Exact triangles in Seiberg-Witten-Floer theory. Part III: proof of exactness

Matilde Marcolli, Bai-Ling Wang

Contents

1 Introduction 2

2 The cobordisms 5 2.1 Splitting of the cobordisms . . . . . . . . . . . . . . . . . . . 8 2.2 Metrics and perturbations on the cobordism . . . . . . . . . . 9 2.3 Finite energy monopoles and virtual dimension . . . . . . . . 15 2.4 Compactification and invariants . . . . . . . . . . . . . . . . . 18

3 Geometric limits 20 3.1 The holomorphic triangles . . . . . . . . . . . . . . . . . . . . 23

4 The chain homomorphisms 24 4.1 The composite map . . . . . . . . . . . . . . . . . . . . . . . 28

5 Injectivity and Surjectivity 30

6 Exactness in the middle term 35 6.1 The moduli space on V × ℝ . . . . . . . . . . . . . . . . . . . 36 6.2 Admissible elements in M̂_{V ×ℝ}(a_{∞}) . . . . . . . . . . . . . . . 41 6.3 Assembling the geometric limits . . . . . . . . . . . . . . . . . 43 6.4 The relation w_{∗}^{0} ∘ w_{∗}^{1} = 0 . . . . . . . . . . . . . . . . . . . . . 51 6.5 The inclusion Ker(w_{∗}^{0}) ⊂ Im(w_{∗}^{1}) . . . . . . . . . . . . . . . . 56

7 The connecting homomorphism 60 7.1 The surgery triangle . . . . . . . . . . . . . . . . . . . . . . . 62

1 Introduction

In the previous parts of this work [6], [17], we considered a homology 3sphere Y and an embedded knot K. We considered manifolds Y_{1} and Y_{0} obtained by 1-surgery and 0-surgery on K, respectively. We proved that the oriented moduli space of solutions of the Seiberg–Witten equations on Y can be described as the union

M_{Y,μ} ≅ M_{Y_{1}} ∪ ⋃_{𝔰_{k}}M_{Y_{0}}(𝔰_{k}),  (1)

where μ is a suitable perturbation that simulates the effect of surgery, and the 𝔰_{k} are the Spin^{c} structures on Y_{0} that reduce to the trivial structure when restricted on the tubular neighbourhood of the knot ν(K) and on the knot complement V. The irreducible components are similarly related

M_{Y,μ}^{∗} ≅ M_{Y_{1}}^{∗} ∪ ⋃_{𝔰_{k}}M_{Y_{0}}(𝔰_{k}),

as shown in [6].

We also proved that there is a way of assigning compatible choices of the relative grading, so that (1) gives an exact sequence of abelian groups

0 → C_{q}(Y_{1}) →^{j_{q}} C_{q}(Y, μ) →^{π_{q}} ⨁_{𝔰_{k}}C_{(q)}(Y_{0}, 𝔰_{k}) → 0.  (2)

Recall that the grading of C_{(q)}(Y_{0}, 𝔰_{k}) denotes a lifting to a ℤ-graded complex of the ℤ_{2i_{k}}-graded complex. This lifting is determined by a compatible choice of grading on (1). This ℤ-graded Floer complex is analyzed in detail in [18]. The maps j_{∗} and π_{∗} are induced by the inclusion

j : M_{Y_{1}} ↪ M_{Y,μ},

and the projection on the quotient, generated by the elements of

∪_{𝔰_{k}}M_{Y_{0}}(𝔰_{k}).

The analysis of the splitting and gluing of the moduli spaces of flow lines in [17] shows that, in general, one should not expect (2) to be an exact sequence of chain complexes: the maps j_{∗} and π_{∗} need not commute with the boundary operators of the Floer complexes. Thus, on the algebraic point of view, the existence of the decomposition (1) simply signifies that the existence of an exact sequence is possible: in fact, the decomposition (2) shows that the ranks are compatible with the existence of the desired exact sequence. The maps that provide the exact sequence are derived from the surgery cobordisms.

In this paper we introduce maps w_{∗}^{1} and w_{∗}^{0}, induced by the surgery cobordisms W_{1} and W_{0} connecting Y_{1} and Y, and Y and Y_{0}, respectively. The maps w_{∗}^{1} and w_{∗}^{0} are defined by a suitable choice of the Spin^{c} structures 𝔰_{ℓ} and 𝔰_{k} on the 4-manifolds W_{1} and W_{0}, that restrict to the assigned Spin^{c} structure at the two ends of the cobordism.

We show that w_{∗}^{1} and w_{∗}^{0} are chain homomorphisms,

C_{q}(Y_{1}) →^{w_{q}^{1}} C_{q}(Y, μ)  (3)

and

C_{q}(Y, μ) →^{w_{q}^{0}} ⨁_{𝔰_{k}}C_{(q)}(Y_{0}, 𝔰_{k}).  (4)

We then show that the map w_{∗}^{0} is surjective and the map w_{∗}^{1} is injective. The main technique is similar to the technique developed in [17] in order to study the behavior of flow lines under the splitting Y = V ∪_{T^{2}} ν(K). Here we consider punctured surgery cobordisms W_{1}\{x_{1}} and W_{0}\{x_{0}} and we stretch product regions T^{2} × [−r, r] × ℝ inside these punctured cobordisms, thus also stretching product regions T^{2} × [−r, r] × [0, ∞) in the S^{3} × [0, ∞) end near the puncture x_{i}. We have geometric limits and a gluing theorem as in [17] and we can show that, when the parameter ϵ in the surgery perturbation is small enough, the solutions on W_{1} and W_{0} have the following behavior. If we consider solutions on W_{1} with asymptotic values a_{1} in M_{Y_{1}} and j(a_{1}) in j(M_{Y_{1}}) ⊂ M_{Y,μ}, and solutions on W_{0} with asymptotic values a ∈ M_{Y,μ}\j(M_{Y_{1}}) and π(a) in ∪_{k}M_{Y_{0}}(𝔰_{k}), then the corresponding components of the maps w_{∗}^{1} and w_{∗}^{0} agree with those of the maps j_{∗} and π_{∗} in (2). In other words, we prove the following relations:

⟨j(a′_{1}), w_{∗}^{1}(a_{1})⟩ = δ_{a′_{1},a_{1}},  (5)

for all a′_{1} and a_{1} in M_{Y_{1}}, and

⟨a_{0}, w_{∗}^{0}(a)⟩ = δ_{a_{0},π(a)},  (6)

for all a ∈ M_{Y,μ}\j(M_{Y_{1}}) and for all a_{0} ∈ ∪_{k}M_{Y_{0}}(𝔰_{k}).

We then prove the relation w_{∗}^{0} ∘ w_{∗}^{1} = 0. Again, we follow the same technique. We compare the geometric limits of zero-dimensional moduli spaces

M_{ℓ}^{W_{1}}(a_{1}, a)

and

M_{k}^{W_{0}}(j(a_{1}), π(a)),

for a ∈ M_{Y,μ}\j(M_{Y_{1}}), regarded as subsets of the moduli space

M_{ℓ,k}^{W}(a_{1}, π(a))

on the composite cobordism W = W_{1} ∪_{Y} W_{0}. We obtain an orientation reversing diffeomorphism

M_{ℓ}^{W_{1}}(a_{1}, a) ≅ M_{k}^{W_{0}}(j(a_{1}), π(a)),

which proves the relation w_{∗}^{0} ∘ w_{∗}^{1} = 0. The main technique consists of identifying the moduli spaces of solutions on the cobordisms with certain pre-gluing data obtained out of the explicit description of the geometric limits developed in Part II [17]. The set of pre-gluing data and the resulting moduli space can be identified up to a diffeomorphism given by the gluing map. The set of pre-gluing data consists of moduli spaces of finite energy monopoles on V ×ℝ together with holomorphic triangles in a covering of the character variety χ(T^{2}) of flat U(1)-connections on T^{2}, with boundary along arcs of Lagrangians ℓ, ℓ_{1}, ℓ_{μ}^{∗} determined by the asymptotic values ∂_{∞}M_{V}^{∗}, and by the flat connections on ν(K), with or without surgery perturbation. In order to use dimensional arguments, we compare the formulae for the Maslov index in the splitting of the spectral flow, with the formulae for the dimension of the moduli space of such holomorphic triangles.

Then we can complete the proof of the exactness of the sequence

0 → C_{q}(Y_{1}) →^{w_{q}^{1}} C_{q}(Y, μ) →^{w_{q}^{0}} ⨁_{𝔰_{k}}C_{(q)}(Y_{0}, 𝔰_{k}) → 0.  (7)

It is enough to show that (5) and (6), together with the inclusion Im(w_{∗}^{1}) ⊂ Ker(w_{∗}^{0}) determine enough relations among the coefficients of the chain maps that force the reverse inclusion Ker(w_{∗}^{0}) ⊂ Im(w_{∗}^{1}) to hold as well.

In the last Section we analyze the connecting homomorphism in the long exact sequence

⋯ → HF_{q}(Y_{1}) → HF_{q}(Y, μ) → ⊕_{k}HF_{(q)}(Y_{0}, 𝔰_{k}) →^{Δ_{q}} HF_{q−1}(Y_{1}) → · · · .

We first show that the coefficients of Δ are given by the component of the boundary on Y that counts flow lines connecting critical points of M_{Y,μ}\j(M_{Y_{1}}) to critical points of j(M_{Y_{1}}), of relative index one. We then proceed to identify this counting with the counting of zero dimensional moduli spaces on another cobordism W̄_{2} connecting Y_{0} and Y_{1}, satisfying the relation

W̄ = W̄_{2}#ℂP^{2},

where W̄ is the composite cobordism W̄ = W̄_{0} ∪_{Y} W̄_{1}. Thus, we obtain the result that the exact triangle for Seiberg-Witten Floer homology is a surgery triangle, that is, the connecting homomorphism in the exact sequence is determined by a chain map w̄_{∗}^{2} induced by the surgery cobordism W̄_{2}, and the resulting diagram

C_{∗}(Y_{1}) →^{w_{∗}^{1}} C_{∗}(Y, μ) →^{w_{∗}^{0}} ⊕_{k}C_{(∗)}(Y_{0}, 𝔰_{k}) →^{w̄_{∗}^{2}} C_{∗}(Y_{1})[−1]

is a distinguished triangle. Acknowledgments. The first author is partially supported by NSF grant DMS-9802480. The second author is partially supported by ARC Fellowship. We thank the Max–Planck–Institut für Mathematik, where a large part of the work was done.

2 The cobordisms

We describe briefly the topology of the cobordisms and then introduce the appropriate perturbed Seiberg–Witten equations, and the corresponding moduli spaces.

The cobordism W_{1} is obtained by removing from the trivial cobordism Y_{1} × I an S^{1} × D ≅ ν(K) × {1}, where D is a disk, and ν(K) is the tubular neighbourhood of the knot in Y_{1}, and then attaching a 2-handle with framing −1. We denote by D_{1} the core disk of the 2-handle in W_{1}. Similarly, the cobordism W_{0} is obtained by removing from the trivial cobordism Y_{0} × I an S^{1}×D ≅ ν(K)×{0} and attaching a 2-handle with framing zero. We denote by D_{0} the core disk of the 2-handle in W_{0}. Attaching the two-handle has the effect of modifying the boundary component Y_{1}×{1} in the trivial cobordism to the boundary component Y × {1} in the non-trivial cobordism W_{1}, or, respectively, the boundary component Y_{0} × {0} in the trivial cobordism to the boundary component Y × {0} in W_{0}.

Lemma 2.1 The cobordisms W_{1} and W_{0} have the following topology:

\begin{matrix}b_{1}(W_{1}) = 0 & b_{2}(W_{1}) = 1 & b_{2}^{+}(W_{1}) = 0 & b_{2}^{−}(W_{1}) = 1 \\ b_{1}(W_{0}) = 0 & b_{2}(W_{0}) = 1 & b_{2}^{+}(W_{0}) = 0 & b_{2}^{−}(W_{0}) = 0. \end{matrix}

The composite cobordism W = W_{0} ∪_{Y} W_{1}, connecting Y_{1} and Y_{0}, can be written as a blow up W = W_{2}#\overline{ℂP^{2}}, where W_{2} satisfies

b_{1}(W_{2}) = 0 b_{2}(W_{2}) = 1 b_{2}^{+}(W_{2}) = 0 b_{2}^{−}(W_{2}) = 0.

Proof. Let σ be a Seifert surface for the knot K in the homology sphere Y, ∂σ = K. In the cobordism W_{1} consider the surface Σ_{1} obtained by attaching the Seifert surface and the core disk D_{1} along the knot K,

Σ_{1} = σ ∪_{K} D_{1}.

The homology H_{2}(W_{1}, ℤ) is generated by the class [Σ_{1}] with self intersection −1. Similarly, consider the surface Σ_{0} in W_{0}, obtained by attaching along K the Seifert surface and the core disk D_{0},

Σ_{0} = σ ∪_{K} D_{0}.

The homology H_{2}(W_{0}, ℤ) is generated by the class [Σ_{0}] with self intersection zero. The surface

Σ = D_{1} ∪_{K} \overline{D_{0}}

in the composite cobordism W has self intersection −1. The homology H_{2}(W, ℤ) is generated by [Σ] = [Σ_{1}]−[Σ_{0}] and [Σ_{0}]. The class [Σ] represents the exceptional divisor E in the blowup. Thus, the blown down cobordism W_{2} has homology H_{2}(W_{2}, ℤ) generated by a class, which we still write [Σ_{0}], with self intersection zero. ⋄

Notice that the surface Σ_{0} is homologous in W_{0} to the generator σ ∪_{K} D of H_{2}(Y_{0}, ℤ). The following simple calculation is useful in classifying the possible Spin^{c} structures on the cobordisms.

Lemma 2.2 We have H_{2}(W_{0}, Y ∪ Y_{0}, ℤ) = ℤ generated by a surface Σ̃_{0} in W_{0} with ∂Σ̃_{0} = γ, with [γ] the generator of H_{1}(Y_{0}, ℤ). The class [Σ_{0}] introduced above, which generates H_{2}(W_{0}, ℤ) is trivial in H_{2}(W_{0}, Y ∪ Y_{0}).

Moreover, we have H_{2}(W_{1}, Y_{1} ∪ Y, ℤ) = ℤ generated by the image of the class [Σ_{1}] which generates H_{2}(W_{1}, ℤ), and H_{2}(W, Y_{1} ∪ Y_{0}, ℤ) = ℤ ⊕ ℤ, generated by the class E of the exceptional divisor of the blow-up and by [Σ̃_{0}]. Again, the class [Σ_{0}], which is non-trivial in H_{2}(W, ℤ), is mapped trivially to H_{2}(W, Y_{1} ∪ Y_{0}, ℤ). Finally, we have H_{2}(W_{2}, Y_{1} ∪ Y_{0}, ℤ) = ℤ generated by [Σ̃_{0}].

Proof. The results simply follow from the exact sequence in homology. We have

H_{3}(W_{0}, Y ∪ Y_{0}, ℤ) ≅ H_{1}(W_{0}) = 0,

hence we have

0 → H_{2}(Y_{0}, ℤ) ⟶^{≅} H_{2}(W_{0}, ℤ) ⟶^{0} H_{2}(W_{0}, Y ∪ Y_{0}, ℤ) ⟶ H_{1}(Y_{0}, ℤ) → 0.

Similarly, we have

0 → H_{2}(Y_{0}, ℤ) ⟶ H_{2}(W, ℤ) ⟶ H_{2}(W, Y_{1} ∪ Y_{0}, ℤ) ⟶ H_{1}(Y_{0}, ℤ) → 0

which is of the form

0 → ℤ → ℤ ⊕ ℤ ⟶^{0⊕I} ℤ ⊕ ℤ → ℤ → 0.

⋄ Thus, we obtain the following result.

Lemma 2.3 We have an identification of the Spin^{c}-structures on W_{0} and W_{1} given by

S(W_{0}) = {𝔰_{k}}_{k∈ℤ},

with

𝔰_{k} ↦ c_{1}(det W_{𝔰_{k}}^{+}) = 2k e_{0} = 2kP D_{W_{0}}[Σ̃_{0}],

with [Σ̃_{0}] the generator of H_{2}(W_{0}, Y ∪ Y_{0}, ℤ). Similarly, we have

S(W_{1}) = {𝔰_{ℓ}}_{ℓ∈ℤ},

with

𝔰_{ℓ} ↦ c_{1}(det W_{𝔰_{ℓ}}^{+}) = (2ℓ + 1) e_{1} = (2ℓ + 1)P D_{W_{1}}[Σ_{1}],

with [Σ_{1}] the generator of H_{2}(W_{1}, Y_{1} ∪ Y, ℤ). Moreover, we have

S(W) = {𝔰_{ℓ,k}}_{ℓ,k∈ℤ},

with

𝔰_{ℓ,k} ↦ c_{1}(det W_{𝔰_{ℓ,k}}^{+}) = (2ℓ + 1) e + 2k e_{0},

where we have e = P D_{W} (E), and E is the image of the class [Σ_{1} − Σ_{0}] in H_{2}(W, Y_{1} ∪ Y, ℤ). Similarly, we have

S(W_{2}) = {𝔰_{k}}_{k∈ℤ}.

Proof. The result follows from the previous Lemma with the additional observation that in W_{1} we have Σ_{1} with self intersection Σ_{1}^{2} = −1, hence the Spin^{c} structures have odd Chern class (2ℓ + 1)P D_{W_{1}}[Σ_{1}]. ⋄

In the case of W_{0}, W_{2}, and W, the Spin^{c} structure 𝔰_{k} or 𝔰_{ℓ,k} restricts to the end Y_{0} × [T_{0}, ∞) of the cobordism to the pullback of the Spin^{c} structure 𝔰_{k} in S(W_{0}). All the Spin^{c} structures of Lemma 2.3 restrict to the trivial Spin^{c} structure on the cylindrical ends modeled on Y or Y_{1}.

2.1 Splitting of the cobordisms

In the following, we shall introduce moduli spaces of Seiberg-Witten equations on the cobordisms. Our purpose is to apply to the moduli spaces on the cobordisms the same techniques we developed in [17], in the study of the moduli spaces of flowlines, that is, of monopoles on the trivial cobordisms. Thus, it is convenient to consider the manifolds W_{1} and W_{0} endowed with infinite cylindrical ends Y_{1} × (−∞, −T_{0}] and Y × [T_{0}, ∞), and Y_{0} × [T_{0}, ∞) and Y ×(−∞, −T_{0}], respectively, with metrics g_{Y} +dt^{2} and g_{Y_{i}} +dt^{2}. Moreover, we shall assume that the 3-manifolds Y_{1}, Y, and Y_{0} are endowed with metrics with a long cylinder T^{2} × [−r, r], as specified in [6].

We can then think of the cobordisms as endowed with a metric which restricts to the flat product metric on the region T^{2} × [−r, r] × ℝ. Moreover, we can identify in the cobordisms W_{i} a product region V × ℝ, on the complement of a tubular neighbourhood of the knot, where the cobordism is trivial. Thus, we can decompose the cobordisms W_{i} as

W_{i} = V × ℝ ∪_{T^{2}×ℝ} T^{2} × [−r, r] × ℝ ∪_{T^{2}×ℝ} W_{i}(ν(K)).  (8)

The non-compact region W_{i}(ν(K)) has the following property. There is a compact set K in W_{i} such that the intersection K ∩W_{i}(ν(K)) is obtained by attaching a 2-handle D × D to the product ν(K) × [−T_{0}, T_{0}], and, outside of K, the region K^{c} ∩ W_{i}(ν(K)) consists of product regions ν(K) × [T_{0}, ∞) and ν(K) × (−∞, −T_{0}], and T^{2} × [r_{0}, r] × [−T_{0}, T_{0}].

In the cobordism W_{i} consider an interior point x_{i} contained in the core disk of the 2-handle, x_{i} ∈ D_{i}. As in [2], we denote by Ŵ_{i} the punctured cobordism Ŵ_{i} = W_{i}\{x_{i}}. Similarly, we can consider the punctured manifold

Ŵ_{i}(ν(K)) = W_{i}(ν(K))\{x_{i}}.

In the manifolds Ŵ_{i}(ν(K)) we can identify a product region

V = ν(K)_{r_{0}} × ℝ ≅ D × (D_{i}\{x_{i}}).  (9)

This corresponds to endowing the manifold Ŵ_{i} with an extra asymptotic end of the form S^{3} × [0, ∞) at the puncture. Thus, we identify the manifold W_{i} with a connected sum

W_{i} = Ŵ_{i}#Q_{i},

with a long cylindrical neck S^{3} × [−T (r), T (r)], and with Q_{i} a 4-ball, as in [2].

Consider the sphere S^{3} decomposed as the union of two solid tori in the standard way, S^{3} = ν ∪ ν̃, with ν ≅ ν̃ ≅ D × S^{1}. Then the product region V of (9) in W_{i} identifies the standard solid torus ν in S^{3} with the neighborhood ν(K) of the knot K in Y, and, similarly, the other solid torus ν̃ in S^{3} is identified with the tubular neighbourhood ν(K) in Y_{i}, after the surgery. This is illustrated in Figure 1.

2.2 Metrics and perturbations on the cobordism

The results of this subsection are based on the metric deformation Lemma that Liviu Nicolaescu kindly communicated to us, [23], and that we enclosed in Part I [6].

Lemma 2.4 Let A be an element in SL(2, ℤ). Suppose given ϵ > 0 sufficiently small. Consider the metric on T^{2} given by

g_{0} = A^{∗}g,

where g is the standard flat metric as before. There exists a constant δ > 0 and a smooth path g(s) of flat metrics on T^{2} with the following properties:

(i) g(s) ≡ \frac{1}{δ^{2}} g_{0}, for all s ≤ ϵ;

(ii) g_{1} = g(1) is a metric of the form

g_{1} = k_{1}du^{2} + k_{2}dv^{2}

Figure 1: The decomposition of the punctured cobordism Ŵ_{1}

for k_{i} positive constants; (iii) g(s) = g_{1} for all s ≥ 1 − ϵ; (iv) The scalar curvature of the metric ĝ = ds^{2} + g(s) on T^{2} × ℝ is

non-negative.

(v) The metric g_{1} can be extended to a metric inside the solid torus ν(K), which we still denote g_{1}, that has non-negative scalar curvature.

The constant δ is given by δ^{2} = g_{0}(∂_{u}, ∂_{u}).

Using this result, when we construct the manifold Y_{1} from Y, by removing a tubular neighborhood ν(K) and gluing it back along T^{2} with the matrix A ∈ SL(2, ℤ) prescribed by the surgery, we can consider the same metric on the knot complement V, with an end isometric to T^{2}×[0, ∞), with the metric g + ds^{2}. On the other hand, on ν(K) we can consider the metric δ^{2}(g(s) + ds^{2}), with g(s) constructed as above, with the parameterization chosen in such a way that we have

g(s) ≡ \frac{1}{δ^{2}} A^{∗}g

on the end T^{2} × [δ^{−2}r_{0}, ∞) and g(s) ≡ g_{1} near T^{2} × {0}, extended to a positive scalar curvature metric inside the solid torus.

With this choice of metrics, we still have the decomposition of the moduli spaces of critical points of the CSD functional, as proved in Part I, [6]. Moreover, this particular choice of metrics allows us to describe the choice of metrics on the cobordisms.

Consider first the trivial cobordism Y_{1} × ℝ. In the limit r → ∞ this splits as V × ℝ, with an end of the form T^{2} × ℝ^{+} × ℝ with metric g + ds^{2} + dt^{2}, and ν(K) × ℝ, with an end of the form T^{2} × ℝ^{+} × ℝ with the metric δ^{2}(g(s)+ds^{2})+dt^{2}, as described above. Now consider the punctured cobordism Ŵ_{1}. This contains a product region ν×ℝ which connects the solid torus ν(K) ⊂ Y with a solid torus ν ⊂ S^{3} at the puncture. On this product region we consider the metric G := g + δ^{2}ds^{2} + dt^{2}, with g the standard flat metric on T^{2} extended to a non-negative scalar curvature metric inside the solid torus as described in [6]. The other product region ν̃ × ℝ connecting the solid torus ν(K) ⊂ Y_{1} with a solid torus ν̃ inside S^{3}, is glued in the punctured 2-handle, along a region T^{2} × ℝ, with framing one. Thus, on this product region we can consider the metric G(s) := δ^{2}(g(s) + ds^{2}) + dt^{2} described above. These regions are illustrated schematically in Figure 2, with a lower dimensional picture of the punctured handle.

Our purpose is to define chain maps between the Floer complexes of the 3-manifolds using the Seiberg–Witten equations on the cobordisms, and to adapt the techniques of [17] to analyze these chain maps, by understanding their asymptotic limits under the splitting of the cobordisms illustrated in the previous section. Thus, we need to introduce a suitable perturbation of the Seiberg–Witten equations on W_{i} which is compatible with the perturbations of the Chern–Simons–Dirac functional on the manifolds Y_{i} and Y, described in [6].

Recall that on Y_{1} we have perturbed flow equations of the form

\{ \begin{matrix}\frac{∂A}{∂t} & = & − ∗ F_{A} + σ(ψ, ψ) + ∑_{j=1}^{N} \frac{∂U_{1}}{∂τ_{j}} μ_{j}^{(1)} \\ \frac{∂ψ}{∂t} & = & −∂_{A}ψ − ∑_{i=1}^{K} \frac{∂V_{1}}{∂ζ_{i}} ν_{i}^{(1)}.ψ, \end{matrix}  (10)

where (U_{1}, V_{1}) is a pair of functions in the class P_{δ} described in [6], that is, it becomes exponentially small along the cylinder T^{2} × [−r, r] inside Y_{1}, and is exponentially small on the solid torus ν(K) ⊂ Y_{1}. All the notation we use here follows [6].

Figure 2: Product regions and metrics on the punctured 2-handle

The equations (10) can be written equivalently as the four dimensional equations on Y_{1} × ℝ,

\{ \begin{matrix}F_{A}^{+} = τ(Ψ, Ψ) + P_{1}^{Y_{1}}(A, Ψ) \\ D_{A}Ψ = P_{2}^{Y_{1}}(A, Ψ), \end{matrix}  (11)

with

P_{1}^{Y_{1}}(A, Ψ) = ∑_{j=1}^{N} \frac{∂U_{1}}{∂τ_{j}} (μ_{j}^{(1)} ∧ dt + ∗_{3}μ_{j}^{(1)}),  (12)

P_{2}^{Y_{1}}(A, Ψ) = ∑_{i=1}^{K} \frac{∂V_{1}}{∂ζ_{i}} ν_{i}^{(1)}.Ψ.  (13)

As in [6], we consider similarly perturbed equations on Y, with the additional surgery perturbation on the solid torus ν(K) ⊂ Y. Thus, on Y we have flow equations

\{ \begin{matrix}\frac{∂A}{∂t} & = & − ∗ F_{A} + σ(ψ, ψ) + ∑_{j=1}^{N} \frac{∂U}{∂τ_{j}} μ_{j} + f′(T_{A(t)}) ∗ μ \\ \frac{∂ψ}{∂t} & = & −∂_{A}ψ − ∑_{i=1}^{K} \frac{∂V}{∂ζ_{i}} ν_{i}.ψ, \end{matrix}  (14)

with

T_{A}(z) = −i ∫_{{z∈D^{2}}×S^{1}}(A − A_{0}).

Here the function f′, which depends on the choice of a small parameter 0 < ϵ ≤ ϵ_{0}, is constructed as in [6], μ is a compactly supported 2-form, and (U, V) is a perturbation in the class P_{δ} on Y.

The equation (14) can be written equivalently as

\{ \begin{matrix}F_{A}^{+} = τ(Ψ, Ψ) + P_{1}^{Y} (A, Ψ) + f′(T_{A})μ^{+} \\ D_{A}Ψ = P_{2}^{Y} (A, Ψ), \end{matrix}  (15)

where μ^{+} is the self-dual part of the pullback of μ along the projection ν(K) × ℝ → ν(K).

We discuss a choice of perturbation on W_{i} which behaves nicely under the splitting of the cobordisms and restricts to the prescribed perturbations on the asymptotic ends. Consider the case of W_{1}. The case of W_{0} is analogous.

The manifold W_{1} has a cylindrical end Y × [T_{0}, ∞) and a cylindrical end Y_{1} × (−∞, −T_{0}]. Consider a cutoff function χ(t) which is χ(t) ≡ 1 for t ≥ T_{0} and χ(t) ≡ 0 for t ≤ T_{0} − 1. In the following we shall use the notation χ̂(t) = χ(−t). Consider, on the cylindrical ends Y × [T_{0}, ∞) and Y_{1} × (−∞, −T_{0}] of W_{1}, the equations (11) and (15), respectively. Now consider the manifold V × ℝ, endowed with an infinite end of the form T^{2} × [0, ∞) × ℝ. Notice that on this region inside the manifold V × ℝ it makes sense to define the temporal gauge condition for pairs (A, Ψ). We denote by (A(t), ψ(t)) a temporal gauge representative of the gauge class of (A, Ψ).

On V we can also introduce perturbations in the class P_{δ}, as described in Section 3 of[6].

This gives a choice of perturbation on V_{r} × ℝ inside W_{1}, for large r ≥ r_{0}. Namely, we consider on V_{r} × ℝ the equations

\{ \begin{matrix}F_{A}^{+} = & τ(Ψ, Ψ) + (1 − χ̂ − χ)P_{1}^{V} (A, Ψ) \\ & +χ̂P_{1}^{Y_{1}}(A, Ψ) + χP_{1}^{Y} (A, Ψ) \\ D_{A}Ψ = & (1 − χ̂ − χ)P_{2}^{V} (A, Ψ) \\ & +χ̂P_{2}^{Y_{1}}(A, Ψ) + χP_{2}^{Y} (A, Ψ). \end{matrix}  (16)

This matches the perturbations (11) and (15) on V × [T_{0}, ∞) and V × (−∞, −T_{0}] in W_{1}.

On the product region V of (9), which connects ν(K) ⊂ Y to the solid torus ν ⊂ S^{3}, we consider the Seiberg–Witten equations with the surgery perturbation, namely the equations

\{ \begin{matrix}F_{A}^{+} = τ(Ψ, Ψ) + f′(T_{A})μ^{+} + χP_{1}^{Y} (A, Ψ) \\ D_{A}Ψ = χP_{2}^{Y} (A, Ψ). \end{matrix}  (17)

On the product region Ṽ inside Ŵ_{1}(ν(K)) that connects the solid torus ν̃ in S^{3} to ν(K) in Y_{1}, we consider equations

\{ \begin{matrix}F_{A}^{+} = τ(Ψ, Ψ) + χ̂P_{1}^{Y_{1}}(A, Ψ) \\ D_{A}Ψ = χ̂P_{2}^{Y_{1}}(A, Ψ). \end{matrix}  (18)

In the case of W_{0}, if we consider Y_{0} with the trivial Spin^{c}-structure, the perturbation along the cylindrical end Y_{0} × [T_{0}, ∞) includes a perturbation ρ_{0} with [∗ρ_{0}] = ηP D_{Y_{0}}(m), with respect to the ∗-operator on Y_{0}, with η > 0 as in [6] and m the generator of the first homology group of Y_{0}.

Throughout the paper, when we consider finite energy solutions of the Seiberg–Witten equations on the cobordisms W_{i}, we will mean finite energy solutions of the Seiberg–Witten equations on the punctured cobordisms Ŵ_{i}, as in Definition 2.5 below, with the perturbations introduced here, and with a removable singularity at the puncture x_{i}, that is, such that they extend to solutions on W_{i}.

In the following, we shall analyze the behavior of finite energy solutions on W_{i}, when stretching r → ∞ in the cobordism

Ŵ_{i}(r) = V_{r} × ℝ ∪ V(r) ∪ Ṽ(r),  (19)

with V(r) ≅ ν_{r}×ℝ and Ṽ(r) ≅ ν̃_{r}×ℝ, the product regions inside Ŵ_{i}(ν(K))_{r}.

In particular, with the choice of perturbation discussed here, a finite energy solution on W_{1} will have an asymptotic value

a_{1} = [A_{1}, ψ_{1}] ∈ M_{Y_{1}}

on the Y_{1} end, and

a = [A, ψ] ∈ M_{Y,μ},

on the other end of the cobordism, with the moduli spaces of solutions of the perturbed equations on Y and Y_{1} as in [6]. Similarly, a finite energy solution on the cobordism W_{0} will have asymptotic values

a = [A, ψ] ∈ M_{Y,μ},
a_{0} = [A_{0}, ψ_{0}] ∈ M_{Y_{0}}(𝔰_{k}).

Thus we can define moduli spaces M_{𝔰_{ℓ}}^{W_{1}}(a_{1}, a) and M_{𝔰_{k}}^{W_{0}}(a, a_{0}) as follows.

2.3 Finite energy monopoles and virtual dimension

We consider finite energy solutions of the perturbed equations introduced above, on the punctured cobordisms Ŵ_{i}, with a removable singularity at the puncture x_{i}, that is, such that they extend to solutions on W_{i}. More precisely, we have the following.

Definition 2.5 Consider the manifold Ŵ_{i}, as a complete Riemannian manifold with infinite cylindrical ends. Consider a fixed choice of the Spin^{c} structure. Write C for any of the cylindrical ends of the manifold Ŵ_{i}. A solution (A, Ψ) of the Seiberg–Witten equations on the manifold Ŵ_{i} is of finite energy, with a removable singularity at the point x_{i}, iff on any of the cylindrical ends C the solutions (A, Ψ) in a temporal gauge satisfies the condition

∫_{C} |∂_{t}A(t)|^{2} + |∂_{t}ψ(t)|^{2} < ∞.

The analysis in Section 3 of [19] of the asymptotics of finite energy solutions on a trivial cobordism Y ×ℝ carries over to the present case and shows that finite energy solutions on the manifolds Ŵ_{i} decay along the cylindrical ends to asymptotic values (A, ψ) satisfying the 3-dimensional Seiberg– Witten equations on the boundary 3-manifolds. Moreover, if the asymptotic value is an irreducible critical point, then the rate of decay is exponential, with the exponent determined by the first eigenvalue of the Hessian at the critical point.

Thus, we define configuration spaces A_{k,δ}(W_{i}) and the group of gauge transformations G_{k+1,δ}(W_{i}) and consider finite energy solutions in the quotient space. The configuration space A_{k,δ}(W_{i}) consists of pairs (A, Ψ) on Ŵ_{i} that are of finite energy, with a removable singularity at x_{i}, and with a rate of decay with exponent δ along the cylindrical ends modeled on Y and Y_{1}. Since the asymptotic limit along the end S^{3} × [0, ∞) at the puncture x_{i} is the unique reducible solution θ_{S^{3}} on S^{3}, we can define moduli spaces

M_{𝔰_{ℓ}}^{W_{1}}(a_{1}, a) and M_{𝔰_{k}}^{W_{0}}(a, a_{0})

of solutions in A_{k,δ}(W_{i}) modulo gauge action, which depend only on the asymptotic limits at the two ends of W_{i} and on the Spin^{c} structure on W_{i}. We can estimate the virtual dimension of these moduli spaces of solutions.

Lemma 2.6 Consider the linearization D_{(A,Ψ)} of the Seiberg–Witten equations at a solution (A, Ψ) on (W_{1}, 𝔰_{ℓ}), with asymptotic values a_{1} = [A_{1}, ψ_{1}] and a = [A, ψ] on the boundary three-manifolds Y_{1} and Y. The index of D_{(A,Ψ)} is given by

Ind(D_{(A,Ψ)}) = \frac{1}{4}(c_{1}(det(𝔰_{ℓ}))^{2} − 2χ(W_{1}) − 3σ(W_{1})) + \frac{ρ}{2}(Y_{1}, a_{1}) − \frac{ρ}{2}(Y, a),

where the last two summands are the APS ρ-invariants [1] of the extended Hessian operators H_{(A_{1},ψ_{1})} and H_{(A,ψ)}. The grading analyzed in [6] satisfies

deg_{Y_{1}}(a_{1}) = \frac{1}{8π^{2}} CSD_{Y_{1}}(a_{1}) + \frac{ρ}{2}(Y_{1}, a_{1}),

up to redefining the CSD functional by a global additive constant.

Proof. In the case of a solution (A, Ψ) on the trivial cobordism Y × I, with asymptotic values a = [A_{a}, ψ_{a}] and b = [A_{b}, ψ_{b}] on Y, we have

deg_{Y} (a) − deg_{Y} (b) = Ind(D_{(A,Ψ)}) = \frac{c_{1}(det(𝔰))^{2}}{4} + \frac{ρ}{2}(Y, a) − \frac{ρ}{2}(Y, b)

= \frac{−1}{16π^{2}} ∫_{Y ×ℝ} F_{A} ∧ F_{A} + \frac{ρ}{2}(Y, a) − \frac{ρ}{2}(Y, b)

= \frac{−1}{8π^{2}} (CSD(A_{b}, ψ_{b}) − CSD(A_{a}, ψ_{a})) + \frac{ρ}{2}(Y, a) − \frac{ρ}{2}(Y, b).

⋄

See the energy estimates in [17] (cf. [1], [21]). This virtual dimension of Seiberg–Witten moduli spaces on four-manifolds with boundary is computed explicitly in [22], in the case of Seifert fibered spaces, where a particular choice of metric makes it possible to compute ρ(Y, a) explicitly.

By the results of [6], the choice of the grading on Y_{1} determines uniquely the grading deg_{Y,μ} on Y, up to changing the functional CSD by a global additive constant. The grading deg_{Y,μ} also determines [6] the choice of the grading deg_{Y_{0}} on Y_{0}. For the properties of the Floer complex on Y_{0} see also [18].

We define the expression

ι(𝔰_{ℓ}, W_{1}, a_{1}, a) = \frac{1}{4}(c_{1}(det(𝔰_{ℓ}))^{2} − 2χ(W_{1}) − 3σ(W_{1})) + \frac{ρ}{2}(Y_{1}, a_{1}) − \frac{ρ}{2}(Y, a)

in the case of W_{1}. Similarly, in the case of W_{0}, we define

ι(𝔰_{k}, W_{0}, a, a_{0}) = \frac{1}{4}(∫_{W_{0}} c_{1}(det(𝔰_{k}))^{2} − L( ˆ∇^{0})) − \frac{1}{2}(χ(W_{0}) + σ(W_{0}))

+\frac{ρ}{2}(Y, a) − \frac{ρ}{2}(Y_{0}, a_{0}),

where ˆ∇^{0} is the metric compatible connection, which along the end Y_{0} × [T_{0}, ∞) has the form dt ⊗ ∂_{t} + ∇^{0}.(This is analogous to the case of [22], but in our setting the connection ∇^{0} is simply the metric connection with no adiabatic rescaling.) The expressions

ι(𝔰_{ℓ}, W_{1}, a_{1}, a) and ι(𝔰_{k}, W_{0}, a, a_{0})

compute the virtual dimensions of the moduli spaces

M_{𝔰_{ℓ}}^{W_{1}}(a_{1}, a) and M_{𝔰_{k}}^{W_{0}}(a, a_{0}).

Notice that, in the case of W_{0}, the Spin^{c}-structure may be non-trivial along the end Y_{0} × [T_{0}, ∞). This has the important consequence that the virtual dimension ι(𝔰_{k}, W_{0}, a, a_{0}) is only defined modulo the integer d(𝔰_{k}), with d(𝔰_{k}) satisfying

c_{1}(det(𝔰_{k}))(H_{2}(Y_{0}, ℤ)) = d(𝔰_{k})ℤ,

that is, in this case, d(𝔰_{k}) = 2k. This ambiguity corresponds to different components of the moduli space of solutions of the Seiberg-Witten equations on W_{0}, with different energies. The minimal energy component corresponds to the minimal non-negative value of ι(𝔰_{k}, W_{0}, a, a_{0}).

2.4 Compactification and invariants

Under a generic choice of the perturbation, we can assume that all the moduli spaces M_{𝔰_{ℓ}}^{W_{1}}(a_{1}, a) and M_{𝔰_{k}}^{W_{0}}(a, a_{0}) are cut out transversely, of dimension ι(𝔰_{ℓ}, W_{1}, a_{1}, a) and ι(𝔰_{k}, W_{0}, a, a_{0}), respectively. Unless otherwise stated, when we write M_{𝔰_{k}}^{W_{0}}(a, a_{0}), we only consider the component of minimal energy, with dimension ι(𝔰_{k}, W_{0}, a, a_{0}).

The following description of the compactification of M_{𝔰_{ℓ}}^{W_{1}}(a_{1}, a) and M_{𝔰_{k}}^{W_{0}}(a, a_{0}) follows from the main gluing theorem proved in [19], together with the results of the previous subsections.

Proposition 2.7 Suppose given a non-empty moduli space M_{𝔰_{k}}^{W_{0}}(a, a_{0}) of dimension

ι(𝔰_{k}, W_{0}, a, a_{0}) = n > 0,

with a ∈ M_{Y,μ}^{∗}. Then M_{𝔰_{k}}^{W_{0}}(a, a_{0}) admits a compactification to a manifold with corners, where the codimension 1 boundary strata consist of

\begin{matrix}⋃_{c∈M_{Y,μ}^{∗}} M̂_{Y,μ}(a, c) × M_{𝔰_{k}}^{W_{0}}(c, a_{0}) \\ ⋃_{c_{0}∈M_{Y_{0}}(𝔰_{k})} M_{𝔰_{k}}^{W_{0}}(a, c_{0}) × M̂_{Y_{0},𝔰_{k}}^{(0)}(c_{0}, a_{0}), \end{matrix}  (20)

and M̂_{Y_{0},𝔰_{k}}^{(0)}(c_{0}, a_{0}) is the minimal energy component of the moduli space M̂_{Y_{0},𝔰_{k}}(c_{0}, a_{0}), as discussed in [18]. If the reducible point c = θ satisfies

deg_{Y,μ}(a) > deg_{Y,μ}(θ),

then we also have an extra component in (20) of the form

M̂_{Y,μ}(a, θ) × U(1) × M_{𝔰_{k}}^{W_{0}}(θ, a_{0})  (21)

with a U(1) gluing parameter.

We have a similar compactification of M_{𝔰_{ℓ}}^{W_{1}}(a_{1}, a) of dimension

ι(ℓ, W_{1}, a_{1}, a) = n > 0,

for a_{1} ∈ M_{Y_{1}}^{∗} and a ∈ M_{Y,μ}^{∗}, with codimension 1 boundary strata of the form

\begin{matrix}⋃_{c∈M_{Y,μ}^{∗}} M_{𝔰_{ℓ}}^{W_{1}}(a_{1}, c) × M̂_{Y,μ}(c, a) \\ ⋃_{c_{1}∈M_{Y_{1}}^{∗}} M̂_{Y_{1}}(a_{1}, c_{1}) × M_{𝔰_{ℓ}}^{W_{1}}(c_{1}, a), \end{matrix}  (22)

and with extra components

\begin{matrix}M̂_{Y_{1}}(a_{1}, θ_{1}) × U(1) × M_{𝔰_{ℓ}}^{W_{1}}(θ_{1}, a) \\ M_{𝔰_{ℓ}}^{W_{1}}(a_{1}, θ) × U(1) × M̂_{Y,μ}(θ, a), \end{matrix}  (23)

when splitting through the reducibles.

The Proposition follows from the main gluing theorem of [19]. The fact that only the minimal energy component, among the components of M̂_{Y_{0}}(λc_{0}, a_{0}), occurs in the compactification of M_{𝔰_{k}}^{W_{0}}(c, a_{0}) is explained in the following, in the proof of Lemma 4.3.

For later use, we also need the following.

Corollary 2.8 If we have ι(𝔰_{ℓ}, W_{1}, a_{1}, a) = 1, or ι(𝔰_{k}, W_{0}, a, a_{0}) = 1, with a_{1} ∈ M_{Y_{1}}^{∗}, a ∈ M_{Y,μ}^{∗}, and a_{0} ∈ M_{Y_{0}}(𝔰_{k}), then the boundary strata are given by (22) and (20), as in the compactification of Proposition 2.7, with c_{1} and c irreducible.

Proof. We need to show that the components (21) and (23) do not occur in the compactification. This follows by dimensional arguments. ⋄

We can define numerical invariants associated to the zero-dimensional components of the moduli spaces M_{𝔰_{ℓ}}^{W_{1}}(a_{1}, a), and M_{𝔰_{k}}^{W_{0}}(a, a_{0}), with a_{1} ∈ M_{Y_{1}}^{∗}, a ∈ M_{Y,μ}^{∗}, and a_{0} ∈ ∪_{k}M_{Y_{0}}(𝔰_{k}).

Recall that these moduli spaces come endowed with an orientation, given by the trivialization of the determinant line bundle of the linearization of the Seiberg–Witten equations. Some care is needed in defining the orientation in the case of non-compact 4-manifolds with infinite cylindrical ends. The necessary details can be found in [22]. The orientation is compatible with the compactification of Proposition 2.7. Throughout this discussion we shall always assume that the perturbations are chosen so that all the moduli spaces are cut out transversely by the equations.

According to Proposition 2.7 of the previous section, in the case of ι(𝔰_{ℓ}, W_{1}, a_{1}, a) = 0, or ι(𝔰_{k}, W_{0}, a, a_{0}) = 0, the moduli spaces M_{𝔰_{ℓ}}^{W_{1}}(a_{1}, a) and M_{𝔰_{k}}^{W_{0}}(a, a_{0}) consist of a finite set of points with an attached sign given by the orientation. Thus, we can define

N_{𝔰_{ℓ}}^{W_{1}}(a_{1}, a) or N_{𝔰_{k}}^{W_{0}}(a, a_{0})

as the algebraic sum of the points in M_{𝔰_{ℓ}}^{W_{1}}(a_{1}, a) and M_{𝔰_{k}}^{W_{0}}(a, a_{0}), respectively. If we have ι(𝔰_{ℓ}, W_{1}, a_{1}, a) < 0 and either a or a_{1} is irreducible, or if we have ι(𝔰_{k}, W_{0}, a, a_{0}) < 0, then the corresponding moduli space is generically empty, so we just set the corresponding invariant equal to zero.

3 Geometric limits

In this section we describe the geometric limits of finite energy solutions on W_{i} when stretching r → ∞ inside the cobordism

Ŵ_{i}(r) = V_{r} × ℝ ∪ V(r) ∪ Ṽ(r).

The analysis is very similar to the analysis of the geometric limits of flow lines in [17].

We give the general description of the geometric limits. We describe the case of the cobordism W_{1}. Simple modifications adapt the argument to W_{0}. We shall omit here the parts of the argument which are completely analogous to the case discussed in [17]. We assume that the metric is chosen as discussed previously.

Proposition 3.1 Consider a family [A_{r}, Ψ_{r}] of finite energy solutions of the Seiberg–Witten equations on the cobordism W_{1}(r), with the choices of perturbation as discussed previously. Assume that the [A_{r}, Ψ_{r}] have asymptotic values a_{1} = [A_{1}, ψ_{1}] and a = [A, ψ] on the ends modeled on Y_{1} and Y. By [6], these asymptotic limits can be written as

(A_{1}, ψ_{1}) = (A′_{1}, ψ′_{1})#_{a^{−}}(a^{−}, 0),

and

(A, ψ) = (A′, ψ′)#_{a^{+}}(a^{+}, 0),

for large r ≥ r_{0}.

Let ϑ_{1} in χ_{0}(T^{2}, Y_{1}) denote the intersection point between the lines {v = 0} and {v − u = 1}.

We have the following types of geometric limits of [A_{r}, Ψ_{r}], as r → ∞.

(a) A finite energy solution (A′, Ψ′) of the perturbed equations (11) on V × ℝ. In radial gauge, this solution decays in the radial direction to a flat connection a′_{∞} on T^{2},

a′_{∞} ∈ χ_{0}(T^{2}, V).

In a temporal gauge on V × ℝ, (A′, Ψ′) converges to elements [A, ψ] and [Ã, ψ̃] in M_{V}^{∗} as t → ±∞, with

∂_{∞}[A, ψ] = a′_{∞} = ∂_{∞}[Ã, ψ̃].

(b) Non-uniform limits on V × ℝ given by paths [A(t), ψ(t)] in M_{V}^{∗} connecting [A, ψ] to [A′_{1}, ψ′_{1}] and [Ã, ψ̃] to [A′, ψ′], and by a function a_{V} :

D^{+} → χ_{0}(T^{2}, V), holomorphic on some neighbourhood of the half disk D^{+}, which agrees on the subset θ ∈ {−π/2, π/2} of the boundary of D^{+} with the asymptotic values ∂_{∞}[A(t), ψ(t)] =: a(t) of the paths in M_{V}^{∗}.

(c) A flat connection a′′_{∞} on T^{2}, obtained as a finite energy solution of the equation

F_{A}^{+} = f′(T_{A})μ^{+},

on the product region V = ν(K) × ℝ with an infinite end T^{2} × [0, ∞) × ℝ.

(d) A flat connection ã′′_{∞} on T^{2}, obtained as a finite energy solution of the unperturbed equations (18) on the region

Ŵ_{1}(ν(K))\V

with an infinite end T^{2} × [0, ∞) × ℝ.

(e) Non-uniform limits on the ends of W_{1}(ν(K)) given by a path ã′′(τ) in

M_{ν(K)}^{red} = {u − v = 1},

for τ ∈ [−1, 1], with ã′′(−1) = a^{−}, ã′′(0) = ã′′_{∞}, and ã′′(1) = ϑ_{1}, and a path a′′(τ) in the perturbed

M_{ν(K),μ}^{red} = {v = f′(u)},

for τ ∈ [−1, 1], with a′′(−1) = ϑ_{1}, a′′(0) = a′′_{∞} and a′′(1) = a^{+}.

(f) We also have a map a_{ν(K)} : D_{ϵ}^{+} → H^{1}(T^{2}, ℝ), holomorphic on some neighbourhood of the domain

D_{ϵ}^{+} = {ρ ∈ [log ϵ, 0], θ ∈ [−π/2, π/2]}.

Upon identifying τ = e^{ρ} sin θ, this map agrees with the path ã′′(τ), for τ ∈ [−1, −ϵ] and with a′′(τ) for τ ∈ [ϵ, 1], on the subset

θ ∈ {−π/2, π/2} and ρ ∈ [log ϵ, 0]

of the boundary of D_{ϵ}^{+}.

(g) A “thin holomorphic triangle” Δ(a′′_{∞}, ã′′_{∞}, ϑ_{1}) in the character variety χ_{0}(T^{2}, V) (cf. [2] pg.234), with vertices {a′′_{∞}, ã′′_{∞}, ϑ_{1}} and with two sides along the lines {v − u = 1} and {v = f′(u)}, with the parameterization ã′′(τ) for τ ∈ [0, 1] and a′′(τ) for τ ∈ [−1, 0].

(h) A limit on compact sets in the region T^{2} × [−r, r] × ℝ, given by a flat connection a_{∞} on T^{2}, and a non-uniform limit after rescaling, given by a map â : D → H^{1}(T^{2}, iℝ), holomorphic up to the boundary, matching the values of a_{V} and a_{ν(K)}, as described in [17].

Proof. The result follows from the analysis of the convergence of flow lines in [17]. The limits (a), (b), (f) and (h) are derived exactly as the analogous cases in [17]. The case (c) describes the limit of the solutions (A_{r}, Ψ_{r}), uniformly on compact sets, in the product region V(r) as r → ∞. Up to gauge transformations, and up to passing to a subsequence, the solutions (A_{r}, Ψ_{r}) converge smoothly on compact sets in V(r) to a finite energy solution of the perturbed abelian ASD equation

F_{A}^{+} = f′(T_{A})μ^{+}

on the strip V with an infinite end T^{2} × [0, ∞) × ℝ. By the analysis of [17], this is (up to gauge) a constant flat connection a′′_{∞} on T^{2}, with holonomies satisfying v = f′(u). Similarly, the case (d) describes the uniform convergence in Ṽ(r). The solutions (A_{r}, Ψ_{r}) converge smoothly on compact sets in Ṽ(r) to a finite energy solution of the unperturbed abelian ASD equation on the region Ṽ with an infinite end T^{2} × [0, ∞) × ℝ. Again, by the analysis of [17] we see that this is up to gauge a constant flat connection ã′′_{∞} on T^{2} contained in the line {v − u = 1} in χ_{0}(T^{2}, Y_{1}). Case (e) describes the non-uniform limits in the regions Ṽ(r) and V(r), away from compact sets, after suitable rescaling as described in [17]. We have Ṽ ≅ ν̃ × ℝ and V ≅ ν × ℝ, connecting the two solid tori ν and ν̃ in the standard Heegaard splitting of S^{3} to the solid tori ν(K) in Y_{1} and Y, respectively. Thus, we can adapt the analysis used in [17] for non-uniform limits on ν(K) × ℝ. After a suitable rescaling, we resulting non-uniform limits in the region Ṽ consist of a path ã′′(τ) along the line {u − v = 1} in χ_{0}(T^{2}, V) and a map ã_{ν} : D^{+} → H^{1}(T^{2}, iℝ), holomorphic in a neighbourhood of D^{+}, which agrees with ã′′ along the subset

θ ∈ {−π/2, π/2} and ρ ∈ (−∞, 0]

of the boundary of D^{+}. Similarly, the non-uniform limits on V, after rescaling, consist of a path a′′(τ) along the curve v = f′(u) in χ_{0}(T^{2}, V) and a map a_{ν} : D^{+} → H^{1}(T^{2}, iℝ), holomorphic in a neighbourhood of D^{+}, which agrees with a′′(τ) along the subset

θ ∈ {−π/2, π/2} and ρ ∈ (−∞, 0]

of the boundary of D^{+}.

The thin holomorphic triangle of Case (g) is obtained by the overlap of these two regions, with the vertex ϑ_{1} at the flat connection on T^{2} = ∂ν = ∂ν̃ which extends to both sides of the standard Heegaard splitting of S^{3} to give the unique reducible solution on S^{3}. That is, we have θ_{S^{3}}|_{T^{2}} = ϑ_{1}. Notice that, in general, the flat connections a′_{∞}, a′′_{∞}, ã′′_{∞}, a_{∞}, a^{+}, and a^{−} on T^{2} are all distinct. ⋄

3.1 The holomorphic triangles

This subsection contains some observations on the holomorphic triangles that appear among the geometric limits of solutions on the cobordisms, as discussed in Proposition 3.1. A better understanding of these triangles will be very useful in analyzing the different properties of the coefficients of the chain maps defined by the cobordisms W_{1}, W_{0}, and W.

In the following, let ϵ be the parameter used in the construction of the surgery perturbation.

Definition 3.2 Consider the unique holomorphic triangle Δ^{ϵ} with vertices {a^{−}, ϑ_{1}, a^{+}} and sides along the curves ℓ_{1} = {v − u = 1}, ℓ_{μ} = {v = f′(u)} and ℓ = ∂_{∞}(M_{V}^{∗}), defined by the geometric limits a_{V}, a_{ν(K)}, â, and Δ(ã′′_{∞}, ϑ_{1}, a′′_{∞}) of Proposition 3.1. We say that the triangle Δ^{ϵ} is degenerate if the holomorphic map obtained as a limit of the triangles Δ^{ϵ}, as ϵ → 0 is a disk Δ with boundary along arcs of ℓ_{1} and ℓ connecting the vertices a^{−} and a^{+}. We say that Δ^{ϵ} is non-degenerate if the holomorphic map obtained as a limit is a triangle Δ with boundary along arcs of ℓ_{1}, ∪_{k}ℓ_{k}, and ℓ, with ∪_{k}ℓ_{k} = {u = 2k, k ∈ ℤ}.

Lemma 3.3 Suppose given a_{1} and ã_{1} in M_{Y_{1}}, a ∈ M_{Y,μ}\j(M_{Y_{1}}), and a_{0} ∈ M_{Y_{0}}(𝔰_{k}). Then the holomorphic triangles that appear in the geometric limits of solutions in the zero-dimensional moduli spaces M_{ℓ}^{W_{1}}(a_{1}, j(ã_{1})) and M_{k}^{W_{0}}(a, a_{0}) are all degenerate in the limit ϵ → 0.

Proof. The triangle Δ^{ϵ} has two sides along arcs of the lines {v − u = 1} and {v = f′(u)} connecting ϑ_{1} and a^{−} and ϑ_{1} and a^{+}, respectively, with the parameterization of Case (e) of Proposition 3.1. If for ϵ → 0 the points {ϑ_{1}, a^{−}, a^{+}} all lie on the same line {v − u = 1}, then the holomorphic map in the limit is a disk with one side along the arc in {v − u = 1} connecting a^{−} and a^{+} and the other side along an arc in ∂_{∞}(M_{V}^{∗}) connecting these same two points. Thus, the limit triangle is degenerate. ⋄

Now observe, instead, that when we consider solutions on the cobordism that intertwine the generators of the Floer complex for Y_{1} with those for Y_{0}, we may have holomorphic triangles that do not degenerate in the limit when ϵ → 0.

Lemma 3.4 Suppose given, as before, critical points a_{1} and ã_{1} in M_{Y_{1}}^{∗}, a_{0}(ϵ) ∈ M_{Y,μ}\j(M_{Y_{1}}), and a_{0} ∈ M_{Y_{0}}(𝔰_{k}). Consider zero-dimensional moduli spaces of the form M_{ℓ}^{W_{1}}(a_{1}, a_{0}(ϵ)) and M_{k}^{W_{0}}(ã_{1}(ϵ), a_{0}), with ã_{1}(ϵ) = j(ã_{1}). Then, in general, the limit holomorphic map will still be a triangle.

Proof.

Suppose given a_{1} in M_{Y_{1}} and a_{0}(ϵ) ∈ M_{Y,μ}\j(M_{Y_{1}}), and consider the geometric limits of solutions in a zero dimensional moduli space M_{ℓ}^{W_{1}}(a_{1}, a). We still may have holomorphic triangles that degenerate in the limit ϵ → 0. By the open mapping theorem, this happens whenever two sides of the boundary of the holomorphic triangle are mapped together as ϵ → 0. However, since the pair of points a_{1} and a_{0} = lim_{ϵ} a_{0}(ϵ), or ã_{1} and a_{0}, are now on two different lines in H^{1}(T^{2}, ℝ), we can also have non-degenerate holomorphic triangles. ⋄

The existence of these non-degenerate holomorphic triangles, that are not “thin” for small ϵ, characterizes the difference between the chain maps induced by the cobordisms and the homomorphisms of abelian groups defined by inclusion and projection, as described in Part I [6], as we shall see when discussing injectivity and surjectivity of the maps in the exact sequence.

4 The chain homomorphisms

In this section we introduce the chain homomorphisms w_{∗}^{1} and w_{∗}^{0}. First we observe that we have an analogue of the formula

N_{X}(−𝔰) = (−1)^{(1+b_{2}^{+}(X)−b_{1}(X))/2}N_{X}(𝔰)  (24)

that holds for compact 4-manifolds. The version we need is given by the following statement.

Lemma 4.1 On the manifold W_{1} we have

N_{ℓ}^{W_{1}}(a_{1}, a) = (−1)^{2 Ind_{ℂ}(D_{A})}N_{−ℓ}^{W_{1}}(a_{1}, a).

Thus, if 𝔰 ∈ S(W_{1}) satisfies

ι(𝔰, W_{1}, a_{1}, a) = 0,

for given a_{1} ∈ M_{Y_{1}}^{∗} and a ∈ M_{Y,μ}^{∗}, then −𝔰 is the unique other Spin^{c} structure in S(W_{1}) which satisfies ι(−𝔰, W_{1}, a_{1}, a) = 0. The corresponding invariant satisfies

N_{−𝔰}^{W_{1}}(a_{1}, a) = N_{𝔰}^{W_{1}}(a_{1}, a).

Proof. The proof for the closed manifold case [20] adapts to this context, with fixed asymptotic values, and compatible choice of admissible sections. In fact, the sign is given by the change of orientation, and the orientation is compatible with the boundary strata. ⋄

Notice that the same argument does not extend to the case of the manifold W_{0}. In fact, in the case of W_{1} changing 𝔰 to −𝔰 does not change the Spin^{c} structure on the boundary Y ∪ Y_{1}, whereas, in the case of W_{0}, changing 𝔰 to −𝔰 amounts to changing the Spin^{c} structure on Y_{0}. On the other hand, on W_{0} or W_{2} we simply do not have solutions in M_{k}^{W_{0}} and in M_{−k}^{W_{0}} with the same asymptotic values.

Let a_{1} = [A_{1}, ψ_{1}] be a class in M_{Y_{1}}^{∗}, and a = [A, ψ] a class in M_{Y,μ}^{∗}, such that the gradings, assigned according to [6], coincide

deg_{Y_{1}}(a_{1}) = deg_{Y,μ}(a).

Then we consider the zero dimensional moduli spaces M_{𝔰_{ℓ}}^{W_{1}}(a_{1}, a) with 𝔰_{ℓ}, ℓ ≥ 0, satisfying ι(𝔰_{ℓ}, W_{1}, a_{1}, a) = 0. Similarly, for a_{0} = [A_{0}, ψ_{0}] in some M_{Y_{0}}(𝔰_{k}) with compatible grading

deg_{Y,μ}(a) = deg_{Y_{0},𝔰_{k}}(a_{0}),

we consider the zero dimensional components

M_{𝔰_{k}}^{W_{0}}(a, a_{0}) with ι(𝔰_{k}, W_{0}, a, a_{0}) = 0.

Recall that we have invariants N_{ℓ}^{W_{1}}(a_{1}, a) and N_{k}^{W_{0}}(a, a_{0}) defined by counting points with the orientation in M_{𝔰_{ℓ}}^{W_{1}}(a_{1}, a) and M_{𝔰_{k}}^{W_{0}}(a, a_{0}), respectively.

Definition 4.2 We define the map w_{∗}^{1} : C_{∗}(Y_{1}) → C_{∗}(Y, μ) with matrix elements

⟨a, w_{∗}^{1}(a_{1})⟩ = N_{ℓ}^{W_{1}}(a_{1}, a),

with ℓ is the unique non-negative Spin^{c} structure satisfying ι(𝔰_{ℓ}, W_{1}, a_{1}, a) = 0.

We define the map w_{∗}^{0} : C_{∗}(Y, μ) → ⊕_{k}C_{(∗)}(Y_{0}, 𝔰_{k}) with the matrix coefficients

⟨a_{0}, w_{∗}^{0}(a)⟩ = N_{k}^{W_{0}}(a, a_{0}).

Thus, we choose to define the map w_{∗}^{1} using only the “positive” Spin^{c} structures, c_{1}(det(𝔰)) = (2ℓ + 1)e_{1}, with ℓ ≥ 0.

Lemma 4.3 The maps w_{∗}^{i} are chain homomorphisms.

Proof. Suppose given a Spin^{c}-structure 𝔰_{ℓ} ∈ S_{1}, with ℓ ≥ 0, satisfying

ι(𝔰_{ℓ}, W_{1}, a_{1}, b) = 1.

We have a compactification of the moduli space M_{𝔰_{ℓ}}^{W_{1}}(a_{1}, b) by boundary strata of the form

⋃_{a∈M_{Y,μ}^{∗}} M_{𝔰_{ℓ}}^{W_{1}}(a_{1}, a) × M̂_{Y} (a, b)

and

⋃_{b_{1}∈M_{Y_{1}}^{∗}} M̂_{Y_{1}}(a_{1}, b_{1}) × M_{𝔰_{ℓ}}^{W_{1}}(b_{1}, b),

as in Proposition 2.7 and Corollary 2.8. Here a and b_{1} satisfy

deg_{Y,μ}(a) − deg_{Y,μ}(b) = 1,
deg_{Y_{1}}(a_{1}) − deg_{Y_{1}}(b_{1}) = 1.

The moduli spaces M̂_{Y_{1}}(a_{1}, b_{1}) and M̂_{Y} (a, b) are gauge classes of flow lines on Y_{1} and Y respectively, modulo the action of ℝ by translation. The counting of boundary points of the 1-dimensional M_{𝔰_{ℓ}}^{W_{1}}(a_{1}, b), with the orientation, gives the relation

∑_{a} N_{𝔰_{ℓ}}^{W_{1}}(a_{1}, a)n_{Y} (a, b) − ∑_{b_{1}} n_{Y_{1}}(a_{1}, b_{1})N_{𝔰_{ℓ}}^{W_{1}}(b_{1}, b) = 0.

Notice that M_{𝔰_{ℓ}}^{W_{1}}(a_{1}, a) and M_{𝔰_{ℓ}}^{W_{1}}(b_{1}, b) are zero-dimensional, thus we have

ι(𝔰_{ℓ}, W_{1}, a_{1}, a) = ι(𝔰_{ℓ}, W_{1}, b_{1}, b) = 0.

Thus, counting points in these moduli spaces gives exactly the counting of the matrix elements of w_{∗}^{1}. This proves the relation

w_{∗}^{1} ∘ ∂_{Y_{1}} = ∂_{Y,μ} ∘ w_{∗}^{1}.

The result for W_{0} is analogous, except for the fact that some care is needed in the case of the trivial Spin^{c}-structure 𝔰_{0}. Let us first consider the case of non-trivial 𝔰_{k} first. Suppose given a 1-dimensional M_{𝔰_{k}}^{W_{0}}(a, b_{0}). In particular, we have

deg_{Y,μ}(a) = deg_{Y_{0}}(b_{0}) + 1,

where deg_{Y_{0}} is the ℤ-lift of the ℤ_{2k}-relative grading on M_{Y_{0}}(𝔰_{k}), induced by the grading on M_{Y,μ}, cf.[6], [18].

The boundary strata in the compactification of M_{𝔰_{k}}^{W_{0}}(a, b_{0}) are given by

\begin{matrix}⋃_{a_{0}∈M_{Y_{0}}(𝔰_{k})} M_{𝔰_{k}}^{W_{0}}(a, a_{0}) × M̂_{Y_{0},𝔰_{k}}^{(0)}(a_{0}, b_{0}) \\ ⋃_{b∈M_{Y,μ}^{∗}} M̂_{Y,μ}(a, b) × M_{𝔰_{k}}^{W_{0}}(b, b_{0}). \end{matrix}  (25)

Here b and a_{0} satisfy

deg_{Y,μ}(a) − deg_{Y,μ}(b) = 1,
deg_{Y_{0}}(a_{0}) − deg_{Y_{0}}(b_{0}) = 1,  (26)

in the integer grading. Notice that, as in [18], (26) can be derived by observing that, in the geometric limits for r → ∞, the flowlines that contribute to the compactification define a contractible path [A′′(t), 0] in M_{ν(K)} ⊂ χ_{0}(T^{2}, ν(K)). Thus, with the notation of [18], we pick only the minimal energy component M_{Y_{0},𝔰_{k}}^{(0)}(a_{0}, b_{0}) in the moduli space M_{Y_{0},𝔰_{k}}(a_{0}, b_{0}). This component of the moduli space is exactly the one which defines the boundary operator on the ℤ-lift C_{(∗)}(Y_{0}, 𝔰_{k}) of the ℤ_{2k} graded complex C_{∗}(Y_{0}, 𝔰_{k}), as analyzed in [18].

The argument used here, based on the geometric limits of solutions, can be generalized to the case of the trivial Spin^{c}-structure 𝔰_{0}. Again, suppose given M_{𝔰_{0}}^{W_{0}}(a, b_{0}), with

ι(𝔰_{0}, W_{0}, a, b_{0}) = 1.

In this case we want to show that the strata

M_{𝔰_{0}}^{W_{0}}(a, a_{0}) × M̂_{Y_{0},𝔰_{0}}(a_{0}, b_{0})

which occur in the compactification, with

deg_{Y_{0},𝔰_{0}}(a_{0}) − deg_{Y_{0},𝔰_{0}}(b_{0}) = 1,

only contain flow lines in the component of minimal energy of M_{Y_{0},𝔰_{0}}(a_{0}, b_{0}). Again, the result follows from the fact that the path [A′′(t), 0] in the geometric limits is contractible, for any flow-line in M_{Y_{0},𝔰_{0}}(a_{0}, b_{0}) that arises in the compactification of M_{𝔰_{0}}^{W_{0}}(a, b_{0}). This precisely identifies the right component of the boundary operator in the Floer complex for (Y_{0}, 𝔰_{0}).

Thus, (25) implies that the map

w_{∗}^{0} : CF_{∗}(Y, μ) → ⊕_{k}CF_{(∗)}(Y_{0}, 𝔰_{k})

is a chain homomorphism precisely when the complex CF_{(∗)}(Y_{0}, 𝔰_{k}) is endowed with the ℤ-grading described in [18] and, in the case of 𝔰_{0}, the Floer homology of Y_{0} is defined as

HF_{(∗)}(Y_{0}, 𝔰_{0}, ℤ[[t]])|_{t=0}

on (Y_{0}, 𝔰_{0}), as described in [18]. ⋄

4.1 The composite map

Let N_{ℓ,k}^{W}(a_{1}, a_{0}) denote the invariant obtained by counting solutions in moduli space M_{ℓ,k}^{W}(a_{1}, a_{0}), for the unique choice of the Spin^{c} structure 𝔰_{ℓ,k} such that M_{ℓ,k}^{W}(a_{1}, a_{0}) is zero dimensional.

Lemma 4.4 Suppose given a_{1} in M_{Y_{1}}^{∗} and a_{0} in M_{Y_{0}}(𝔰_{k}). Let ℓ ≥ 0 be the unique non-negative Spin^{c}-structure satisfying

ι(𝔰_{ℓ,k}, W, a_{1}, a_{0}) = 0.

Then the composite map w_{∗}^{0} ∘ w_{∗}^{1} is given by

⟨w_{∗}^{0} ∘ w_{∗}^{1}(a_{1}), a_{0}⟩ = N_{ℓ,k}^{W}(a_{1}, a_{0}).

Proof.

The composite map w_{∗}^{0} ∘ w_{∗}^{1} has matrix elements

∑_{a∈Θ} N_{ℓ}^{W_{1}}(a_{1}, a)N_{k}^{W_{0}}(a, a_{0}),

where the sum is over the set

Θ = {a ∈ M_{Y,μ}^{∗}|ι(𝔰_{ℓ}, W_{1}, a_{1}, a) = 0, ι(𝔰_{k}, W_{0}, a, a_{0}) = 0,
deg_{Y_{1}}(a_{1}) = deg_{Y,μ}(a) = deg_{Y_{0}}(a_{0})}.

On the other hand, we have

N_{ℓ,k}^{W}(a_{1}, a_{0}) = ∑_{a∈Θ_{1}} N_{ℓ}^{W_{1}}(a_{1}, a)N_{k}^{W_{0}}(a, a_{0}),

where now the sum is over the set

Θ_{1} = {a ∈ M_{Y,μ}^{∗}|ι(𝔰_{ℓ}, W_{1}, a_{1}, a) = 0, ι(𝔰_{k}, W_{0}, a, a_{0}) = 0}.

Notice that, given any 𝔰_{ℓ,k} on W which satisfies

ι(𝔰_{ℓ,k}, W, a_{1}, a_{0}) = 0,

and such that we have

M_{𝔰_{ℓ,k}}^{W}(a_{1}, a_{0}) ≠ ∅,

there exists some a ∈ M_{Y,μ}^{∗} such that we have

ι(𝔰_{ℓ}, W_{1}, a_{1}, a) = 0 and ι(𝔰_{k}, W_{0}, a, a_{0}) = 0.

This follows by stretching the cylinder Y × [−T, T] in the composite cobordism: the condition

M_{𝔰_{ℓ,k}}^{W}(a_{1}, a_{0}) ≠ ∅

ensures the existence of a limiting translation invariant solution on Y × ℝ. The argument is similar to the one used in [15].

Thus, we only need to prove that we have

∑_{a∈Θ_{1}\Θ} N_{ℓ}^{W_{1}}(a_{1}, a)N_{k}^{W_{0}}(a, a_{0}) = 0.

Suppose not. Then we have an element a ∈ Θ_{1}\Θ such that the moduli spaces

M_{ℓ}^{W_{1}}(a_{1}, a) and M_{k}^{W_{0}}(a, a_{0})

are zero-dimensional and non-empty. The point a will be in j(M_{Y_{1}}^{∗}), or in M_{Y,μ}^{∗}\j(M_{Y_{1}}^{∗}). We consider the first case. The proof in the second case is completely analogous. We have a = j(ã_{1}) and deg_{Y_{1}}(a_{1}) ≠ deg_{Y_{1}}(ã_{1}). Consider the geometric limits of the solutions in M_{ℓ}^{W_{1}}(a_{1}, j(ã_{1})). As we discuss in greater detail in Lemma 5.1 and Lemma 5.3, as we let ϵ → 0, the geometric limits of solutions in M_{ℓ}^{W_{1}}(a_{1}, j(ã_{1})) define geometric limits of solutions in M_{Y_{1}}(a_{1}, ã_{1}). In particular, if the moduli space M_{ℓ}^{W_{1}}(a_{1}, j(ã_{1})) is non-empty and zero-dimensional, also the moduli space M_{Y_{1}}(a_{1}, ã_{1}) will be non-empty and zero-dimensional, which contradicts the assumption that deg_{Y_{1}}(a_{1}) ≠ deg_{Y_{1}}(ã_{1}).

This implies that we have an identification of the zero-dimensional components

M_{𝔰_{ℓ,k}}^{W}(a_{1}, a_{0}) ≅ ∪_{a∈Θ}M_{𝔰_{ℓ}}^{W_{1}}(a_{1}, a) × M_{𝔰_{k}}^{W_{0}}(a, a_{0}),  (27)

for large T ≥ T_{0} in the cylinder Y × [−(T − T_{0}), (T − T_{0})] in the composite cobordism W. This follows from the gluing theorem of [19], cf. [5] [18], together with the previous argument. ⋄

5 Injectivity and Surjectivity

In the previous section we have constructed a sequence

0 → C_{∗}(Y_{1}) →^{w_{∗}^{1}} C_{∗}(Y, μ) →^{w_{∗}^{0}} ⊕_{k}C_{(∗)}(Y_{0}, 𝔰_{k}) → 0.

We now proceed to show exactness in the first and last place, namely injectivity of w_{∗}^{1} and surjectivity of w_{∗}^{0}.

We want to give a better description of the components

⟨j(a′_{1}), w_{∗}^{1}(a_{1})⟩,

with a_{1} and a′_{1} in M_{Y_{1}}^{∗}, and

⟨a_{0}, w_{∗}^{0}(a)⟩,

a ∈ M_{Y,μ}^{∗}\j(M_{Y_{1}}^{∗}), and a_{0} ∈ ∪_{k}M_{Y_{0}}(𝔰_{k}). For simplicity, let us introduce

the following notation. Let {a_{i}^{(1)}}_{i=1,...m} be the elements in M_{Y_{1}}^{∗} and let {a_{j}^{(0)}}_{i=m+1,...n} be the elements in ∪_{𝔰_{i_{k}}}M_{Y_{0}}(𝔰_{i_{k}}). Then the n elements in M_{Y,μ}^{∗} can be identified with the union of these two sets of points. More precisely, if ϵ > 0 is the parameter used in the definition of the function f′ in the construction of the surgery perturbation, then the moduli space M_{Y,μ}^{∗} can be identified with a collection of points

M_{Y,μ}^{∗} = {a_{i}^{(1)}(ϵ)}_{i=1,...m} ∪ {a_{j}^{(0)}(ϵ)}_{j=m+1,...n}.

For ϵ small enough, there is a bijection

M_{Y,μ}^{∗} ≅ {a_{i}^{(1)}}_{i=1,...m} ∪ {a_{j}^{(0)}}_{j=m+1,...n},  (28)

which is compatible with the grading [6].

In the following we shall use the notation (28) which identifies the elements in M_{Y,μ} with elements in the other two moduli spaces.Whenever it is crucial to distinguish between these moduli spaces, we shall use the notation a_{1} ∈ M_{Y_{1}}^{∗}, a_{0} ∈ M_{Y_{0}}(𝔰_{k}), and a ∈ M_{Y,μ}^{∗}, with a = j(a_{1}) or π(a) = a_{0}. We hope this will not cause any confusion.

We want to describe solutions in the moduli spaces

M_{ℓ}^{W_{1}}(a_{i}^{(1)}, a_{q}^{(1)})

and

M_{k}^{W_{0}}(a_{j}^{(0)}, a_{p}^{(0)}),

by describing these moduli spaces as a gluing of solutions on the trivial cobordism V × ℝ and solutions on the regions Ŵ_{i}(ν(K)). The techniques involved in the splitting and gluing of solutions on the cobordisms are analogous to the ones developed in [17] to analyze the splitting and gluing of flow lines on the trivial cobordism.

We are assuming here that the asymptotic values also satisfy the condition

deg_{Y_{1}}(a_{i}^{(1)}) = deg_{Y,μ}(a_{q}^{(1)}),

deg_{Y,μ}(a_{j}^{(0)}) = deg_{Y_{0},𝔰_{k}}(a_{p}^{(0)}). (29)

In particular, we are going to prove the fundamental relations

⟨a_{i}^{(1)}, w_{∗}^{1}(a_{q}^{(1)}⟩ = δ_{iq}

for all a_{i}^{(1)} and a_{q}^{(1)} in M_{Y_{1}}^{∗} and

⟨a_{j}^{(0)}, w_{∗}^{0}(a_{p}^{(0)})⟩ = δ_{jp}

for all a_{j}^{(0)} and a_{p}^{(0)} in ∪_{k}M_{Y_{0}}(𝔰_{k}).

In the following we shall discuss the case of the manifold W_{1}. The case of W_{0} is analogous. Consider the manifold

Ŵ_{1}(r) = (V_{r} × ℝ) ∪ V(r) ∪ V̂(r),

as in (19).

We use the description of the geometric limits of solutions on W_{1} given in Proposition 3.1, together with the analysis of [17] of the geometric limits of flow lines, in order to analyst solutions in M_{ℓ}^{W_{1}}(a_{i}^{(1)}, a_{i}^{(1)}) and in M_{k}^{W_{0}}(a_{j}^{(0)}, a_{j}^{(0)}).

Suppose given a_{1}, an element of M_{Y_{1}}^{∗}. For large r ≥ r_{0} we represent a_{1} as in [6],

a_{1}(r) = [(Ã′, ψ̃′)#_{ã′′_{∞}}^{r}(ã′′_{∞}, 0)].

Lemma 5.1 There is a unique solution [A_{r}, Ψ_{r}] in the zero-dimensional moduli space

M_{ℓ}^{W_{1}(r)}(a_{1}(r), j(a_{1}(r))),

for large r ≥ r_{0} and for small enough ϵ > 0, where ϵ is the parameter used in the definition of the surgery perturbation.

Proof. We write a_{1}(r) = [A(r), ψ(r)], with

[A(r), ψ(r)] = [(Ã′, ψ̃′)#_{ã′′_{∞}}^{r}(ã′′_{∞}, 0)]

on Y_{1}(r), for large r ≥ r_{0}, and

j(a_{1}(r)) = j[A(r), ψ(r)] = [(A′, ψ′)#_{a′′_{∞}}^{r}(a′′_{∞}, 0)].

In the limit ϵ → 0, the asymptotic values a′′_{∞} and ã′′_{∞} coincide, and the elements [A′, ψ′] and [Ã′, ψ̃′] in M_{V}^{∗} also coincide, because of the result of Lemma 3.3, which shows that the holomorphic triangle degenerates. Thus, when ϵ → 0 there is a unique finite energy solution on V × ℝ with the same asymptotic value [A′, ψ′] at t → ±∞, given by the constant flow [A′, ψ′].

In fact, if we had a non-constant finite energy solution (A′, Ψ′) on V × ℝ with the same limits at t → ±∞, then, using the gluing theorem of [17], we could glue this (A′, Ψ′) along the flat connection a′′_{∞} to a solution on Y_{1} × ℝ, extending it as the reducible solution (a′′_{∞}, 0) on ν(K) × ℝ. This solution would be a non-trivial flow line connecting the critical point a_{1} to itself in the configuration space on Y_{1}, but the moduli space M̂_{Y_{1}}(a_{1}, a_{1}) is generically empty for dimensional reasons.

Thus, in the limit ϵ → 0, there is a unique solution [A_{r}, Ψ_{r}] on W_{1}(r) with the same asymptotic value at the two ends, obtained by gluing along the asymptotic value a′′_{∞} the constant flow [A′, ψ′] with the unique perturbed ASD equation on W_{1}(ν(K)) which extends a_{∞}.

Since the moduli space

M_{ℓ}^{W_{1}(r)}(a_{1}(r), j(a_{1}(r)))

is discrete for all 0 < ϵ ≤ ϵ_{0}, we obtain that, for ϵ small enough, there is a unique solution. ⋄

Corollary 5.2 We have

⟨a_{i}^{(1)}, w_{∗}^{1}(a_{i}^{(1)})⟩ = 1

for all a_{i}^{(1)} in M_{Y_{1}}^{∗}, and the analogous

⟨a_{j}^{(0)}, w_{∗}^{0}(a_{j}^{(0)})⟩ = 1

for all a_{j}^{(0)} in ∪_{k}M_{Y_{0}}(𝔰_{k}).

We also have the following result.

Lemma 5.3 Consider the moduli space M_{ℓ}^{W_{1}}(ã_{1}, j(a_{1})), for large enough r ≥ r_{0} and with two critical points a_{1} ≠ ã_{1} in M_{Y_{1}}^{∗}. Assume as before that 0 < ϵ ≤ ϵ_{0} is the parameter used in the construction of the surgery perturbation. Then, for ϵ small enough, we have

M_{ℓ}^{W_{1}}(ã_{1}, j(a_{1})) = ∅

if deg_{Y_{1}}(a_{1}) = deg_{Y_{1}}(ã_{1}).

Proof. Suppose given a solution [A_{r}, Ψ_{r}] in

M_{ℓ}^{W_{1}}(ã_{1}, j(a_{1})),

with a_{1} ≠ ã_{1} in M_{Y_{1}}^{∗} and

deg_{Y_{1}}(a_{1}) = deg_{Y_{1}}(ã_{1}).

According to the result of Proposition 3.1, we have geometric limits as r → ∞. We show that we can construct from these geometric limits a non-trivial flow line in the configuration space over Y_{1} which connects the two points a_{1} and ã_{1} in M_{Y_{1}}^{∗}. This will contradict the assumption that

deg_{Y_{1}}(a_{1}) = deg_{Y_{1}}(ã_{1}).

In fact, this assumption implies that the moduli space of flow lines is generically empty for dimensional reasons.

We can write the endpoints a_{1} and ã_{1} as

j(a_{1}) = [(A′, ψ′)#_{a_{∞}}^{r}(a_{∞}, 0)],

and

ã_{1} = [(Ã′, ψ̃′)#r ; ã_{∞}(ã_{∞}, 0)].

Here a_{∞} is a flat connection on T^{2} satisfying the relation v = f′(u), with f′ depending on the choice of a small ϵ > 0, and ã_{∞} is a flat connection on T^{2} satisfying the relation v − u = 1.

Suppose that we have M_{ℓ}^{W_{1}}(ã_{1}, j(a_{1})) ≠ ∅. Then, in the limit ϵ → 0, the geometric limits on Ŵ_{1}(ν(K)) consist of the cases (c), (d), and (e) of Proposition 3.1, where the connections a′′_{∞} and ã′′_{∞} coincide, because of the result of Lemma 3.3 on the degenerate holomorphic triangles. Therefore, these geometric limits determine a preglued solution on ν(K) × ℝ inside Y_{1}×ℝ which can be glued to the geometric limits on V ×ℝ to determine a flow line on Y_{1}(r) × ℝ. Since we know that M_{Y_{1}(r)×ℝ}(ã_{1}, a_{1}) = ∅ if deg_{Y_{1}}(a_{1}) = deg_{Y_{1}}(ã_{1}), we have shown that, for ϵ → 0, we also have

M_{ℓ}^{W_{1}}(ã_{1}, j(a_{1})) = ∅,

hence the relation

⟨w_{∗}^{1}(ã_{1}), j(a_{1})⟩ = 0

holds for small enough ϵ, if a_{1} ≠ ã_{1} in M_{Y_{1}}^{∗}. ⋄

We derive from Lemma 5.3 the second fundamental relation.

Corollary 5.4 We have

N_{ℓ}^{W_{1}}(a_{i}^{(1)}, a_{q}^{(1)}) = 0

whenever i ≠ q and

deg_{Y_{1}}(a_{i}^{(1)}) = deg_{Y_{1}}(a_{q}^{(1)}).

Similarly, for W_{0} we have

N_{k}^{W_{0}}(a_{j}^{(0)}, a_{p}^{(0)}) = 0

whenever j ≠ p and

deg_{Y_{0}}(a_{j}^{(0)}) = deg_{Y_{0}}(a_{p}^{(0)}).

Notice how the previous results do not give any information about the components of the maps w_{∗}^{1} and w_{∗}^{0} that interchange the critical points a_{i}^{(1)} and a_{j}^{(0)}. In fact, in this case there are in general non-degenerate triangles, as ϵ → 0 in the surgery perturbation, cf. the result of Lemma 3.4. In other words, the presence of these non-degenerate holomorphic triangles measures the difference between the chain maps w_{∗}^{1} and w_{∗}^{0} and the group homomorphisms j and π defined by the inclusion and projection, in the identification of the generators on Y with generators on Y_{1} and Y_{0}.

6 Exactness in the middle term

Recall that the results of [6] imply that the ranks of the Floer complexes in the sequence

0 → C_{∗}(Y_{1}) → C_{∗}(Y, μ) → ⊕_{k}C_{(∗)}(Y_{0}, 𝔰_{k}) → 0

are as prescribed for the existence of an exact sequence. Moreover, in the previous section we have proved injectivity of the first map and surjectivity of the last. Now we analyze the middle term.

The result of Lemma 4.4, together with Corollary 5.2 and Corollary 5.4, yields the following.

Lemma 6.1 Suppose given a ∈ M_{Y,μ}\j(M_{Y_{1}}) and a_{1} ∈ M_{Y_{1}} and let π be the identification π : M_{Y,μ}\j(M_{Y_{1}}) → ∪_{k}M_{Y_{0}}(𝔰_{k}). The coefficients of the composite map w_{∗}^{0} ∘ w_{∗}^{1} satisfy the relation

N_{ℓ,k}^{W}(a_{1}, π(a)) = N_{ℓ}^{W_{1}}(a_{1}, a) + N_{k}^{W_{0}}(j(a_{1}), π(a)).  (30)

The main purpose of this section is to show that the counting in (30) is zero. We then verify that this is sufficient to prove exactness in the middle term. Establishing the relation w_{∗}^{0} ∘ w_{∗}^{1} = 0 depends again essentially on the analysis of the geometric limits of solutions on the cobordisms, following the technique of [17]. We need a preliminary discussion of the geometric limits on V × ℝ which completes the results of Part II, [17].

6.1 The moduli space on V × ℝ

Consider elements a_{i}^{(1)} ∈ M_{Y_{1}}^{∗} and a_{j}^{(0)}(ϵ) ∈ M_{Y,μ}^{∗}, which we can write as

a_{i}^{(1)} = [(A_{i}^{−}, ψ_{i}^{−})#(a_{∞,i}^{−}, 0)]
a_{j}^{(0)}(ϵ) = [(A_{j}^{+}(ϵ), ψ_{j}^{+}(ϵ))#(a_{∞,j}^{+}, 0)].

Assume that we have solutions [A_{1}(r), Ψ_{1}(r)] in M_{ℓ}^{W_{1}(r)}(a_{i}^{(1)}, a_{j}^{(0)}(ϵ)), for all sufficiently large r ≥ r_{0}. Then these solutions define geometric limits as in Proposition 3.1 (a) - (h).

In particular, we list here separately the limits on V × ℝ. Our purpose now is to simplify and group together in a more efficient way the information on the geometric limits given in Proposition 3.1.

Remark 6.2 A family of solutions [A_{1}(r), Ψ_{1}(r)] in M_{ℓ}^{W_{1}(r)}(a_{i}^{(1)}, a_{j}^{(0)}(ϵ)) defines the following limits on V × ℝ: (a). A finite energy solution [A′, Ψ′]^{ϵ} of the perturbed equations (11) on V ×ℝ, with a radial limit a_{∞}(ϵ) in ∂_{∞}(M_{V}^{∗}) ⊂ χ_{0}(T^{2}, V), and with temporal limits [A, ψ]_{1}^{ϵ} and [Ã, ψ̃]_{1}^{ϵ} in

∂_{∞}^{−1}(a_{∞}(ϵ)) ⊂ M_{V}^{∗} .

(b). Two paths [A(t), ψ(t)]_{1}^{ϵ} in M_{V}^{∗}, for t ∈ [−1, 0) and t ∈ (0, 1], with

[A(−1), ψ(−1)]_{1}^{ϵ} = [A_{i}^{−}, ψ_{i}^{−}]

lim_{t→0−}[A(t), ψ(t)]_{1}^{ϵ} = [A, ψ]_{1}^{ϵ}

lim_{t→0+}[A(t), ψ(t)]_{1}^{ϵ} = [Ã, ψ̃]_{1}^{ϵ}

[A(1), ψ(1)]_{1}^{ϵ} = [A_{j}^{+}(ϵ), ψ_{j}^{+}(ϵ)].

These paths induce a continuous, piecewise smooth path

a_{1}^{ϵ}(t) ⊂ ∂_{∞}(M_{V}^{∗}) ⊂ χ_{0}(T^{2}, V)

satisfying

a_{1}^{ϵ}(t) = ∂_{∞}[A(t), ψ(t)]_{1}^{ϵ},

with

a_{1}^{ϵ}(−1) = a_{∞,i}^{−} a_{1}^{ϵ}(0) = a_{∞}(ϵ) a_{1}^{ϵ}(1) = a_{∞,j}^{+}(ϵ).

As ϵ → 0, these geometric limits define paths [A(t), ψ(t)] and a(t) with similar properties, and with

a(−1) = a_{∞,i}^{−} a(0) = a_{∞} a(1) = a_{∞,j}^{+},

in ∂_{∞}(M_{V}^{∗}) ⊂ χ_{0}(T^{2}, V), with a_{∞} = lim_{ϵ} a_{∞}(ϵ).

(c) Moreover, we have a holomorphic triangle in H^{1}(T^{2}, ℝ) with vertices

{a_{∞,i}^{−}, ϑ_{1}, a_{∞,j}^{+}(ϵ)}

and sides given by parameterized arcs along the lines ℓ_{1} = {v − u = 1}, ℓ_{μ} = {v = f′(u)} and by

{a_{1}^{ϵ}(t)} ⊂ ℓ = ∂_{∞}(M_{V}^{∗}).

This is obtained from the non-uniform limits of Proposition 3.1.

We now describe how to assemble together these geometric limits in a suitable moduli space. This will be useful in the following subsection, in the proof of Theorem 6.9 that establishes the exactness in the middle term.

Let a_{∞} be an element in χ_{0}(T^{2}, V). We define the configuration space

A_{k,δ}(V × ℝ, a_{∞})

as follows.

We can write a pair (A, Ψ) of a U(1)-connection and a spinor on V × ℝ in the form

A = a(w, s, t) + f(w, s, t)ds + h(w, s, t)dt

in the region T^{2} × [0, ∞) × ℝ inside V × ℝ. A pair (A, Ψ) as above is in the configuration space A_{k,δ}(V × ℝ, a_{∞}) if (A, Ψ) is in L_{k}^{2} on V × ℝ. Moreover, we also require that, after the change of coordinates s + it = e^{ρ+iθ}, and the corresponding change of variables

a(w, ρ, θ) = a(w, e^{ρ+iθ})
f(w, ρ, θ) = e^{−ρ} cos θ h(w, e^{ρ+iθ}) − e^{−ρ} sin θ f(w, e^{ρ+iθ})
h(w, ρ, θ) = e^{−ρ} cos θ f(w, e^{ρ+iθ}) + e^{−ρ} sin θ h(w, e^{ρ+iθ}),

(31)

we have

(a − a_{∞}, h, f − f_{0}, α, β) ∈ L_{k,δ}^{2}(T^{2} × [−π/2, π/2] × [ρ_{0}, ∞)),

where f_{0} is a constant. The L_{k,δ}^{2} norm we consider is defined by

∥F ∥_{L_{k,δ}^{2}} = ∥e_{δ} · F ∥_{L_{k}^{2}},

where e_{δ} is a smooth non-negative function satisfying

e_{δ}(w, ρ, θ) = exp(δe^{ρ}),

for (w, ρ, θ) in the range

T^{2} × [ρ_{0}, ∞) × [−π/2, π/2].

We have a group of gauge transformations acting on this configuration space, namely the group

G_{V ×ℝ,k+2,δ}

given by maps λ : V × ℝ → U(1) of the form λ = e^{iℓ} with ℓ : V × ℝ → ℝ, such that ℓ − ℓ_{∞} is in L_{k+2,δ}^{2} on the domain

T^{2} × [ρ_{0}, ∞) × [−π/2, π/2] ⊂ V × ℝ.  (32)

Here λ_{∞} = e^{iℓ_{∞}} is a gauge transformation on T^{2} that extends to V, that is, λ_{∞} ∈ G_{V}.

Recall that, by the analysis of [17], any finite energy solution (A, Ψ) on V × ℝ has the property that in radial gauge, (A, Ψ) has limits

a_{∞}(θ) = λ(θ)a_{∞},

with θ ∈ [−π/2, π/2] and with λ(θ) a family of gauge transformations on T^{2} that extend to gauge transformations on V, and [a_{∞}] a fixed gauge class of flat connections on T^{2}. With a slight abuse of notation, we write a_{∞} for this gauge class. Thus, we can represent all finite energy solutions by elements in some configuration space A_{k,δ}(V × ℝ, a_{∞}).

Given a fixed element Γ_{a_{∞}} = (A, Ψ) in the configuration space A_{k,δ}(V × ℝ, a_{∞}), we define the slice at Γ_{a_{∞}} as

S_{Γ_{a∞}} = {(ã, Φ̃) ∈ A_{k,δ}(V × ℝ, a_{∞})|G_{Γ_{a∞}}^{∗} (ã, Φ̃) = 0},

where the operator G_{Γ_{a∞}}^{∗} is the L_{δ}^{2} adjoint of the infinitesimal gauge action G_{Γ_{a∞}}. In the region (32) the elements (ã, Φ̃) can be written in the form

(ã, Φ̃) = (a, h, f, α, β),

with

ã = a + f ds + h dt and Φ̃ = (α, β),

as above.

We define the moduli space M_{V ×ℝ}(a_{∞}) as the set of solutions of the Seiberg-Witten equations in A_{k,δ}(V ×ℝ, a_{∞}), modulo the action of the gauge group G_{V ×ℝ,k+2,δ}. We denote by M̂_{V ×ℝ}(a_{∞}) the balanced energy moduli space, namely the elements in M_{V ×ℝ}(a_{∞}) satisfying

\begin{matrix}∫_{−∞}^{0}(∥∂_{t}A(t)∥_{L^{2}(V)}^{2} + ∥∂_{t}ψ(t)∥_{L^{2}(V)}^{2})dt = \\ ∫_{0}^{+∞}(∥∂_{t}A(t)∥_{L^{2}(V)}^{2} + ∥∂_{t}ψ(t)∥_{L^{2}(V)}^{2})dt, \end{matrix}

for a temporal gauge representative (A(t), ψ(t)).

The linearization L_{(A,Ψ)} at a solution (A, Ψ) in the slice S_{Γ_{a∞}} is given by the operator

L_{(A,Ψ)}(ã, Φ̃) = \{ \begin{matrix}d^{+}ã − \frac{1}{2}Im(Ψ · Φ̃) \\ D_{A}Φ̃ + ã.Ψ \\ G_{Γ_{a∞}}^{∗} (ã, Φ̃) \end{matrix}  (33)

The virtual dimension of the moduli space M̂_{V ×ℝ}(a_{∞}) at a solution (A, Ψ) in the slice S_{Γ_{a∞}} is given by

Index(L_{(A,Ψ)}).

Assuming that the element a_{∞} ≠ ϑ is away from the bad point ϑ in the character variety of T^{2}, we know, by the result of Section 3.2 of [17], that all finite energy solutions (A, Ψ) have a uniform exponential decay in radial gauge, hence they can be regarded as elements of the moduli spaces M̂_{V ×ℝ}(a_{∞}) introduced here, for some δ which depends only on a_{∞}.

Recall also that in Proposition 3.4 of Part II we proved that, given a finite energy solution (A, Ψ) in A_{k,δ}(V × ℝ, a_{∞}) of the Seiberg-Witten equations on V × ℝ, with asymptotic value in the gauge class of a_{∞}, by applying a gauge transformation λ(θ) in G_{V ×ℝ,k+2,δ}, we obtain a solution λ(A, Ψ) with h ≡ 0, and f − f_{0} ≡ 0. In particular, when inverting the change of variables (31), the condition that λ(A, Ψ) is in L_{k}^{2} gives f_{0} ≡ 0. In particular the resulting solution is in a temporal gauge, hence we can write λ(A, Ψ) as (A(t), ψ(t)). Thus, given a solution (A, Ψ) in A_{k,δ}(V × ℝ, a_{∞}), up to gauge transformations, we obtain two classes [A, ψ] and [Ã, ψ̃] in

∂_{∞}^{−1}(a_{∞}) ⊂ M_{V}^{∗}

defined by the asymptotic values as t → ±∞ of the temporal gauge representative (A(t), ψ(t)) = λ(A, Ψ), cf. the result of Lemma 3.9 of [17].

Thus, we can break the moduli space M̂_{V ×ℝ}(a_{∞}) into a union of components

M̂_{V ×ℝ}(a_{∞}) = ⋃_{[A,ψ],[Ã,ψ̃]} M̂_{V ×ℝ}([A, ψ], [Ã, ψ̃], a_{∞}),

with

[A, ψ], [Ã, ψ̃] ∈ ∂_{∞}^{−1}(a_{∞}) ⊂ M_{V}^{∗} .

We can rephrase the calculation of the virtual dimension as follows.

Proposition 6.3 Let [A, Ψ] be a gauge class in

M̂_{V ×ℝ}(a_{∞}).

Let (A(t), ψ(t)) be a temporal gauge representative of [A, Ψ], which satisfies

lim_{t→−∞}[A(t), ψ(t)] = [A, ψ],

lim_{t→∞}[A(t), ψ(t)] = [Ã, ψ̃].

Let H_{A(t),ψ(t)} be the operator

H_{A(t),ψ(t)} = \{ \begin{matrix}L_{A(t),ψ(t)}(α, ϕ) + G_{A(t),ψ(t)}(f) \\ G_{A(t),ψ(t)}^{∗}(α, ϕ), \end{matrix}

with

L_{A(t),ψ(t)}(α, ϕ) = (∗_{3}dα − 2iIm(ψ(t), ϕ), ∂_{A(t)}ϕ + α · ψ(t)),
G_{A(t),ψ(t)}(f) = (−df, e^{if}ψ(t)),

and G_{A(t),ψ(t)}^{∗} is the L_{δ}^{2} adjoint of G_{A(t),ψ(t)}. Let Q_{a_{∞}} be the asymptotic operator of H_{A(t),ψ(t)}, for each t, given by

Q_{a_{∞}}(a, α, β) = (0, −i¯∂_{a_{∞}}^{∗}β, i¯∂_{a_{∞}}α),

as in [6].

Each moduli space

M_{V ×ℝ}([A, ψ], [Ã, ψ̃], a_{∞}),

for a fixed choice of [A, ψ] and [Ã, ψ̃] in ∂_{∞}^{−1}(a_{∞}) in M_{V}^{∗}, is a smooth finite dimensional oriented manifold, of dimension given by the spectral flow

SF (H_{A(t),ψ(t)}) = Index(∂_{t} + H_{A(t),ψ(t)}).

Thus, for the balanced energy case, we have virtual dimension

virtdimM̂_{V ×ℝ}([A, ψ], [Ã, ψ̃], a_{∞}) = SF (H_{A(t),ψ(t)}) − 1.

An orientation of M̂_{V ×ℝ}([A, ψ], [Ã, ψ̃], a_{∞}) is obtained by considering the determinant line bundle of the operator L_{A,Ψ} = ∂_{t} + H_{A(t),ψ(t)}.

6.2 Admissible elements in M̂_{V ×ℝ}(a_{∞})

Now consider a(t) a regular parameterization of an arc in ∂_{∞}(M_{V}^{∗}) inside χ_{0}(T^{2}, V), which satisfies

a(−1) = a^{−} a(0) = a_{∞} a(1) = a^{+},

with given

a^{−} ∈ ∂_{∞}(M_{V}^{∗}) ∩ {u − v = 1}
a^{+} ∈ ∂_{∞}(M_{V}^{∗}) ∩ {v = f′(u)}.

Let us assume that the path a(t) also satisfies the condition a(t) ≠ ϑ, for all t ∈ [−1, 1], and that the path a(t) avoids all the boundary points of ∂_{∞}(M_{V}^{∗}) on χ(V)\ϑ, cf. [6]. By the analysis of [6], we know then that the fiber

∂_{∞}^{−1}(a(t)) ⊂ M_{V}^{∗}

is a finite set of points, for each fixed t. Moreover, under the current hypotheses, the set

∪_{t∈[−1,1]}∂_{∞}^{−1}(a(t))

describes a cobordism between

∂_{∞}^{−1}(a(−1)) and ∂_{∞}^{−1}(a(0))

and between

∂_{∞}^{−1}(a(0)) and ∂_{∞}^{−1}(a(1)).

Now suppose given two assigned elements

[A^{−}, ψ^{−}] ∈ ∂_{∞}^{−1}(a^{−})
[A^{+}, ψ^{+}] ∈ ∂_{∞}^{−1}(a^{+}),

and elements

[A, ψ], [Ã, ψ̃] ∈ ∂_{∞}^{−1}(a_{∞}).

In the cobordism

∪_{t∈[−1,1]}∂_{∞}^{−1}(a(t)),

there is at most one path [A(t), ψ(t)] in M_{V}^{∗}, for t ∈ [−1, 0] that satisfies ∂_{∞}[A(t), ψ(t)] = a(t) and [A(−1), ψ(−1)] = [A^{−}, ψ^{−}]. Similarly, there is at most one path [A(t), ψ(t)] in M_{V}^{∗}, for t ∈ [0, 1] that satsifies ∂_{∞}[A(t), ψ(t)] = a(t) and [A(1), ψ(1)] = [A^{+}, ψ^{+}], cf. the Figure 3. The case (d) of Figure 3 illustrates an example of a parameterization a(t) and a choice of [A^{−}, ψ^{−}] and [A^{+}, ψ^{+}] for which a path with the desired properties does not exist. Such case does not arise as a geometric limit.

Thus, we can introduce the following notation, to distinguish which choices of elements ([A, ψ], [Ã, ψ̃], a_{∞}) can arise as part of the geometric limits of solutions in M_{ℓ}^{W_{1}(r)}(a_{1}, a) (or M_{k}^{W_{0}(r)}(a_{1}, a_{0}), or M_{Y (r)×ℝ}(a, b)).

Definition 6.4 We say that a triple ([A, ψ], [Ã, ψ̃], a_{∞}) is admissible, with respect to the endpoints (a_{1}, a), if the following conditions hold. The element a_{∞} lies on a path component of ∂_{∞}(M_{V}^{∗}) connecting a^{−} and a^{+}. Moreover, there exists a smooth regular parameterization a(t), for t ∈ [−1, 1] of the path in ∂_{∞}(M_{V}^{∗}) connecting a^{−} and a^{+}, such that a(0) = a_{∞}, and corresponding smooth paths [A(t), ψ(t)] in M_{V}^{∗}, for t ∈ [−1, 0) and t ∈ (0, 1], satisfying ∂_{∞}[A(t), ψ(t)] = a(t), and with

[A(−1), ψ(−1)] = [A^{−}, ψ^{−}] and lim_{t→0_{−}}[A(t), ψ(t)] = [A, ψ]

lim_{t→0_{+}}[A(t), ψ(t)] = [Ã, ψ̃] and [A(1), ψ(1)] = [A^{+}, ψ^{+}],

with [A, ψ] and [Ã, ψ̃] in ∂_{∞}^{−1}(a_{∞}) in M_{V}^{∗} and

a_{1} = [(A^{−}, ψ^{−})#(a^{−}, 0)]
a = [(A^{+}, ψ^{+})#(a^{+}, 0)].

In the examples of Figure 3, the cases (a)-(b) represent admissible elements, and case (c) is not an admissible element, because the parameterization a(t) = ∂_{∞}[A(t), ψ(t)] is not regular (∂_{t}a(t) = 0 for some t), and the case (d) is also non admissible because no path [A(t), ψ(t)] with the desired properties exists. An element ([A, ψ], [Ã, ψ̃], a_{∞}) can appear as part of the geometric limits of solutions on W_{1}(r) (or W_{0}(r), or Y (r) × ℝ, etc.) only if it is admissible with respect to the endpoints (a_{1}, a) (or (a, a_{0}), or (a, b)).

Notice that, in general, if we consider different solutions in M_{ℓ}^{W_{1}(r)}(a_{1}, a) (or M_{k}^{W_{0}(r)}(a_{1}, a_{0}), or M_{Y (r)×ℝ}(a, b)), these will give rise to geometric limits with different parameterizations a(t), and in general different a_{∞}. In the next subsection we describe how to assemble the various geometric limits.

6.3 Assembling the geometric limits

Let us first return to the setting of [17] and consider moduli spaces of flowlines. We shall then generalize our statements to the case of finite energy solutions on one of the surgery cobordisms.

Consider first the case of a zero-dimensional moduli space of flow lines, M̂_{Y ×ℝ,μ}(a, b), with a, b ∈ M_{Y,μ}^{∗} satisfying deg_{Y,μ}(a) − deg_{Y,μ}(b) = 1. We recall the results on the splitting of the spectral flow that we used in Part I [6] in order to compare the relative gradings, cf. [7].

We can describe the critical points a and b in M_{Y,μ}^{∗} as

a = [(A^{−}, ψ^{−})#_{r}(a^{−}, 0)]
b = [(A^{+}, ψ^{+})#_{r}(a^{+}, 0)].

Figure 3: The paths [A(t), ψ(t)] in M_{V}^{∗}

The relative grading is given by

1 = deg_{Y,μ}(a) − deg_{Y,μ}(b) = \frac{1}{r^{2}} SF_{Y (r)}(H_{(A_{r}(t),ψ_{r}(t))}),

with the notation as in [6], [3]. Then the spectral flow can be written as

\begin{matrix}\frac{1}{r^{2}} SF_{Y (r)}(H_{(A_{r}(t),ψ_{r}(t))}) = ϵSF_{V (r)}(H_{(A(t),ψ(t)),ℓ̃_{V} (t)})+ \\ ϵSF_{ν(K)(r)}(H_{(a′′(t),0),ℓ̃_{ν}(t)}) + Maslov(ℓ̃_{V} (t), ℓ̃_{ν}(t)) \end{matrix}  (34)

Here we follow the same convention of [3] regarding the definition of the ϵspectral flow. The boundary conditions are prescribed by assigning a choice of Lagrangian subspaces (ℓ̃_{V} (t), ℓ̃_{ν}(t)) in H^{1}(T^{2}, ℝ).

We consider the following Lagrangians, which we already introduced in Part I, [6]. Let ℓ_{μ}^{∗} be the piecewise smooth Lagrangian submanifold of χ_{0}(T^{2}, Y) described in Part I, [6], and let ℓ̃_{μ} be the path of Lagrangian subspaces of H^{1}(T^{2}, ℝ) defined as in Part I, [6], which is given by the tangent spaces of ℓ_{μ}^{∗} where the latter is smooth, completed with a specific choice of paths at the singular points, as discussed in [6]. Let ℓ be the union of the arcs in the Lagrangian submanifold with boundary ∂_{∞}(M_{V}^{∗}) in χ_{0}(T^{2}, Y) that connect the points a^{−} and a^{+}. By ∂_{∞}(M_{V}^{∗}) ⊂ χ_{0}(T^{2}, Y) we mean the pullback of ∂_{∞}(M_{V}^{∗}) ⊂ χ_{0}(T^{2}, V) under the covering map χ_{0}(T^{2}, Y) → χ_{0}(T^{2}, V). Under the assumption that these arcs avoid the boundary points ∂_{∞}(∂M_{V}^{∗}), we can consider regular parameterizations a′(t) and a′′(t) of the arcs of ℓ_{μ}^{∗} and ℓ, respectively, connecting a^{−} to a^{+}. We consider the corresponding paths of Lagrangians ℓ̃(t) = T_{a′(t)}ℓ and ℓ̃_{μ}(t), for t ∈ [0, 1]. We can also assume that ℓ_{μ}^{∗} and ℓ intersect transversely.

Now recall that, in Part II, we identified the geometric limits of flowlines in M̂_{Y (r)×ℝ}(a, b) with finite energy solutions in M̂_{V ×ℝ}([A, ψ], [Ã, ψ̃], a_{∞}), together with paths [A(t), ψ(t)] ∈ M_{V}^{∗}, and a holomorphic disk

Δ : (D^{2}, ∂D^{2}) → (χ_{0}(T^{2}, Y), ℓ ∪ ℓ_{μ}^{∗}),

which determines the regular parameterization a(t) of the arcs of ℓ and ℓ_{μ}^{∗}, hence the admissible data ([A, ψ], [Ã, ψ̃], a_{∞}). Thus, we are interested in understanding the space of inequivalent holomorphic disks Δ, which can appear in the geometric limits. We have the following result, see for instance [12] pg.3. The set of equivalence classes under Aut(D^{2}) ≃ P SL(2, ℝ) of holomorphic disks

Δ : (D^{2}, ∂D^{2}) → (χ_{0}(T^{2}, Y), ℓ ∪ ℓ_{μ}^{∗}),

in a given homotopy class β = [Δ] in π_{2}(χ_{0}(T^{2}, Y), ℓ ∪ ℓ_{μ}^{∗}) has virtual dimension

μ(β) − 1,

with μ(β) the Maslov class, defined as in [12] pg. 3. The original formula given in [12] for this virtual dimension is n + μ(β) − 2, where 2n is the real dimension of the ambient symplectic manifold. In our case, we have n = 1. We can describe the Maslov class μ(β) in terms of the Maslov index Maslov(ℓ̃(t), ℓ̃_{μ}(t)) as follows.

Lemma 6.5 Consider the piecewise smooth Lagrangian submanifolds ℓ and ℓ_{μ}^{∗} in χ_{0}(T^{2}, Y), defined as above. Consider all possible holomorphic disks

Δ : (D^{2}, ∂D^{2}) → (χ_{0}(T^{2}, Y), ℓ ∪ ℓ_{μ}^{∗}),

up to automorphisms Aut(D^{2}) ≃ P SL(2, ℝ), which map the boundary to arcs of the Lagrangians connecting two points a^{−} and a^{+} in ℓ ∩ ℓ_{μ}^{∗}, in the homotopy class β specified by the regular parameterizations a′(t) and a′′(t) of these arcs of Lagrangians. Then the Maslov class satisfies

μ(β) = Maslov(ℓ̃(t), ℓ̃_{μ}(t)) + 1.

Proof. Here we follow the notation of Section 13 of [4]. First let us define

ν_{±}^{a^{±}}(T_{a^{±}}ℓ_{1}, T_{a^{±}}ℓ_{2})

as the path of Lagrangian subspaces that connects the two Lagrangian specified subspaces (T_{a^{+}}ℓ_{1}, T_{a^{+}}ℓ_{2}), rotating in the positive or negative direction, according to sign. Recall that, given a pair of Lagrangian paths f(t) = (ℓ̃_{1}(t), ℓ̃_{2}(t)), we use the notation f̂(t) = f(1 − t), and

f_{flip}(t) = (ℓ̃_{2}(t), ℓ̃_{1}(t)).

Thus, given f(t) = (ℓ̃_{1}(t), ℓ̃_{2}(t)) with ℓ_{1}(0) = ℓ_{2}(0) = a^{−} and ℓ_{1}(1) = ℓ_{2}(1) = a^{+}, we obtain four possible loops f_{±±}(t), given as the concatenation of paths

f_{ij}(t) = ℓ̃_{1}(t) ∗ ν_{i}^{a^{+}}(t) ∗ ˆ ℓ̃_{2}(t) ∗ (ν_{j}^{a^{−}}(t))_{flip},

with i, j = ±.

The definition of the Maslov class given in [12] pg. 3, for one smooth Lagrangian submanifold ℓ inside a symplectic manifold X and disks Δ : (D^{2}, ∂D^{2}) → (X, ℓ), coincides with the notion of Maslov index for closed loops à la Floer [10]. Using the results of Section 13 of [4], we can relate this to the Maslov index, as described in the Lemma. Since the arguments of Part II, [17] actually ensure the existence of a holomorphic disk on a slightly larger domain, which maps a subdomain homeomorphic to (D^{2}, ∂D^{2}) to a disk filling a region bounded by arcs of the Lagrangians ℓ∪ℓ_{μ}^{∗} with endpoints a^{−} and a^{+}, we obtain that the Maslov class is computed by

μ(β) = μ(f_{++}(t)),

with the notation as above, and μ(f_{++}(t)) the Maslov index of a loop of Lagrangians à la Floer, cf. Section 13 of [4], where we have

f(t) = (ℓ̃(t), ℓ̃_{μ}(t)).

Now the result of Section 13 of [4] gives

μ(f_{++}(t)) = Maslov(ℓ̃(t), ℓ̃_{μ}(t)) + 1.

⋄

Notice that we can think of these holomorphic disks equivalently as equivalence classes of

Δ : (D^{2}, ∂D^{2}) → (χ_{0}(T^{2}, Y), ℓ ∪ ℓ_{μ}^{∗})

that map Δ(−1) = a^{−} and Δ(+1) = a^{+}, modulo the subgroup of Aut(D^{2}) that fixes the points ±1, or then again, equivalently, as holomorphic maps Δ of an infinite strip

ℝ × [0, 1] → χ_{0}(T^{2}, Y)

that map the boundary ℝ × {0, 1} to arcs of the Lagrangians ℓ and ℓ_{μ}^{∗}, and with asymptotic values along t ∈ ℝ, as t → ∓∞, equal to a^{−} and a^{+}. In this case, we consider classes modulo the action of ℝ by reparameterizations.

Thus, the solutions in the zero-dimensional moduli space M̂_{Y ×ℝ,μ}(a, b) are obtained by gluing solutions in a zero-dimensional moduli space

M̂_{V ×ℝ}([A, ψ], [Ã, ψ̃], a_{∞})

with a holomorphic disk

Δ : (D^{2}, ∂D^{2}) → (χ_{0}(T^{2}, Y), ℓ ∪ ℓ_{μ}^{∗}),

in a fixed homotopy class β = [Δ] in π_{2}(χ_{0}(T^{2}, Y), ℓ ∪ ℓ_{μ}^{∗}). The admissible data ([A, ψ], [Ã, ψ̃], a_{∞}) is determined by the holomorphic disk Δ, as we discuss in the following Lemma.

Lemma 6.6 Suppose given a zero-dimensional moduli space M̂_{Y ×ℝ,μ}(a, b). The space of pre-glued solutions, obtained by pasting together the geometric limits of solutions in M̂_{Y ×ℝ,μ}(a, b), as described in [17], is given by

⋃_{Δ∈Ξ} M̂_{V ×ℝ}([A, ψ], [Ã, ψ̃], a_{∞}).

where Δ is a choice of one particular representative in each equivalence class in the set Ξ of classes of holomorphic disks

Δ : (D^{2}, ∂D^{2}) → (χ_{0}(T^{2}, Y), ℓ ∪ ℓ_{μ}^{∗}),

in the fixed homotopy class [Δ] in π_{2}(χ_{0}(T^{2}, Y), ℓ ∪ ℓ_{μ}^{∗}), modulo the action of Aut(D^{2}) ≃ P SL(2, ℝ). Each such representative Δ determines the corresponding admissible element ([A, ψ], [Ã, ψ̃], a_{∞}).

Proof. The statement of the Lemma follows from the previous discussion. In order to see the dependence of the data ([A, ψ], [Ã, ψ̃], a_{∞}) on the holomorphic disk Δ, recall that the choice of Δ in particular fixes the parameterization a(t) of the arc of ℓ connecting a^{−} and a^{+}. This determines the point a_{∞} = a(0). However, the choice of the parameterization a(t) also determines the choice of the points [A, ψ] and [Ã, ψ̃] in ∂_{∞}^{−1}(a_{∞}). Notice that each solution (A(r), Ψ(r)), representing an element in M̂_{Y (r)×ℝ,μ}(a, b), determines a holomorphic disk

Δ = Δ(A(r), Ψ(r))

in the geometric limits. Each such disk determines a possibly different parameterization a(t), hence different admissible data

([A, ψ], [Ã, ψ̃], a_{∞}).

Thus, the set of pre-glued solutions can be written as

⋃_{Δ} M̂_{V ×ℝ}([A, ψ], [Ã, ψ̃], a_{∞}),

where Δ varies in the set of holomorphic disks

Δ : (D^{2}, ∂D^{2}) → (χ_{0}(T^{2}, Y), ℓ ∪ ℓ_{μ}^{∗}),

in the fixed homotopy class [Δ] in π_{2}(χ_{0}(T^{2}, Y), ℓ ∪ ℓ_{μ}^{∗}). However, we only count the disks Δ up to Aut(D^{2}) ≃ P SL(2, ℝ).

For geometric limits of a zero-dimensional moduli space, the set of such holomorphic disks, up to Aut(D^{2}) is also a zero-dimensional moduli space. This follows by the result of Lemma 6.5, in the case where we have

Maslov(ℓ̃(t), ℓ̃_{μ}(t)) = 0.

In fact, in our case, with disks in χ_{0}(T^{2}, Y), the homotopy class is fixed by the choice of the arcs of the Lagrangians ℓ and ℓ_{μ}^{∗} connecting a^{−} and a^{+}, and the representatives in each class only differ by reparameterizations of the boundary. A choice of a representative Δ in each class identifies uniquely a corresponding moduli space

M̂_{V ×ℝ}([A, ψ], [Ã, ψ̃], a_{∞}).

Thus, with the notation as above, we obtain

n_{Y,μ}(a, b) = ∑_{Δ} #M̂_{V ×ℝ}([A, ψ], [Ã, ψ̃], a_{∞}).

⋄

Thus, we can rephrase the gluing theorem of [17] as the following statement. Let us introduce the following notation:

M̂_{V ×ℝ}([A^{−}, ψ^{−}], [A^{+}, ψ^{+}]) := ⋃_{Δ} M̂_{V ×ℝ}([A, ψ], [Ã, ψ̃], a_{∞}),

with the admissible data ([A, ψ], [Ã, ψ̃], a_{∞}) and the corresponding holomorphic disk Δ as in Lemma 6.6.

Proposition 6.7 The gluing map gives an orientation preserving diffeomorphism

# : M̂_{V ×ℝ}([A^{−}, ψ^{−}], [A^{+}, ψ^{+}]) ≅ M̂_{Y ×ℝ}(a, b),

with

a = [(A^{−}, ψ^{−})#_{a^{−}}(a^{−}, 0)]
b = [(A^{+}, ψ^{+})#_{a^{+}}(a^{+}, 0)].

Now we can rephrase the result in the case of moduli spaces of finite energy solutions of the Seiberg–Witten equations on one of the surgery cobordisms. In this case, we can assemble the geometric limits in a similar way, to obtain spaces

M_{V ×ℝ}([A^{−}, ψ^{−}], [A^{+}(ϵ), ψ^{+}(ϵ)]) := ⋃_{Δ_{1}^{ϵ}} M_{V ×ℝ}([A, ψ]_{1}^{ϵ}, [Ã, ψ̃]_{1}^{ϵ}, a_{∞}^{1}(ϵ)), (35)

in the case of W_{1}, with the admissible data

([A, ψ]_{1}^{ϵ}, [Ã, ψ̃]_{1}^{ϵ}, a_{∞}^{1}(ϵ))

determined by the holomorphic triangle Δ_{1}^{ϵ}, or

M_{V ×ℝ}([A^{−}(ϵ), ψ^{−}(ϵ)], [A^{+}, ψ^{+}]) := ⋃_{Δ_{0}^{ϵ}} M_{V ×ℝ}([A, ψ]_{0}^{ϵ}, [Ã, ψ̃]_{0}^{ϵ}, a_{∞}^{0}(ϵ)), (36)

in the case of W_{0}, with the admissible data

([A, ψ]_{0}^{ϵ}, [Ã, ψ̃]_{0}^{ϵ}, a_{∞}^{0}(ϵ))

determined by the holomorphic triangle Δ_{0}^{ϵ}. Here the Δ_{i}^{ϵ} vary in the set of inequivalent holomorphic triangles in H^{1}(T^{2}, ℝ), namely

Δ_{1}^{ϵ} : (D^{2}, ∂D^{2}) → (H^{1}(T^{2}, ℝ), ℓ ∪ ℓ_{1} ∪ ℓ_{μ}),

in the case of W_{1}, or

Δ_{0}^{ϵ} : (D^{2}, ∂D^{2}) → (H^{1}(T^{2}, ℝ), ℓ ∪ ℓ_{μ} ∪_{k} ℓ_{k}),

in the case of W_{0}, with ℓ_{1} = {u − v = 1}, ℓ_{μ} = {v = f′(u)}, and ℓ_{k} = {u = 2k} or {u = η} in the case of ℓ_{0}, as in [6]. The image of each Δ_{i}^{ϵ} describes a triangle in H^{1}(T^{2}, ℝ) with vertices {a^{−}, ϑ_{i}, a^{+}(ϵ)} or {a^{−}(ϵ), ϑ_{i}, a^{+}} and sides along the Lagrangians, as specified. Here the points ϑ_{i} are the intersection of the lines ℓ_{1} and ℓ_{μ} for W_{1} and ℓ_{k} and ℓ_{μ} for W_{0}, that is, the restriction to T^{2} = ∂ν = ∂ν̃ of the unique reducible point θ_{S^{3}} at the puncture in the cobordism.

Thus, we obtain the following result on the gluing theorem for the moduli spaces M_{ℓ}^{W_{1}}(a_{1}, a), or for the minimal energy component of M_{k}^{W_{0}}(a, a_{0}). We state the result in the case of W_{1}.

Lemma 6.8 Suppose given a pair a_{1} ∈ M_{Y_{1}}, and a ∈ M_{Y,μ}. Suppose that we have the decomposition

a_{1} = [(A^{−}, ψ^{−})#_{a^{−}}(a^{−}, 0)]
a = [(A^{+}(ϵ), ψ^{+}(ϵ))#_{a^{+}(ϵ)}(a^{+}(ϵ), 0)].

Then the gluing map gives an orientation preserving diffeomorphism

# : M_{V ×ℝ}([A^{−}, ψ^{−}], [A^{+}(ϵ), ψ^{+}(ϵ)]) → M_{ℓ}^{W_{1}}(a_{1}, a),

where the first moduli space is defined as in (35), with the union over inequivalent holomorphic triangles Δ with vertices {a^{−}, ϑ_{1}, a^{+}(ϵ)} and sides along the union of Lagrangians ℓ ∪ ℓ_{1} ∪ ℓ_{μ}, with ℓ defined by the asymptotic values ∂_{∞}(M_{V}^{∗}). Different choices of the Spin^{c}-structure 𝔰_{ℓ} correspond to moduli spaces of different dimension. If M_{ℓ}^{W_{1}}(a_{1}, a) is non-empty, then, under the gluing map we obtain

ι(W_{1}, 𝔰_{ℓ}, a_{1}, a) = μ([Δ]) − 1 + virtdimM_{V ×ℝ}([A, ψ], [Ã, ψ̃], a_{∞}).

Again we can observe that each holomorphic triangle fixes a parameterization a(t), hence a choice of the admissible data ([A, ψ], [Ã, ψ̃], a_{∞}). The argument then proceeds as in the case of flow lines. In the case of the moduli spaces M_{k}^{W_{0}}(a, a_{0}), the minimal energy condition ensures that the path a(t) is contractible in χ_{0}(T^{2}, V), so that the path along the union of Lagrangians can be filled by a holomorphic triangle of minimal energy.

We now return to our analysis of the sequence

0 → C_{∗}(Y_{1}) →^{w_{∗}^{1}} C_{∗}(Y, μ) →^{w_{∗}^{0}} ⨁_{k} C_{(∗)}(Y_{0}, 𝔰_{k}) → 0.

In the next subsection, we use the results obtained in this section on the moduli spaces M_{V ×ℝ}([A^{−}, ψ^{−}], [A^{+}, ψ^{+}]) to prove exactness in the middle term.

6.4 The relation w_{∗}^{0} ∘ w_{∗}^{1} = 0

By Lemma 6.1, we can show that the composite map w_{∗}^{0} ∘ w_{∗}^{1} is trivial by proving the following result.

Theorem 6.9 For small enough ϵ and large r ≥ r_{0}, there is an orientation reversing diffeomorphism

M_{ℓ}^{W_{1}(r)}(a_{i}^{(1)}, a_{j}^{(0)}(ϵ)) ≅ M_{k}^{W_{0}(r), (0)}(a_{i}^{(1)}(ϵ), a_{j}^{(0)}),

where ℓ is the unique positive Spin^{c} structure on W_{1}(r) such that the moduli space M_{ℓ}^{W_{1}(r)}(a_{i}^{(1)}, a_{j}^{(0)}(ϵ)) is zero dimensional, and M_{k}^{W_{0}(r), (0)}(a_{i}^{(1)}(ϵ), a_{j}^{(0)})

is the zero-dimensional components of M_{k}^{W_{0}(r)}(a_{i}^{(1)}(ϵ), a_{j}^{(0)}).

Proof. By our assumptions, the moduli spaces M_{ℓ}^{W_{1}(r)}(a_{i}^{(1)}, a_{j}^{(0)}(ϵ)) and

M_{k}^{W_{0}(r), (0)}(a_{i}^{(1)}(ϵ), a_{j}^{(0)}) are smooth, compact, oriented 0-dimensional man-

ifolds. We write the critical points a_{i}^{(1)}, a_{j}^{(0)}(ϵ) and a_{i}^{(1)}(ϵ), a_{j}^{(0)} according to the following decomposition:

a_{i}^{(1)} = [(A_{i}(r), ψ_{i}(r))] = [(A_{i}^{−}, ψ_{i}^{−})#_{a_{∞,i}^{−}}(a_{∞,i}^{−}, 0)]

on Y_{1}(r),

\begin{matrix}a_{i}^{(1)}(ϵ) = j([(A_{i}(r), ψ_{i}(r))]) = [(A_{i}^{−}(ϵ), ψ_{i}^{−}(ϵ))#_{a_{∞,i}^{−}(ϵ)}(a_{∞,i}^{−}(ϵ), 0)] \\ a_{j}^{(0)}(ϵ) = [(A_{j}^{+}(ϵ), ψ_{j}^{+}(ϵ))#_{a_{∞,j}^{+}(ϵ)}(a_{∞,j}^{+}(ϵ), 0)] \end{matrix}

on Y (r), and

a_{j}^{(0)} = [(A_{j}^{+}, ψ_{j}^{+})#_{a_{∞,j}^{+}}^{r}(a_{∞,j}^{+}, 0)]

on Y_{0}(r). In the limit ϵ → 0, the asymptotic values satisfy

lim_{ϵ→0} a_{∞,i}^{−}(ϵ) = a_{∞,i}^{−}

lim_{ϵ→0} a_{∞,j}^{+}(ϵ) = a_{∞,j}^{+},

in H^{1}(T^{2}, iℝ) and

lim_{ϵ→0}[A_{i}^{−}(ϵ), ψ_{i}^{−}(ϵ)] = [A_{i}^{−}, ψ_{i}^{−}]

lim_{ϵ→0}[A_{j}^{+}(ϵ), ψ_{j}^{+}(ϵ)] = [A_{j}^{+}, ψ_{j}^{+}]

in M_{V}^{∗}.

We now apply the geometric limits results of Proposition 3.1 to study the moduli spaces M_{ℓ}^{W_{1}(r)}(a_{i}^{(1)}, a_{j}^{(0)}(ϵ)) and M_{k}^{W_{0}(r), (0)}(a_{i}^{(1)}(ϵ), a_{j}^{(0)}).

Let [A_{1}(r), Ψ_{1}(r)] be solutions in M_{ℓ}^{W_{1}(r)}(a_{i}^{(1)}, a_{j}^{(0)}(ϵ)), for sufficiently large r ≥ r_{0}. Then these solutions define geometric limits as recalled in Remark 6.2, cf. Proposition 3.1.

For all sufficiently small ϵ, the geometric limits listed in Proposition 3.1 can be grouped together to give a holomorphic triangle

Δ_{1}^{ϵ} : (D^{2}, ∂D^{2}) → (H^{1}(T^{2}, ℝ), ℓ ∪ ℓ_{1} ∪ ℓ_{μ})

with vertices {a_{∞,i}^{−}, ϑ_{1}, a_{∞,j}^{+}(ϵ)} and sides along the Lagrangians ℓ_{1} = {v − u = 1}, ℓ = ∂_{∞}M_{V}^{∗}, and ℓ_{μ} = {v = f′(u)}, and an element in a moduli space

M_{V ×ℝ}([A, ψ]_{1}^{ϵ}, [Ã, ψ̃]_{1}^{ϵ}, a_{∞}^{1}(ϵ)),

for some admissible data

([A, ψ]_{1}^{ϵ}, [Ã, ψ̃]_{1}^{ϵ}, a_{∞}^{1}(ϵ))

determined by the holomorphic triangle Δ_{1}^{ϵ}.

We use the notation Δ_{1}^{ϵ} = Δ^{ϵ}(A_{1}(r), Ψ_{1}(r)), for any such holomorphic triangle determined by a family of solutions (A_{1}(r), Ψ_{1}(r)) representing an element in

M_{ℓ}^{W_{1}(r)}(a_{i}^{(1)}, a_{j}^{(0)}(ϵ)).

The orientation of this triangle is the same as the orientation of the region of non-uniform convergence. The orientation of

M̂_{V ×ℝ}([A, ψ]_{1}^{ϵ}, [Ã, ψ̃]_{1}^{ϵ}, a_{∞}^{1}(ϵ))

at a solution [A′, Ψ′]^{ϵ} is given by the determinant line bundle of the linearization L_{(A′,Ψ^{′ϵ})}, as discussed previously.

Now we can proceed to compare the geometric limits of solutions in M_{ℓ}^{W_{1}(r)}(a_{i}^{(1)}, a_{j}^{(0)}(ϵ)) and M_{k}^{W_{0}(r), (0)}(a_{i}^{(1)}(ϵ), a_{j}^{(0)}).

Recall that, as discussed previously (cf. also [17]), we can pre-glue the geometric limits to form an approximate monopole on W_{1}(r), for sufficiently large r ≥ r_{0}. That is, as discussed in the previous subsection, a holomorphic triangle Δ determines the regular parameterizations of the arcs of Lagrangians, hence the admissible data ([A, ψ], [Ã, ψ̃], a_{∞}). Therefore, the space of pre-glued monopoles can be identified with the moduli space

M_{V ×ℝ}([A_{i}^{−}, ψ_{i}^{−}], [A_{j}^{+}(ϵ), ψ_{j}^{+}(ϵ)]) = ⋃_{Δ_{1}^{ϵ}} M_{V ×ℝ}([A, ψ]_{1}^{ϵ}, [Ã, ψ̃]_{1}^{ϵ}, a_{∞}^{1}(ϵ)),

with the admissible data specified by

Δ_{1}^{ϵ} = Δ^{ϵ}(A_{1}(r), Ψ_{1}(r)),

ranging over the set of inequivalent holomorphic triangles in H^{1}(T^{2}, ℝ), with vertices {a_{∞,i}^{−}, ϑ_{1}, a_{∞,j}^{+}(ϵ)} and sides along arcs of the Lagrangians ℓ, ℓ_{1}, and ℓ_{μ} connecting these points. The orientation on the moduli space M̂_{V ×ℝ}([A_{i}^{−}, ψ_{i}^{−}], [A_{j}^{+}(ϵ), ψ_{j}^{+}(ϵ)]) is the product orientation of the pairs

([A′, Ψ′]^{ϵ}, Δ_{1}^{ϵ})

with [A′, Ψ′]^{ϵ} an element of the moduli space

M̂_{V ×ℝ}([A, ψ]_{1}^{ϵ}, [Ã, ψ̃]_{1}^{ϵ}, a_{∞}^{1}(ϵ))

determined by the corresponding triangle Δ_{1}^{ϵ}. The gluing map gives an orientation preserving diffeomorphism

M_{ℓ}^{W_{1}(r)}(a_{i}^{(1)}, a_{j}^{(0)}(ϵ)) ≅ M_{V ×ℝ}([A_{i}^{−}, ψ_{i}^{−}], [A_{j}^{+}(ϵ), ψ_{j}^{+}(ϵ)]),  (37)

as in Lemma 6.8 in the previous subsection. Moreover, by an analogous argument, we have a similar result for the zero-dimensional component M_{k}^{W_{0}(r), (0)}(a_{i}^{(1)}(ϵ), a_{j}^{(0)}).

Claim: For M_{k}^{W_{0}(r), (0)}(a_{i}^{(1)}(ϵ), a_{j}^{(0)}), the zero-dimensional component of

M_{k}^{W_{0}(r)}(a_{i}^{(1)}(ϵ), a_{j}^{(0)}), there is no energy loss in the process of stretching along the r-direction. In this case, we have a similar orientation preserving diffeomorphism

M_{k}^{W_{0}(r), (0)}(a_{i}^{(1)}(ϵ), a_{j}^{(0)}) ≅ M̂_{V (∞)×ℝ}([A_{i}^{−}(ϵ), ψ_{i}^{−}(ϵ)], [A_{j}^{+}, ψ_{j}^{+}]). (38)

Here the moduli space of pre-glued solutions is given by

M_{V ×ℝ}([A_{i}^{−}(ϵ), ψ_{i}^{−}(ϵ)], [A_{j}^{+}, ψ_{j}^{+}] = ⋃_{Δ_{0}^{ϵ}} M_{V ×ℝ}([A, ψ]_{0}^{ϵ}, [Ã, ψ̃]_{0}^{ϵ}, a_{∞}^{0}(ϵ)),

with the admissible data determined by

Δ_{0}^{ϵ} = Δ^{ϵ}(A_{0}(r), Ψ_{0}(r)),

ranging over the set of inequivalent holomorphic triangles in H^{1}(T^{2}, ℝ), with vertices {a_{∞,i}^{−}(ϵ), ϑ_{0}, a_{∞,j}^{+}} and sides along arcs of the Lagrangians ℓ, ℓ_{μ}, and ∪_{k}ℓ_{k} connecting these points. Claim: As ϵ → 0, we can identify the moduli spaces

M_{V ×ℝ}([A_{i}^{−}, ψ_{i}^{−}], [A_{j}^{+}(ϵ), ψ_{j}^{+}(ϵ)])

and

M_{V ×ℝ}([A_{i}^{−}(ϵ), ψ_{i}^{−}(ϵ)], [A_{j}^{+}, ψ_{j}^{+}])

as sets of points.

In fact, if the triangles Δ_{1}^{ϵ} and Δ_{0}^{ϵ} are non-degenerate, for ϵ → 0, then these holomorphic triangles reduce to holomorphic triangles

Δ : (D^{2}, ∂D^{2}) → (H^{1}(T^{2}, ℝ), ℓ ∪ ℓ_{1} ∪_{k} ℓ_{k}),

in the same homotopy class in π_{2}(H^{1}(T^{2}, ℝ), ℓ ∪ ℓ_{1} ∪_{k} ℓ_{k}), with sides along arcs of the Lagrangians ℓ ∪ ℓ_{1} ∪_{k} ℓ_{k} connecting the points a_{∞,i}^{−}, a_{∞,j}^{+} and the intersection point between the lines ℓ_{1} and ℓ_{k}. Notice that the set theoretic difference between the triangles Δ_{1}^{ϵ} and Δ_{0}^{ϵ} shrinks to zero size as ϵ → 0, hence they define the same triangle Δ as a limit. Again we can count these up to automorphisms of D^{2}. Thus, in the limit as ϵ → 0 we can identify both moduli spaces with

M_{V ×ℝ}([A_{i}^{−}, ψ_{i}^{−}], [A_{j}^{+}, ψ_{j}^{+}]) = ⋃_{Δ} M_{V ×ℝ}([A, ψ], [Ã, ψ̃], a_{∞}),

with the admissible data determined by Δ. Notice that the parameterizations a(t) determined by the holomorphic triangles Δ are sufficiently close in the C^{∞} topology to the parameterizations a^{1}(t) and a^{0}(t) determined by the Δ_{1}^{ϵ} and Δ_{0}^{ϵ}, for sufficiently small ϵ, and the corresponding admissible data ([A, ψ], [Ã, ψ̃], a_{∞}) are also sufficiently close to the admissible data

([A, ψ]_{1}^{ϵ}, [Ã, ψ̃]_{1}^{ϵ}, a_{∞}^{1}(ϵ)) and ([A, ψ]_{0}^{ϵ}, [Ã, ψ̃]_{0}^{ϵ}, a_{∞}^{0}(ϵ)).

Thus, for sufficiently close data, the arguments of the previous subsections show that the moduli spaces

M_{V ×ℝ}([A, ψ], [Ã, ψ̃], a_{∞}) ≅
M_{V ×ℝ}([A, ψ]_{1}^{ϵ}, [Ã, ψ̃]_{1}^{ϵ}, a_{∞}^{1}(ϵ)) ≅
M_{V ×ℝ}([A, ψ]_{0}^{ϵ}, [Ã, ψ̃]_{0}^{ϵ}, a_{∞}^{0}(ϵ))

can also be identified.

Thus, we have obtained that the moduli spaces M_{ℓ}^{W_{1}(r)}(a_{i}^{(1)}, a_{j}^{(0)}(ϵ)) and

M_{k}^{W_{0}(r), (0)}(a_{i}^{(1)}(ϵ), a_{j}^{(0)}) can be identified as sets of points. This implies that the composite map satisfies w_{∗}^{0}∘w_{∗}^{1} ≡ 0 (mod 2). In order to obtain the result over the integers, we need to compare the orientations of these moduli spaces as they appear in the decomposition of M_{ℓ,k}^{W (r)}(a_{i}^{(1)}, a_{j}^{(0)}) in Lemma 4.4. Claim: In the decomposition

M_{ℓ,k}^{W (r)}(a_{i}^{(1)}, a_{j}^{(0)}) ≅ M_{ℓ}^{W_{1}(r)}(a_{i}^{(1)}, a_{j}^{(0)}(ϵ)) ∪ M_{k}^{W_{0}(r), (0)}(a_{i}^{(1)}(ϵ), a_{j}^{(0)}),

as in Lemma 4.4, the moduli spaces

M_{ℓ}^{W_{1}(r)}(a_{i}^{(1)}, a_{j}^{(0)}(ϵ)) and M_{k}^{W_{0}(r), (0)}(a_{i}^{(1)}(ϵ), a_{j}^{(0)})

have opposite orientations.

In fact, it is sufficient to notice that, when we glue the punctured cobordisms Ŵ_{1}(r) and Ŵ_{0}(r) along the common boundary Y, to obtain

Ŵ (r) = Ŵ_{1}(r) ∪_{Y (r)} Ŵ_{0}(r),

there is a long product region from a solid torus in Q_{1} to a solid torus in Q_{0}. Thus, the orientations of the two rescaled regions of non-uniform convergence, that is, of the domains of the two holomorphic triangles

Δ_{1}^{ϵ} = Δ^{ϵ}(A_{1}(r), Ψ_{1}(r)) and Δ_{0}^{ϵ} = Δ^{ϵ}(A_{0}(r), Ψ_{0}(r))

have opposite orientations. This implies that, when regarded as solutions in

M_{ℓ,k}^{W (r)}(a_{i}^{(1)}, a_{j}^{(0)}),

one of the triangles is antiholomorphic (holomorphic up to a change of orientation). In fact, the triangles in H^{1}(T^{2}, ℝ) have the same orientation, whereas the rescaled regions of non-uniform convergence for the triangles on Ŵ_{1} and Ŵ_{0} differ by a change of orientations when compared inside the composite cobordism W.

In other words, we obtain an orientation reversing diffeomorphism

M_{ℓ}^{W_{1}(r)}(a_{i}^{(1)}, a_{j}^{(0)}(ϵ)) ≅ M_{k}^{W_{0}(r), (0)}(a_{i}^{(1)}(ϵ), a_{j}^{(0)}),

when both moduli spaces are identified with solutions in

M_{ℓ,k}^{W (r)}(a_{i}^{(1)}, a_{j}^{(0)}).

⋄ We now derive from Theorem 6.9 the third fundamental relation.

Corollary 6.10 Let a_{i}^{(1)} and a_{j}^{(0)} be critical points on Y_{1}(r) and Y_{0}(r) respectively with

deg_{Y_{1}}(a_{i}^{(1)}) = deg_{Y_{0},𝔰_{k}}(a_{j}^{(0)}),

and let a_{i}^{(1)}(ϵ) and a_{j}^{(0)}(ϵ) be the corresponding critical points on Y (r), for sufficiently small ϵ and large r ≥ r_{0}, then we have

N_{ℓ}^{W_{1}(r)}(a_{i}^{(1)}, a_{j}^{(0)}(ϵ)) = −N_{k}^{W_{0}(r)}(a_{i}^{(1)}(ϵ), a_{j}^{(0)}).

6.5 The inclusion Ker(w_{∗}^{0}) ⊂ Im(w_{∗}^{1})

Recall that, for a fixed pair of asymptotic values (a_{i}^{(1)}, a_{j}^{(0)}), there exists a unique non-negative Spin^{c} structure 𝔰_{ℓ} satisfying

ι(𝔰_{ℓ}, W_{1}, a_{i}^{(1)}, a_{j}^{(0)}) = 0.

We denote the corresponding ℓ with ℓ_{ij}. The coefficient of the map w_{∗}^{1} is then given by

N_{ℓ_{ij}}^{W_{1}}(a_{i}^{(1)}, a_{j}^{(0)}).

Using the results of Section 6, we can write explicitly the condition that a Floer chain on Y is in the image of w_{∗}^{1}.

Lemma 6.11 Suppose given a Floer chain a ∈ C_{∗}(Y, μ),

a = ∑_{i=1}^{m} n_{i}a_{i}^{(1)} + ∑_{j=m+1}^{n} n_{j}a_{j}^{(0)},

with coefficients n_{i} and n_{j} in ℤ. If we assume that the parameter ϵ > 0 in the surgery perturbation is sufficiently small, then the chain a ∈ C_{∗}(Y, μ) is in the image Im(w_{∗}^{1}) under the morphism

w_{∗}^{1} : C_{∗}(Y_{1}) → C_{∗}(Y, μ)

if and only if the coefficients satisfy the relation

n_{j} = ∑_{i=1}^{m} n_{i}N_{ℓ_{ij}}^{W_{1}}(a_{i}^{(1)}, a_{j}^{(0)}),

for all j = m + 1, …, n.

Proof. Assume that the element

a = ∑_{i=1}^{m} n_{i}a_{i}^{(1)} + ∑_{j=m+1}^{n} n_{j}a_{j}^{(0)}

is in Im(w_{∗}^{1}). Then it satisfies

a = w_{∗}^{1}(∑_{i=1}^{m} p_{i}a_{i}^{(1)}),

for some coefficients p_{i} ∈ ℤ. For sufficiently small ϵ > 0, we know from the results of Section 6 that the relation

⟨a_{i}^{(1)}, w_{∗}^{1}(a_{k}^{(1)})⟩ = δ_{ik}

holds. This gives

a = ∑_{i=1}^{m} p_{i}a_{i}^{(1)} + ∑_{j=m+1}^{n} ∑_{i=1}^{m} p_{i}N_{ℓ_{ij}}^{W_{1}}(a_{i}^{(1)}, a_{j}^{(0)}).

Thus, we obtain

n_{i} = p_{i} for all i = 1, …, m

and

n_{j} = ∑_{i=1}^{m} p_{i}N_{ℓ_{ij}}^{W_{1}}(a_{i}^{(1)}, a_{j}^{(0)}).

⋄

On the other hand, we can also write explicitly the condition that a Floer chain on Y is in the kernel of w_{∗}^{0}, using the expression for the map w_{∗}^{0} for sufficiently small ϵ > 0 in the surgery perturbation.

Lemma 6.12 Suppose given a Floer chain a ∈ C_{∗}(Y, μ),

a = ∑_{i=1}^{m} n_{i}a_{i}^{(1)} + ∑_{j=m+1}^{n} n_{j}a_{j}^{(0)},

with coefficients n_{i} and n_{j} in ℤ. For sufficiently small ϵ > 0 in the surgery perturbation, we have that a ∈ C_{∗}(Y, μ) is in the kernel Ker(w_{∗}^{0}) if and only if the coefficients satisfy

n_{j} = − ∑_{i=1}^{m} n_{i}N_{k_{j}}^{W_{0}}(a_{i}^{(1)}, a_{j}^{(0)}),

for all j = m + 1, …, n, with

a_{j}^{(0)} ∈ M_{Y_{0}}(𝔰_{k_{j}}).

Proof. Assume that the element

a = ∑_{i=1}^{m} n_{i}a_{i}^{(1)} + ∑_{j=m+1}^{n} n_{j}a_{j}^{(0)}

is in Ker(w_{∗}^{0}). Then it satisfies

(∑_{i=1}^{m} ∑_{j=m+1}^{n} n_{i}N_{k_{j}}^{W_{0}}(a_{i}^{(1)}, a_{j}^{(0)}) + ∑_{j=m+1}^{n} n_{j}) a_{j}^{(0)} = 0.

Here we use the condition, proved in Section 6, that for sufficiently small ϵ > 0 we have

⟨a_{j}^{(0)}, w_{∗}^{0}(a_{k}^{(0)})⟩ = δ_{ik}.

The above condition can be rewritten as

n_{j} = − ∑_{i=1}^{m} n_{i}N_{k_{j}}^{W_{0}}(a_{i}^{(1)}, a_{j}^{(0)}) for all j = m + 1, …, n.

Notice that this means that we can choose arbitrarily the first {n_{i}}_{i=1,...m} coefficients and the remaining {n_{j}}_{j=m+1,...,n} are determined by the above relation. ⋄

We then derive the exactness result as follows.

Proposition 6.13 The inclusion Ker(w_{∗}^{0}) ⊂ Im(w_{∗}^{1}) is satisfied.

Proof. Suppose given an element a ∈ Ker(w_{∗}^{0}). By Lemma 6.12, for sufficiently small ϵ > 0 in the surgery perturbation, we can write

a = ∑_{i=1}^{m} n_{i}a_{i}^{(1)} + ∑_{j=m+1}^{n} n_{j}a_{j}^{(0)}

with n_{i} and n_{j} in ℤ, satisfying

n_{j} = − ∑_{i=1}^{m} n_{i}N_{k_{j}}^{W_{0}}(a_{i}^{(1)}, a_{j}^{(0)}).

Consider the element

a_{1} = ∑_{i=1}^{m} n_{i}a_{i}^{(1)},

with the coefficients n_{i} as above. We have

w_{∗}^{1}(a_{1}) = ∑_{i=1}^{m} n_{i}a_{i}^{(1)} + ∑_{j=m+1}^{n} ∑_{i=1}^{m} n_{i}N_{ℓ_{ij}}^{W_{1}}(a_{i}^{(1)}, a_{j}^{(0)}).

The relation

N_{ℓ_{ij}}^{W_{1}}(a_{i}^{(1)}, a_{j}^{(0)}) = −N_{k_{j}}^{W_{0}}(a_{i}^{(1)}, a_{j}^{(0)})

implies that we have

a = w_{∗}^{1}(a_{1}).

⋄

The result of Proposition 6.13, together with the relation w_{∗}^{0} ∘ w_{∗}^{1} = 0 proved in the previous section, implies that we have exactness in the middle term of the sequence

0 → C_{q}(Y_{1}) →^{w_{q}^{1}} C_{q}(Y, μ) →^{w_{q}^{0}} ⨁_{𝔰_{k}}C_{(q)}(Y_{0}, 𝔰_{k}) → 0.

7 The connecting homomorphism

We have established the exactness of the sequence

0 → C_{∗}(Y_{1}) →^{w_{∗}^{1}} C_{∗}(Y, μ) →^{w_{∗}^{0}} ⊕_{k}C_{(∗)}(Y_{0}, 𝔰_{k}) → 0.

Now we can give a more precise description of the connecting homomorphism in the induced long exact sequence of Floer homologies. As we are going to see, the connecting homomorphism is described in terms of the discrepancy between the boundary operator ∂_{Y} of the Floer complex C_{∗}(Y, μ) and the operator ∂_{Y_{1}} ⊕ ∂_{Y_{0}}, with ∂_{Y_{0}} = ⊕_{k}∂_{Y_{0},k} on

C_{∗}(Y_{1}) ⊕ ⊕_{k}C_{(∗)}(Y_{0}, 𝔰_{k}).

More explicitly, let us identify again the points of M_{Y,μ}^{∗} with

M_{Y,μ}^{∗} = {a_{i}^{(1)}}_{i=1,...m} ∪ {a_{j}^{(0)}}_{i=m+1,...n},

as in (28). Consider a cycle ∑_{j} x_{j}a_{j}^{(0)} in ⊕_{k}C_{(∗)}(Y_{0}, 𝔰_{k}). We have

∂_{Y_{0}}(∑_{j} x_{j}a_{j}^{(0)}) = 0.

Now consider the corresponding element ∑_{j} x_{j}a_{j}^{(0)}(ϵ) in C_{∗}(Y, μ). We have the following result.

Lemma 7.1 Given a cycle ∑_{j} x_{j}a_{j}^{(0)} in C_{(∗)}(Y_{0}, 𝔰_{k}), we have

\begin{matrix}∂_{Y,μ}(∑_{j} x_{j}a_{j}^{(0)}(ϵ)) = \\ ∑_{i,j} x_{j}n_{Y,μ}(a_{j}^{(0)}(ϵ), a_{i}^{(1)}(ϵ))a_{i}^{(1)}(ϵ) \\ − ∑_{r,j,p} x_{j}n_{Y,μ}(a_{j}^{(0)}(ϵ), a_{r}^{(1)}(ϵ))N_{k}^{W_{0}}(a_{r}^{(1)}(ϵ), a_{p}^{(0)})a_{p}^{(0)}(ϵ). \end{matrix}  (39)

Here the first sum is over a_{i}^{(1)} in M_{Y_{1}}^{∗} with

deg_{Y,μ}(a_{j}^{(0)}(ϵ)) − deg_{Y,μ}(a_{i}^{(1)}(ϵ)) = 1

and the second sum is over a_{p}^{(0)} in M_{Y_{0}}(𝔰_{k}) satisfying

deg_{Y_{0},𝔰_{k}}(a_{j}^{(0)}) − deg_{Y_{0},𝔰_{k}}(a_{p}^{(0)}) = 1

and a_{r}^{(1)} in M_{Y_{1}}^{∗} satisfying

deg_{Y,μ}(a_{j}^{(0)}(ϵ)) − deg_{Y,μ}(a_{r}^{(1)}(ϵ)) = 1

and

ι(W_{0}, 𝔰_{k}, a_{i}^{(1)}(ϵ), a_{p}^{(0)}) = 0.

Proof. It is sufficient to check that we have

\begin{matrix}π ∘ ∂_{Y,μ}(a_{j}^{(0)}(ϵ)) = ∂_{Y_{0}}(a_{j}^{(0)}) \\ − ∑_{r,j,p} x_{j}n_{Y,μ}(a_{j}^{(0)}(ϵ), a_{r}^{(1)}(ϵ))N_{k}^{W_{0}}(a_{r}^{(1)}(ϵ), a_{p}^{(0)})a_{p}^{(0)}(ϵ), \end{matrix}

where π : C_{q}(Y, μ) → ⊕_{k}C_{(q)}(Y_{0}, 𝔰_{k}) is the projection induced from the identification of the moduli spaces in [6]. In order to prove this relation, consider 1-dimensional moduli spaces M_{k}^{W_{0}}(a_{j}^{(0)}(ϵ), a_{p}^{(0)}) and their compactification. We have boundary strata

∂M_{k}^{W_{0}}(a_{j}^{(0)}(ϵ), a_{p}^{(0)}) =
⋃_{a_{s}^{(0)}} M̂_{Y,μ}(a_{j}^{(0)}(ϵ), a_{s}^{(0)}(ϵ)) × M_{k}^{W_{0}}(a_{s}^{(0)}(ϵ), a_{p}^{(0)})∪
⋃_{a_{i}^{(1)}} M_{Y,μ}(a_{j}^{(0)}(ϵ), a_{i}^{(1)}(ϵ)) × M_{k}^{W_{0}}(a_{i}^{(1)}(ϵ), a_{p}^{(0)})∪
⋃_{a_{s}^{(0)}} M_{k}^{W_{0}}(a_{j}^{(0)}(ϵ), a_{s}^{(0)}) × M̂_{Y_{0},𝔰_{k}}(a_{s}^{(0)}, a_{p}^{(0)}).

Using the results of Corollary 5.2 and 5.4 we can identify these expressions with

M_{Y,μ}(a_{j}^{(0)}(ϵ), a_{p}^{(0)}(ϵ))∪
⋃_{a_{i}^{(1)}} M_{Y,μ}(a_{j}^{(0)}(ϵ), a_{i}^{(1)}(ϵ)) × M_{k}^{W_{0}}(a_{i}^{(1)}(ϵ), a_{p}^{(0)})∪
M̂_{Y_{0},𝔰_{k}}(a_{j}^{(0)}, a_{p}^{(0)}).

When keeping into account the orientations, this yields the desired formula. ⋄

Using this result we obtain the following.

Lemma 7.2 Suppose given a cycle in ∑_{j} x_{j}a_{j}^{(0)} in C_{(∗)}(Y_{0}, 𝔰_{k}). The image

of ∑_{j} x_{j}a_{j}^{(0)} under the connecting homomorphism Δ is given by

Δ(∑_{j} x_{j}a_{j}^{(0)}) = ∑_{j} x_{j}n_{Y,μ}(a_{j}^{(0)}(ϵ), a_{i}^{(1)}(ϵ))a_{i}^{(1)},

that is, by the algebraic counting of flowlines in

M̂_{Y ×ℝ,μ}(a_{j}^{(0)}(ϵ), a_{i}^{(1)}(ϵ)),

for deg_{Y,μ}(a_{j}^{(0)}(ϵ)) − deg_{Y,μ}(a_{i}^{(1)}(ϵ)) = 1.

Proof. Using the result of Theorem 6.9 we can write (39) equivalently as

∂_{Y,μ}(∑_{j} x_{j}a_{j}^{(0)}(ϵ) =
∑_{i,j} x_{j}n_{Y,μ}(a_{j}^{(0)}, a_{i}^{(1)})a_{i}^{(1)}(ϵ)
+ ∑_{i,j,p} x_{j}n_{Y,μ}(a_{j}^{(0)}(ϵ), a_{i}^{(1)}(ϵ))N_{ℓ}^{W_{1}}(a_{i}^{(1)}, a_{p}^{(0)}(ϵ))a_{p}^{(0)}(ϵ).

By comparing this expression with

w_{∗}^{1}(∑_{j} x_{j}n_{Y,μ}(a_{j}^{(0)}(ϵ), a_{i}^{(1)}(ϵ))a_{i}^{(1)}) =
∑_{j} x_{j}n_{Y,μ}(a_{j}^{(0)}(ϵ), a_{i}^{(1)}(ϵ))a_{i}^{(1)}(ϵ)+
∑_{j} x_{j}n_{Y,μ}(a_{j}^{(0)}(ϵ), a_{i}^{(1)}(ϵ))N_{ℓ}^{W_{1}}(a_{i}^{(1)}, a_{p}^{(0)}(ϵ))a_{p}^{(0)}(ϵ),

we obtain

Δ(∑_{j} x_{j}a_{j}^{(0)}) = ∑_{j} x_{j}n_{Y,μ}(a_{j}^{(0)}(ϵ), a_{i}^{(1)}(ϵ))a_{i}^{(1)}.

This completes the proof. ⋄

7.1 The surgery triangle

In this section we give a different description of the connecting homomorphism. This will prove that, in the ϵ → 0 limit of the surgery perturbation, the exact triangle for Seiberg-Witten Floer homology is a surgery triangle, that is, the connecting homomorphism in the exact sequence can also be described as a map w̄_{∗}^{2} induced by a surgery cobordism W̄_{2}, and the resulting diagram

C_{∗}(Y_{1}) →^{w_{∗}^{1}} C_{∗}(Y, μ) →^{w_{∗}^{0}} ⊕_{k}C_{(∗)}(Y_{0}, 𝔰_{k}) →^{w̄_{∗}^{2}} C_{∗}(Y_{1})[−1]

is a distinguished triangle, cf. [9], [2], [24].

Proposition 7.3 The connecting homomorphism Δ in the exact triangle is given by the following expression,

Δ(∑_{j} x_{j}a_{j}^{(0)}) = w̄_{∗}^{2}(∑_{j} x_{j}a_{j}^{(0)}),

for any cycle ∑_{j} x_{j}a_{j}^{(0)} in C_{(∗)}(Y_{0}, 𝔰_{k}), for some 𝔰_{k}, where

w̄_{∗}^{2} : ⊕_{k}C_{(∗)}(Y_{0}, 𝔰_{k}) → C_{∗}(Y, μ)

is the homomorphism defined by counting solutions in the zero-dimensional components of the moduli spaces

MW̄_{2} ; k(a_{j}^{(0)}, a_{i}^{(1)}),

over the cobordism W̄_{2}.

Proof. We only need to prove that the algebraic counting of flow lines in

M̂_{Y (r)×ℝ,μ}(a_{j}^{(0)}(ϵ), a_{i}^{(1)}(ϵ)),

agrees with the algebraic counting of monopoles in the zero-dimensional

components of MW̄_{2} ; k(a_{j}^{(0)}, a_{i}^{(1)}).

We begin with a few observations on the topology of the cobordisms. Recall that we have W̄ = W̄_{0}∪_{Y} W̄_{1}. Moreover, we can write W̄ = W̄_{2}#ℂP^{2}, where the generator of the homology of the ℂP^{2} summand is the surface S^{2} = D_{0}^{2} ∪_{K} D_{1}^{2}, with D_{i}^{2} the core handles of the cobordisms W̄_{0} and W̄_{1}. In our analysis of the geometric limits of solutions, we have considered the punctured cobordisms Ŵ_{1} and Ŵ_{0}. It is convenient here to consider the punctured \overline{Ŵ_{1}} = W̄_{1}\\{x_{1}\} and \overline{Ŵ_{0}} = W̄_{0}\\{x_{0}\}, obtained by removing points {x_{i}} inside the core disks D_{i}^{2}. Now consider the doubly punctured space

W̄ \\{x_{0}, x_{1}\} ≅ \overline{Ŵ_{1}} ∪_{Y} \overline{Ŵ_{0}},

with a metric that has cylindrical ends S^{3} × [0, ∞) at the punctures and stretched regions T^{2}×[−r, r] inside the standard Heegaard splittings of these S^{3} and along the product regions connecting the solid tori in the Heegaard splittings of S^{3} to the tubular neighborhoods of the knot in Y_{1}, Y, and Y_{0}. Then the sphere S^{2} = D_{0}^{2} ∪_{K} D_{1}^{2} corresponds to a cylinder

S^{2}\{x_{0}, x_{1}} ≅ S^{1} × ℝ

in W̄ \{x_{0}, x_{1}}.

Thus, when removing the points x_{0} and x_{1}, the ℂP^{2} summand in W̄ becomes a disk bundle S over the cylinder S^{1} × ℝ, which, with respect to a fixed trivialization at the two ends, has Euler number +1. In turn, the doubly punctured cobordism W̄_{2}\{x_{0}, x_{1}} is obtained by replacing this disk bundle with the trivial disk bundle S_{0} = D^{2} × S^{1} × ℝ.

Now we proceed to compare the geometric limits of solutions in

M_{Y ×ℝ,μ}(a_{j}^{(0)}(ϵ), a_{i}^{(1)}(ϵ)) and MW̄_{2} ; k(a_{j}^{(0)}, a_{i}^{(1)}).

There are two distinct cases: these correspond to (d) and (e) of Figure 4. Case 1. Consider geometric limits of solutions on the doubly punctured cobordism W̄ \{x_{0}, x_{1}}. By the previous analysis of the geometric limits of solutions on the punctured cobordisms W_{1} and W_{0}, together with the gluing theorem for MW̄ ; ℓ,k(a_{j}^{(0)}, a_{i}^{(1)}), which in this case gives

MW̄ ; ℓ,k(a_{j}^{(0)}, a_{i}^{(1)}) ≅

MW̄_{0} ; k(a_{j}^{(0)}, a_{i}^{(1)}(ϵ)) × MW̄_{1} ; ℓ(a_{i}^{(1)}(ϵ), a_{i}^{(1)})∪

MW̄_{0} ; k(a_{j}^{(0)}, a_{j}^{(0)}(ϵ)) × MW̄_{1} ; ℓ(a_{j}^{(0)}(ϵ), a_{i}^{(1)})

for zero-dimensional moduli spaces, we obtain geometric limits as in Figure 4, (a), (b). These reduce to solutions on the once punctured W̄_{2}\{x_{2}} that have geometric limits as in Figure 4 (d). The counting

N W̄_{0} ; k(a_{j}^{(0)}, a_{i}^{(1)}(ϵ)) + N W̄_{1} ; ℓ(a_{j}^{(0)}(ϵ), a_{i}^{(1)}) = 0

implies that the total counting of these solutions in the zero-dimensional moduli space MW̄_{2} ; k(a_{j}^{(0)}, a_{i}^{(1)}) is zero. Notice that, in this case we have

M_{Y (r)×ℝ,μ}(a_{j}^{(0)}(ϵ), a_{i}^{(1)}(ϵ)) = ∅,

therefore we also have n_{Y,μ}(a_{j}^{(0)}(ϵ), a_{i}^{(1)}(ϵ)) = 0.

Case 2. In this case, the zero-dimensional moduli space MW̄_{2} ; k(a_{j}^{(0)}, a_{i}^{(1)}) is obtained from approximate solutions of the form

M̂_{V ×ℝ}([A, ψ], [Ã, ψ̃], a_{∞})#Δ,

with Δ a triangle as in Figure 4 (e), and ([A, ψ], [Ã, ψ̃], a_{∞}) is the admissible triple with respect to (a_{j}^{(0)}, a_{i}^{(1)}). (The corresponding holomorphic triangle for geometric limits on W̄ − {x_{0}, x_{1}} is illustrated in Figure 4 (c).) This means that, in forming the preglued solutions, elements in the moduli spaces

M_{V ×ℝ}([A, ψ], [Ã, ψ̃], a_{∞})

that differ by a translation will give rise to the same solution on W̄_{2}. Notice that geometric limits for M_{Y (r)×ℝ}(a_{j}^{(0)}(ϵ), a_{i}^{(1)}(ϵ)) are given by

M_{V ×ℝ}([A, ψ]^{ϵ}, [Ã, ψ̃]^{ϵ}, a_{∞})#Δ^{ϵ}

with Δ^{ϵ} a holomorphic disk in H^{1}(T^{2}, ℝ) with boundary along arcs of ℓ_{μ} and ℓ connecting the asymptotic limits of a_{j}^{(0)}(ϵ), a_{i}^{(1)}(ϵ) respectively, the

corresponding admissible triple is ([A, ψ]^{ϵ}, [Ã, ψ̃]^{ϵ}, a_{∞}) for (a_{j}^{(0)}(ϵ), a_{i}^{(1)}(ϵ)). As we let ϵ → 0 these geometric limits approximate geometric limits

M̂_{V ×ℝ}([A, ψ], [Ã, ψ̃], a_{∞})#Δ

of MW̄_{2} ; k(a_{j}^{(0)}, a_{i}^{(1)}). Under the gluing map, we obtain that the algebraic

counting of solutions in the zero-dimensional moduli space MW̄_{2} ; k(a_{j}^{(0)}, a_{i}^{(1)})

agrees with the algebraic counting of flow lines in M̂_{Y ×ℝ,μ}(a_{j}^{(0)}(ϵ), a_{i}^{(1)}(ϵ)),

N W̄_{2} ; k(a_{j}^{(0)}, a_{i}^{(1)}) = n_{Y,μ}(a_{j}^{(0)}(ϵ), a_{i}^{(1)}(ϵ)).

This completes the proof. ⋄

Let us recall briefly the following definitions of homological algebra, cf. [13].

A triangle (in a category of complexes) is a diagram of the form

K_{•} →^{u} L_{•} →^{v} M_{•} →^{w} K_{•}[−1].

Figure 4: triangles on W̄_{2}

Here we take complexes with differentials of degree (−1) for consistency with our notation. K_{•}[n] is the n-th shift with K_{i}[n] = K_{i+n}, and d_{K[n]} = (−1)^{n}d_{K}. A triangle is distinguished if it is quasi–isomorphic to a diagram of the form

K_{•} →^{ũ} Cyl(u) →^{π} C(u) →^{δ} K_{•}[−1].

Here Cyl(u) and C(u) are the cylinder and the cone on the morphism u, defined as in [13] pg.154 (up to adjusting notations to differentials of degree (−1)), and δ is the connecting homomorphism.

With this notation and the previous results, we can rephrase our main result as the following Proposition.

Proposition 7.4 For sufficiently small ϵ ≤ ϵ_{0} in the surgery perturbation μ, the sequence

C_{∗}(Y_{1}) →^{w_{∗}^{1}} C_{∗}(Y, μ) →^{w_{∗}^{0}} ⊕_{k}C_{(∗)}(Y_{0}, 𝔰_{k}) →^{w̄_{∗}^{2}} C_{∗}(Y_{1})[−1]

is a distinguished triangle.

Proof. The result follows from the previous results, together with Proposition 5, pg.157 of [13]. In fact, we have proven that, as ϵ → 0, the sequence

0 → C_{∗}(Y_{1}) →^{w_{∗}^{1}} C_{∗}(Y, μ) →^{w_{∗}^{0}} ⊕_{k}C_{(∗)}(Y_{0}, 𝔰_{k}) → 0

is an exact triple of complexes ([13] pg. 42). Proposition 5, pg.157 of [13] then gives the result. ⋄

References

[1] M.F. Atiyah, V.K. Patodi, I.M. Singer, Spectral asymmetry and Riemannian geometry, I,II,III; Math. Proc. Cambridge Phil. Soc. 77 (1975) 43-69; 78 (1975) 405-432; 79 (1976) 71-99.

[2] P.J. Braam, S.K. Donaldson, Floer’s work on instanton homology, knots and surgery, Floer Memorial Volume, Progress in Mathematics, Vol. 133; Birkhäuser 1995, 195-256.

[3] S.E. Cappell, R. Lee, E.Y. Miller, Self-adjoint elliptic operators and manifold decompositions. II: Spectral flow and Maslov index, Comm. Pure Appl. Math., 49, N.9, (1996) 869-909.

[4] S.E. Cappell, R. Lee, E.Y. Miller, On the Maslov index, Comm. Pure Appl. Math., Vol. XLVII (1994) 121-186.

[5] A. Carey, B.L. Wang, Seiberg-Witten-Floer homology and gluing formulae, to appear in Asian Journal of Mathematics.

[6] A. Carey, M. Marcolli, B.L. Wang, Exact triangles in Seiberg-Witten Floer theory. Part I: the geometric triangle, preprint.

[7] M. Daniel, P.Kirk, with an appendix by K.P. Wojciechowski, A general splitting formula for the spectral flow, Michigan Math. J. 46 (1999), no. 3, 589–617.

[8] S. Donaldson, The Seiberg–Witten equations and 4-manifold topology, Bull. AMS, Vol.33 N.1 (1996) 45-70.

[9] A. Floer, Instanton homology and Dehn surgery, Floer Memorial Volume, Progress in Mathematics, Vol. 133; Birkhäuser 1995, 77-97.

[10] A. Floer, Morse theory for Lagrangian intersections, J. Diff. Geom. 28 (1988) 513-547.

[11] K.A. Froyshov, The Seiberg–Witten equations and four-manifolds with boundary, Math. Res. Lett. 3 (1996), no. 3, 373–390.

[12] K. Fukaya, Floer homology and mirror symmetry, I, preprint.

[13] S.I. Gelfand, Yu.I. Manin, Methods of homological algebra, Springer Verlag, 1996.

[14] P.B. Kronheimer, Embedded surfaces and gauge theory in three and four dimensions, Surveys in differential geometry, Vol. III (1996), 243–298.

[15] P.B. Kronheimer, T.S. Mrowka, The genus of embedded surfaces in the projective plane, Math. Research Letters 1 (1994) 797-808.

[16] M. Marcolli, Seiberg-Witten gauge theory, Texts and Readings in Mathematics, Vol. 17, Hindustan Book Agency, 1999.

[17] M. Marcolli, B.L. Wang, Exact triangles in Seiberg-Witten Floer theory. Part II: geometric limits of flow lines, preprint.

[18] M. Marcolli, B.L. Wang, Exact triangles in Seiberg-Witten Floer theory. Part IV: ℤ-graded monopole homology, to appear in Asian Journal of Mathematics.

[19] M. Marcolli, B. L. Wang Equivariant Seiberg-Witten-Floer homology, preprint.

[20] J.W. Morgan, The Seiberg-Witten equations and applications to the topology of smooth four-manifolds, Princeton 1996.

[21] J.W. Morgan, T.S. Mrowka and D. Ruberman, The L^{2}-moduli space and a vanishing theorem for Donaldson polynomial invariants, Monographs in Geometry and Topology, Vol 2, 1994.

[22] L.I. Nicolaescu, Eta invariants of Dirac operators on circle bundles over Riemann surfaces and virtual dimensions of finite energy Seiberg–Witten moduli spaces, preprint, math.DG/9805046.

[23] L.I. Nicolaescu, private communication.

[24] H.Y. Wang, The exactness theorem for Floer homology, Publ. RIMS, Kyoto Univ. 33 (1997) 713-750.

Matilde Marcolli, Max–Planck–Institut für Mathematik, D-53111 Bonn, Germany. marcolli@mpim-bonn.mpg.de

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