Exact triangles in Seiberg-Witten-Floer theory. Part IV: ℤ-graded monopole homology

Matilde Marcolli and Bai-Ling Wang

1 Introduction

The content of this paper is part of our ongoing project of establishing the surgery formulae for Seiberg-Witten-Floer theory. As explained in the introductory part of [2], if K is a knot in a homology 3-sphere Y, we expect the Floer homologies of Y and of the manifolds Y_{1} and Y_{0} obtained by Dehn surgery on K with framing 1 and 0, respectively, to be related by an exact triangle

\xymatrix{HF_{∗}^{SW} (Y_{1}, g_{1}) \ar[rr]^{I_{∗}} &  & HF_{∗}^{SW} (Y, g) \ar[dl]_{L_{∗}} \\  & ⨁_{k} HF_{(∗)}^{SW} (Y_{0}, 𝔰_{k}) \ar@{-->}[ul]_{Δ_{(∗)}}}

There are several subtleties involved in the precise definition of the term

⨁_{k} HF_{(∗)}^{SW} (Y_{0}, 𝔰_{k})

in the previous diagram. In this paper we explain the precise meaning of this term.

More specifically, we analyze some aspects of the construction of the Seiberg-Witten-Floer homology of the manifold Y_{0}, obtained by 0-surgery on the knot K in the homology sphere Y. The manifold Y_{0} has the homology of S^{1} × S^{2}. For manifolds with b_{1}(Y_{0}) > 0 we know (cf. [3], [10], [11], [20]) that there is a well defined ℤ_{ℓ} graded Seiberg-Witten-Floer homology HF_{∗}^{SW} (Y_{0}, 𝔰), for every choice of a non-trivial Spin^{c} structure 𝔰 ∈ S(Y_{0}), where the integer ℓ = ℓ(𝔰) satisfies

c_{1}(L)(H_{2}(Y_{0}, ℤ)) = ℓℤ,

and S(Y_{0}) denotes the set of Spin^{c} structures on Y_{0}. Here L = det W is the determinant line bundle of the spinor bundle W associated to the Spin^{c} structure 𝔰. We use the equivalent notation L = det 𝔰. Here we are assuming that c_{1}(L) is non-torsion, hence ℓ ≠ 0.

In this paper we shall discuss several issues regarding the Floer homology HF_{∗}^{SW} (Y_{0}, 𝔰). The first is connected to the integer lift of the ℤ_{ℓ} graded Floer homology. Analogous constructions of integer lifts of Floer homologies were derived by Fintushel and Stern [4], in the case of instanton homology, and by Weiping Li [7], in the case of symplectic Floer homology. We follow closely the construction of [4] and show that, in our case, there is a well defined integer lift of HF_{∗,(ω)}^{SW}(Y_{0}, 𝔰) of the ℤ_{ℓ}-graded Floer homology, here ω ∈ ℝ is a regular value of the Chern-Simons-Dirac functional on the infinite cyclic cover space of the gauge equivalence classes of connections and spinor sections. By studying the Chern-Simons-Dirac function on this infinite cyclic cover space, we will define an integer lift 𝔦_{Y_{0}}^{(ω)} of the indices of the critical points. We thus form a chain complex C_{∗}^{(ω)}(Y_{0}, 𝔰) depending on ω ∈ ℝ. For any n ∈ ℤ_{ℓ}, the original ℤ_{ℓ}-graded Seiberg–Witten–Floer chain complex satisfies C_{n}(Y_{0}, 𝔰) = ⨁_{k∈ℤ} C_{j+kℓ}^{(ω)}(Y_{0}, 𝔰) where j = n(modℓ).

In general, after defining a suitable boundary operator on C_{∗}^{(ω)}(Y_{0}, 𝔰), we observe that the resulting homology groups HF_{∗,(ω)}^{SW}(Y_{0}, 𝔰) do not satisfy the simple relation ⨁_{k∈ℤ} HF_{∗+kℓ,(ω)}^{SW}(Y_{0}, 𝔰) = HF_{∗}^{SW} (Y_{0}, 𝔰). However, the Floer homologies HF_{∗+kℓ,(ω)}^{SW}(Y_{0}, 𝔰) and HF_{∗}^{SW} (Y_{0}, 𝔰) are related via a spectral sequence determined by a filtration of the chain complex C_{∗}(Y_{0}, 𝔰). This spectral sequence converges to HF_{∗}^{SW} (Y_{0}, 𝔰), and the E^{1} term coincides with HF_{∗,(ω)}^{SW}(Y_{0}, 𝔰). For instance, in the particular case where all the higher differentials in the spectral sequence are zero, then we simply have, for m ∈ {0, 1, ⋯, ℓ − 1},

HF_{m}^{SW} (Y_{0}, 𝔰) = ⨁_{n∈ℤ}HF_{m+nℓ,(ω)}^{SW}(Y_{0}, 𝔰).

The arguments we develop in this section extend easily to the general case of any 3-manifold (Y_{0}, 𝔰) with non-trivial rational homology and with a Spin^{c}- structure satisfying c_{1}(𝔰)(H_{2}(Y_{0}, ℤ)) = ℓℤ ≠ 0.

The second main issue discussed in this paper is a different construction of a ℤ-graded Seiberg–Witten–Floer homology, where the integer lift of the original ℤ_{ℓ}-graded Seiberg–Witten–Floer homology is determined by the exact triangle, as derived in [2] [12] [13]. Under the surgery identifications of the Seiberg–Witten monopoles on Y and Y_{0}, proved in [2], there exists a one-to-one map

j :  ⨆_{𝔰}M_{Y_{0}}(𝔰) ⟶ M_{Y,μ}\M_{Y_{1}}.  (1)

Then the ℤ-valued indices on M_{Y,μ} define a ℤ-lifting 𝔦_{Y_{0}}^{(Y)} of the grading M_{Y_{0}}(𝔰). The corresponding chain complex C_{(n)}(Y_{0}, 𝔰) is defined to be ℤ{a ∈ M_{Y_{0}}(𝔰) : 𝔦_{Y_{0}}^{(Y)}(a) = n}. We will show that the restriction of the boundary operator on C_{∗}(Y_{0}, 𝔰) to C_{(n)}(Y_{0}, 𝔰) is a well-defined boundary operator, and the resulting homology groups HF_{(∗)}(Y_{0}, 𝔰) satisfy

HF_{n}^{SW} (Y_{0}, 𝔰) = ⨁_{k∈ℤ}HF_{(n+kℓ)}^{SW}(Y_{0}, 𝔰).  (2)

The analysis of the splitting and gluing of the moduli spaces of flow lines in [12] play an essential role in the proof of the direct sum formula (2). It is precisely these ℤ-graded monopole homology groups that we use in the exact triangles [13].

The third main issue that we discuss in this paper is the construction of the Seiberg-Witten-Floer homology on a 3-manifold Y_{0} with b_{1}(Y_{0}) > 0, in the case of a Spin^{c} structure 𝔰_{0} with torsion c_{1}(det 𝔰_{0}). This case was not included in our original paper [11]. We show that, in this case, the boundary operator of the Floer complex is well defined only after separating the components of uniform energy in the moduli space M(a, b). This is done by introducing a Novikov complex with coefficients in ℤ[[t]], which depends on the non-trivial cohomology class of the perturbation for the Chern-Simons-Dirac functional. The component of this Novikov-Floer homology HF_{∗}(Y_{0}, 𝔰_{0}, ℤ[[t]]) that we are interested in, for the purpose of the exact triangle, is the evaluation

HF_{(∗)}(Y_{0}, 𝔰_{0}) = HF_{∗}(Y_{0}, 𝔰_{0}, ℤ[[t]])|_{t=0}.

We also generalize the construction of HF_{∗}(Y_{0}, 𝔰_{0}, ℤ[[t]]) to a more natural setting over the coefficient field of formal Laurent series ℚ((t)) = ℚ[[t]][t^{−1}]. One reason for this definition is that, in this setting, we can associate a relative Seiberg–Witten invariant for any 4-manifold (X, 𝔰) with boundary (Y_{0}, 𝔰_{0}), where c_{1}(𝔰|_{Y_{0}}) is a torsion class. Here we introduce a non-trivial 1-cycle Γ in Y_{0}, which depends on the non-trivial cohomology class of the perturbation. The corresponding Seiberg-Witten-Floer homology is denoted by HF_{Γ,∗}^{SW} (Y_{0}, 𝔰_{0}, ℚ((t))).

Suppose given 4-manifold (X, 𝔰) with a cylindrical end modeled on (Y_{0}, 𝔰_{0} = 𝔰|_{Y_{0}}), where c_{1}(𝔰_{0}) is a torsion cohomology class in H^{2}(Y_{0}.ℤ).

We study the perturbed 4-dimensional Seiberg-Witten equations on (X, 𝔰), the perturbation for (X, 𝔰) is compactible with the perturbation on (Y_{0}, 𝔰_{0}) used to the construction of HF_{Γ,∗}^{SW} (Y_{0}, 𝔰_{0}, ℚ((t))). For simplicity, we assume that Γ bounds a relative 2-cycle in X, then the relative Seiberg–Witten invariant SW_{X}(𝔰) takes values in HF_{Γ,∗}^{SW} (Y_{0}, 𝔰_{0}, ℚ((t))). We derive the gluing formulae for this case. When Γ does not bound any relative 2-cycle in X, the relative invariant takes values in the Seiberg-Witten-Floer homology (Y_{0}, 𝔰_{0}) over the coefficient field of formal Laurent series on H^{1}(Y_{0}, ℤ). The gluing formulae along the torsion Spin^{c} structure can be formulated similarly.

As an example, our gluing formula can be applied to show that, for certain particular choice of metric and perturbation on the three torus S^{1} × T^{2} with the trivial Spin^{c} structure 𝔰_{0}, we have

HF_{∗}(S^{1} × T^{2}, 𝔰_{0}, ℚ((t))) = ℚ((t)),

and for D^{2} × T^{2} with the cylindrical end modeled on S^{1} × T^{2}, we get

SW_{D^{2}×T^{2}}(𝔰_{0}) = \frac{1}{t − t^{−1}} .

In the more general setting, these gluing formulae provide useful tools in the study of the structure of the Seiberg–Witten–Floer homology groups (cf. [17]). The relative invariant SW_{D^{2}×T^{2}}(𝔰_{0}) can also be applied to knot surgery of 4-manifolds to give some of the gluing formulae obtained in [5] and [16].

Acknowledgments Both the authors benefited of conversations with Ron Wang on some of the issues discussed in this paper. We thank Vicente Muñoz for his help in formulating the gluing formulae. We also thank the referee for useful comments. We thank the Max Planck Institut für Mathematik for the kind hospitality and for support. The first author is partially supported by NSF grant DMS-9802480. The second author is partially supported by ARC Fellowship.

2 Seiberg-Witten-Floer homology

In the following two sections we present two different constructions of an integer lift of the Seiberg–Witten–Floer homology. The first case applies to the general setting where Y_{0} is a compact 3-manifold with b_{1}(Y_{0}) > 0, and the Spin^{c}-structure is non-torsion. The second applies to the special case where Y_{0} is obtained by zero-surgery on a knot in a homology 3-sphere Y, again with a non-torsion Spin^{c}-structure. In this particular case, the integer lift of the Seiberg–Witten–Floer homology that we consider appears naturally in the context of the exact triangle formula.

We give some preliminary definitions.

Let M_{Y_{0}}(𝔰) be the moduli space of gauge classes of solutions of suitably perturbed Seiberg–Witten equations on a 3-manifold (Y_{0}, 𝔰), with b_{1}(Y_{0}) > 0 and c_{1}(𝔰) a non-torsion element. A suitable class of perturbations that achieves transversality has been introduced in [2]. The perturbation can be chosen so that all the solutions in M_{Y_{0}}(𝔰) are irreducible critical points. Thus, under the choice of a generic perturbation, M_{Y_{0}}(𝔰) is a compact, oriented, 0-dimensional manifold, cut out transversely by the equations inside the configuration space B = A/G, where A is the space of pairs formed by a U(1)-connection on det(𝔰) and a spinor section of W, and G is the group of gauge transformations on (Y_{0}, 𝔰).

Recall that we have the Chern–Simons–Dirac functional on A, defined as

\begin{aligned}CSD(A, ψ) & = & −\frac{1}{2} ∫_{Y_{0}}(A − A_{0}) ∧ (F_{A} + F_{A_{0}} − 2\sqrt{−1} ∗ ρ) \\ & & +_{∫Y_{0}}⟨ψ, ∂_{A}ψ⟩dvol_{Y_{0}}, \end{aligned}

with ρ a co-closed 1-form on Y_{0}. In order to achieve transversality in the moduli spaces of flow lines, we consider an additional perturbation of CSD by functions U(τ_{1}, …, τ_{N}) + V (ζ_{1}, …, ζ_{K}) as explained in [2]. Here τ_{j} and ζ_{j} are functions on A respectively associated to a complete basis {μ_{j}}_{j=1}^{∞} for the co-closed L^{2} one-forms on Y_{0} and a complete basis {ν_{j}}_{j=1}^{∞} for the imaginary-valued one-forms on Y_{0},

\begin{matrix}τ_{j}(A, ψ) = τ_{j}(A − A_{0}) = _{∫Y_{0}}(A − A_{0}) ∧ ∗\sqrt{−1}μ_{j} \\ ζ_{j}(A, ψ) = ζ_{j}(ψ, ψ) = _{∫Y_{0}}⟨ν_{j}.ψ, ψ⟩dvol_{Y_{0}}. \end{matrix}

The pair (U, V) is chosen from a subspace of functions in

⋃_{N≥b_{1},K>0} C^{∞}(ℝ^{N}, ℝ) × C^{∞}(ℝ^{K}, ℝ)

completed to a Banach space in the Floer ϵ-norm (as specified in [2]). This perturbation term is gauge invariant by construction, and CSD changes by

CSD(λ.(A, ψ)) − CSD(A, ψ) = ⟨(8π^{2}c_{1}(L) + 4π[∗ρ]) ∪ [λ], [Y_{0}]⟩,

where [λ] ∈ H^{1}(Y_{0}, ℤ) determines the connected component of G that contains λ, and is represented by the closed 1-form \frac{1}{2π\sqrt{−1}}λ^{−1}dλ.

With this choice of perturbation, the Seiberg–Witten equations are of the form

\{ \begin{matrix}∗F_{A} = σ(ψ, ψ) + ∑_{j=1}^{N} \frac{∂U}{∂τ_{j}} μ_{j} \\ ∂_{A}(ψ) + ∑_{j=1}^{K} \frac{∂V}{∂ζ_{j}} ν_{j}.ψ = 0, \end{matrix}

for the critical points of the Chern–Simons–Dirac functional, and

\{ \begin{matrix}\frac{∂A}{∂t} & = & − ∗ F_{A} + σ(ψ, ψ) + ∑_{j=1}^{N} \frac{∂U}{∂τ_{j}} μ_{j} \\ \frac{∂ψ}{∂t} & = & −∂_{A}ψ − ∑_{j=1}^{K} \frac{∂V}{∂ζ_{j}} ν_{j}.ψ \end{matrix}

for the corresponding downward gradient flow.

The Seiberg–Witten–Floer homology groups HF_{∗}^{SW} (Y_{0}, 𝔰) (cf. [3], [10], [11]) are the homology groups of (C_{∗}(Y_{0}, 𝔰), ∂), where C_{∗}(Y_{0}, 𝔰) is generated by the critical points of the perturbed CSD on B. The entries of the boundary operator ∂ are defined by counting the points in the zero dimensional components of the moduli space of unparameterized flow lines of the perturbed functional CSD on B. A considerable amount of technical work goes into checking that the moduli spaces of flow lines have all the desired properties that make this definition rigorous, and we refer the reader to [11] for a precise account. The resulting Floer homology HF_{∗}^{SW} (Y_{0}, 𝔰) is ℤ_{ℓ}-graded with ℓℤ = c_{1}(𝔰)(H_{2}(Y_{0}, ℤ)). This is due to an ambiguity in the index formula computed in terms of the spectral flow of the Hessian operator for CSD around a loop in B. To understand this ambiguity, let (A, ψ) represent a critical point a of CSD on B, let λ be a gauge transformation whose class [λ] ∈ H^{1}(Y_{0}, ℤ) is non-trivial. Then the spectral flow of the Hessian operator H from λ.(A, ψ) to (A, ψ) can be calculated from the index formula for the linearization of the 4-dimensional Seiberg–Witten equations on Y_{0} × S^{1}. This gives

SF (H)_{λ.(A,ψ)}^{(A,ψ)} = ⟨[λ] ∪ c_{1}(𝔰), [Y_{0}]⟩ ∈ ℓℤ.  (3)

Remark 2.1. Notice that the periodicity ℓ is an even number (cf. [2]). In fact, the map S(Y_{0}) → H^{2}(Y_{0}, ℤ), given by 𝔰 ↦ c_{1}(det 𝔰), is equivariant with respect to the action of H^{2}(Y_{0}, ℤ), so that we have det(W ⊗ H) = L ⊗ H^{2}, with L = det(W) = det(𝔰) and H ∈ H^{2}(Y_{0}, ℤ), cf. [9].

There is a natural and non-trivial way to lift these ℤ_{ℓ}-graded homology groups to ℤ-graded homology groups, so that the Euler characteristic number agrees with the original one. Following the idea of [4], we will discuss this integer lift and construct a spectral sequence which has the integer lift as E^{1} term and converges to the original ℤ_{ℓ}-graded Floer homology. We shall then consider a different lifting of the ℤ_{ℓ}-graded homology groups which is relevant to the exact triangles considered in [2], [13], and [14].

3 ℤ-graded homology groups and the spectral sequence

The construction we present in this section holds in the general case of a 3-manifold Y_{0} with b_{1}(Y_{0}) > 0 and a Spin^{c}-structure with non-torsion class c_{1}(L).

As explained in [11], there is a cyclic covering of B, obtained by taking the quotient of A, the space of U(1)-connections and spinors, with respect to the subgroup G_{ℓ} of the gauge group G given by

G_{ℓ} = {λ ∈ G | ⟨c_{1}(L) ∪ [λ], [Y_{0}]⟩ = 0},

where ℓ satisfies ℓℤ = c_{1}(L)(H_{2}(Y_{0}, ℤ)). This subgroup depends on c_{1}(L).

The space B has the homotopy type of ℂP^{∞} × K(H^{1}(Y_{0}, ℤ), 1), hence it has a universal covering obtained by taking the quotient of A by the identity component of the gauge group. The resulting space B̃ covers B with fibers H^{1}(Y_{0}, ℤ). Define H_{ℓ} to be

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

The group H_{ℓ} also depends on c_{1}(L). Then the space B_{ℓ} = A/G_{ℓ} is a covering of B with fiber H^{1}(Y_{0}, ℤ)/H_{ℓ} ≅ ℤ. Hence B_{ℓ} is an infinite cyclic covering space of B.

The perturbed Chern–Simons–Dirac functional is a real valued functional CSD : B_{ℓ} → ℝ on the covering space B_{ℓ}. The critical manifold M̃_{Y_{0}}(𝔰) is a ℤ-covering of M_{Y_{0}}(𝔰). The critical values form a discrete set in ℝ, which is a finite set mod ℤ. Let Ω ⊂ ℝ denote the set of regular values. Let ω ∈ Ω be a regular value. Given any point a ∈ M_{Y_{0}}(𝔰), there is a unique element a^{ω} in the fiber π^{−1}(a) in M̃_{Y_{0}}(𝔰) that satisfies CSD(a^{ω}) ∈ (ω, ω + 8π^{2}ℓ).

We have the following Lemma which shows that the relative indices on M̃_{Y_{0}}(𝔰), defined by the spectral flow of the Hessian operator, take values in ℤ.

Lemma 3.1. 1. Consider the Hessian operator H_{A(t),ψ(t)} defined in [2]. The spectral flow of the operator H_{A(t),ψ(t)} around a loop in B_{ℓ} is zero, hence the relative index of any two points ã and b̃ in M̃_{Y_{0}}(𝔰) is well defined in ℤ.

2. Let ã be a critical point in M̃_{Y_{0}}(𝔰) ∈ B_{ℓ}, and let λ ∈ G/G_{ℓ} be a gauge transformation. We have the following identities:

\begin{matrix}CSD(λ(ã)) − CSD(ã) = 8π^{2}⟨[λ] ∪ c_{1}(𝔰), [Y_{0}]⟩; \\ SF (H_{(A(t),ψ(t))})|ã ; λ(ã) = ⟨[λ] ∪ c_{1}(𝔰), [Y_{0}]⟩, \end{matrix}

with [λ] = [\frac{1}{2π\sqrt{−1}}λ^{−1}dλ]. Here SF (H_{(A(t),ψ(t))})|ã ; λ(ã) denotes the spec-

tral flow of the Hessian operator H_{(A(t),ψ(t))} along a path from λ(ã) to ã in B_{ℓ}.

Proof. By the Atiyah-Patodi-Singer index theorem, we have

SF (H_{(A(t),ψ(t))})|ã ; λ(ã) = Index(\frac{∂}{∂t} + H_{(A(t),ψ(t))}).

This index calculates the virtual dimension of the 4-dimensional Seiberg– Witten monopole moduli space on Y_{0}×S^{1}, with the Spin^{c} structure obtained by gluing the Spin^{c} structure 𝔰 on Y_{0} × ℝ along the two ends with the gauge transformation λ. By the index formula for the Seiberg–Witten monopoles, we obtain

Index(\frac{∂}{∂t} + H_{(A(t),ψ(t))})

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

The remaining claims are direct consequence of this index formula.

Upon fixing a base point ã_{0} in M̃_{Y_{0}}(𝔰), we can define the following ℤlifting of the ℤ_{ℓ}-grading of the elements of M_{Y_{0}}(𝔰).

Definition 3.2. We define the grading of elements in M_{Y_{0}}(𝔰) as the relative index of the ω-lifting in M̃_{Y_{0}}(𝔰),

𝔦_{Y_{0}}^{(ω)}(a) = 𝔦_{Y_{0}}(a^{ω}) = SF (H_{(A(t),ψ(t))})|ã_{0} ; a^{ω}.

This definition of the grading depends on the choice of the base point ã_{0} and on the choice of a regular value ω. Notice that we can reduce the choice of ã_{0} to the choice of a base point a_{0} in M_{Y_{0}}(𝔰). From now on, we fix a base point a_{0} in M_{Y_{0}}(𝔰). Then ã_{0} is chosen to be the unique critical point in π^{−1}(a_{0}) with CSD(ã_{0}) ∈ (ω, ω + 8π^{2}). It is easy to see, from the definition, that we have 𝔦_{Y_{0}}^{(ω)} = 𝔦_{Y_{0}}^{(ω′)} whenever ω and ω′ are connected by a

path in the set Ω of regular values. Moreover, we have 𝔦_{Y_{0}}^{(ω+8π^{2}kℓ)} = 𝔦_{Y_{0}}^{(ω)}, for any k ∈ ℤ.

Definition 3.3. For any q ∈ ℤ, we define

M_{Y_{0},q}^{(ω)}(𝔰) = {a ∈ M_{Y_{0}}(𝔰) |𝔦_{Y_{0}}^{(ω)}(a) = q}.

The q-chains in the ℤ-graded Floer complex are the elements of the abelian group C_{q}^{(ω)}(Y_{0}, 𝔰) generated by the monopoles in M_{Y_{0},q}^{(ω)}(𝔰). The boundary operator

∂_{q}^{(ω)} : C_{q}^{(ω)}(Y_{0}, 𝔰) → C_{q−1}^{(ω)}(Y_{0}, 𝔰)

is defined as

∂_{q}^{(ω)}(a) = ∑_{b∈M_{Y_{0},q−1}^{(ω)}(𝔰)}#(M̂^{0}(a, b))b,

where M̂^{0}(a, b) is the zero dimensional components of the moduli space of unparameterized flow lines in M̂(a, b). The compactness theorem (cf. [11]) tells us that M̂^{0}(a, b) is an oriented, compact, 0-dimensional manifold. Thus, the coefficient #(M̂^{0}(a, b)) is well-defined as the algebraic sum of points in M̂^{0}(a, b).

The following Lemma shows that we have ∂_{q−1}^{(ω)} ∘ ∂_{q}^{(ω)} = 0. The resulting homology groups are denoted as HF_{∗,(ω)}^{SW}(Y_{0}, 𝔰), ∗ ∈ ℤ.

Lemma 3.4. 1. For n ∈ ℤ_{ℓ} and q ∈ ℤ with q = n(mod ℓ), we have

C_{n}(Y_{0}, 𝔰) = ⨁_{k∈ℤ}C_{q+kℓ}^{(ω)}(Y_{0}, 𝔰).

2. Under the decomposition as above, the boundary operator ∂_{n} on

C_{n}(Y_{0}, 𝔰) can be expressed as

∂_{n} = ( \begin{matrix}∂_{∗}^{(ω)} & ∂_{∗,1}^{(ω)} & ∂_{∗,2}^{(ω)} & ⋯ & ⋯ & ∂_{∗,∗}^{(ω)} \\ 0 & ∂_{∗+ℓ}^{(ω)} & ∂_{∗+ℓ,1}^{(ω)} & ⋯ & ⋯ & ∂_{∗+ℓ,∗}^{(ω)} \\ 0 & 0 & ∂_{∗+2ℓ}^{(ω)} & ⋯ & ⋯ & ∂_{∗+2ℓ,∗}^{(ω)} \\ 0 & 0 & 0 & ... & & ∂_{∗+3ℓ,∗}^{(ω)} \\ ⋮ & ⋮ & ⋮ & & ... & ⋮ \\ 0 & 0 & 0 & ⋯ & ⋯ & ∂_{∗+∗ℓ}^{(ω)} \end{matrix} ) .

The meaning of this matrix notation can be explained more precisely as follows. Assume that a is a generator in C_{q}^{(ω)}(Y_{0}, 𝔰). Upon regarding a as a generator of C_{n}(Y_{0}, 𝔰) for n = q(mod ℓ), we obtain

∂_{n}(a) = ∂_{q}^{(ω)}(a) + ∑_{k>0} ∂_{q,k}^{(ω)}(a),

with ∂_{q,k}^{(ω)} : C_{q}^{(ω)}(Y_{0}, 𝔰) → C_{q−1+kℓ}^{(ω)} for k > 0. In particular, the relation

∂_{n−1} ∘ ∂_{n} = 0 implies that ∂_{n−1}^{(ω)} ∘ ∂_{n}^{(ω)} = 0 is also satisfied.

Proof. The first statement about the decomposition of the chain complex is obvious from the definition. Now we study the boundary operator under this decomposition.

For any k < 0, a ∈ M_{Y_{0},q}^{(ω)}(𝔰), and b ∈ M_{Y_{0},q−1+kℓ}^{(ω)}(𝔰), we shall prove that the 0-dimensional components M̂^{0}(a, b) in M̂(a, b) is empty, hence the entry of the boundary operator is trivial, ⟨b, ∂_{n}(a)⟩ = 0.

Notice that we have

𝔦_{Y_{0}}^{(ω)}(a) = 𝔦_{Y_{0}}(a^{ω}) = q and 𝔦_{Y_{0}}^{(ω)}(b) = 𝔦_{Y_{0}}(b^{ω}) = q − 1 + kℓ.

Thus, the moduli space of flow lines on B_{ℓ}, M̂(a^{ω}, b^{ω}) has virtual dimension −kl > 0. There exists a unique element [λ] ∈ G/G_{ℓ}, such that

⟨[λ] ∪ c_{1}(L), [Y_{0}]⟩ = −kl.

This implies that we have 𝔦_{Y_{0}}(λ(b^{ω})) = SF (H_{(A(t),ψ(t))})ã_{0} ; λ(b^{ω}) = q − 1, and CSD(λ(b^{ω})) = CSD(b^{ω}) − 8π^{2}kℓ, see Lemma 3.1.

When non-empty, M̂^{0}(a, b) is isomorphic to M̂(a^{ω}, λ(b^{ω})). We can prove that M̂(a^{ω}, λ(b^{ω})) is empty, since the CSD functional is non-increasing along the gradient flow lines and the difference of CSD between a^{ω} and λ(b^{ω}) is negative:

CSD(a^{ω}) − CSD(λ(b^{ω}))
= CSD(a^{ω}) − CSD(b^{ω}) − (CSD(λ(b^{ω})) − CSD(b^{ω}))
= CSD(a^{ω}) − CSD(b^{ω}) + 8π^{2}kℓ < 0,

as k < 0 and |CSD(a^{ω}) − CSD(b^{ω})| < 8π^{2}ℓ. This proves that the entries below the diagonal are always zero.

For a ∈ M_{Y_{0},q}^{(ω)}(𝔰) and b ∈ M_{Y_{0},q−1}^{(ω)}(𝔰), it is easy to see that we have

⟨∂_{n}(a), b⟩ = ⟨∂_{q}^{(ω)}(a), b⟩.

For a ∈ M_{Y_{0},q}^{(ω)}(𝔰) and b ∈ M_{Y_{0},q−1+kℓ}^{(ω)}(𝔰) with (k > 0), we can define

⟨∂_{q,k}^{(ω)}(a), b⟩ = ⟨∂_{n}(a), b⟩ = #(M̂^{0}(a, b)).

This counts the points in the zero dimensional components M̂^{0}(a, b) in the moduli space of trajectories on B from a to b. Equivalently, we have

⟨∂_{q,k}^{(ω)}(a), b⟩ = #(M̂(a^{ω}, λ(b^{ω}))),

where M̂(a^{ω}, λ(b^{ω})) is the moduli space of trajectories on B_{ℓ} from a^{ω} to λ(b^{ω}) and λ represents the unique element [λ] in G/G_{ℓ} such that

⟨[λ] ∪ c_{1}(L), [Y_{0}]⟩ = −kl.

Thus we have 𝔦_{Y_{0}}(λ(b^{ω})) = q − 1. Therefore, if non-empty, the moduli space M̂(a^{ω}, λ(b^{ω})) is an oriented, compact 0-dimensional manifold with energy given by

8π^{2}kℓ + CSD(a^{ω}) − CSD(b^{ω}) > 8π^{2}(k − 1)ℓ > 0.

This completes the proof of the Lemma.

The expression of ∂_{n} and the appearance of ∂_{q,k}^{(ω)} in Lemma 3.4 lead us naturally to introduce a filtration of of the ℤ_{ℓ} graded complex C_{∗}(Y_{0}, 𝔰). The filtration is given by

F_{q}^{(ω)}C_{n} = ⨁_{k≥0} C_{q+kℓ}^{(ω)}(Y_{0}, 𝔰).

for n ∈ ℤ_{ℓ}, and q ∈ ℤ with q ≡ n(mod ℓ). Thus, we have

⋯ ⊂ F_{q+ℓ}^{(ω)}C_{n} ⊂ F_{q}^{(ω)}C_{n} ⊂ F_{q−ℓ}^{(ω)}C_{n} ⊂ ⋯ ⊂ C_{n}(Y_{0}, 𝔰),  (4)

a finite length decreasing filtration of the ℤ-graded Seiberg–Witten–Floer chain complex. From Lemma 3.4, we see that the boundary operator

∂_{n} : F_{q}^{(ω)}C_{n} ⟶ F_{q−1}^{(ω)}C_{n−1}

preserves the filtration. Let F_{q}^{(ω)}H_{n} denote the homology of the complex

⋯ →^{∂} F_{q}^{(ω)}C_{n} ⟶^{∂} F_{q−1}^{(ω)}C_{n−1} →^{∂} · · · .

Define

F_{q}^{(ω)}HF_{n}^{SW} (Y_{0}, 𝔰) = Im(F_{q}^{(ω)}H_{n} → HF_{n}^{SW} (Y_{0}, 𝔰)).

We thus obtain a bounded filtration on HF_{n}^{SW} (Y_{0}, 𝔰),

\begin{matrix}⋯ ⊂ F_{q+ℓ}^{(ω)}HF_{n}^{SW} (Y_{0}, 𝔰) ⊂ F_{q}^{(ω)}HF_{n}^{SW} (Y_{0}, 𝔰) \\ ⊂ F_{q−ℓ}^{(ω)}HF_{n}^{SW} (Y_{0}, 𝔰) ⊂ ⋯ ⊂ HF_{n}^{SW} (Y_{0}, 𝔰). \end{matrix}

The standard procedure of constructing the spectral sequence for a filtration [18] gives the following theorem on the relation between the ℤ-graded and the ℤ_{ℓ}-graded homology groups.

Theorem 3.5. There exists a spectral sequence (E_{q,n}^{k}(Y_{0}, 𝔰), d^{k}) with

E_{q,n}^{1}(Y_{0}, 𝔰) ≅ HF_{q,(ω)}^{SW}(Y_{0}, 𝔰)

for n ∈ ℤ_{ℓ} and q ∈ ℤ with q ≡ n(mod ℓ). The higher differentials

d^{k} : E_{q,n}^{k}(Y_{0}, 𝔰) ⟶ E_{q−1+kℓ,n−1}^{k}(Y_{0}, 𝔰)

are induced by the maps ∂_{q,k}^{(ω)} defined in Lemma 3.4. Furthermore, the spectral sequence (E_{q,n}^{k}(Y_{0}, 𝔰), d^{k}) converges to the ℤ_{ℓ}-graded homology groups HF_{n}^{SW} (Y_{0}, 𝔰).

Proof. By construction, the ℤ_{ℓ}-graded chain complex C_{n}(Y_{0}, 𝔰) has a bounded filtration (4) with the associated graded complex given by

F_{q}^{(ω)}C_{n}/F_{q+ℓ}^{(ω)}C_{n} = C_{q}^{(ω)}(Y_{0}, 𝔰).

Then by the standard technique of [18], we derive the existence of a spectral sequence

(E_{q,n}^{k}(Y_{0}, 𝔰), d^{k})

with E_{q,n}^{1}(Y_{0}, 𝔰) ≅ HF_{q,(ω)}^{SW}(Y_{0}, 𝔰), and

\begin{matrix}Z_{q,n}^{k}(Y_{0}, 𝔰) = \{a ∈ F_{q}^{(ω)}C_{n}| ∂_{n}(a) ∈ F_{q−1+kℓ}^{(ω)}C_{n−1}\} \\ E_{q,n}^{k}(Y_{0}, 𝔰) = Z_{q,n}^{k}(Y_{0}, 𝔰)/(Z_{q+ℓ,n}^{k−1}(Y_{0}, 𝔰) + ∂_{n+1}Z_{q+1−(k−1)ℓ,n+1}^{k−1}(Y_{0}, 𝔰)). \end{matrix}

The higher differentials are induced by ∂_{n}. The expression of ∂_{n}, as discussed in Lemma 3.4, tells us that the higher differentials acting on E_{q,n}^{k}(Y_{0}, 𝔰) are defined by ∂_{q,k}^{(ω)}.

4 ℤ-graded homology for exact triangles

In this section, we present a different construction which applies to the special case where Y_{0} is obtained by Dehn surgery with framing 0 on a knot K in a ℤ-homology 3-sphere Y. Again we assume that the Spin^{c}-structure is non-torsion. The torsion case will be analyzed in the next Section.

Under our assumptions, Y_{0} has the homology of S^{1} × S^{2}. We denote by Y_{1} the ℤ-homology 3-sphere obtained by Dehn surgery on K with framing 1. With minor modifications, the arguments in this subsection can be extended to the case of a knot K representing a zero homology class in H_{1}(Y, ℤ) in a general 3-manifold Y.

In [2], we have identified the (perturbed) irreducible Seiberg–Witten monopoles (as critical points of the perturbed CSD functional) on (Y, g), with monopoles on (Y_{1}, g_{1}) and on Y_{0}, with

M_{Y,g,μ}^{∗} ≅ M_{Y_{1},g_{1}}^{∗} ∪ ⋃_{𝔰}M_{Y_{0}}(𝔰).  (5)

The metric and perturbation on Y have been carefully chosen to met the above identification. In particular, the perturbation μ is constructed as in [2] so as to “simulate the effect of surgery”. Namely, it is obtained as a deformation of the circle of flat connections on the tubular neighborhood of the knot ν(K) in Y, so that it approximates the union of the lines of flat connections on ν(K) inside Y_{1} and Y_{0}.

For each Spin^{c} structure 𝔰 on Y_{0}, with non-empty moduli space M_{Y_{0}}(𝔰), we have an injective map

j_{𝔰} :  M_{Y_{0}}(𝔰) ⟶ M_{Y,g,μ}  (6)

defined by the identification (5).

Definition 4.1. Assume that c_{1}(𝔰) is a non-torsion class in H^{2}(Y_{0}, ℤ). We define the ℤ-grading of elements in M_{Y_{0}}(𝔰) as induced from the grading on M_{Y,g,μ},

𝔦_{Y_{0}}^{(Y)}(a) = 𝔦_{Y} (j_{𝔰}(a))

for any a ∈ M_{Y_{0}}(𝔰).

Recall that the grading on M_{Y,g,μ} is defined by the spectral flow of the Hessian operator of CSD along a path from the critical point in M_{Y,g,μ} to the unique reducible point ϑ in M_{Y,g,μ}. The relative index 𝔦_{Y_{0}}(a, b) of two points a and b in M_{Y_{0}}(𝔰), defined as the spectral flow of the Hessian operator of CSD function for (Y_{0}, 𝔰), agrees with the relative index 𝔦_{Y} (j_{𝔰}(a), j_{𝔰}(b)), cf. [2]. Thus, we have

𝔦_{Y_{0}}^{(Y)}(a) − 𝔦_{Y_{0}}^{(Y)}(b) = 𝔦_{Y_{0}}(a, b) (mod ℓ)

where ℓ ∈ ℤ satisfies c_{1}(𝔰)(H_{2}(Y_{0}, ℤ)) = ℓℤ. Thus, 𝔦_{Y_{0}}^{(Y)} is a ℤ-lifting of the ℤ_{ℓ}-grading on M_{Y_{0}}(𝔰).

Definition 4.2. For any q ∈ ℤ, we define

M_{Y_{0}}^{(q)}(𝔰) = {a ∈ M_{Y_{0}}(𝔰) | 𝔦_{Y_{0}}^{(Y)}(a) = q}.

The q-chains in the corresponding ℤ-graded Floer complex are the elements of the abelian group C_{(q)}(Y_{0}, 𝔰) generated by the points of M_{Y_{0}}^{(q)}(𝔰). The boundary operator

∂_{(q)} : C_{(q)}(Y_{0}, 𝔰) → C_{(q−1)}(Y_{0}, 𝔰)

is defined as

∂_{(q)}(a) = ∑_{b∈M_{Y_{0}}^{(q−1)}(𝔰)}#(M̂^{0}(a, b))b,

where M̂^{0}(a, b) is the zero dimensional components of the moduli space of unparameterized flow lines in M̂(a, b). The compactness theorem tells us that M̂^{0}(a, b) is an oriented, compact 0-dimensional manifold. Thus, the coefficient #(M̂^{0}(a, b)) is well-defined as the algebraic sum of points in M̂^{0}(a, b).

In order to prove that we have ∂_{(q)} ∘ ∂_{(q−1)} = 0, we need the following Lemma to know when the zero-dimensional moduli space of flow lines between a and b over (Y_{0}(r), 𝔰) is non-empty. Here we follow the notation of [2], where Y_{0}(r) and Y (r) denote the 3-manifolds Y_{0} and Y endowed with long cylinders T^{2} × [−r, r] along the boundary of the tubular neighborhood ν(K) around the knot K.

Lemma 4.3. Let 𝔰 be a Spin^{c} structure on Y_{0} with non-empty M_{Y_{0}}(𝔰), and c_{1}(𝔰) is a non-trivial class in H^{2}(Y_{0}, ℚ). Then the moduli spaces of flow lines can be described as follows. Assume that a, b are two critical points in M_{Y_{0}}(𝔰) of relative index 1 modulo ℓ, here ℓ ∈ ℤ satisfies c_{1}(𝔰)(H_{2}(Y_{0}, ℤ)) = ℓℤ. If for sufficiently large r, the 1-dimensional component M_{Y_{0}(r)×ℝ}^{1}(a, b) of M_{Y_{0}(r)×ℝ}(a, b), is non-empty, then we have

𝔦_{Y_{0}}^{(Y)}(a) − 𝔦_{Y_{0}}^{(Y)}(b) = 1.

Proof. The result is the consequence of the results on the geometric limits in [12] [13], together with the dimension formulae for various moduli spaces.

Suppose given a solution [A_{r}(t), Ψ_{r}(t)] in a one-dimensional component of M_{Y_{0}(r)×ℝ}(a, b), with asymptotic values

[A_{r}(±∞), Ψ_{r}(±∞)] = [A′_{±∞}, Ψ′_{±∞}]#_{a_{±}}^{r}[a^{±}, 0],

for t → ±∞. The geometric limits as r → ∞ are determined in [12]. Among these geometric limits, there are two parameterized oriented paths a_{V} (τ), a_{0}(τ) on ∂_{∞}(M_{V}^{∗}) and M_{ν(K)⊂Y_{0}} = L_{Y_{0}}, respectively, where both paths connect [a^{±}, 0], consistently with the orientations. Here M_{ν(K)∈Y_{0}} is the circle of flat U(1)-connections on ν(K) ⊂ Y_{0} modulo gauge equivalence. There is also a holomorphic map a_{D} from the unit half-disc to χ(T^{2}, Y_{0}) with boundary along the paths a_{V} (τ) and a_{0}(τ).

First we show that, if the moduli space M_{Y (r)×ℝ}(j_{𝔰}(a), j_{𝔰}(b)) is nonempty, then we have

M_{Y (r)×ℝ}(j_{𝔰}(a), j_{𝔰}(b)) ≅ M_{Y_{0}(r)×ℝ}^{1}(a, b).

Under this assumption, the geometric limits of a family of solutions [A_{r}(t), Ψ_{r}(t)] in M_{Y_{0}(r)×ℝ}(a, b), for r → ∞, can be deformed by a homotopy in H^{1}(T^{2}, iℝ) to geometric limits for a monopole in M_{Y (r)×ℝ}(j_{𝔰}(a), j_{𝔰}(b)). In particular, this can happen if and only if the path a_{0}(τ) = [A(τ), 0] along L_{Y_{0}} can be homotopically deformed to a path along the curve ∂_{∞}M_{ν(K),μ}, by a homotopy that moves the endpoints a^{±} to the corresponding points a^{±}(ϵ), along a_{V} (τ). (We use here the same notation as in [2], [12], [13].) Both paths a_{V} (τ) and a_{0}(τ) must be contractible in χ(T^{2}, Y_{0}) = S^{1} × ℝ. In fact, the geometric limits of any solution in M_{Y (r)×ℝ}(j_{𝔰}(a), j_{𝔰}(b)) define an approximate solution to the monopole equations on Y_{0}(r) × ℝ, which can be deformed to a solution in the minimal energy component of M_{Y_{0}(r)×ℝ}(a, b). Conversely, in this case, the geometric limits of [A_{r}(t), Ψ_{r}(t)] can be spliced together to form a solution to the monopole equations on (Y (r) × ℝ, μ, g) between the critical points j_{𝔰}(a) and j_{𝔰}(b). Thus the identification of the moduli spaces implies that

𝔦_{Y_{0}}^{(Y)}(a) − 𝔦_{Y_{0}}^{(Y)}(b) = 1.

Now assume that the moduli space M_{Y (r)×ℝ}(j_{𝔰}(a), j_{𝔰}(b)) is empty. Notice that oriented paths a_{V} (τ), a_{0}(τ) which are non-contractible in the cylinder χ_{0}(T^{2}, ν(K)) ≅ S^{1} × ℝ correspond to flowlines in the components with higher energy. Here

χ(T^{2}, Y_{0}) = χ_{0}(T^{2}, ν(K))

is a covering of the torus χ(T^{2}) of flat U(1)-connections on T^{2} modulo gauge equivalence, obtained by taking the quotient only by those gauge transformations on T^{2} that extend to Y_{0}(r), cf. [2]. Since we are considering the flowlines in the minimal energy components M_{Y_{0}(r)×ℝ}^{1}(a, b), the paths a_{V} (τ) and a_{0}(τ) are also contractible in χ(T^{2}, Y_{0}) = S^{1} × ℝ. The analysis of the moduli space on the cobordant manifold W_{0} with cylindrical ends Y_{0}(r) × (−∞, 0] and Y (r) × [0, ∞) in [13] shows that the holomorphic halfdisc a_{D}, from the geometric limits of [A_{r}(t), Ψ_{r}(t)] in the one-dimensional components of M_{Y_{0}(r)×ℝ}(a, b), is the degenerate limit of the assembled holomorphic triangles Δ^{ϵ} as ϵ → 0, where ϵ is the parameter in the perturbation on Y (r) that “simulates the effect of surgery”. In particular, we have the following identification

M_{Y_{0}(r)×ℝ}^{1}(a, b) ≅ ⋃_{c∈M_{Y_{1}}^{∗}} M_{Y (r)×ℝ}(j_{𝔰}(a), i(c)) × M^{W_{0},(0)}(i(c), b).

Here i(c) is the corresponding monopole in M_{Y,μ,g}^{∗}, under (5), satisfying 𝔦_{Y} (j_{𝔰}(a)) − 𝔦_{Y} (i(c)) = 1, and M^{W_{0},(0)}(i(c), b) is given by the non-empty 0dimensional components of M^{W_{0},(0)}(i(c), b). Here we follow the notation of [13]. Then the dimension formulae developed in [13] can be applied to show that

𝔦_{Y} (i(c)) − 𝔦_{Y_{0}}^{(Y)}(b) = 0.

Thus we have 𝔦_{Y_{0}}^{(Y)}(a)−𝔦_{Y_{0}}^{Y}(b) = 1. This completes the proof of the statement.

Using the result of Lemma 4.3, we now show that ∂_{(q−1)} ∘ ∂_{(q)} = 0, and we relate the resulting homology groups to the original ℤ_{ℓ}-graded Floer homology groups for (Y_{0}, 𝔰).

Proposition 4.4. 1. For n ∈ ℤ_{ℓ} and q ∈ ℤ with q = n(mod ℓ), then

C_{n}(Y_{0}, 𝔰) = ⨁_{k∈ℤ}C_{(q+kℓ)}(Y_{0}, 𝔰).

2. Under the decomposition as above, the boundary operator ∂_{n} on C_{n}(Y_{0}, 𝔰) can be expressed as ⊕_{k∈ℤ}∂_{(q+kℓ)}. Thus, in particular, we have ∂_{(q−1)} ∘ ∂_{(q)} = 0 and

HF_{n}^{SW} (Y_{0}, 𝔰) = ⨁_{k∈ℤ}HF_{(q+kℓ)}^{SW}(Y_{0}, 𝔰),

where HF_{(q)}^{SW} (Y_{0}, 𝔰) is the q-th homology group of the chain complex (C_{(q)}(Y_{0}, 𝔰), ∂_{(q)}).

Proof. We only need to prove the second statement. For any generator a ∈ C_{(q)}(Y_{0}, 𝔰), as an element in C_{n}(Y_{0}, 𝔰), we shall verify that ∂_{n}(a) = ∂_{(q)}(a). From the definition, we know that

∂_{n}(a) = ∑_{b∈M_{Y_{0}}(𝔰):𝔦_{Y_{0}}(a,b)=1(mod ℓ)}#(M̂_{Y_{0}×ℝ}^{0}(a, b))b,

where M̂_{Y_{0}×ℝ}^{0}(a, b) is the zero dimensional component of the moduli space of the unparameterized flow lines in M̂_{Y_{0}×ℝ}(a, b). By the result of Lemma 4.3, we know that the critical points b ∈ M_{Y_{0}}(𝔰) with non-empty zero dimensional component in M̂_{Y_{0}×ℝ}(a, b) are contained in the set

M_{Y_{0}}^{(q−1)}(𝔰) = {b ∈ M_{Y_{0}}(𝔰) | 𝔦_{Y_{0}}^{(Y)}(b) = q − 1}.

This implies that ∂_{n}(a) = ∂_{(q)}(a) for a ∈ C_{(q)}(Y_{0}, 𝔰), hence ∂_{n} = ⊕_{k∈ℤ}∂_{(q+kℓ)}. Then the statement follows from standard arguments in homological algebra.

In the exact triangle for the Seiberg–Witten–Floer homologies relating HF_{∗}^{SW} (Y, g), HF_{∗}^{SW} (Y_{1}, g_{1}), and HF_{(∗)}^{SW} (Y_{0}, 𝔰_{k}), as studied in [12][13], the ℤ-graded Seiberg–Witten–Floer homology for (Y_{0}.𝔰) with 𝔰 a non-trivial Spin^{c} structure, is precisely the ℤ-graded HF_{(∗)}^{SW} (Y_{0}, 𝔰) described in this subsection.

Remark 4.5. In the case where Y_{0} is obtained as 0-surgery on a knot K in a homology 3-sphere, both the constructions of ℤ-graded Floer homology are possible.

5 Monopole homology for trivial Spin^{c}-structures

In this section we consider again the general case where Y_{0} is a compact 3-manifold with b_{1}(Y_{0}) > 0. We analyze the Seiberg–Witten Floer theory in the case of a Spin^{c}-structure 𝔰_{0} with c_{1}(det(𝔰_{0})) torsion. The results we obtain specialize to the case of Y_{0} a 0-surgery on a knot in a homology 3-sphere, with the trivial Spin^{c}-structure with c_{1}(det(𝔰_{0})) = 0.

A cohomologically trivial perturbation would introduce a torus of reducible monopoles among the critical points of the Chern–Simons–Dirac functional given by the flat U(1)-connections H^{1}(Y_{0}, ℝ)/H^{1}(Y_{0}, ℤ). In order to avoid the torus of reducibles, we need to introduce a small cohomologically non-trivial perturbation of the Seiberg–Witten equations on Y_{0} given by a 1-form ρ satisfying [∗ρ] ∈ H^{2}(Y_{0}, ℝ) non-trivial. With this perturbation, M_{Y_{0}}(𝔰_{0}) consists of a finite set of points in B, cut out transversely by the 3-dimensional (perturbed) Seiberg–Witten monopole equations.

The different nature of the case c_{1}(𝔰_{0}) torsion can be regarded as a result of the following phenomenon. In the case analyzed previously, with c_{1}(𝔰) non-torsion, we studied the critical points of CSD on the infinite cyclic covering space B_{ℓ} of B. The main point of our argument has been the following: the non-trivial covering B_{ℓ} is the essential tool in separating the various components of M̂(a, b) of different dimensions, with each of them separately admitting a nice compactification to a smooth manifold with corners as in [11], [15].

In the case with c_{1}(𝔰_{0}) torsion, however, the relative index 𝔦(a) − 𝔦(b) is already well defined as an integer in B. Thus, the moduli space M̂(a, b) does not break into components of different dimensions. In fact, all components in M̂(a, b) have constant dimension given by the relative index 𝔦(a) − 𝔦(b) ∈ ℤ. In order to define the Floer homology, we still need a nice compactification of the moduli spaces M̂(a, b). However, we lack the uniform energy bound on M̂(a, b), since the CSD functional is only circle-valued, due to the presence of the non-trivial cohomology class of the perturbation [∗ρ]. This lack of a uniform energy estimate implies that we cannot define the usual boundary operator

⟨∂a, b⟩ = n_{ab}

of the Floer complex by setting

n_{ab} = #M̂(a, b),

since M̂(a, b) may be non-compact in this case.

Thus, we need to separate M̂(a, b) into components of uniform energy and define a boundary operator in terms of components of a fixed energy. We identify M̂(a, b) with a subset of M(a, b) consisting of those trajectories with equal energy distributions on (−∞, 0] and [0, ∞), that is,

\begin{aligned}& ∫_{−∞}^{0}(∥∂_{t}A(t)∥_{L^{2}(Y_{0})}^{2} + ∥∂_{t}ψ(t)∥_{L^{2}(Y_{0})}^{2})dt \\ = & ∫_{0}^{∞}(∥∂_{t}A(t)∥_{L^{2}(Y_{0})}^{2} + ∥∂_{t}ψ(t)∥_{L^{2}(Y_{0})}^{2})dt. \end{aligned}  (7)

Every non-constant trajectory [A(t), ψ(t)] in a non-empty connected component of M̂(a, b) has positive energy

∫ℝ(∥∂_{t}A(t)∥_{L^{2}(Y)}^{2} + ∥∂_{t}ψ(t)∥_{L^{2}(Y)}^{2})dt > 0.

The energy is constant, independent of the trajectory, in each connected component of M̂(a, b). This energy agrees with the variation of the perturbed CSD functional along [A(t), ψ(t)]. In fact, we have

−∥∇CSD_{U,V} ∥_{L^{2}}^{2} = ⟨∂_{t}(A, ψ), ∇CSD_{U,V} ⟩,

if CSD_{U,V} = CSD + U + V denotes the perturbed CSD functional, and (A, ψ) is a solution of the flow equation

∂_{t}(A, ψ) = −∇CSD_{U,V} .

This gives

∫_{t}^{∞} ∥∇CSD_{U,V} (A(τ), ψ(τ))∥_{L^{2}(Y)}^{2}dτ =
CSD_{U,V} (A(t), ψ(t)) − CSD_{U,V} (A_{b}, ψ_{b}).

Thus, we can define the energy function over M̂(a, b) as

\begin{matrix}E :  M̂(a, b) ⟶ ℝ \\ E([A(t), ψ(t)]) = _{∫ℝ}(∥∂_{t}A(t)∥_{L^{2}(Y)}^{2} + ∥∂_{t}ψ(t)∥_{L^{2}(Y)}^{2})dt. \end{matrix}  (8)

The image of E is positive and discrete in ℝ. The energies of any two connected components may differ by some multiple of a positive number

e_{ρ} = min{|[∗ρ]([Σ])| : [Σ] ∈ H_{2}(Y_{0}, ℤ)}.

Denote e_{min}(a, b) the minimal energy on M̂(a, b). Then, for any n ≥ 0 in ℤ, we can define

M̂^{(n)}(a, b) = E^{−1}(e_{min}(a, b) + ne_{ρ}).

We have the following compactness theorem for the moduli space M̂^{(n)}(a, b) with the fixed energy e_{min}(a, b) + ne_{ρ}, for some n ≥ 0.

Proposition 5.1. Let (Y_{0}, 𝔰_{0}) be a 3-manifold with b_{1}(Y) > 0 and with c_{1}(𝔰_{0}) torsion. Let M̂^{(n)}(a, b) be the component of M̂(a, b) defined as above, for two critical points a and b of relative index q +1. Then M̂^{(n)}(a, b) can be compactified to be a smooth manifold of dimension q with corner structure, by adding lower dimensional boundary strata. The codimension k boundary strata are given by

⋃_{a_{1},··· ,a_{k}} M̂^{(n_{0})}(a, a_{1}) × M̂^{(n_{1})}(a_{1}, a_{2}) × ⋯ × M̂^{(n_{k})}(a_{k}, b).

Here the union is over all possible sequences a_{0} = a, a_{1}, ⋯ a_{k}, a_{k+1} = b with strictly decreasing indices and with ∑_{j=0}^{k} n_{j} = n. In particular, we have the following results.

1. The component of fixed energy in M̂(a, b) with relative index 1 is compact.

2. If a and c are two critical points of relative index 𝔦(a) − 𝔦(c) = 2, then the compactified moduli space M̂^{(n)}(a, c) is an oriented, smooth manifolds with boundary points given by

⋃_{b,i+j=n} M̂^{(i)}(a, b) × M̂^{(j)}(b, c),

where b runs over all critical points with relative index 𝔦(a) − 𝔦(b) = 1.

Proof. The convergence and gluing theorems developed in Section 4 of [11] can be applied to this case to give M̂^{(n)}(a, b) the structure of a smooth manifold structure with corners. In particular, the local diffeomorphism provided by the convergence and gluing theorem in [11] is energy preserving. Then the above claims follow from the arguments of Section 4 of [11]. Notice that we have both inequalities

e_{min}(a, b) ≤ e_{min}(a, a_{1}) + ⋯ + e_{min}(a_{k}, b)

and

e_{min}(a, b) ≥ e_{min}(a, a_{1}) + ⋯ + e_{min}(a_{k}, b),

where we use essentially the identification between energy and variation of the perturbed CSD functional.

We now introduce a chain complex over the ring ℤ[[t]] of formal power series as follows. We define the chain complex C_{∗}(Y_{0}, 𝔰_{0}, ℤ[[t]]) to be the ℤ[[t]]-module generated by the critical points in M_{Y_{0}}(𝔰_{0}), with the grading given by the relative grading 𝔦(a) − 𝔦(a_{0}), where a_{0} is a fixed critical point in M_{Y_{0}}(𝔰_{0}). The boundary operator ∂ is defined by

∂_{[t]}(a) = ∑_{b:𝔦(a)−𝔦(b)=1} ∑_{k≥0} #(M̂^{(n)}(a, b))t^{n}b.  (9)

By the result of Proposition 5.1, we know that the boundary operator ∂_{[t]} in (9) is well-defined and satisfies ∂^{2} = 0. The corresponding homology groups are the homology groups of the chain complex (C_{∗}(Y_{0}, 𝔰_{0}, ℤ[[t]]), ∂_{[t]}),

HF_{∗}(Y_{0}, 𝔰_{0}, ℤ[[t]]) = H_{∗}(C_{∗}(Y_{0}, 𝔰_{0}, ℤ[[t]]), ∂_{[t]}).  (10)

In the construction of the exact triangle in [2], [12], [13], we set

HF_{(∗)}(Y_{0}, 𝔰_{0}) = HF_{∗}(Y_{0}, 𝔰_{0}, ℤ[[t]])|_{t=0}.

Here the grading on M_{Y_{0}}(𝔰_{0}) is induced from the injective map

j_{𝔰_{0}} : M_{Y_{0}}(𝔰_{0}) ⟶ M_{Y,g,μ},

that is, for any a ∈ M_{Y_{0}}(𝔰_{0}), we define the induced grading from j_{𝔰_{0}} as

𝔦_{Y_{0}}^{(Y)}(a) = 𝔦_{Y} (j_{𝔰_{0}}(a)).

This induced grading on M_{Y_{0}}(𝔰_{0}) satisfies (cf. [2])

𝔦_{Y_{0}}^{(Y)}(a) − 𝔦_{Y_{0}}^{(Y)}(b) = 𝔦_{Y_{0}}(a) − 𝔦_{Y_{0}}(b).

After a suitable choice of a base point a_{0} in M_{Y_{0}}(𝔰_{0}), we can assume that

𝔦_{Y_{0}}^{(Y)}(a) = 𝔦_{Y_{0}}(a) = 𝔦_{Y} (j_{𝔰_{0}}(a)).  (11)

Notice that, in the case of the 0-surgery Y_{0} that has b_{1}(Y_{0}) = 1, the Floer homology HF_{∗}(Y_{0}, 𝔰_{0}, ℤ[[t]]) defined as in (10) depends on the cohomology class [∗ρ] ∈ H^{2}(Y_{0}, ℝ). This dependence can be seen from the fact that, when the value [∗ρ] ∈ ℝ crosses zero, some irreducible solutions in M_{Y_{0},ρ}(𝔰_{0}) may hit the circle of reducibles, cf. [8].

We have the analog of Lemma 4.3, in the case of a 3-manifold Y_{0} that is obtained as a 0-surgery on a knot in a homology 3-sphere Y.

Lemma 5.2. Let 𝔰_{0} be a trivial Spin^{c} structure on Y_{0}, where Y_{0} is obtained as a 0-surgery on a knot in a homology 3-sphere Y. Let a, b be two critical points in M_{Y_{0}}(𝔰_{0}) with relative index 1. Then if M̂_{Y_{0}(r)×ℝ}^{(0)}(a, b) is nonempty, for sufficiently large r the oriented paths a_{0}(τ), a_{V} (τ) in the geometric limits of any flow line in the component of minimal energy are contractible in χ_{0}(T^{2}, Y_{0}). Here the moduli spaces M_{Y_{0}}(𝔰_{0}) and M̂_{Y_{0}(r)×ℝ}^{(0)}(a, b) are considered with an additional perturbation ρ with [∗ρ] = η > 0 in H^{2}(Y_{0}, ℝ) = ℝ, as in [2], cf. Remark 5.3.

Proof. This is again a direct applications of the geometric limit results in [12] and [13]. The argument proceeds as in the proof of Lemma 4.3.

The homology groups for (Y_{0}, 𝔰_{0}), which appear in the exact triangle, are precisely given by the homology groups

HF_{(∗)}(Y_{0}, 𝔰_{0}) = HF_{∗}(Y_{0}, 𝔰_{0}, ℤ[[t]])|_{t=0},

after a possible global index shifting.

Remark 5.3. The Seiberg–Witten–Floer homology for (Y_{0}, 𝔰_{0}) depends on the perturbation [∗ρ], as the energy function E and the Novikov-type chain complex all depend on the choice of [∗ρ]. In the exact triangle, we use a particular choice of [∗ρ], satisfying

[∗ρ] = ηP D_{Y_{0}}(m) ∈ H^{2}(Y_{0}, ℝ),

with m the meridian of the knot K.

6 Gluing formulae along the trivial Spin^{c} structure

Motivated by the general ideas of topological field theory, one can associate to any 4-manifold (X_{i}, 𝔰_{i}) with a boundary (Y_{0}, 𝔱) Seiberg–Witten invariants with values in the monopole homology groups HF_{∗}^{SW} (Y_{0}, 𝔱),

SW_{X_{i}}(𝔰_{i}, ·) : 𝔸(X_{i}) ⟶ HF_{∗}^{SW} (Y_{0}, 𝔱),

where 𝔸(X_{i}) = Sym^{∗}(H_{0}(X_{i})) ⊗ Λ^{∗}(H_{1}(X_{i})) is the free graded algebra generated by elements in H_{0}(X_{i}) of degree 2 and cycles in H_{1}(X_{i}) of degree 1. Moreover, if a closed 4-manifold X = X_{1} ∪_{Y_{0}} X_{2} splits in two components X_{i}, i = 1, 2, along a 3-manifold Y_{0}, then the natural pairing between HF_{∗}^{SW} (Y_{0}, 𝔱) and HF_{∗}^{SW} (−Y_{0}, −𝔱) can be applied to give gluing formulae for the Seiberg–Witten invariants of X in terms of the SW_{X_{i}}. For any 3manifold (Y_{0}, 𝔱) with b_{1}(Y) > 0 and c_{1}(𝔱) non-torsion, these gluing formulae were developed in [3]. For a manifold Y with a torsion Spin^{c} structure 𝔱, we discuss the corresponding gluing formulae in this section.

To the purpose of gluing relative invariants, it is natural to construct the monopole homology groups over the field ℚ((t)) of formal Laurent series and use another definition of the energy in the construction of monopole homology groups.

We assume that the perturbation term [∗ρ] is Poincaré dual to ϵ[Γ] for some 1-cycle Γ in Y_{0} and a small ϵ ∈ ℝ^{+}. Here Γ is an integer cycle, that is, it defines a (non-torsion) integer homology class. The moduli space of critical points with this perturbation is denoted by M_{Y_{0}}(𝔱, Γ). Let a, b be two critical points in M_{Y_{0}}(𝔱, Γ) of relative index k + 1 > 0. We know that M̂(a, b) has infinite many components of dimension k. Note that, if we identify elements in M̂(a, b) with trajectories in M(a, b) satisfying the equal energy condition (7), we can now define the following map on M̂(a, b):

\begin{matrix}E_{ρ} :  M̂(a, b) ⟶ ℤ \\ E_{ρ}([A(t), ψ(t)]) = ⟨[\frac{\sqrt{−1}}{2π}F_{A(t)}], [Γ × ℝ]⟩. \end{matrix}  (12)

Here F_{A(t)} is the 4-dimensional curvature of A(t). E_{ρ} essentially measures the variation of the Chern-Simons-Dirac functional on Y_{0}×ℝ up to a positive scalar. We denote by

M̂^{[n]}(a, b) = E_{ρ}^{−1}(n)

the preimage of n under the energy map E_{ρ}.

Lemma 6.1. Let Y be a 3-manifold with b_{1}(Y) > 0, endowed with a Spin^{c} structure 𝔱 with c_{1}(𝔱) = 0. Let a, b be two critical points in M_{Y_{0}}(𝔱, Γ) of relative index k + 1 > 0. We have the following results.

1. M̂^{[n]}(a, b) is empty for n << 0;

2. For any non-empty M̂^{[n]}(a, b), there exists a compactification of M̂^{[n]}(a, b) to a smooth manifold with corners, obtained by adding lower dimensional strata of the form

⋃_{a_{1},··· ,a_{k}} M̂^{[n_{0}]}(a, a_{1}) × M̂^{[n_{1}]}(a_{1}, a_{2}) × ⋯ × M̂^{[n_{k}]}(a_{k}, b).

Here the union is over all possible sequences a_{0} = a, a_{1}, ⋯ a_{k}, a_{k+1} = b in M_{Y_{0}}(𝔱, Γ) with strictly decreasing indices and with ∑_{j=0}^{k} n_{j} = n.

In particular, given any two critical points a and b of relative index 1, if the moduli space M̂^{[n]}(a, b) is non-empty, then it is compact. For any two critical points a and c of relative index 2, the compactified moduli space M̂^{[n]}(a, c) is an oriented, smooth manifold with boundary points given by

⋃_{b,i+j=n} M̂^{[i]}(a, b) × M̂^{[j]}(b, c).

Here b runs over all critical points with relative index 𝔦(a) − 𝔦(b) = 1.

Proof. Any non-empty M̂^{[n]}(a, b) has positive and constant energy defined by E (8). This implies that M̂^{[n]}(a, b) must be empty for n << 0. The argument for the compactification is analogous to Proposition 5.1.

Definition 6.2. Define the chain complex C_{Γ,∗}(Y_{0}, 𝔱, ℚ((t))) as the vector space generated by the critical points in M_{Y_{0}}(𝔱, Γ), over the coefficient field of formal Laurent series ℚ((t)) = ℚ[[t]][t^{−1}], with the grading given by the relative index. The boundary operator is given by

∂(a) = ∑_{b:𝔦_{Y_{0}}(a)−𝔦_{Y_{0}}(b)=1} ∑_{n∈Z} #(M̂^{[n]}(a, b))t^{n}b.

The fact that ∂^{2} = 0 follows from Lemma 6.1. The homology groups of the chain complex C_{∗}(Y_{0}, 𝔱, ℚ((t))) are denoted by HF_{Γ,∗}^{SW} (Y_{0}, 𝔱, ℚ((t))).

By the construction of HF_{Γ,∗}^{SW} (Y_{0}, 𝔱, ℚ((t))), we know that these homology groups depend on the choice of the non-trivial cohomology class [∗ρ], which in turn depends on the choice of the one-cycle Γ. There exists a natural isomorphism HF_{Γ,∗}^{SW} (Y_{0}, 𝔱, ℚ((t))) ≅ HF_{−Γ}^{SW,∗}(−Y_{0}, −𝔱, ℚ((t))), where −Y_{0} is the manifold Y_{0} with the reversed orientation. For the definition of Seiberg–Witten–Floer cohomology, in the equivariant setting, the reader can refer to Section 5 of [11]. This yields a natural pairing

⟨, ⟩ : HF_{Γ,∗}^{SW} (Y_{0}, 𝔱, ℚ((t))) × HF_{−Γ,∗}^{SW}(−Y_{0}, −𝔱, ℚ((t))) ⟶ ℚ((t)). (13)

6.1 Relative Seiberg–Witten invariants and gluing formulae

Let X_{1} be an oriented Riemannian 4-manifold with a cylindrical end modeled on Y_{0} × [0, ∞). Let 𝔰_{1} be a Spin^{c} structure on X_{1}, such that the induced Spin^{c} structure 𝔱 on Y_{0} satisfies c_{1}(𝔱) = 0 in H_{2}(Y_{0}, ℚ). Denote by M_{Y_{0}}(𝔱, Γ) the moduli space of Seiberg–Witten monopoles on (Y_{0}, 𝔱), with respect to a perturbation ρ satisfying [∗ρ] = ϵP D([Γ]), for a non-trivial 1-cycle Γ in Y_{0}. Here again we assume that Γ defines a non-torsion integer homology class. Moreover, for simplicity, we assume that Γ lies in the image of the map H_{2}(X_{1}, Y_{0}; ℤ) → H_{1}(Y, ℤ). As we discuss in this section, the choice of Σ with ∂Σ = Γ will provide a convenient way of separating the moduli spaces on (X_{1}, 𝔰_{1}) into components of uniform energy, which admit a nice compactification by adding lower dimensional strata.

We assume also that we are working with an additional generic perturbation (U, V) as in section 2, in order to achieve transversality for the moduli space of trajectories. We write η_{0} = ρ+∑_{j} \frac{∂U}{∂τ_{j}} μ_{j} and ν_{0} = ∑_{j} \frac{∂V}{∂ζ_{j}} ν_{j}. Choose c : X_{1} → ℝ^{+} to be a cut-off function supported away from a compact set and equaling 1 on the end. Choose μ ∈ Λ^{2,+}(X_{1}, iℝ), satisfying μ − c\sqrt{−1}η_{0}^{+} ∈ C_{δ}^{k}. Here C_{δ}^{k} denotes the space of δ-decaying C^{k}-forms for some large integer k and η_{0}^{+} is the self-dual 2-form obtained by η_{0}. Next we choose an imaginary valued 1-form ν on X_{1}, with ν − cν_{0} ∈ C_{δ}^{k}. Over the cylindrical end, we write (A, ψ) in temporal gauge. When restricted to the cylindrical end, μ and ν can be thought of as functions of (A, ψ) on the cylindrical end. The relative Seiberg–Witten invariant for X_{1} is obtained from the perturbed Seiberg–Witten moduli space on X_{1} with asymptotic limit representing a critical point in M_{Y_{0}}(𝔱, Γ).

For each a in M_{Y_{0}}(𝔱, Γ), define M_{X_{1}}(𝔰, a) to be the moduli space of the Seiberg–Witten equations on X_{1} with asymptotic limit in the class of a, modulo the group of gauge transformations. The function spaces for the monopole equations modeled on a are L_{1}^{2}-Sobolev spaces with ‘δ-decay’ where δ > 0 is determined by the least absolute eigenvalue of the Hessian of the CSD at a. Let (A_{a}, ψ_{a}) be a Spin^{c} connection and a spinor on X_{1} which agrees with the pull-back of some gauge representatives of a on the cylindrical end. Note that the gauge class of (A_{a}, ψ_{a}) can be identified with the set H^{1}(Y_{0}, ℤ)/Im(H^{1}(X_{1}, ℤ) → H^{1}(Y_{0}, ℤ)).

Let M_{X_{1}}(𝔰_{1}, a) be the moduli space of solutions to the perturbed Seiberg–Witten equations on (X_{1}, 𝔰):

\{ \begin{matrix}F_{A}^{+} = q(ψ) + μ \\ D_{A}ψ + ν.ψ = 0, \end{matrix}  (14)

for (A, ψ) in the weighted Sobolev space

{(A, ψ)|A − A_{a} ∈ L_{k,δ}^{2}, ψ − ψ_{a} ∈ L_{l,δ,A_{a}}^{2}},

and with gauge group given by {u : X → ℂ||u| = 1, 1−u ∈ L_{l+1,δ}^{2}}. Here we fix a U(1)-connection A_{a} on the spinor bundle to define the covariant derivatives on the spinor sections. For generic (μ, ν), the moduli space M_{X_{1}}(𝔰_{1}, a), if non-empty, is a smooth, oriented manifold whose dimension is given by the index of the deformation complex for (14). Denote this index by 𝔦_{X_{1}}(a). We then have

𝔦_{X_{1}}(a) − 𝔦_{X_{1}}(b) = 𝔦_{Y_{0}}(a) − 𝔦_{Y_{0}}(b).

In general, we cannot compactify M_{X_{1}}(𝔰_{1}, a) naturally by adding lower dimensional strata. Instead, we define some energy map on M_{X_{1}}(𝔰_{1}, a). Assume that the 1-cycle Γ is bounded by a 2-cycle Σ in X_{1} with ∂Σ = Γ. Then the energy map is defined as the relative Chern class of [A, ψ] restricted to (Σ, Γ):

\begin{matrix}E_{Σ} :  M_{X_{1}}(𝔰_{1}, a) ⟶ H^{2}(Σ, Γ; ℤ) \\ E_{Σ}([A, ψ]) = ⟨[\frac{\sqrt{−1}}{2π}F_{A}], [Σ]⟩. \end{matrix}  (15)

Denote by M_{X_{1}}^{[n]}(𝔰_{1}, a) the preimage of n ∈ ℤ under the energy map E_{Σ}. Then we can compactify M_{X_{1}}^{[n]}(𝔰_{1}, a) to a smooth manifold with corners. The following proposition can be proved by adapting the convergence and gluing theorems of [11] to the present setting.

Proposition 6.3. Assume that the moduli space M_{X_{1}}(𝔰_{1}, a) is non-empty. Then M_{X_{1}}^{[n]}(𝔰_{1}, a) is empty for n << 0. Moreover, if non-empty, M_{X_{1}}^{[n]}(𝔰_{1}, a) has a compactification to a smooth manifold with corners, with lower dimensional strata of the form

⋃_{a_{1},··· ,a_{k}} M_{X_{1}}^{[n_{0}]}(𝔰_{1}, a_{1}) × M̂^{[n_{1}]}(a_{1}, a_{2}) × ⋯ × M̂^{[n_{k}]}(a_{k}, a).

Here the union is over all possible sequences a_{1}, ⋯ a_{k}, a_{k+1} = a in M_{Y_{0}}(𝔱, Γ) with strictly decreasing indices and with ∑_{j=0}^{k} n_{j} = n.

Proof. From the Weitzenböck formula and the Seiberg–Witten equations, we know that E_{Σ} is bounded from below. Thus, M_{X_{1}}^{[n]}(𝔰_{1}, a) is empty for n << 0. The remaining statements follow from the convergence and gluing theorems of [11].

Denote by 𝔸(X_{1}) the free graded algebra

Sym^{∗}(H_{0}(X_{1})) ⊗ Λ^{∗}(H_{1}(X_{1}))

where the degree of the elements in H_{0}(X_{1}) is 2 and the degree of the elements in H_{1}(X_{1}) is 1. For any monomial z = [x]^{n}γ_{1}∧γ_{2}∧⋯∧γ_{k} in 𝔸(X_{1}), and for each m ∈ ℤ, we need to consider all the components of dimension 2n + k in M_{X_{1}}^{[m]}(𝔰_{1}) = ∪_{a}M_{X_{1}}^{[m]}(𝔰_{1}, a). Suppose that M_{X_{1}}^{[m]}(𝔰_{1}, a_{(2n+k)}) is the union of components of dimension 2n+k and has energy m under the energy map (15). Here the {a_{(2n+k)}} denote all those 3-dimensional monopoles (with perturbations) on (Y_{0}, 𝔱) which satisfy

𝔦_{X_{1}}(a_{(2n+k)}) = 2n + k.

Thus, any non-empty component of M_{X_{1}}(𝔰_{1}, a_{(2n+k)}) has dimension 2n + k. Choose smooth loops which represent 1-cycles γ_{1}, ⋯, γ_{k}. The holonomy map of the [A]-part in [A, ψ] around these k loops defines maps

hol : M_{X_{1}}^{[m]}(𝔰_{1}, a_{(2n+k)}) ↦ U(1)^{k}.

These maps can be extended naturally to the compactification of the moduli spaces M_{X_{1}}^{[m]}(𝔰_{1}, a_{(2n+k)}), as described by Proposition 6.3. The generic fiber of hol defines a smooth submanifold V_{γ_{1},··· ,γ_{k}} of dimension 2n in M_{X_{1}}^{[m]}(𝔰_{1}, a_{(2n+k)}). This submanifold V_{γ_{1},··· ,γ_{k}} can be compactified by the corresponding fiber of hol on the boundary strata of M_{X_{1}}^{[m]}(𝔰_{1}, a_{(2n+k)}).

On M_{X_{1}}^{[m]}(𝔰_{1}, a_{(2n+k)}), there is a canonical U(1) bundle, defined by the based monopole moduli space M̃_{X_{1}}^{[m]}(𝔰_{1}, a_{(2n+k)}), which consists of those solutions representing points in M_{X_{1}}^{[m]}(𝔰_{1}, a_{(2n+k)}) but considered modulo only those gauge transformations which fix the fiber of the spinor bundle over x. A different choice of base point provides an isomorphic U(1) bundle. Therefore, we have an associated complex rank n vector bundle over M_{X_{1}}^{[m]}(𝔰_{1}, a_{(2n+k)})

E_{n}(a_{(2n+k)}) = M̃_{X_{1}}^{[m]}(𝔰_{1}, a_{(2n+k)}) ×_{U(1)} ℂ^{n}.

This vector bundle E_{n}(a_{(2n+k)}) is compatible with the boundary strata of M_{X_{1}}^{[m]}(𝔰_{1}, a_{(2n+k)}) in the sense that E_{n}(a_{(2n+k)}) has a natural extension to the boundary strata of M_{X_{1}}^{[m]}(𝔰_{1}, a_{(2n+k)}), to a vector bundle in the category of manifolds with corners (cf. [15]). Note that E_{n}(a_{(2n+k)}) is isomorphic to ⊕_{i=1}^{n}L(x_{i}), where L(x_{i}) is complex line bundle associated with the U(1)- bundle M̃_{X_{1}}^{[m]}(𝔰_{1}, a_{(2n+k)}) with the base point at x_{i} ∈ X_{1}. By construction, the line bundle L(x_{i}) extends to a line bundle in the category of manifolds with corners over the compactification. Since ⊕ is well defined in the category of manifolds with corners, so does E_{n}(a_{(2n+k)}).

Now consider the restriction of E_{n}(a_{(2n+k)}) to the 2n-dimensional submanifold V_{γ_{1},··· ,γ_{k}}, which is also compatible with its compactification.

Choose a generic transversal section σ_{0}, which is also transversal along the boundary strata, i.e. a strata transverse section of a bundle over a manifold with corners. By the transversality along the boundary strata, we know that all the zeroes of this section in V_{γ_{1},··· ,γ_{k}} lie within a compact set and consist of finitely many points with an orientation. Counting these points gives a number which is denoted by SW_{X_{1}}^{[m]}(𝔰_{1}, z; a_{(2n+k)}). The relative Seiberg–Witten invariant is defined to be

SW_{X_{1}}(𝔰_{1}, z) = ∑_{m∈ℤ,a_{(deg(z))}} SW_{X_{1}}^{[m]}(𝔰_{1}, z; a_{(deg(z))}) < a_{(deg(z))} > t^{m},

as an element in the Seiberg–Witten–Floer complex. It is a routine to prove that this element is in fact a cycle, thus defining an element in the Seiberg– Witten–Floer homology groups HF_{Γ,∗}^{SW} (Y_{0}, 𝔱, ℚ((t))). We also have the analog of Proposition 3.3 in [3], stating that SW_{X_{1}}(𝔰_{1}, z), as an element in HF_{Γ,∗}^{SW} (Y_{0}, 𝔱, ℚ((t))), is independent of the choice of V_{γ_{1},··· ,γ_{k}} and the choice of the strata transversal section of E_{n}. Note that, if we choose another relative 2-cycle Σ in the definition of the energy map (15), the relative invariant SW_{X_{1}}(𝔰_{1}, z) only changes by multiplication by a certain power of t.

Proposition 6.4. SW_{X_{1}}(𝔰_{1}, z) ∈ HF_{Γ,∗}^{SW} (Y_{0}, 𝔱, ℚ((t))) is well-defined, that is, it is independent of the choices of V_{γ_{1},··· ,γ_{k}} and the choice of the strata transversal section σ_{0} of E_{n}.

Proof. For any m ∈ ℤ, let V′_{γ_{1},··· ,γ_{k}} be another 2n-dimensional submanifold in M_{X_{1}}^{[m]}(𝔰_{1}, a_{(2n+k)}) obtained from the construction of the holonomy map and let σ_{1} be a strata transversal section of E_{n} over V′_{γ_{1},··· ,γ_{k}}. We will prove that the difference between #σ_{0}^{−1}(0) and #σ_{1}^{−1}(0) defines an element in C_{Γ,∗}(Y_{0}, 𝔱, ℚ((t))) which is homologous to zero.

Over M_{X_{1}}^{[m]}(𝔰_{1}, a_{(2n+k)}) × [0, 1], we can choose a strata transverse section σ of E_{0}, with σ(·, 0) = σ_{0} and σ(·, 1) = σ_{1}. We also choose a codimension k submanifold V in M_{X_{1}}^{[m]}(𝔰_{1}, a_{(2n+k)}) × [0, 1], whose intersection with M_{X_{1}}^{[m]}(𝔰_{1}, a_{(2n+k)}) × {0} is V_{γ_{1},··· ,γ_{k}}, and whose intersection with M_{X_{1}}^{[m]}(𝔰_{1}, a_{(2n+k)}) × {1} is V′_{γ_{1},··· ,γ_{k}}, respectively. V is also compatible with the boundary strata of M_{X_{1}}^{[m]}(𝔰_{1}, a_{(2n+k)}) × [0, 1].

From this construction, the zero set of σ over V is a 1-manifold with boundary which consists of three parts: (1) the zero set of σ_{0} in V_{γ_{1},··· ,γ_{k}}; (2) the zero set of σ_{1} in V′_{γ_{1},··· ,γ_{k}}; (3) the zero set of σ in the intersection of V with ∂^{1}M_{X_{1}}^{[m]}(𝔰_{1}, a_{(2n+k)}) × [0, 1], where ∂^{1}M_{X_{1}}^{[m]}(𝔰_{1}, a_{(2n+k)}) is the codimension one boundary of M_{X_{1}}^{[m]}(𝔰_{1}, a_{(2n+k)}).

By choosing the base points x in X_{1} away from the cylindrical end for the U(1)-fibration M̃_{X_{1}}^{[m]}(𝔰_{1}, a_{(2n+k)}), and noticing that [x] is the generator of H_{0}(X_{1}, ℤ), we know that the contribution of (3) from the codimension one boundary of M_{X_{1}}^{[m]}(𝔰_{1}, a_{(2n+k)}) times [0, 1] only comes from

⋃_{a_{(2n+k−1)}} ⋃_{m_{1}∈ℤ}M_{X_{1}}^{[m_{1}]}(𝔰_{1}, a_{(2n+k−1)}) × M̂^{[m−m_{1}]}(a_{(2n+k−1)}, a_{(2n+k)})

where a_{(2n+k−1)} runs over all the possible critical points with index 𝔦_{X_{1}}(a_{(2n+k−1)}) = 2n + k − 1. Now the transversality condition over the boundary strata over the intersection of V with M_{X_{1}}^{[m_{1}]}(𝔰_{1}, a_{(2n+k−1)}) × [0, 1] provides a set of finitely many oriented points. Counting these points gives a number, denoted by H_{a_{(2n+k−1)}; m_{1}}. Then the difference of the two choices defines a number

∑_{a_{(2n+k−1)}} H_{a_{(2n+k−1)}; m_{1}}#(M^{[m−m_{1}]}(a_{(2n+k−1)}, a_{(2n+k)})),

thus,

\begin{matrix}∑_{{a_{(2n+k−1)} \atop m∈ℤ;m_{1}∈ℤ}} H_{a_{(2n+k−1)}; m_{1}}#(M^{[m−m_{1}]}(a_{(2n+k−1)}, a_{(2n+k)}))t^{m}a_{(2n+k)} \\ = ∂(∑_{a_{(2n+k−1)};m_{1}∈ℤ} H_{a_{(2n+k−1)}; m_{1}}t^{m_{1}}a_{(2n+k−1)}), \end{matrix}

is homologous to zero. This proves that the two choices define the same homology class in HF_{Γ,∗}^{SW} (Y_{0}, 𝔱, ℚ((t))).

Consider a closed 4-manifold (X, 𝔰) with b_{2}^{+} ≥ 1 and a Spin^{c} structure 𝔰. Suppose that X has a decomposition X_{1} ∪_{Y_{0}} X_{2} along a 3-dimensional submanifold Y_{0} with b_{1}(Y_{0}) > 0, and with a Spin^{c} structure 𝔰|_{Y_{0}} = 𝔱. We assume that the class c_{1}(𝔰|_{Y_{0}}) = c_{1}(𝔱) is a torsion element in H^{2}(Y_{0}, ℤ). Here we need to choose a metric g_{T} on X such that X contains a very long ‘neck’ Y_{0} × [−T, T]. For b_{2}^{+} = 1, when the Seiberg–Witten invariant has a chamber structure, our choice of the perturbation compatible with Γ gives rise to a fixed chamber in the space of metrics and perturbations. Notice that, since we are dealing with the case of a torsion element c_{1}(𝔱), in the case where b_{1}(Y_{0}) = 1 the chamber also depends on the class [∗ρ] in H^{2}(Y_{0}, ℝ) Now we can state the gluing formulae of the Seiberg–Witten invariants on X as follows.

Theorem 6.5. Let X be a closed manifold with b^{+} ≥ 1 which is written as X = X_{1} ∪_{Y_{0}} X_{2}, with X_{1} and X_{2} two 4-manifolds with boundary, and with ∂X_{1} = −∂X_{2} = Y_{0}. Suppose that we have Spin^{c} structures 𝔰_{1} and 𝔰_{2}, on X_{1} and X_{2} respectively, such that 𝔰_{1}|_{Y_{0}} ≅ 𝔰_{2}|_{Y_{0}} ≅ 𝔱 is a torsion Spin^{c} structure on Y_{0}. Suppose given 1-cycle Γ in Y_{0} bounds two relative 2-cycles Σ_{i} in X_{i} respectively. Then the relative Seiberg-Witten invariants SW_{X_{1}}(𝔰_{1}) and SW_{X_{2}}(𝔰_{1}) take values in HF_{Γ,∗}^{SW} (Y_{0}, 𝔱, ℚ((t))) and HF_{−Γ,∗}^{SW}(−Y, −𝔱, ℚ((t))) respectively. Let Σ = Σ_{1} + Σ_{2} ∈ H_{2}(X, ℤ), under the natural pairing (13), we have the following gluing formula for z_{i} ∈ A(X_{i}), i = 1, 2,

∑_{{𝔰:𝔰|_{X_{i}}=𝔰_{i}}} SW_{X,𝔰}(z_{1}z_{2})t^{c_{1}(𝔰)·Σ} = ⟨SW_{X_{1}}(𝔰_{1}, z_{1}), SW_{X_{2}}(𝔰_{2}, z_{2})⟩.

When b^{+} = 1, the Seiberg–Witten invariants correspond to a fixed chamber determined by the perturbation which is compatible with Γ.

Proof. Let i_{k} denote the boundary embedding maps of Y_{0} into X_{k}. Then the set of Spin^{c}-structures on X which agree with 𝔰_{i} over X_{i}, denoted by Spin^{c}(X; 𝔰_{1}, 𝔰_{2}), form an affine space over

H^{1}(Y_{0}, ℤ)/(Imi_{1}^{∗} + Imi_{2}^{∗}),

with i_{1}^{∗} : H^{1}(X_{1}, ℤ) → H^{1}(Y_{0}, ℤ) and i_{2}^{∗} : H^{1}(X_{2}, ℤ) → H^{1}(Y_{0}, ℤ). For any 𝔰 ∈ Spin^{c}(X; 𝔰_{1}, 𝔰_{2}), after a generic perturbation, the moduli space M_{X}(𝔰) is an oriented, compact, smooth manifold with dimension given by

\frac{1}{4}(c_{1}(𝔰)^{2} − (2χ + 3σ)) = d_{X}(𝔰) ≥ 0.

The Seiberg–Witten invariant is a linear functional

SW_{X}(𝔰, ·) : 𝔸(X) ⟶ ℤ

where 𝔸(X) = Sym^{∗}(H_{0}(X)) ⊗ Λ^{∗}(H_{1}(X)). We summarize this definition briefly. We shall only consider monomials z = [x]^{n}γ_{1} ∧ ⋯ ∧ γ_{k} ∈ 𝔸(X) with 2n + k = d_{X}(𝔰). For any other degree of z, we assign SW_{X}(𝔰, z) to be zero. For any monomial as above, we can choose smooth 1-dimensional submanifolds representing γ_{1}, ⋯, γ_{k}. Then the holonomy along these loops defines a map: M_{X}(𝔰) ↦ U(1)^{k}, whose generic fiber is a closed codimension k submanifold V_{γ_{1},··· ,γ_{k}} in M_{X}(𝔰). The based monopoles define a U(1)-fiber bundle M̃_{X}(𝔰) over M_{X}(𝔰). Then SW_{X}(𝔰, z) is defined to be the result of integration of the n^{th} power of the first Chern class of this U(1) bundle over V_{γ_{1},··· ,γ_{k}}. Equivalently, we can consider the associated rank n complex vector bundle E over the 2n-dimensional manifold V_{γ_{1},··· ,γ_{k}}. Then SW_{X}(𝔰, z) is obtained by counting points with the orientation in the zero set of a generic strata transverse section of E over V_{γ_{1},··· ,γ_{k}}.

We can adapt the gluing theory for Seiberg–Witten monopoles developed in [11] to the setting of [6] and [19]. Thus, when T is sufficiently large, we obtain a gluing theorem for Seiberg–Witten monopoles, which identifies the 4-dimensional monopoles on X with the following product:

⋃_{𝔰∈Spin^{c}(X,𝔰_{1},𝔰_{2})} M_{X}(𝔰) ≅ ⋃_{a} M_{X_{1}}(𝔰_{1}, a) × M_{X_{2}}(𝔰_{2}, a).  (16)

Notice that, for [A, Ψ] ∈ M_{X}(𝔰), we write [A, Ψ] as [A_{1}, Ψ_{1}]#^{T} [A_{2}, Ψ_{2}] under the gluing map (16). We have

E_{Σ_{1}}([A_{1}, Ψ_{1}]) + E_{Σ_{2}}([A_{2}, Ψ_{2}]) = c_{1}(𝔰) · Σ.

Now the gluing formulae follow from the definition of the relative invariants SW_{X_{1}}(𝔰_{1}, z_{1}) and SW_{X_{2}}(𝔰_{2}, z_{2}) for z_{i} ∈ 𝔸(X_{i}), with values in HF_{Γ,∗}^{SW} (Y_{0}, 𝔱, ℚ((t))) and HF_{−Γ,∗}^{SW}(−Y_{0}, −𝔱) respectively.

Note that, in the case of b_{2}^{+}(X) = 1, the choice of the 1-cycle Γ fixes the choice of the chamber. In fact, the moduli space M_{Y_{0}}(𝔱, Γ), with the perturbation [∗ρ] = ηP D(Γ), contains no reducibles. Thus, in particular, since the relative invariants take values in HF_{Γ,∗}^{SW} (Y_{0}, 𝔱, ℚ((t))), once Γ is fixed, there are no wall crossing terms for SW_{X,𝔰} when we change the perturbation and the metric.

We conclude with the following example.

Example 6.6. ([17]) Let (Y_{0}, 𝔰_{0}) be the 3-manifold Σ_{g} × S^{1} with the trivial Spin^{c} structure 𝔰_{0}, where Σ_{g} is a closed Riemann surface of genus g. Then the gluing formulae in Theorem 6.5 for 4-manifold Σ_{g} × S^{2} can be applied to study HF_{Γ,∗}^{SW} (Σ_{g} × S^{1}, 𝔰_{0}; ℚ((t))) and its ring structure. Here we have Γ = [pt × S^{1}], with the surface Σ of Theorem 6.5 corresponding to pt × S^{2}. It turns out that, as a Sp(2g, ℤ)-equivariant vector space, we have

HF_{Γ,∗}^{SW} (Σ_{g} × S^{1}, 𝔰_{0}; ℚ((t))) ≅ H_{∗}(Sym^{g−1}(Σ), ℚ) ⊗ ℚ((t)).

In particular, when Σ_{g} is a torus, then

HF_{Γ,∗}^{SW} (T^{2} × S^{1}, 𝔰_{0}; ℚ((t))) ≅ ℚ((t)).

Upon choosing the metric on T^{2} × S^{2} such that it has a long neck around T^{2} × S^{1}, the Seiberg-Witten invariant is computed as the wall-crossing term from the chamber corresponding to the unperturbed equations for the positive scalar curvature which has trivial Seiberg-Witten invariant, and the other chamber determined by the choice of Γ, then we obtain the following result from the gluing formulae:

⟨SW_{T^{2}×D^{2}}(𝔰_{0}, 1), SW_{T^{2}×D^{2}}(𝔰_{0}, 1)⟩
= ∑_{n≥1} SW_{T^{2}×S^{2},2nP D([T^{2}])}(1)t^{2n}
= ∑_{n≥1} nt^{2n}.

Thus, up to sign, we can express the relative Seiberg–Witten invariant of T^{2} × D^{2} with the trivial Spin^{c} structure as

SW_{T^{2}×D^{2}}(𝔰_{0}, 1) = \frac{1}{t − t^{−1}} .

For z ∈ 𝔸(T^{2} × D^{2}) of non-zero degree, we have SW_{T^{2}×D^{2}}(𝔰_{0}, z) = 0.

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Matilde Marcolli, Department of Mathematics, Massachusetts Institute of Technology, 2-275. Cambridge, MA 02139, USA Max-Planck-Institut für Mathematik, Vivatsgasse 7, D-53111 Bonn, Germany. matilde@math.mit.edu, marcolli@mpim-bonn.mpg.de

Bai-Ling Wang, Department of Pure Mathematics, University of Adelaide, Adelaide SA 5005, Australia. bwang@maths.adelaide.edu.au Max-Planck-Institut für Mathematik, Vivatsgasse 7, D-53111 Bonn, Germany. bwang@mpim-bonn.mpg.de