germany

   E-mail address: feehan@mpim-bonn.mpg.de

Department of Mathematics, Florida International University, Miami, FL 33199

   E-mail address: lenesst@fiu.edu
   URL: http://www.fiu.edu/∼lenesst/