Exact triangles in monopole homology and the Casson-Walker invariant

Matilde Marcolli and Bai-Ling Wang

Contents

1 Introduction 1 2 The geometric triangle 7 3 The relative gradings 15 4 Geometric limits and the holomorphic triangles 18

4.1 Surgery cobordisms . . . . . . . . . . . . . . . . . . . . . . . . 18 4.2 Geometric limits and holomorphic triangles . . . . . . . . . . 20

5 Proof of exactness and the surgery triangle 25 5.1 The chain homomorphisms . . . . . . . . . . . . . . . . . . . 25 5.2 Proof of exactness . . . . . . . . . . . . . . . . . . . . . . . . 27 5.3 The surgery triangle . . . . . . . . . . . . . . . . . . . . . . . 30

6 Seiberg–Witten and Casson–Walker invariant 31

1 Introduction

In the series of papers [2] [5] [6] [7], we studied how the Seiberg-Witten-Floer homology for an integral homology 3-sphere behaves under surgery: there exists a surgery triangle for the Seiberg-Witten-Floer theory. If K is a knot in a homology 3-sphere Y, the Seiberg-Witten-Floer homologies of Y and of the manifolds Y_{1} and Y_{0} obtained by Dehn surgery on K with framing 1 and 0, respectively, are related by an exact triangle

\xymatrix{HF_{∗}^{SW} (Y_{1}, g_{1}) \ar[r]^{w_{∗}^{1}} & HF_{∗}^{SW} (Y, g_{0}, μ) \ar[dl]_{w_{∗}^{0}} \\ ⨁_{k∈ℤ} HF_{(∗)}^{SW} (Y_{0}, 𝔰_{k}) \ar[u]_{w_{(∗)}}}

In this triangle the maps w_{∗}^{1}, w_{∗}^{0} and w_{(∗)} are induced by the surgery cobordisms connecting Y_{1} and Y, Y and Y_{0}, Y_{0} and Y_{1}, respectively, and μ is the surgery perturbation that simulates the effect of Dehn surgery. In this paper, we extend this exact triangle to any closed oriented 3-manifold with a smooth embedded knot.

Before we state the main theorem of this paper, we introduce some notation. Let Y be a closed oriented 3-manifold, with a smooth embedded knot K, let ν(K) be the tubular neighbourhood of K in Y. Choose an identification of ν(K) with D^{2} × S^{1}:

ν(K) ≅ D^{2} × S^{1},  (1)

where K is mapped to the core of the solid torus D^{2} × S^{1}. Under the identification (1), on the boundary T^{2}, we fix a basis m, l of H_{1}(T^{2}, ℤ) such that l is the longitude (parallel to K under the identification (1)) and m is the right-handed meridian (intersecting l once), the orientation determined by m ∧ l coincides with the orientation induced from Y. The corresponding longitude and meridian in the knot complement

V = Y \ν(K)

are denoted by m′, l′ respectively. Similarly, let m′′ and l′′ be the meridian and longitude in the tubular neighbourhood of the knot ν(K). The meridian m′′ bounds a disk D^{2} in ν(K), and l′′ generates H_{1}(ν(K), ℤ) and parallels to K. Let p and q be two relatively prime integers, the Dehn surgery with coefficient p/q ∈ ℚ ∪ {∞} on K is the operation of removing ν(K) and gluing in D^{2} × S^{1} by an orientation reversing diffeomorphism f_{p/q} of T^{2} that satisfies

f_{p/q}(m′′) = pm′ − ql′.

The resulting manifold is denoted by Y_{p/q}. Note that in general Y_{p/q} depends on the choice of the identification (1) if K is not a null-homologous knot.

Let 𝔰 be a Spin^{c} structure on Y. We shall see that, with a suitable choice of metrics and perturbation, (Y, 𝔰) has non-empty monopole moduli space only if 𝔰|_{ν(K)} has trivial determinant, hence we shall always assume that the Spin^{c} structure 𝔰 is trivial around K. If K represents a trivial homology class in H_{1}(Y, ℤ), then there is only one Spin^{c} structure on Y which agrees with 𝔰 over Y − ν(K) and ν(K). Suppose that K represents a torsion element of order n in H_{1}(Y, ℤ), which means,

\frac{|Torsion(H_{1}(Y, ℤ))|}{|Torsion(H_{1}(Y − ν(K), ℤ))|} = n.  (2)

In other words, n is the minimal positive integer such that n[K] is homologous to zero in H_{1}(Y, ℤ). Then there exists a ℤ_{n}-family of Spin^{c} structures

𝔰 ⊗ L_{k} (k = 1, ⋯, n)

which agree with 𝔰 over Y − ν(K) and ν(K), where L_{k} is a complex line bundle whose Euler class is given by kP D([K]). If [K] ≠ 0 in H_{1}(Y, ℚ), then there exists a ℤ-family of Spin^{c} structures

𝔰 ⊗ L_{k} (k ∈ ℤ)

which agree with 𝔰 over Y − ν(K) and ν(K), where L_{k} is a complex line bundle whose Euler class is given by kP D([K]).

Let Y_{1} be the (+1)-surgery on K, obtained via an orientation reversing diffeomorphism f_{1} of T^{2} that satisfies

(f_{1})_{∗}(m′′ = [pt × ∂D^{2}]) = m′ − l′ (f_{1})_{∗}(l′′) = l′.

Let Y_{0} be the 0-surgery on K, obtained via an orientation reversing diffeomorphism f_{0} of T^{2} that satisfies

(f_{0})_{∗}(m′′) = l′ (f_{0})_{∗}(l′′) = m′.

We shall consider separately the following cases.

• First we assume that Y is a rational homology 3-sphere with a smoothly embedded knot K representing a torsion element of order n in H_{1}(Y, ℤ) in the sense of (2). Then Y_{1} is a rational homology 3-sphere, and Y_{0} is a rational homology S^{1} ×S^{2}. Let 𝔰 be a Spin^{c} structure on Y, which is trivial on ν(K). Gluing the Spin^{c} structures s|_{Y −ν(K)} and 𝔰|_{ν(K)} along T^{2} via different gauge transformations on T^{2} results in a ℤ_{n}-family of Spin^{c} structures on Y and Y_{1}, and a ℤ-family of Spin^{c} structures on Y_{0}. Without confusion, thinking K as the core of the attaching solid torus ν(K), we denote these structures on Y, Y_{1} and Y_{0} all by

𝔰 ⊗ L_{k} (k ∈ Z),

where L_{k} is a complex line bundle of Euler class kP D([K]). With this notation, it is understood that for Y and Y_{1}, and with n[K] = 0, there is only a ℤ_{n}-family of Spin^{c} structures. Then, with a careful choice of metrics and perturbations on Y, Y_{1} and Y_{0} as in Part I [2], we will obtain the following exact triangles for the Seiberg-Witten-Floer homologies:

\xymatrix{HF_{∗}^{SW} (Y_{1}, 𝔰 ⊗ L_{m}, g_{1}) \ar[r]^{w_{∗}^{1}} & HF_{∗}^{SW} (Y, 𝔰 ⊗ L_{m}, g, μ) \ar@{-->}[dl]_{w_{∗}^{0}} \\ ⨁_{k∈ℤ} HF_{(∗)}^{SW} (Y_{0}, 𝔰 ⊗ L_{nk+p}) \ar[u]_{w_{∗}}}

which holds for any fixed choice of m and p in {0, …, n − 1}, and for a corresponding choice of perturbation, and

\xymatrix{⨁_{k=1}^{n} HF_{∗}^{SW} (Y_{1}, 𝔰 ⊗ L_{k}, g_{1}) \ar[r]^{w_{∗}^{1}} & ⨁_{k=1}^{n} HF_{∗}^{SW} (Y, 𝔰 ⊗ L_{k}, g, μ) \ar[dl]_{w_{∗}^{0}} \\ ⨁_{k∈ℤ} HF_{(∗)}^{SW} (Y_{0}, 𝔰 ⊗ L_{k}) \ar[u]_{w_{∗}}}

In both cases the homomorphisms w_{∗}^{1}, w_{∗}^{0} and w_{∗} are induced by the surgery cobordisms. These exact triangles generalize the results of Part I-IV [2] [5][6] [7], in the sense that the above exact triangle reduces to the exact triangle for an integral homology 3-sphere when n = 1.

• Now we assume that (Y, 𝔰) is a closed oriented 3-manifold of b_{1}(Y) > 0 and with a Spin^{c} structure 𝔰. Let K be a knot representing a torsion homology class in H_{1}(Y, ℤ) of order n in the sense of (2). If 𝔰 has non-trivial determinant in the sense that c_{1}(𝔰) ≠ 0 in H^{2}(Y, ℚ), then the Seiberg-Witten-Floer homology for (Y, 𝔰) is ℤ_{2ℓ}-graded, where 2ℓ is the multiplicity of c_{1}(𝔰) in H^{2}(Y, ℤ)/Torsion, i.e, c_{1}(𝔰)(H_{2}(Y, ℤ)/Torsion) = 2ℓ. The gluing of the Spin^{c} structures corresponding to these three different surgeries gives rise to a ℤ_{n}-family of Spin^{c} structures on Y, Y_{1} and a ℤ-family of Spin^{c} structures on Y_{0}, which we denote by

𝔰 ⊗ L_{k} (k ∈ ℤ),

with the convention as before. Then, for any Spin^{c} structure 𝔰 ⊗ L_{k} (k = 1, ⋯, n) on Y and Y_{1}, the corresponding Seiberg-Witten-Floer homologies are all ℤ_{2ℓ}-graded, while for any Spin^{c} structure 𝔰 ⊗ L_{k} (k ∈ ℤ) on Y_{0}, the Seiberg-Witten-Floer homology HF_{∗}^{SW} (Y_{0}, 𝔰 ⊗ L_{k}) is ℤ_{ℓ_{[k]}}-graded, where ℓ_{[k]} is the greatest common factor in 2ℓ and 2k. Similar to Part IV [7], the ℤ_{ℓ_{[k]}}-graded homology HF_{∗}^{SW} (Y_{0}, 𝔰 ⊗ L_{k}) can be lifted to a ℤ_{ℓ}-graded homology HF_{(∗)}^{SW} (Y_{0}, 𝔰 ⊗ L_{k}). For this case, we derive in this paper the following exact triangles:

\xymatrix{HF_{∗}^{SW} (Y_{1}, 𝔰 ⊗ L_{m}, g_{1}) \ar[r]^{w_{∗}^{1}} & HF_{∗}^{SW} (Y, 𝔰 ⊗ L_{m}, g, μ) \ar[dl]_{w_{∗}^{0}} \\ ⨁_{k∈ℤ} HF_{(∗)}^{SW} (Y_{0}, 𝔰 ⊗ L_{nk+p}). \ar[u]_{w_{(∗)}}}  (3)

which holds for any choice of m and p in {0, …, n − 1}, and for a corresponding choice of perturbation, and

\xymatrix{⨁_{k=1}^{n} HF_{∗}^{SW} (Y_{1}, 𝔰 ⊗ L_{k}, g_{1}) \ar[r]^{w_{∗}^{1}} & ⨁_{k=1}^{n} HF_{∗}^{SW} (Y, 𝔰 ⊗ L_{k}, g, μ) \ar[dl]_{w_{∗}^{0}} \\ ⨁_{k∈ℤ} HF_{(∗)}^{SW} (Y_{0}, 𝔰 ⊗ L_{k}). \ar[u]_{w_{(∗)}}}  (4)

• If 𝔰 is a torsion Spin^{c} structure on Y, then for any k = 1, ⋯, n, HF_{∗}^{SW} (Y, 𝔰⊗L_{k}, ℤ[[t]]) and HF_{∗}^{SW} (Y_{1}, 𝔰⊗L_{k}, ℤ[[t]]) are ℤ-graded with ℤ[[t]]- coefficient. The reduced versions HF_{∗}^{SW} (Y, 𝔰 ⊗ L_{k}) and HF_{∗}^{SW} (Y_{1}, 𝔰 ⊗ L_{k}) are obtained by setting t = 0. The Seiberg-Witten-Floer homology HF_{∗}^{SW} (Y_{0}, 𝔰 ⊗ L_{k}) is ℤ_{2k}-graded, and can be lifted to a ℤ-graded version, denoted by HF_{(∗)}^{SW} (Y_{0}, 𝔰⊗L_{k}). Then the exact triangle takes the same form as (4).

• The remaining case is when a smoothly embedded knot K represents a non-trivial homology class in H_{1}(Y, ℚ). Let n be the minimal positive integer such that there is a 2-cycle in H_{2}(Y, ℤ) intersecting K at n points. Then the 0-surgery on Y along K yields Y_{0} satisfying b_{1}(Y_{0}) = b_{1}(Y) − 1. More precisely, H_{1}(Y_{0}, ℤ) is obtained by replacing the ℤ-component ℤ⟨[K]⟩ of H_{1}(Y, ℤ) by ℤ_{n}⟨[m′′] = [l′]⟩. Notice that Y can be thought of as the manifold obtained by 0-surgery on m′ ⊂ Y − ν(K) ⊂ Y_{0}, and Y_{1} can be thought as the result of (+1)-surgery on m′ ⊂ Y − ν(K) ⊂ Y_{0}. Since m′ represents a torsion element of order n in H_{1}(Y_{0}, ℤ) in the sense of (2), we have the exact triangle for (Y, 𝔰, K) obtained from the corresponding exact triangle for (Y_{0}, 𝔰, m′) in the form of (4). Thus, it is enough to establish the exact triangle for a general closed oriented 3-manifold Y with a smoothly embedded knot representing a torsion element of order n in H_{1}(Y, ℤ).

We now summarize the main theorem of this paper.

Theorem 1.1. Let (Y, 𝔰) be a closed oriented 3-manifold with a Spin^{c} structure which is trivial around a smoothly embedded knot K. Assume that K represents a torsion element of order n in H_{1}(Y, ℤ) in the sense of (2). Assume that the canonical framing of K is given by an identification of D^{2}×S^{1} with the tubular neighbourhood ν(K) such that K is given by {0}×S^{1}. Here the parallel simple curve on T^{2} provides the longitude of K and the right handed meridian is given by ∂(D^{2}) × {pt}. The orientation determined by m ∧ l coincides with the orientation induced from Y. Let Y_{1} and Y_{0} be the manifolds obtained, respectively, by (+1) and 0 surgery along K in Y. With a careful choice of metrics and perturbations, we obtain the following exact triangle induced by the surgery cobordisms after possible grading shifts:

\xymatrix{⨁_{k=1}^{n} HF_{∗}^{SW} (Y_{1}, 𝔰 ⊗ L_{k}) \ar[r]^{w_{∗}^{1}} & ⨁_{k=1}^{n} HF_{∗}^{SW} (Y, 𝔰 ⊗ L_{k}) \ar[dl]_{w_{∗}^{0}} \\ ⨁_{k∈ℤ} HF_{(∗)}^{SW} (Y_{0}, 𝔰 ⊗ L_{k}) \ar[u]_{w_{(∗)}}}

Here 𝔰 ⊗ L_{k} is the Spin^{c} structure obtained by tensoring a Spin^{c} structure 𝔰 with a complex line bundle L_{k} of Euler class kP D([K]). Moreover, for any fixed m, p ∈ {0, ⋯, n − 1}, we have the following more refined version of the exact triangle:

\xymatrix{HF_{∗}^{SW} (Y_{1}, 𝔰 ⊗ L_{m}) \ar[r]^{w_{∗}^{1}} & HF_{∗}^{SW} (Y, 𝔰 ⊗ L_{m}) \ar@{-->}[dl]_{w_{∗}^{0}} \\ ⨁_{k∈ℤ} HF_{(∗)}^{SW} (Y_{0}, 𝔰 ⊗ L_{nk+p}) \ar[u]_{w_{(∗)}}}

Here again the maps are induced by the surgery cobordisms, possibly after a shift in the grading.

The major technical analysis required in the proof has already been developed in [2][5][6][7]. Thus, in this paper, we shall address only those issues that are relevant to this general exact triangle, while we refer to the previous papers for the general setting and the basic results.

In the last section of this paper, we will show that a modified version of the Seiberg-Witten invariant agrees with the Casson-Walker invariants, for any closed and oriented rational homology 3-sphere. For any rational homology 3-sphere (Y, 𝔰, g) with a Spin^{c} structure 𝔰 and a Riemannian metric, the counting of the irreducible Seiberg-Witten monopoles defines the Seiberg-Witten invariant

SW_{Y} (𝔰, g) = #(M_{Y}^{∗} (𝔰, g)),

where each irreducible monopole in M_{Y}^{∗} (𝔰, g) has a natural orientation from the linearization of the Seiberg-Witten equations. As studied in [4], SW_{Y} (𝔰, g) depends on the metric and perturbation used in the definition, in order to obtain a topological invariant, we can modify SW_{Y} (𝔰, g) by a metric and perturbation dependent correction term as follows. Choose any four manifold X with boundary Y, such that X is endowed with a cylindricalend metric modeled on (Y, g_{Y}). Choose a Spin^{c} structure 𝔰_{X} on X which agrees with 𝔰 on Y over the end, and choose a connection A on (X, 𝔰_{X}) which extends the unique reducible on (Y, 𝔰). Then we set

ξ_{Y} (𝔰, g) = Ind_{ℂ}(\begin{matrix}/ \\ D \end{matrix}_{A}^{X}) − \frac{1}{8}(c_{1}(𝔰_{X}^{2}) − σ(X)),  (5)

where Ind_{ℂ}(\begin{matrix}/ \\ D \end{matrix}_{A}^{X}) is the complex index of the Dirac operator on (X, 𝔰_{X}) twisted with the extending Spin^{c} connection A and σ(X) is the signature of X. By the Atiyah-Patodi-Singer index theorem, ξ_{Y} (𝔰, g) is independent of the choice of (X, 𝔰_{X}) and A. Moreover, ξ_{Y} (𝔰, g) can be expressed as a combination of the Atiyah-Patodi-Singer eta invariants for the Dirac operator and signature operator on (Y, 𝔰). The modified version of the Seiberg-Witten invariant is defined as

ŜW_{Y} (𝔰) = SW_{Y} (𝔰, g) − ξ_{Y} (𝔰, g).  (6)

Then we prove the following equivalence between ŜW_{Y} and the Casson-Walker invariant.

Theorem 1.2. Let Y be a rational homology 3-sphere. Then,

∑_{𝔰∈Spin^{c}(Y)} ŜW_{Y} (𝔰) = \frac{1}{2}|H_{1}(Y, ℤ)|λ(Y),

where λ(Y) is the Casson-Walker invariant of Y (cf. [11]).

Acknowledgements We like to acknowledge the inspiring paper of Ozsváth and Szabó [10] on the theta divisor and the Casson-Walker invariant which lead us to our proof of the equivalence of ŜW_{Y} and the Casson-Walker invariant, hence proving the conjecture formulated in [10] on the equivalent between ŜW_{Y} and their θ̂ invariant for all rational homology 3-sphere. BLW is partially supported by Australia Research Council Fellowship.

2 The geometric triangle

In this section, we identify the monopoles on Y with monopoles on Y_{1} and Y_{0}. Suppose given a smoothly embedded knot K in (Y, 𝔰), which represents a torsion element of order n in H_{1}(Y, ℤ). We can split Y along a torus as in [2],

Y = V ∪_{T^{2}} ν(K).

We choose a metric on Y with a long cylinder [−r, r] × T^{2}, and denote the resulting manifold as

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

We additionally require that the chosen metric on Y satisfies the condition of Lemma 3.18 in [2] in the neighbourhood ν(K) of the knot: it has nonnegative scalar curvature, strictly positive away from the boundary. This induces a natural metric on Y_{0}(r). On Y_{1}(r), we need to choose a metric which agrees with the original metric on Y (r) when restricted to the knot complement V. The induced metric from Y_{1}(r) in the torus neighbourhood of ν(K) is the metric described in Lemma 3.21 [2].

With this choice of the metric, the moduli space of monopoles on (Y, 𝔰) is non-empty only if 𝔰|_{ν(K)} is trivial. Gluing the two Spin^{c} structures 𝔰|_{V} and 𝔰|_{ν(K)} along T^{2} by a gauge transformation on T^{2} gives rise to a ℤ_{n}- family of Spin^{c} structures on Y and Y_{1}, and to a ℤ-family of Spin^{c} structures on Y_{0}. The resulting Spin^{c} structures can be classified as the result of tensoring the original Spin^{c} structure 𝔰 with complex line bundles L_{k}(k ∈ ℤ) whose Euler class is given by kP D([K]). The gluing theorem for 3-dimensional monopoles and the perturbation μ on ν(K), “simulating the effect of surgery”, provide the decomposition of the moduli space for

∪_{k∈{1,··· ,n}}M_{Y} (𝔰 ⊗ L_{k}).

Theorem 2.1. With the choice of perturbations and metrics on Y, Y_{1} and Y_{0} described above, we have the following relation between the critical sets of the Chern-Simons-Dirac functional on the manifolds Y, Y_{1} and Y_{0}:

\begin{aligned}& ⋃_{k∈\{1,··· ,n\}} M_{Y} (𝔰 ⊗ L_{k}) \\ = & ⋃_{k∈\{1,··· ,n\}} M_{Y_{1}}(𝔰 ⊗ L_{k}) ∪ ⋃_{k∈ℤ} M_{Y_{0}}(𝔰 ⊗ L_{k}). \end{aligned}

Proof. First we assume that Y is a rational homology 3-sphere. When stretching the neck in Y (r), as r → ∞, we get two manifolds, each endowed with an infinite cylindrical end,

V (∞) = V ∪_{T^{2}} ([0, ∞) × T^{2})

ν(K)(∞) = ν(K) ∪_{T^{2}} ((−∞, 0] × T^{2}).

The Seiberg-Witten monopole moduli space of Y (r), for sufficiently large r, can be described in terms of the moduli spaces of V (∞) and ν(K)(∞) as analyzed in [2].

Notice that the moduli spaces of flat connections on T^{2}, modulo the subgroups of the gauge transformations on T^{2} which can be extended to the whole manifolds V or ν(K), define the following character varieties:

χ_{0}(T^{2}, V)
= H^{1}(T^{2}, ℝ)/2Im(H^{1}(V, ℤ) → H^{1}(T^{2}, ℤ)
= H^{1}(T^{2}, ℝ)/2nℤ⟨P D([l])⟩,

χ_{0}(T^{2}, ν(K))
= H^{1}(T^{2}, ℝ)/2Im(H^{1}(ν(K), ℤ) → H^{1}(T^{2}, ℤ)
= H^{1}(T^{2}, ℝ)/2ℤ⟨P D([m])⟩.

Thus the character variety χ_{0}(T^{2}, Y) is

χ_{0}(T^{2}, Y) = H^{1}(T^{2}, ℝ).

The covering maps between these character varieties are illustrated as follows

\xymatrix{2ℤ⟨P D([m])⟩ \ar[dr] &  & 2nℤ⟨P D([l])⟩ \ar[dl] \\  & χ_{0}(T^{2}, Y) \ar[dl]^{π_{1}} \ar[dr]^{π_{2}} \\ χ_{0}(T^{2}, V) \ar[dr] &  & χ_{0}(T^{2}, ν(K)) \ar[dl] \\  & χ(T^{2})}

where the maps π_{i} are the covering maps with fibers as indicated.

Based on the analysis of the space M_{V}^{∗} (𝔰|_{V}) of irreducible monopoles on a 3-manifold with a cylindrical end modelled on T^{2}, as in [2], we see that the asymptotic limit map defines a continuous map:

M_{V}^{∗} (𝔰|_{V}) →^{∂_{∞}} χ_{0}(T^{2}, V).

The reducibles on V embed in the character variety χ_{0}(T^{2}, V)

M_{V}^{red} → χ_{0}(T^{2}, V)

as the connections whose holonomy around the meridian is trivial. Then, for a sufficiently large r, the gluing theorem gives a diffeomorphism:

#_{Y} : M_{V}^{∗} \∂_{∞}^{−1}(U_{θ}) ×_{χ_{0}(T^{2},Y)} χ(ν(K)) → ⋃_{k=1}^{n} M_{Y (r)}^{∗}(𝔰 ⊗ L_{k}),

here U_{θ} is a small neighbourhood of the “bad points” θ in χ_{0}(T^{2}, Y), and χ(ν(K)) are the reducible lines in χ_{0}(T^{2}, Y). The above fibred product is obtained (cf.[2]) by taking the pullbacks of the images of the boundary value maps under the projections

\xymatrix{ & χ_{0}(T^{2}, Y) \ar[dl]^{π_{1}} \ar[dr]^{π_{2}} \\ χ_{0}(T^{2}, V) &  & χ_{0}(T^{2}, ν(K)).}

Let (u, v) be the coordinates on χ_{0}(T^{2}, Y) ≅ H^{1}(T^{2}, ℝ) corresponding to the holonomy around the longitude l and the meridian m respectively. In the gluing map #_{Y} above, χ(ν(K)) corresponds to the lines {v = 2k, k = 1, ⋯, n}. For each line {v = 2k}, the image of the gluing map gives a diffeomorphism onto M_{Y (r)}^{∗}(𝔰 ⊗ L_{k}).

For each Spin^{c} structure 𝔰 ⊗ L_{k}, there is a unique reducible monopole on Y (r), which is given by the intersection of χ(V, 𝔰), the flat connections on (V, 𝔰), with the line {v = 2k} in χ_{0}(T^{2}, Y):

θ_{Y} (k) = {u = u(𝔰)} ×_{χ_{0}(T^{2},Y)} {v = 2k}.

Here u(𝔰) is the holonomy of the flat connections in χ(V, 𝔰) around the longitude l′.

Similarly, we have gluing maps for monopoles on Y_{1}(r) and Y_{0}(r), respectively. In the gluing map #_{Y_{1}} for M_{Y_{1}(r)}^{∗}(𝔰 ⊗ L_{k}),

#_{Y_{1}} : M_{V}^{∗} \∂_{∞}^{−1}(U_{θ}) ×_{χ_{0}(T^{2},Y_{1})} χ(ν(K)) → ⋃_{k=1}^{n} M_{Y_{1}(r)}^{∗}(𝔰 ⊗ L_{k}),

χ(ν(K)) is identified with v − u = 2k + 1. Similarly, we have the gluing map #_{Y_{0}} for M_{Y_{0}(r)}^{∗}(𝔰 ⊗ L_{k}),

#_{Y_{0}} : M_{V}^{∗} \∂_{∞}^{−1}(U_{θ}) ×_{χ_{0}(T^{2},Y_{0})} χ(ν(K)) → ⋃_{k∈ℤ}M_{Y_{0}(r)}^{∗}(𝔰 ⊗ L_{k}),

where χ(ν(K)) is given by u = 2k. For each k ∈ {1, ⋯, n}, the reducible monopole for (Y_{1}, 𝔰 ⊗ L_{k}), consists of the unique point

θ_{Y_{1}}(k) = {u = u(𝔰)} ×_{χ_{0}(T^{2},Y)} {v = u + 2k + 1}.

For Y_{0} with a non-trivial Spin^{c} structure 𝔰⊗L_{k} (k ∈ ℤ, k ≠ 0), the set of reducibles is empty for any generic perturbation, and for 𝔰 ⊗ L_{0} = 𝔰, it consists of one circle of reducibles u = 0 in the cylinder χ_{0}(T^{2}, Y_{0}) = χ_{0}(T^{2}, V), which can be perturbed away by introducing a small perturbation as in Theorem 6.13 [2].

Now we apply the perturbation to simulate the effect of Dehn surgery. This amounts to a careful choice of perturbation as in Section 6 [2], which we now briefly describe.

Choose a compactly supported 2-form μ representing the generator of H_{cpt}^{2}(D^{2} × S^{1}), defined as in Lemma 3.18 [2], such that we have

∫_{D^{2}×{pt}} μ = 1  (7)

for any point on S^{1}. Under the isomorphism H_{cpt}^{2}(ν(K)) ≅ H_{1}(ν(K)), given by Poincaré duality, this form corresponds to the generator [μ] = P D_{ν(K)}(l). The class of μ in H^{2}(D^{2} × S^{1}) is trivial, and we can write μ = dν, where ν is a 1-form satisfying ∫_{S^{1}×{pt}} ν = 1, i.e. ν = P D_{T^{2}}(l). Choose on ν(K) a metric as in Lemma 3.18 [2].

Fix a U(1)-connection A_{0} representing the trivial connection on T^{2}, For any U(1)-connection A, define T_{A} to be

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

For any given ϵ > 0, we can choose a function f̂ : ℝ → ℝ with the following properties.

(a) f̂ is continuously differentiable on (−1, 1) and satisfies the periodicity f̂(t + 2) = f̂(t)

(b) the derivative f̂′ has range f̂′(t) ∈ [−1, 1] for all t ∈ [−1, 1], and satisfies f̂′(1 − t) = f̂′(1 + t) for t ∈ ℝ.

(c) the following estimate holds:

sup_{t∈[−1+ϵ,1−ϵ]} |f̂′(t) − t| < ϵ.

Now, for the Spin^{c} structure 𝔰 ⊗ L_{k} (k = 1, ⋯, n), consider the function f′_{k}(t) = f̂′(t + 1) + 2k and define a perturbation of the Seiberg-Witten equations on (Y, 𝔰 ⊗ L_{k}) in the following way:

\{ \begin{matrix}F_{A} = ∗σ(ψ, ψ) + f′_{k}(T_{A})μ \\ / ∂_{A}(ψ) = 0 \end{matrix} .  (8)

With respect to the chosen metric on ν(K), with sufficiently large positive scalar curvature on the support of μ as specified in Lemma 3.18, the only solutions of the perturbed monopole equations are reducibles (A, 0), that satisfy

F_{A} = f′_{k}(T_{A})μ.  (9)

In addition to this surgery perturbation, we consider another perturbation of the Seiberg–Witten equations on the tubular neighbourhood ν(K) in Y, Y_{1}, and Y_{0}. This perturbation has the effect of producing a global shift in the character variety to avoid the bad points on H^{1}(T^{2}, ℝ) when we deform the unperturbed geometric triangles in χ(T^{2}) to the perturbed geometric triangles in χ(T^{2}).

Let μ be a compactly supported 2-form in D^{2} × S^{1} satisfying (7). Let η > 0 be some small real parameter. Consider an additional perturbation

F_{A} = ∗σ(ψ, ψ) ± ημ  (10)

of the curvature equation on ν(K) inside Y and inside Y_{0}.

This perturbation has the effect of shifting the asymptotic values by an amount η. We choose the sign so that the line of reducibles on ν(K) ⊂ Y for 𝔰 ⊗ L_{k} becomes {(u, v)|v = 2k + η}, the line of reducibles on ν(K) ⊂ Y_{1} for 𝔰 ⊗ L_{k} remains the same {(u, v)|v − u = 2k + 1}, and the line of reducibles on ν(K) ⊂ Y_{0} for 𝔰 ⊗ L_{k} becomes {(u, v)|u = 2k + η}.

On ν(K) inside Y for 𝔰 ⊗ L_{k} we shall consider the perturbed curvature equation

F_{A} = ∗σ(ψ, ψ) + (f′_{k}(T_{A} − η) + η)μ.  (11)

Therefore, we can partition the moduli spaces for (Y, 𝔰 ⊗ L_{k}) (k = 1, ⋯, n) into the union of the moduli spaces for (Y_{1}, 𝔰 ⊗ L_{k}) (k = 1, ⋯, n) and (Y, 𝔰⊗L_{k}) (k ∈ ℤ), as in Theorem 6.3 [2]. This completes the proof of the theorem for the case of rational homology 3-sphere Y with a knot K representing a torsion element of order n in H_{1}(Y, ℤ).

For a general 3-manifold Y with a smoothly embedded knot K representing a torsion element of order n in H_{1}(Y, ℤ), the proof is essentially the same as the case of rational homology spheres discussed above, and we omit the details here.

The perturbation can be illustrated as in Figure 1 where n = 4. Here we used the notation

L_{Y} (𝔰_{k}) = {(u, v)| v = f′_{k}(u − η) + η}
L_{Y_{1}}(𝔰_{k}) = {(u, v)| v = u + 2k + 1}
L_{Y_{0}}(𝔰_{k}) = {(u, v)| v = 2k + η}.

Figure 1: The perturbed geometric triangle as in Theorem 2.1

With a more careful study of the perturbed geometric triangles, we have the following decomposition of 3-dimensional monopoles on Y under the Dehn surgery.

Theorem 2.2. With the perturbations and metrics on Y, Y_{1} and Y_{0} as in Theorem 2.1, and for any fixed m, p ∈ {0, ⋯, n − 1}, there exists a further perturbation on Y, such that we have the following relation between the critical sets of the Chern-Simons-Dirac functional on the manifolds Y, Y_{1} and Y_{0}:

M_{Y} (𝔰 ⊗ L_{m}) = M_{Y_{1}}(𝔰 ⊗ L_{m}) ∪ ⋃_{k∈ℤ}M_{Y_{0}}(𝔰 ⊗ L_{nk+p}).  (12)

Proof. In the proof of Theorem 2.1, we know that the perturbed Seiberg-Witten monopoles on Y, Y_{1} and Y_{0} are given by the following gluing models (here we assume that Y is a rational homology 3-sphere):

M_{Y (r)}^{∗}(𝔰 ⊗ L_{k}) ≅ M_{V}^{∗} \∂_{∞}^{−1}(U_{θ}) ×_{χ_{0}(T^{2},Y)} {v = 2k},

M_{Y_{1}(r)}^{∗}(𝔰 ⊗ L_{k}) ≅ M_{V}^{∗} \∂_{∞}^{−1}(U_{θ}) ×_{χ_{0}(T^{2},Y_{1})} {v = u + 2k + 1},

M_{Y_{0}(r)}^{∗}(𝔰 ⊗ L_{k}) ≅ M_{V}^{∗} \∂_{∞}^{−1}(U_{θ}) ×_{χ_{0}(T^{2},Y_{0})} {u = 2k}.

Note that the additional perturbation (10) of the curvature equation on ν(K) inside Y and inside Y_{0} introduce a shift of coordinates (u, v) to (u+η, v +η). We can introduce these new coordinates, still denoted by (u, v). Then the reducible line for V is given by u = −η in H^{1}(T^{2}, ℝ), and the set of “bad points” is given by {(−η, −η + 2k) : k ∈ ℤ}.

We will show that there exists a further surgery perturbation on Y that suits the purpose of identifying monopoles on Y, Y_{1} and Y_{0} as stated in the Theorem. Recall that, in the case of rational homology 3-spheres, the Seiberg-Witten-Floer homology is metric and perturbation dependent (cf. [4]). A change in the Floer homology occurs whenever the restriction to χ_{0}(T^{2}, Y) of the perturbed reducible on Y passes through one of the “bad points” {(−η, −η + 2k) : k ∈ ℤ} along the line u = −η. We will see later that no wall-crossing takes place, as long as we take the same m 𝔰 ⊗ L_{m} on Y and Y_{1}. Without loss of the generality, after possible coordinates change, we can assume that m = 0. Fix p ∈ {0, ⋯, n − 1}. We will construct a function f′_{0} : ℝ → ℝ, which depends on a small ϵ > 0, such that, as ϵ → 0, the curve v = f′_{0}(u) approaches the union of lines

L_{Y_{1}}(𝔰 ⊗ L_{0}) ∪ ⋃_{k∈ℤ}L_{Y_{0}}(𝔰 ⊗ L_{nk+p})

with

L_{Y_{1}}(𝔰 ⊗ L_{0}) = {v = u + 1}

and

L_{Y_{0}}(𝔰 ⊗ L_{nk+p}) = {u = 2nk + 2p}.

We identify χ_{0}(T^{2}, V) with the fundamental domain

{u ∈ ℝ} × {0 ≤ v < 2n},

in H^{1}(T^{2}, ℝ). The asymptotic values ∂_{∞}(M_{V}^{∗}) ⊂ χ_{0}(T^{2}, V) can be lifted to H^{1}(T^{2}, ℝ) periodically. Using this 2n–periodicity, we only need to construct a function f_{0} : [−1, 2n − 1] → [0, 2n], which depends on ϵ, such that, for any given ϵ ≤ ϵ_{0}, we have

sup_{t∈[−1,2n−1]\[−ϵ,ϵ]} |f′_{0}(t) − t| < ϵ.

Such function can be easily constructed as in the proof of Theorem 2.1. Then over [2nk − 1, 2nk + 2n − 1], f′_{0}(t) is defined to be f′_{0}(t) + 2nk. See Figure 2 where the perturbation is illustrated in the cases with n = 4, m = p = 0 and with n = 4, m = 0, p = 2, respectively. The general case can be proved by a similar method.

Figure 2: The perturbed geometric triangle as in Theorem 2.2

3 The relative gradings

In the previous section, we have the following decomposition:

M_{Y,μ}(𝔰 ⊗ L_{m}) = M_{Y_{1}}(𝔰 ⊗ L_{m}) ∪ ⋃_{k∈ℤ}M_{Y_{0}}(𝔰 ⊗ L_{nk+p})

for any fixed m, p ∈ {0, ⋯, n − 1}. Assume that Y is a rational homology 3-sphere. First we fix a grading on M_{Y,μ}(𝔰⊗L_{0}) defined in terms of the spectral flow of the linearization operator for the 3-dimensional Seiberg-Witten equations of a path connecting an irreducible monopole in M_{Y,μ}(𝔰 ⊗ L_{0}) to the unique reducible θ_{Y} (0) in the configuration space for (Y, 𝔰 ⊗ L_{0}). Then the analysis of the relative grading in Part I section 7 [2] can be applied to induce a compatible grading on M_{Y_{1}}(𝔰 ⊗ L_{0}) ∪ ⋃_{k∈ℤ} M_{Y_{0}}(𝔰 ⊗ L_{nk+p}) as follows, cf. Proposition 7.3 – Corollary 7.7 in [2].

Proposition 3.1. Let Y be a rational homology 3-sphere. For any fixed p ∈ {0, ⋯, n − 1}, the Floer complexes

C_{∗}(Y, 𝔰 ⊗ L_{0}, μ) = ⊕_{a∈M_{Y,μ}(𝔰⊗L_{0})}ℤ⟨a⟩,

C_{∗}(Y_{1}, 𝔰 ⊗ L_{0}) = ⊕_{a∈M_{Y_{1}}(𝔰⊗L_{0})}ℤ⟨a⟩,

C_{∗}(Y_{0}, 𝔰 ⊗ L_{nk+p}) = ⊕_{a∈M_{Y_{0}}(𝔰⊗L_{nk+p})}ℤ⟨a⟩,

have a compatible relative grading of generators in the following sense.

1. Suppose given two irreducible critical points a, b in M_{Y_{1}}^{∗}(𝔰 ⊗ L_{0}), and the corresponding elements a^{ϵ}, b^{ϵ} in M_{Y,μ}^{∗}(𝔰 ⊗ L_{0}) under the above decomposition (12). Then

deg_{Y,μ}(a^{ϵ}) − deg_{Y,μ}(b^{ϵ}) = deg_{Y_{1}}(a) − deg_{Y_{1}}(b).

2. Suppose given two monopoles a, b in M_{Y_{0}}^{∗}(𝔰 ⊗ L_{nk+p}), and the corresponding elements a^{ϵ}, b^{ϵ} in M_{Y,μ}^{∗}(𝔰⊗L_{0}) under the above decomposition (12). Then

deg_{Y_{0},𝔰_{k}}(a) − deg_{Y_{0},𝔰_{k}}(b) = deg_{Y,μ}(a^{ϵ}) − deg_{Y,μ}(b^{ϵ}) mod (2nk + 2p).

Therefore, the grading deg_{Y,μ} defines a ℤ-valued lift of the ℤ_{2nk+2p}- valued relative index on M_{Y_{0}}^{∗}(𝔰 ⊗ L_{nk+p}) under the decomposition of M_{Y,μ}^{∗}(𝔰 ⊗ L_{0}).

For a general 3-manifold (Y, 𝔰) with b_{1}(Y) > 0 and a knot K representing a torsion element of order n, if 𝔰 is a torsion Spin^{c} structure, we know that, for any k ∈ {0, ⋯, n−1}, the Spin^{c} structure 𝔰⊗L_{k} also has a torsion class c_{1}(𝔰⊗L_{k}). Thus, after a small perturbation to get rid of the (S^{1})^{b_{1}(Y)}-family of reducibles, we have a ℤ-graded

M_{Y,μ}^{∗}(𝔰 ⊗ L_{k}) ≅ M_{Y,μ}(𝔰 ⊗ L_{k}).

Then it is easy to see that the results in Part I section 7 [2] hold in this case as well without any substantial change.

Now assume that c_{1}(𝔰) is a non-torsion element, with multiplicity 2ℓ in H^{2}(Y, ℤ)/Torsion. Then, for any k ∈ {0, ⋯, n − 1}, the set of generators

M_{Y,μ}^{∗}(𝔰 ⊗ L_{k}) ≅ M_{Y,μ}(𝔰 ⊗ L_{k})

is 2ℓ-graded, and so is

M_{Y_{1}}^{∗}(𝔰 ⊗ L_{k}) ≅ M_{Y_{1}}(𝔰 ⊗ L_{k}).

Then for any k ∈ ℤ, any non-empty moduli space

M_{Y,μ}^{∗}(𝔰 ⊗ L_{k}) ≅ M_{Y,μ}(𝔰 ⊗ L_{k})

is ℤ_{2ℓ_{[k]}}-graded, where 2ℓ_{[k]} is the maximum common factor of 2ℓ and 2k. We can choose a relative grading on M_{Y,μ}^{∗}(𝔰⊗L_{0}) by the spectral flow of the linearization operator along a path connecting any irreducible monopole in M_{Y}^{∗} (𝔰⊗L_{0}) to a fixed monopole a_{0} in M_{Y,μ}^{∗}(𝔰⊗L_{0}). Note that this grading is 2ℓ-graded. Then the analysis in Part I section 7 can also be applied to obtain the following proposition.

Proposition 3.2. For p ∈ {0, ⋯, n − 1} and k ∈ ℤ, the Floer complexes C_{∗}(Y, 𝔰⊗L_{0}), C_{∗}(Y_{1}, 𝔰⊗L_{0}), and C_{∗}(Y_{0}, 𝔰⊗L_{nk+p}) have a compatible relative grading of generators in the following sense.

1. Suppose given two irreducible critical points a, b in M_{Y_{1}}^{∗}(𝔰 ⊗ L_{0}), and the corresponding elements a^{ϵ}, b^{ϵ} in M_{Y,μ}^{∗}(𝔰 ⊗ L_{0}), then

deg_{Y,μ}(a^{ϵ}) − deg_{Y,μ}(b^{ϵ}) = deg_{Y_{1}}(a) − deg_{Y_{1}}(b),

as a ℤ_{ℓ}-valued function.

2. Suppose given two monopoles a, b in M_{Y_{0}}^{∗}(𝔰 ⊗ L_{nk+p}), and the corresponding elements a^{ϵ}, b^{ϵ} in M_{Y,μ}^{∗}(𝔰 ⊗ L_{0}), then

deg_{Y_{0},𝔰_{k}}(a) − deg_{Y_{0},𝔰_{k}}(b) = deg_{Y,μ}(a^{ϵ}) − deg_{Y,μ}(b^{ϵ}) mod (2ℓ_{[nk+p]}).

Thus, the induced grading deg_{Y,μ} on M_{Y_{0}}^{∗}(𝔰⊗L_{nk+p}), under the decomposition (12), defines a choice of a ℤ_{2ℓ}-valued lifting of the ℤ_{2ℓ_{[nk+p]}}- valued relative index.

In the rest of this section we discuss the induced gradings on the various M_{Y}^{∗} (𝔰 ⊗ L_{m}), for m ∈ {1, ⋯, n − 1}. Notice that, in the case of a rational homology 3-sphere Y, the perturbed line L_{Y} (𝔰⊗L_{m}) is a deformation of the line L_{Y} (𝔰 ⊗ L_{0}). Along this deformation, the corresponding parameterized reducibles form a path connecting θ_{Y} (0) to θ_{Y} (m). As the following Lemma shows, this deformation can be realized as a perturbation of the monopole equations for (Y, 𝔰 ⊗ L_{0}).

Lemma 3.3. For any m ∈ {1, ⋯, n − 1}, there exists a perturbation μ_{m} for the Seiberg-Witten monopole equations on (Y, 𝔰 ⊗ L_{0}), such that there is a diffeomorphism:

M_{Y,μ_{m}}(𝔰 ⊗ L_{0}) ≅ M_{Y} (𝔰 ⊗ L_{m}).

Moreover, if Y is a rational homology 3-sphere, the grading on M_{Y}^{∗} (𝔰 ⊗ L_{m}) induces a grading on M_{Y,μ_{m}}^{∗}(𝔰 ⊗ L_{0}). Under the above identification, the resulting grading differs from the original grading on M_{Y}^{∗} (𝔰 ⊗ L_{0}) by the wall-crossing formulae studied in [4] for (Y, 𝔰 ⊗ L_{0}). If b_{1}(Y) > 0, then the induced grading on M_{Y} (𝔰⊗L_{m}) from M_{Y,μ_{m}}(𝔰⊗L_{0}) agrees with the relative grading on M_{Y} (𝔰 ⊗ L_{m}).

Proof. The first claim follows from the gluing models of the monopoles in M_{Y} (𝔰 ⊗ L_{0}) and M_{Y} (𝔰 ⊗ L_{m}). Then, using the results in [4], we know that, in the case of the rational homology 3-sphere, the spectral flow of the twisted Dirac operator along the path of reducibles gives the index shift on M_{Y}^{∗} (𝔰 ⊗ L_{0}) and M_{Y,μ_{m}}^{∗}(𝔰 ⊗ L_{0}) according to the wall-crossing formulae derived in [4]. Again by the results of [4], for Y with b_{1}(Y) > 0, the induced grading from the parametrized spectral flow is same as the original relative index on M_{Y} (𝔰 ⊗ L_{m}).

Thus, Lemma 3.3 provides a consistent way of assigning a choice of absolute grading on the various M_{Y} (𝔰 ⊗ L_{m}). In fact, notice that, in the case of rational homology 3-sphere Y, for different values p ∈ {0, ⋯, n−1}, the additional perturbations on (Y, 𝔰 ⊗ L_{0}) to achieve the decomposition

M_{Y}^{∗} (𝔰 ⊗ L_{0}) ≅ M_{Y_{1}}^{∗}(𝔰 ⊗ L_{0}) ∪ ⋃ M_{Y_{0}}^{∗}(𝔰 ⊗ L_{nk+p}),

does not cross the wall where the grading would jump.

The degree shift of Lemma 3.3 can be described as follows. Let θ_{Y} (t) be the path of reducibles on (Y, 𝔰 ⊗ L_{0}) for the family of perturbations connecting M_{Y}^{∗} (𝔰 ⊗ L_{0}) to M_{Y,μ_{m}}^{∗}(𝔰 ⊗ L_{0}), then the wall crossing formulae in [4] tell us that the index shift is given by the spectral flow

SF (/ ∂_{θ_{Y} (t)}).  (13)

4 Geometric limits and the holomorphic triangles

4.1 Surgery cobordisms

We first briefly describe the surgery cobordisms from Y_{1} to Y, from Y to Y_{0}, and from Y_{0} to Y_{1}, respectively, as in [6]. The cobordism W_{1}, from Y_{1} to Y, is obtained by removing from the trivial cobordism Y_{1} × [0, 1] 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}, form Y to Y_{0}, is obtained by removing from the trivial cobordism Y_{0} × [0, 1] 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 nontrivial cobordism W_{1}, or, respectively, the boundary component Y_{0} × {0} in the trivial cobordism to the boundary component Y × {0} in W_{0}. The cobordism W̄_{2} connecting Y_{0} and Y_{1}, satisfies the relation

W̄ = W̄_{2}#ℂℙ^{2},

where W̄ is the composite cobordism W̄ = W̄_{0} ∪_{Y} W̄_{1}.

We 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 section 2 (see also [2]). We 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. As in [6], 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)).  (14)

The 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}].

As in [6], consider an interior point x_{i} contained in the core disk of the 2-handle, x_{i} ∈ D_{i}, and 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)), endowed with an extra asymptotic end of the form S^{3} × [0, ∞) at the puncture, we can identify a product region

V = ν(K)_{r_{0}} × ℝ ≅ D × (D_{i}\{x_{i}}).  (15)

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, where S^{3} is 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 (15) in W_{i} identifies the standard solid torus ν in S^{3} with the neighbourhood ν(K) of the knot K in Y. Similarly, there is a product region Ṽ which identifies the other solid torus ν̃ in S^{3} with the tubular neighbourhood ν(K) in Y_{i}, after the surgery. The resulting punctured cobordism can be written as

Ŵ_{i}(r) = (V_{r} × ℝ) ∪ V(r) ∪ Ṽ(r).  (16)

We now impose a choice of metrics and perturbations for the Seiberg-Witten equations on the cobordisms as in subsection 2.2 of [6]. Then we can adopt the results of [5][6] to understand the asymptotic limits of finite energy monopoles, under the splitting of the punctured cobordisms as r → ∞, as in (16).

4.2 Geometric limits and holomorphic triangles

We assume that Y and Y_{1} are endowed with the Spin^{c} structure 𝔰⊗L_{0}. The other Spin^{c} structures can be studied analogously.

On the surgery cobordism W_{1}(r), there is a ℤ-family of Spin^{c} structures whose restrictions to the two ends agree with 𝔰⊗L_{0} on Y and Y_{1} respectively. We will study the moduli spaces of monopoles on W_{1}(r) with asymptotic values in M_{Y_{1}}(𝔰 ⊗ L_{0}) and M_{Y,μ}(𝔰 ⊗ L_{0}) at the two ends, where μ is the surgery perturbation defined by f′_{0} as in the proof of Theorem 2.2. In the case b_{1}(Y) > 0, we only consider the components of minimal energy, as defined in [6] [7], among all the possible moduli spaces with different Spin^{c} structures and with the given asymptotic values. For a rational homology 3-sphere Y, we only consider the moduli spaces of minimal dimension among the ℤ-family of Spin^{c} structures.

With this convention understood, we denote by M^{W_{1}}(a_{1}, a) the moduli space with asymptotic values a_{1} ∈ M_{Y_{1}}(𝔰 ⊗ L_{0}) and a ∈ M_{Y,μ}(𝔰 ⊗ L_{0}). Similarly, we denote by M^{W_{0}}(a, a_{0}) the moduli space with asymptotic values a ∈ M_{Y,μ}(𝔰 ⊗ L_{0}) and a_{0} ∈ M_{Y_{0}}(⊗L_{nk+p}) for k ∈ ℤ, for a fixed p ∈ {0, …, n − 1}. Under a generic choice of the perturbation, all these moduli spaces M^{W_{1}}(a_{1}, a) and M^{W_{0}}(a, a_{0}) are cut out transversely and of the expected dimension.

The convergence and gluing arguments developed in [4] can be applied to this case as well, to give the following compactifications of M^{W_{1}}(a_{1}, a) and M^{W_{0}}(a, a_{0}).

Proposition 4.1. Suppose that M^{W_{1}}(a_{1}, a) is non-empty, then M^{W_{1}}(a_{1}, a) admits a compactification to a manifold with corners, where the codimension 1 boundary strata consist of

\begin{matrix}⋃_{c∈M_{Y,μ}^{∗}(𝔰⊗L_{0})} M^{W_{1}}(a_{1}, c) × M̂_{Y,μ}(c, a) \\ ∪ ⋃_{c_{1}∈M_{Y_{1}}^{∗}(𝔰⊗L_{0})} M̂_{Y_{1}}(a_{1}, c_{1}) × M^{W_{1}}(c_{1}, a), \end{matrix}  (17)

and with extra components

\begin{matrix}M̂_{Y_{1}}(a_{1}, θ_{1}) × U(1) × M^{W_{1}}(θ_{1}, a) \\ M^{W_{1}}(a_{1}, θ) × U(1) × M̂_{Y,μ}(θ, a), \end{matrix}  (18)

when splitting through the reducibles θ_{1} and θ in M_{Y_{1}}^{∗}(𝔰 ⊗ L_{0}) and M_{Y,μ}^{∗}(𝔰 ⊗ L_{0}) respectively . We also have the similar compactification for M^{W_{0}}(a, a_{0}).

Now we can describe the geometric limits of monopoles in M^{W_{1}}(a_{1}, a) and M^{W_{0}}(a, a_{0}) when stretching r → ∞ inside

Ŵ_{i}(r) = V_{r} × ℝ ∪ V(r) ∪ Ṽ(r).

We only describe the case of Ŵ_{1}(r). With similar arguments we have the corresponding geometric limits for Ŵ_{0}(r). The proof of the results stated below on these geometric limits follows from the same arguments of [5][6].

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

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

with

[A_{i}^{−}, ψ_{i}^{−}], [A_{j}^{+}(ϵ), ψ_{j}^{+}(ϵ)] ∈ M_{V}^{∗} (𝔰 ⊗ L_{0}|_{V})

and

a_{∞,i}^{−} ∈ L_{Y_{1}}(𝔰 ⊗ L_{0})
a_{∞,j}^{+}(ϵ) ∈ L_{Y,μ}(𝔰 ⊗ L_{0}).

Theorem 4.2. (Proposition 3.1 and Remark 6.2 in [6]) Assume that M^{W_{1}(r)}(a_{i}^{(1)}, a_{j}(ϵ)) is non-empty for all sufficiently large r ≥ r_{0}. Then a family of solutions [A_{1}(r), Ψ_{1}(r)] in M^{W_{1}(r)}(a_{i}^{(1)}, a_{j}(ϵ)) defines the following geometric limits on

V_{r} × ℝ ∪ V(r) ∪ Ṽ(r)

as r → ∞. (a). A finite energy solution [A′, Ψ′]^{ϵ} of the perturbed Seiberg-Witten equations 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′_{0}(u)} and by

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

Here we have

ϑ_{1} = L_{Y_{1}}(𝔰 ⊗ L_{0}) ∩ L_{Y,μ}(𝔰 ⊗ L_{0}).

This Theorem shows that the moduli space M^{W_{1}(r)}(a_{i}^{(1)}, a_{j}(ϵ)) is characterized by the geometric limits on V × ℝ from (a) and the holomorphic triangles in (c). Two typical holomorphic triangles for M^{W_{1}(r)}(a_{i}^{(1)}, a_{j}(ϵ)) or M^{W_{0}(r)}(a_{i}(ϵ), a_{j}^{(0)}) are illustrated in Figure 3 where n = 4, m = k = 0 and p = 2. Here the points ϑ_{i} are the intersection points

ϑ_{1} = L_{Y_{1}}(𝔰 ⊗ L_{0}) ∩ L_{Y,μ}(𝔰 ⊗ L_{0})

for W_{1} and

ϑ_{0} = L_{Y,μ}(𝔰 ⊗ L_{0}) ∩ L_{Y_{0}}(𝔰 ⊗ L_{nk+p})

in the case of W_{0}. In other words, ϑ_{i} is the restriction to T^{2} = ∂ν = ∂ν̃ of the unique reducible point θ_{S^{3}} at the puncture in the cobordism.

Figure 3: Holomorphic triangles for W_{1}(r) and W_{0}(r)

In order to glue back the geometric limits and the holomorphic triangles, there are some admissible conditions for the geometric limits on V ×ℝ, which are studied in section 6.1, section 6.2 and section 6.3 [6]. As in section 6 [6], assuming that the element a_{∞} is away from the bad points in the character variety of T^{2}, we denote by M̂_{V ×ℝ}(a_{∞}) the balanced energy moduli space of the Seiberg-Witten equations on V × ℝ with asymptotic value a_{∞} in the radial direction. Then we have

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}^{∗} .

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 the expected dimension, where the orientation is given by the corresponding determinant line bundle of the linearization operator for the monopole equations on V × ℝ.

Remember that we are only considering the components with minimal energy or minimal dimension among all the moduli spaces of finite energy monopoles with fixed asymptotic values. Recall that there is a notion of admissible triples (cf. Definition 6.4 [6]), which singles out those elements ([A, ψ], [Ã, ψ̃], a_{∞}) which arise as part of the geometric limits of solutions in M^{W_{1}(r)}(a_{1}, a) (or M^{W_{0}(r)}(a_{1}, a_{0}), or M_{Y (r)×ℝ}(a, b) etc). For example, in the case of M^{W_{1}(r)}(a_{1}, a), with

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

an admissible triple ([A, ψ], [Ã, ψ̃], a_{∞}) must satisfy the following conditions: 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^{+}].

By the results of [6], we know that the possible choices of a_{∞} and of the admissible data in M̂_{V ×ℝ}(a_{∞}) are uniquely determined by the inequivalent holomorphic triangles Δ with vertices {a^{−}, ϑ_{1}, a^{+}} and sides along the union of Lagrangians ℓ ∪ ℓ_{1} ∪ ℓ_{μ}, with ℓ defined by the asymptotic values ∂_{∞}(M_{V}^{∗}). We denote by Ξ_{W_{1}}(a_{1}, a) the set of such inequivalent holomorphic triangles. Moreover, under the identification of the choice of admissible triples ([A, ψ], [Ã, ψ̃], a_{∞}) with the choice of inequivalent oriented holomorphic triangles in Ξ_{W_{1}}(a_{1}, a), the gluing map gives an orientation preserving diffeomorphism

#_{W_{1}} : ⋃_{([A,ψ],[Ã,ψ̃],a_{∞})∈Ξ_{W_{1}}(a_{1},a)} M_{V ×ℝ}([A, ψ], [Ã, ψ̃], a_{∞}) → M^{W_{1}}(a_{1}, a). (19)

Similarly, there are orientation preserving diffeomorphisms given by the gluing maps for W_{0}(r), (Y (r) × ℝ, 𝔰 ⊗ L_{0}), (Y_{1}(r) × ℝ, 𝔰 ⊗ L_{0}) and (Y_{0}(r) × ℝ, 𝔰 ⊗ L_{nk+p}) defined over the set of admissible triples, which are in turn determined by the corresponding inequivalent oriented holomorphic triangles or holomorphic discs. The corresponding sets are denoted by Ξ_{W_{0}}(a, a_{0}), Ξ_{Y} (a, b), Ξ_{Y_{1}}(a_{1}, b_{1}) and Ξ_{Y_{0}}(a_{0}, b_{0}), respectively. We summarize all the gluing theorems for the geometric limits and holomorphic triangles (discs) as follows:

Theorem 4.3. Suppose given a pair of monopoles a^{(1)}, b^{(1)} in M_{Y_{1}}(𝔰⊗L_{0}), and a pair of monopoles a(ϵ), b(ϵ) in M_{Y_{1},μ}(𝔰 ⊗ L_{0}), where the surgery perturbation μ determined by f′_{0} depends on a small parameter ϵ. Suppose given a pair of monopoles a^{(0)}, b^{(0)} in M_{Y_{0}}(𝔰 ⊗ L_{nk+p}) for some k ∈ ℤ and p ∈ {0, ⋯, n − 1}. Then, for sufficiently large r, the gluing maps give the following orientation preserving diffeomorphisms

⋃_{Θ∈Ξ_{Y} (a(ϵ),b(ϵ))} M_{V ×ℝ}(Θ) ⟶^{#_{Y (r)×ℝ}} M_{Y ×ℝ}(a(ϵ), b(ϵ)),

⋃_{Θ∈Ξ_{Y_{1}}(a^{(1)},b^{(1)})} M_{V ×ℝ}(Θ) ⟶^{#_{Y_{1}(r)×ℝ}} M_{Y_{1}×ℝ}(a^{(1)}, b^{(1)}),

⋃_{Θ∈Ξ_{Y_{0}}(a^{(0)},b^{(0)})} M_{V ×ℝ}(Θ) ⟶^{#_{Y_{0}(r)×ℝ}} M_{Y_{0}×ℝ}(a^{(0)}, b^{(0)}),

⋃_{Θ∈Ξ_{W_{1}}(a^{(1)},a(ϵ))} M_{V ×ℝ}(Θ) ⟶^{#_{W_{1}}} M^{W_{1}}(a^{(1)}, a(ϵ)),

⋃_{Θ∈Ξ_{W_{0}}(a(ϵ),a^{(0)})} M_{V ×ℝ}(Θ) ⟶^{#_{W_{0}}} M^{W_{0}}(a(ϵ), a^{(0)}),

where, for simplicity, we denoted a triple ([A, ψ], [Ã, ψ̃], a_{∞}) by Θ.

5 Proof of exactness and the surgery triangle

In this section, we will prove the main theorem of this paper. Notice that we only prove the refined exact triangle in Theorem 1.1 for m = 0. The arguments for this case can be adapted to give an analogous proof for m ≠ 0.

5.1 The chain homomorphisms

Recall that the moduli spaces for W_{1} and W_{0} which we consider are only the components of minimal energy (or dimension). They are smooth and oriented manifolds of the expected dimension, and they can be compactified according to Proposition 4.1. In particular, whenever one such moduli space is 0-dimensional, we have a counting of points with the orientation. Thus, as in [6], we can define the chain homomorphisms between the Floer chain groups of (Y, μ, 𝔰 ⊗ L_{0}), (Y_{1}, 𝔰 ⊗ L_{0}) and (Y_{0}, 𝔰 ⊗ L_{nk+p}) as follows.

Definition 5.1. We define the map w_{∗}^{1} : C_{∗}(Y_{1}) → C_{∗}(Y, μ) by assigning the matrix elements

⟨a, w_{∗}^{1}(a_{1})⟩ = #M^{W_{1}}(a, a_{1}),

where the right hand side is a counting of points with the orientation if M^{W_{1}}(a, a_{1}) is 0-dimensional and non-empty, and it is ⟨a, w_{∗}^{1}(a_{1})⟩ = 0 otherwise. Similarly, we define the map w_{∗}^{0} : C_{∗}(Y, μ) → ⊕_{k}C_{(∗)}(Y_{0}, 𝔰⊗L_{nk+p}), with the the matrix coefficients

⟨a_{0}, w_{∗}^{0}(a)⟩ = #M^{W_{0}}(a, a_{0})

if M^{W_{0}}(a, a_{0})) is 0-dimensional and non-empty, and ⟨a_{0}, w_{∗}^{0}(a)⟩ = 0 otherwise. Here the chain groups C_{(∗)}(Y_{0}, 𝔰 ⊗ L_{nk+p}) are equipped with the lifting of the relative grading on M_{Y_{0}}(𝔰 ⊗ L_{nk+p}) as described in Proposition 3.2.

Notice that on W_{1} there is a ℤ-family of Spin^{c} structures which agree with 𝔰⊗L_{0} when restricted to the two ends. From Lemma 4.1 in [6], the choice of components with minimal energy and dimension essentially excludes other Spin^{c} structures when we consider the moduli space M^{W_{1}}(a_{1}, a) with fixed asymptotic values a_{1} ∈ M_{Y_{1}}^{∗}(𝔰 ⊗ L_{0}) and a ∈ M_{Y,μ}^{∗}(𝔰 ⊗ L_{0}).

In the case of b_{1}(Y) > 0 and c_{1}(𝔰 ⊗ L_{0}) is non-torsion, when both M_{Y,μ}(𝔰 ⊗ L_{0}) and M_{Y_{1}}(𝔰 ⊗ L_{0}) are ℤ_{2ℓ}-graded, with 2ℓ the multiplicity of c_{1}(𝔰 ⊗ L_{0}) in H^{2}(Y, ℤ)/Torsion, then Proposition 3.2 ensures that there exists a compatible grading on M_{Y_{1}}(𝔰 ⊗ L_{0}) and a ℤ_{2ℓ}–lifting of the relative grading on any M_{Y_{0}}(𝔰 ⊗ L_{nk+p}). For those Spin^{c} structures 𝔰 ⊗ L_{0} with torsion c_{1}(𝔰 ⊗ L_{0}), the corresponding Floer homology is defined to be

HF_{∗}^{SW} (Y, 𝔰 ⊗ L_{0}, ℤ[[t]])|t = 0,

as described in [7]. Then it is easy to see that the choice of components with minimal energy and dimension makes w_{∗}^{1} and w_{∗}^{0} well-defined.

Then, with the help of the compactifications in Proposition 4.1, the proofs of Lemma 4.3 and Lemma 4.4 in [6] go through without any substantial change to give the following Lemma.

Lemma 5.2. 1. The maps w_{∗}^{i} are chain homomorphisms.

2. Suppose given a_{1} in M_{Y_{1}}^{∗}(𝔰⊗L_{0}) and a_{0} in M_{Y_{0}}(𝔰⊗L_{nk+p}), such that the relative index induced from M_{Y,μ}^{∗}(𝔰 ⊗ L_{0}) as in Proposition 3.2 is zero, then the composite map w_{∗}^{0} ∘ w_{∗}^{1} is given by

⟨w_{∗}^{0} ∘ w_{∗}^{1}(a_{1}), a_{0}⟩ = #M^{W} (a_{1}, a_{0}).

Here we use the same convention on the choice of the moduli spaces for W = W_{1}#_{Y} W_{0}.

Thus, we have obtained a sequence of chain complexes induced by the surgery cobordisms:

0 → C_{∗}(Y_{1}, 𝔰 ⊗ L_{0}) ⟶^{w_{∗}^{1}} C_{∗}(Y, μ, 𝔰 ⊗ L_{0}) ⟶^{w_{∗}^{0}} ⊕_{k∈ℤ}C_{(∗)}(Y_{0}, 𝔰 ⊗ L_{nk+p}) → 0,

for any fixed p ∈ {0, ⋯, n − 1}. We shall prove that this sequence is exact and that the corresponding exact triangle is the surgery triangle, namely the connecting homomorphism is also defined via a surgery cobordism. The gluing theorems 4.3 for the admissible geometric limits on V × ℝ and the corresponding holomorphic triangles (or discs) will play a crucial role in the proof of these statements.

5.2 Proof of exactness

We first prove that w_{∗}^{1} is injective and w_{∗}^{0} is surjective, then we show that as in [6], w_{∗}^{0} ∘ w_{∗}^{1} = 0. This, together with the specific properties of the maps w_{∗}^{1} and w_{∗}^{0} will be sufficient to prove the exactness in the middle term of the sequence:

0 → C_{∗}(Y_{1}, 𝔰 ⊗ L_{0}) ⟶^{w_{∗}^{1}} C_{∗}(Y, μ, 𝔰 ⊗ L_{0}) ⟶^{w_{∗}^{0}} ⊕_{k∈ℤ}C_{(∗)}(Y_{0}, 𝔰 ⊗ L_{nk+p}) → 0.

We can partition the moduli space M_{Y,μ}(𝔰 ⊗ L_{0}) according to the decomposition (12) of Theorem 2.2,

M_{Y,μ}^{∗}(𝔰 ⊗ L_{0}) ≅ M_{Y_{1}}^{∗}(𝔰 ⊗ L_{0}) ∪ ⋃_{k∈ℤ}M_{Y_{0}}(𝔰 ⊗ L_{nk+p}),

where μ represents the surgery perturbation determined by f′_{0} as in the proof of Theorem 2.2. Here f′_{0} depends on a small parameter ϵ > 0. That is, we identify M_{Y,μ}^{∗}(𝔰 ⊗ L_{0}) with a collection of points

M_{Y,μ}^{∗}(𝔰 ⊗ L_{0}) = {a_{i}^{(1)}(ϵ)}_{i=1,...r} ∪ {a_{j}^{(0)}(ϵ)}_{j=r+1,... ,s},  (20)

so that, as we let ϵ → 0, the points {a_{i}^{(1)}(ϵ)}_{i=1,...r} get identified with the corresponding elements {a_{i}^{(1)}}_{i=1,...r} in M_{Y_{1}}^{∗}(𝔰 ⊗ L_{0}) and, similarly, the points {a_{j}^{(0)}(ϵ)}_{j=r+1,... ,s} get identified with the corresponding elements

{a_{j}^{(0)}}_{j=r+1,... ,s} in ∪_{k∈ℤ}M_{Y_{0}}(𝔰 ⊗ L_{nk+p}).

Lemma 5.3. The coefficients of the maps w_{∗}^{1} and w_{∗}^{0} satisfy

1. ⟨a_{i}^{(1)}(ϵ), w_{∗}^{1}(a_{j}^{(1)}⟩ = δ_{ij};

2. ⟨a_{i}^{(0)}, w_{∗}^{0}(a_{j}^{(0)}(ϵ)⟩ = δ_{ij}.

Thus, w_{∗}^{1} is injective and w_{∗}^{0} is surjective.

Proof. Using the gluing Theorem 4.3, we can describe the moduli space M^{W_{1}}(a_{i}^{(1)}, a_{j}^{(1)}(ϵ)) for two monopoles a_{i}^{(1)}, a_{j}^{(1)}(ϵ) of relative index 0. With our convention on the choice of components for these moduli spaces, we know that M^{W_{1}}(a_{i}^{(1)}, a_{j}^{(1)}(ϵ)) is zero-dimensional, if non-empty, and obtained as the gluing of the admissible geometric limits on V ×ℝ and the corresponding holomorphic triangles.

As we let ϵ → 0, it is easy to see that the holomorphic triangles degenerate to certain holomorphic discs, and the admissible geometric limits on V × ℝ for M^{W_{1}}(a_{i}^{(1)}, a_{j}^{(1)}(ϵ)) are identified with the admissible geometric

limits on V × ℝ for M_{Y_{1}×ℝ}(a_{i}^{(1)}, a_{j}^{(1)}). Since the relative index of a_{i}^{(1)}, a_{j}^{(1)} is

zero, M_{Y_{1}×ℝ}(a_{i}^{(1)}, a_{j}^{(1)}), being a zero-dimensional moduli space of minimal

energy, is empty unless a_{i}^{(1)} = a_{j}^{(1)}, in which case M_{Y_{1}×ℝ}(a_{i}^{(1)}, a_{i}^{(1)}) consists of a unique solution. This proves that

⟨a_{i}^{(1)}(ϵ), w_{∗}^{1}(a_{j}^{(1)}⟩ = δ_{ij}.

Similarly, we obtain ⟨a_{i}^{(0)}, w_{∗}^{0}(a_{j}^{(1)}(ϵ)⟩ = δ_{ij}.

Now we prove the exactness in the middle term. We proceed as in [6] to show that w_{∗}^{0} ∘ w_{∗}^{1} = 0, which, together with Lemma 5.3 is sufficient to establish the exact triangle. Again, we will use heavily the gluing theorem 4.3 to analyze the moduli spaces on the cobordisms. The following Lemma is the direct consequence of the results of Lemma 5.2 and Lemma 5.3.

Lemma 5.4. Suppose given a_{i}^{(1)} ∈ M_{Y_{1}}(𝔰⊗L_{0}) and a_{j}^{(0)}(ϵ) ∈ M_{Y,μ}(𝔰⊗L_{0})

which corresponds to a_{j}^{(0)} ∈ ∪_{k∈ℤ}M_{Y_{0}}(𝔰 ⊗ L_{nk+p}). The coefficients of the composition map w_{∗}^{0} ∘ w_{∗}^{1} satisfy

⟨a_{j}^{(0)}, w_{∗}^{0} ∘ w_{∗}^{1}(a_{i}^{(1)})⟩ = ⟨a_{j}^{(0)}(ϵ), w_{∗}^{1}(a_{i}^{(1)})⟩ + ⟨a_{j}^{(0)}, w_{∗}^{0}(a_{i}^{(1)}(ϵ))⟩.

Since the coefficient ⟨a_{j}^{(0)}(ϵ), w_{∗}^{1}(a_{i}^{(1)})⟩ is given by the counting of

monopoles in M^{W_{1}}(a_{i}^{(1)}, a_{j}^{(0)}(ϵ)) with the orientation, and ⟨a_{j}^{(0)}, w_{∗}^{0}(a_{i}^{(1)}(ϵ))⟩

is given by the counting of monopoles in M^{W_{0}}(a_{i}^{(1)}(ϵ), a_{j}^{(0)}) with the orientation, the gluing Theorem 4.3, for the admissible geometric limits on V ×ℝ and the corresponding holomorphic triangles, yields the following Lemma (cf. Theorem 6.9 [6]).

Lemma 5.5. For small enough ϵ and large r ≥ r_{0}, there is an orientation reversing diffeomorphism

M^{W_{1}(r)}(a_{i}^{(1)}, a_{j}^{(0)}(ϵ)) ≅ M^{W_{0}(r)}(a_{i}^{(1)}(ϵ), a_{j}^{(0)}).

Hence, we have w_{∗}^{0} ∘ w_{∗}^{1} = 0.

The Lemma corresponds to the fact that, in the two cases, the same triangles are counted with the reverse orientation.

With these Lemmata at hand, the arguments in Section 6.5 [6] yield the following exact triangle for any p ∈ {0, ⋯, n − 1}:

\xymatrix{HF_{∗}^{SW} (Y_{1}, 𝔰 ⊗ L_{0}) \ar[r]^{w_{∗}^{1}} & HF_{∗}^{SW} (Y, 𝔰 ⊗ L_{0}) \ar[dl]_{w_{∗}^{0}} \\ ⨁_{k∈ℤ} HF_{(∗)}^{SW} (Y_{0}, 𝔰 ⊗ L_{nk+p}) \ar[u]_{Δ_{(∗)}}}  (21)

After a possible shift of relative index, we obtain the general exact triangle for any m, p ∈ {0, ⋯, n − 1}:

\xymatrix{HF_{∗}^{SW} (Y_{1}, 𝔰 ⊗ L_{0}) \ar[r]^{w_{∗}^{1}} & HF_{∗}^{SW} (Y, 𝔰 ⊗ L_{0}) \ar[dl]_{w_{∗}^{0}} \\ ⨁_{k∈ℤ} HF_{(∗)}^{SW} (Y_{0}, 𝔰 ⊗ L_{nk+p}) \ar[u]_{Δ_{(∗)}}}

This completes the proof of our main result, Theorem 1.1, except for the claim that the connecting homomorphisms Δ_{(∗)} are induced by the surgery cobordism connecting Y_{0} to Y_{1}. We shall discuss this statement in the next subsection.

5.3 The surgery triangle

To give a precise description of the connecting homomorphism Δ_{(∗)}, we need to study the discrepancy between the boundary operator ∂_{Y} of the Floer complex C_{∗}(Y, 𝔰 ⊗ L_{0}, μ) and the operator ∂_{Y_{1}} ⊕ ⨁_{k∈ℤ} ∂_{Y_{0},k} on

C_{∗}(Y_{1}, 𝔰 ⊗ L_{0}) ⊕ ⨁_{k∈ℤ}C_{(∗)}(Y_{0}, 𝔰 ⊗ L_{nk+p}).

Let us identify again the points of M_{Y,μ}^{∗}(𝔰 ⊗ L_{0}) as in (20) with

M_{Y,μ}^{∗}(𝔰 ⊗ L_{0}) = {a_{i}^{(1)}(ϵ)}_{i=1,...r} ∪ {a_{j}^{(0)}(ϵ)}_{j=r+1,... ,s}.

Then we have the following Lemma which gives the connecting homomorphism Δ_{(∗)}.

Lemma 5.6. Suppose given a cycle in ∑_{i} x_{i}a_{i}^{(0)} in C_{(∗)}(Y_{0}, 𝔰⊗L_{nk+p}). The image of ∑_{i} x_{i}a_{i}^{(0)} under the connecting homomorphism Δ is given by

Δ(∑_{i} x_{i}a_{i}^{(0)}) = ∑_{i,j} x_{i}#(M_{Y ×ℝ}(a_{i}^{(0)}(ϵ), a_{j}^{(1)}(ϵ)))a_{j}^{(1)}.

Proof. (Lemma 7.1, Lemma 7.2 in [6]) Using the diagram chasing, Lemma 5.3 and Lemma 5.5, we see that we have

∂_{Y,μ}(∑_{i} x_{i}a_{i}^{(0)}(ϵ))
= ∑_{i,j} x_{i}#(M_{Y ×ℝ}(a_{i}^{(0)}(ϵ), a_{j}^{(1)}(ϵ)))a_{j}^{(1)}(ϵ)

− ∑_{i,j,k} x_{i}#(M_{Y ×ℝ}(a_{i}^{(0)}(ϵ), a_{j}^{(1)}(ϵ)))#(M^{W_{0}}(a_{j}^{(1)}(ϵ), a_{k}^{(0)})))a_{k}^{(0)}(ϵ)

= ∑_{i,j} x_{i}#(M_{Y ×ℝ}(a_{i}^{(0)}(ϵ), a_{j}^{(1)}(ϵ)))a_{j}^{(1)}(ϵ)

+ ∑_{i,j,k} x_{i}#(M_{Y ×ℝ}(a_{i}^{(0)}(ϵ), a_{j}^{(1)}(ϵ)))#(M^{W_{1}}(a_{j}^{(1)}, a_{k}^{(0)})(ϵ)))a_{k}^{(0)}(ϵ).

By comparing this expression with

w_{∗}^{1}(∑_{i} x_{i}#(M_{Y ×ℝ}(a_{i}^{(0)}(ϵ), a_{j}^{(1)}(ϵ)))a_{j}^{(1)})
= ∑_{j} x_{j}#(M_{Y ×ℝ}(a_{j}^{(0)}(ϵ), a_{i}^{(1)}(ϵ)))a_{i}^{(1)}(ϵ)

+ ∑_{i,j,k} x_{i}#(M_{Y ×ℝ}(a_{i}^{(0)}(ϵ), a_{j}^{(1)}(ϵ)))#(M^{W_{1}}(a_{j}^{(1)}, a_{k}^{(0)}(ϵ))a_{k}^{(0)}(ϵ)),

we obtain

Δ(∑_{i} x_{i}a_{i}^{(0)}) = ∑_{i,j} x_{i}#(M_{Y ×ℝ}(a_{i}^{(0)}(ϵ), a_{j}^{(1)}(ϵ))a_{j}^{(1)}.

This completes the proof.

In next proposition, we show that the connecting homomorphism in the exact sequence can also be described as a map w_{(∗)}, induced by a surgery cobordism W̄_{2} connecting Y_{0} to Y_{1}, which satisfies W_{1}#_{Y} W_{0} = W_{2}#ℂ̄ℙ^{2}. The resulting diagram

C_{∗}(Y_{1}, 𝔰⊗L_{0}) →^{w_{∗}^{1}} C_{∗}(Y, 𝔰⊗L_{0}, μ) →^{w_{∗}^{0}} ⊕_{k}C_{(∗)}(Y_{0}, 𝔰⊗L_{nk+p}) →^{w_{(∗)}} C_{∗}(Y_{1}, 𝔰⊗L_{0})[−1]

is therefore a distinguished triangle, the surgery triangle.

Proposition 5.7. (Proposition 7.3 [6]) The connecting homomorphism Δ_{(∗)} in the exact triangle is given by the following expression,

Δ_{(∗)}(∑_{i} x_{i}a_{i}^{(0)}) = w_{(∗)}(∑_{i} x_{i}a_{i}^{(0)}),

for any cycle ∑_{i} x_{i}a_{i}^{(0)} in ⊕_{k}C_{(∗)}(Y_{0}, 𝔰 ⊗ L_{nk+p}) for any fixed p ∈ {0, ⋯, n − 1}, where

w_{(∗)} : ⊕_{k}C_{(∗)}(Y_{0}, 𝔰 ⊗ L_{nk+p}) → C_{∗}(Y_{1}, 𝔰 ⊗ L_{0})[−1]

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

MW̄_{2}(a_{j}^{(0)}, a_{i}^{(1)}),

over the cobordism W̄_{2}.

Proof. The argument is exactly the same as in the proof of Proposition 7.3 in [6], therefore we shall omit the proof here.

6 Seiberg–Witten and Casson–Walker invariant

In this section, we derive the relation between the topologically invariant version of the Seiberg-Witten invariant and the Casson-Walker invariant for rational homology 3-spheres. Together with the equivalence between the Casson-Walker invariant and the theta invariant introduced by Ozsváth and Szabó in [10], our result proves their conjecture relating the Seiberg-Witten invariant and their theta invariant.

Let Y be a rational homology 3-sphere with a smoothly embedded knot K representing a torsion element of order n in H_{1}(Y, ℤ),

\frac{|H_{1}(Y, ℤ)|}{|Torsion(H_{1}(Y − ν(K), ℤ))|} = n,

and endowed with the canonical framing (m, l) in ν(K) ≅ D^{2} × S^{1}. Let p and q be relatively prime integers. The Dehn surgery with coefficient p/q ∈ ℚ ∪ {∞} on K gives rise to another closed manifold Y_{p/q}.

Denote by Spin^{c}(V) the set of equivalence classes of Spin^{c} structures on V = Y \ν(K) with trivial restriction to the boundary T^{2}. Then, for any Y_{p/q}, there is a surjective map:

ι_{Y_{p/q}} : Spin^{c}(Y_{p/q}) → Spin^{c}(V),

where, for any 𝔰 ∈ Spin^{c}(Y_{p/q}), ι_{Y_{p/q}}(𝔰) is given by the restriction to V ⊂ Y_{p/q}. The fiber of ι_{Y_{p/q}} is given by a cyclic group generated by the Poincaré dual of the core of Y_{p/q}\V. Both V and ν(K) are equipped with a metric with a cylindrical end modeled on T^{2}. Formally, for ι_{Y} (𝔰) with 𝔰 ∈ Spin^{c}(Y), we identify the fiber of ι_{Y} with the following set of Spin^{c} structures

ι_{Y}^{−1}(ι_{Y} (𝔰) = ⋃_{m=0,··· ,n−1}{𝔰 ⊗ L_{m}|c_{1}(L_{m}) = mP D([K]) ∈ H^{2}(Y, ℤ)}.

Similarly the fiber of ι_{Y_{p/q}} is given by

⋃_{m=0,··· ,np−1}{𝔰 ⊗ L_{m}|c_{1}(L_{m}) = mP D([K]) ∈ H^{2}(Y_{p/q}, ℤ)},

and the fiber of ι_{Y_{0}} is given by

⋃_{m∈ℤ}{𝔰 ⊗ L_{m}|c_{1}(L_{m}) = mP D([K]) ∈ H^{2}(Y_{0}, ℤ)}.

Here we use the same notation 𝔰 on Y_{p/q} (Y, or Y_{0}) as the corresponding Spin^{c} structure obtained by gluing 𝔰 ∈ Spin^{c}(V) with the trivial Spin^{c} structure on ν(K) by the trivial gauge transformation on T^{2}. We hope this notation will not cause any confusion.

Let 𝔰 be a Spin^{c} structure on Y. By the result of [2] on the moduli space of finite energy monopoles on (V, ι_{Y} (𝔰)), we know that the irreducible part, denoted M_{V}^{∗} (𝔰), is a smooth, oriented 1-dimensional manifold. The asymptotic values along the cylindrical end define a continuous map:

∂_{∞} : M_{V}^{∗} (𝔰) → χ_{0}(T^{2}, V),

where χ_{0}(T^{2}, V) is a ℤ × ℤ_{n}-covering of the character torus χ(T^{2}). Sometime it is convenient to compose the above asymptotic value map with this covering map and define a boundary value map:

∂_{∞} :  M_{V}^{∗} (𝔰) → χ(T^{2}).  (22)

Notice that the reducible part χ(V) of the moduli space on (V, ι_{Y} (𝔰)) is an embedded circle χ(V) ⊂ χ_{0}(T^{2}, V) under the asymptotic value map. This becomes a circle of multiplicity n in χ(T^{2}). There is a “bad point” in χ(T^{2}), given by the flat connections such that the corresponding twisted Dirac operator has a non-trivial kernel. We can endow χ(T^{2}) with a coordinate system (u, v) defined by the holonomy around the longitude l and the meridian m, respectively, so that the bad point corresponds to (u, v) = (1, 1). The reducible circle χ(V), with the holonomy around the longitude l of order n, is given by u = u(𝔰), with u(𝔰) ∈ {0, 2/n, ⋯, 2(n − 1)/n}. After a suitable perturbation, and a corresponding shift of coordinates, as discussed in [2], we can assume that the bad point does not lie on any of the n possible circles u = u(𝔰) of reducibles χ(V).

From the result in [2], we know that, under the map ∂_{∞} in (22), the boundary points ∂(M_{V}^{∗} (𝔰)) are either mapped to the bad point in χ(T^{2}) or mapped to the reducible circle u = u(𝔰) on χ(T^{2}).

Let χ(ν(K) ⊂ Y_{p/q}) be the reducible circle on χ(T^{2}), which maps to a closed curve on χ(T^{2}) with slope p/q in the (u, v)-coordinates: parallel to pv = qu. Looking at the induced Spin structure on T^{2} ⊂ Y_{p/q}, we know that the curve χ(ν(K) ⊂ Y_{p/q}) goes through (0, 1) if q is odd or goes through (0, 0) if q is even, cf.[2]. Again, after a suitable perturbation, and the corresponding shift of coordinates, we can assume that this p/q-curve is away from the bad point on χ(T^{2}) and does not meet u = u(𝔰) along the coordinate line v = 0. Then we know that u = u(𝔰) intersects χ(ν(K) ⊂ Y_{p/q}) inside χ(T^{2}) at p points, which are denoted by θ_{1}, ⋯, θ_{p}, ordered according the orientation of u = u(𝔰) ⊂ χ(T^{2}). They can be lifted to pn points in χ_{0}(T^{2}, V). We denote these points by θ_{1}^{(k)}, ⋯, θ_{p}^{(k)}, (k = 0, 1, ⋯, n − 1) according to the order. Denote by θ_{0} the intersection point of u = u(𝔰) with v = 0 in χ(T^{2}). This can be lifted to n-points θ_{0}^{(0)}, θ_{0}^{(1)}, ⋯, θ_{0}^{(n−1)} on χ_{0}(T^{2}, V). Moreover, we can assume that the map ∂_{∞} in (22) is transverse to the curves u = u(𝔰), v = 0 and χ(ν(K) ⊂ Y_{p/q}), by a suitable perturbation of the Seiberg-Witten equations on V as in [2]. We can also assume that the image ∂_{∞}M_{V}^{∗} (𝔰) does not meet the points θ_{0}, θ_{1}, ⋯ θ_{p} in χ(T^{2}), again by suitable perturbation, as discussed in [2].

Let I be any open interval in

χ(V) = {u = u(𝔰)} ⊂ χ_{0}(T^{2}, V).

We denote by SF_{I}(/ ∂^{V}) the complex spectral flow of the Dirac operator on V twisted with the path of reducible connections I on V. From the analysis in [2] and [4], we know that

#(∂_{∞}|_{∂M_{V}^{∗} (𝔰)})^{−1}(I) = SF_{I}(/ ∂^{V}).  (23)

For convenience, we define

SF_{[θ_{i},θ_{j}]}(/ ∂^{V}) = ∑_{k=0}^{n−1} SF_{[θ_{i}^{(k)},θ_{j}^{(k)}]}(/ ∂^{V}).  (24)

With this notation understood, we can state the following proposition relating the Seiberg-Witten invariants on Y_{p/q}, Y and Y_{0}.

Proposition 6.1. Consider generic compatible small perturbations of the Seiberg-Witten equations on Y_{p/q}, Y and Y_{0}, such that the map ∂_{∞} as in (22) is transverse to the curves u = u(𝔰), v = 0 and χ(ν(K) ⊂ Y_{p/q}) and misses the points θ_{0}, θ_{1}, ⋯, θ_{p} in χ(T^{2}). Then we have the following relation:

∑_{k=0}^{pn−1} SW_{Y_{p/q}}(𝔰 ⊗ L_{k}, g_{Y_{p/q}})
= p ∑_{k=0}^{n−1} SW_{Y} (𝔰 ⊗ L_{k}, g_{Y}) + q ∑_{k∈ℤ} SW_{Y_{0}}(𝔰 ⊗ L_{k})
+ ∑_{i=1}^{p} SF_{[θ_{0},θ_{i}]}(/ ∂^{V}).

Proof. By the gluing theorem for 3-dimensional monopoles as in [2], we have

⋃_{k=0}^{pn−1} M_{Y_{p/q}}^{∗}(𝔰 ⊗ L_{k}) = M_{V}^{∗} (𝔰) ×_{χ(T^{2})} χ(ν(K) ⊂ Y_{p/q}),

where χ(ν(K) ⊂ Y_{p/q}) is the p/q-curve on χ(T^{2}). Thus, we obtain

∑_{k=0}^{pn−1} SW_{Y_{p/q}}(𝔰 ⊗ L_{k}, g_{Y_{p/q}}) = #(M_{V}^{∗} (𝔰) ×_{χ(T^{2})} χ(ν(K) ⊂ Y_{p/q})). (25)

Notice that the set {θ_{1}^{(k)}, ⋯, θ_{p}^{(k)} : k = 0, 1, ⋯, n−1} consists of the unique reducible monopole for each (Y_{p/q}, 𝔰 ⊗ L_{k}).

Similarly, we have

∑_{k=0}^{n−1} SW_{Y} (𝔰 ⊗ L_{k}, g_{Y}) = #(M_{V}^{∗} (𝔰) ×_{χ(T^{2})} {v = 0}).  (26)

Here the reducible set consists of {θ_{0}^{(0)}, θ_{0}^{(1)}, ⋯, θ_{0}^{(n−1)}}.

In order to avoid the circle of reducibles on (Y_{0}, 𝔰 ⊗ L_{0}), we need to introduce a small perturbation such that χ(ν(K) ⊂ Y_{0}) on χ(T^{2}) is a small parallel shifting of u = u(𝔰) such that the bad point is not contained in the narrow strip bounded by these two parallel curves. We denote this small shift of u = u(𝔰) by u = u(𝔰) + η, where η is a sufficiently small positive number. This can be achieved by a perturbation of the equations as in [2]. Then we have

∑_{k∈ℤ}SW_{Y_{0}}(𝔰 ⊗ L_{k}) = #(M_{V}^{∗} (𝔰) ×_{χ(T^{2})} {u = u(𝔰) + η}).  (27)

In order to compare the three countings in (25) – (27), we need to choose an oriented 2-chain C in χ(T^{2}) whose boundary 1-chain is given by

\begin{aligned}& χ(ν(K) ⊂ Y_{p/q}) − pχ(ν(K) ⊂ Y) − qχ(ν(K) ⊂ Y_{0}) \\ = & χ(ν(K) ⊂ Y_{p/q}) − p\{v = 0\} − q\{u = u(𝔰) + η\}, \end{aligned}

and such that C does not contain the bad point in χ(T^{2}). Then, counting the boundary points of ∂_{∞}^{−1}(C), as a 0-chain, we obtain

\begin{aligned}& #(∂_{∞}^{−1}(χ(ν(K) ⊂ Y_{p/q})) \\ = & p#(∂_{∞}^{−1}(\{v = 0\})) + q#(∂_{∞}^{−1}(\{u = u(𝔰) + η\})) \\ & +#(∂_{∞} |_{∂(M_{V}^{∗} (𝔰))})^{−1}(C). \end{aligned}  (28)

As C does not contain the bad points, we know that the possible points of ∂_{∞}(∂(M_{V}^{∗} (𝔰))) ∩ C all lie on the curve u = u(𝔰), which misses the points θ_{0}, θ_{1}, ⋯, θ_{p}. It is easy to see that C covers the intervals of u = u(𝔰) between two consecutive points θ_{i} with different multiplicities: the multiplicities are p, p − 1, ⋯, 1, 0, for the intervals

[θ_{0}, θ_{1}], [θ_{1}, θ_{2}], ⋯, [θ_{p−1}, θ_{p}], [θ_{p}, θ_{0}],

respectively. By the identity (23) and the definition (24), we know that

#(∂_{∞}|_{∂(M_{V}^{∗} (𝔰))})^{−1}(C) = ∑_{i=1}^{p} SF_{[θ_{0},θ_{i}]}(/ ∂^{V}).  (29)

Combining all the identities in (25), (26), (27), (28) and (29), we obtain the proof of the proposition.

The Seiberg-Witten invariant for any rational homology 3-sphere depends on metric and perturbation (cf.[4]). We now consider the correction term (5) as defined in the introduction. We have the following proposition relating the correction terms for Y_{p/q} and Y.

Proposition 6.2. 1. For any rational homology 3-sphere Y with a Spin^{c} structure 𝔰 and a Riemannian metric g_{Y},

ŜW_{Y} (𝔰) = SW_{Y} (𝔰, g_{Y}) − ξ(𝔰, g_{Y})

is a well-defined topological invariant.

2. For any relatively prime integers p and q, a positive integer n, and u ∈ {0, 2/n, …, 2(n − 1)/n}, we have that

∑_{k=0}^{pn−1} ξ_{Y_{p/q}}(𝔰 ⊗ L_{k}, g_{Y_{p/q}}) − p ∑_{k=0}^{n−1} ξ_{Y} (𝔰 ⊗ L_{k}, g_{Y}) − ∑_{i=1}^{p} SF_{[θ_{0},θ_{i}]}(/ ∂^{V})

is independent of the manifold Y and depends only on p, q, n, and u(𝔰) ∈ {0, 2/n, …, 2(n − 1)/n}.

Proof. Claim (1) follows from the wall-crossing formulae in [4] and the Atiyah-Patodi-Singer index theorem. The proof of claim (2) is analogous to the proof of Proposition 7.9 in [10]. We adapt their arguments to our situation. We write the standard surgery cobordism between S^{3} and the Lens space L(p, q) as

W (S^{3}, L(p, q)) = ([0, 1] × S^{1} × D^{2}) ∪_{[0,1]×S^{1}×S^{1}} X_{p/q},

Then the surgery cobordism between Y and Y_{p/q} can be identified as

W_{p/q} = ([0, 1] × V ) ∪_{[0,1]×S^{1}×S^{1}} X_{p/q}.

We fix a metric on W_{p/q} which respects the product structure [0, 1] × V and [0, 1] × S^{1} × S^{1}, and agrees with g_{Y} and g_{Y_{p/q}} on the boundaries Y and Y_{p/q}, respectively.

For a Spin^{c} structure 𝔰 ⊗ L_{i}^{(m)} in {𝔰 ⊗ L_{k} : k = 0, ⋯, pn − 1} on Y_{p/q}, whose reducible monopole corresponds to θ_{i}^{(m)} (with i ∈ {1, ⋯, p} and m ∈ {0, ⋯ n − 1}), we consider the Spin^{c} structure 𝔰 ⊗ L_{m} on Y whose reducible monopole is θ_{0}^{(m)}. Then we claim that

ξ_{Y_{p/q}}(𝔰 ⊗ L_{i}^{(m)}, g_{Y_{p/q}}) − ξ_{Y} (𝔰 ⊗ L_{m}, g_{Y}) − SF_{[θ_{0}^{(m)},θ_{i}^{(m)}]}(/ ∂^{V})  (30)

is independent of Y and depends only on p, q, n and on u(𝔰) ∈ {0, 2/n, ⋯, 2(n − 1)/n}.

To prove this claim, we choose a Spin^{c} structure 𝔰̃ on W_{p/q} whose restriction to Y and Y_{p/q} is given by 𝔰 ⊗ L_{m} and 𝔰 ⊗ L_{i}^{(m)}, respectively, and such that c_{1}(𝔰̃)^{2} = 1. On (W_{p/q}, 𝔰̃), we choose a connection A, whose restriction to V ×[0, 1] is the path of reducibles connecting θ_{0}^{(m)} to θ_{i}^{(m)} along the curve χ(V) ⊂ χ_{0}(T^{2}, V). Then we have

\begin{aligned}& ξ_{Y_{p/q}}(𝔰 ⊗ L_{i}^{(m)}, g_{Y_{p/q}}) − ξ_{Y} (𝔰 ⊗ L_{m}, g_{Y}) \\ = & Ind_{ℂ}(\begin{matrix}/ \\ D \end{matrix}_{A}^{W_{p/q}}) − (\frac{c_{1}(𝔰̃)^{2} − σ(W_{p/q})}{8}) \\ = & Ind_{ℂ}(\begin{matrix}/ \\ D \end{matrix}_{A}^{W_{p/q}}) \\ = & Ind_{ℂ}(\begin{matrix}/ \\ D \end{matrix}_{A}^{[0,1]×V}) + Ind_{ℂ}(\begin{matrix}/ \\ D \end{matrix}_{A}^{X_{p/q}}) \end{aligned}  (31)

where the third equality follows from the splitting principle for the index, as the Dirac operator has no kernel on the various boundaries and corners [1]. Notice that we have

Ind_{ℂ}(\begin{matrix}/ \\ D \end{matrix}_{A}^{[0,1]×V}) = SF_{[θ_{0}^{(m)},θ_{i}^{(m)}]}(/ ∂^{V}),

and the connection A|_{X_{p/q}} extends to connection A_{0} on W (S^{3}, L(p, q)) by a flat connection, whose index on [0, 1] × S^{1} × D^{2} satisfies

Ind_{ℂ}(\begin{matrix}/ \\ D \end{matrix}_{A_{0}}^{[0,1]×S^{1}×D^{2}}) = 0.

In fact, we can choose the metric on W (S^{3}, L(p, q)) with a positive scalar curvature metric on [0, 1] × S^{1} × D^{2}. Therefore, we have

Ind_{ℂ}(\begin{matrix}/ \\ D \end{matrix}_{A}^{X_{p/q}}) = Ind_{ℂ}(\begin{matrix}/ \\ D \end{matrix}_{A_{0}}^{W (S^{3},L(p,q))}),

which depends only on p, q, n and u(𝔰), and so does the quantity

ξ_{Y_{p/q}}(𝔰 ⊗ L_{i}^{(m)}, g_{Y_{p/q}}) − ξ_{Y} (𝔰 ⊗ L_{m}, g_{Y}) − SF_{[θ_{0}^{(m)},θ_{i}^{(m)}]}(/ ∂^{V}).  (32)

When summing the identity (32) over i ∈ {1, ⋯, p} and m ∈ {0, ⋯, n−1}, notice that the term ξ_{Y} (𝔰 ⊗ L_{m}, g_{Y}) is independent of i ∈ {1, ⋯, p}, hence we obtain the proof of the claim (2) by using the definition (24).

With these two propositions in place, we now have the following surgery formula for the modified version of the Seiberg-Witten invariant.

Theorem 6.3. Given any two relatively prime integers p and q, a positive integer n and u ∈ {0, 2/n, 2(n − 1)/n}, there is a rational valued function s(p, q, n, u), depending only on p, q, n and u, satisfying the following property. Let Y be a rational homology 3-sphere with a smoothly embedded knot and a canonical framing (m, l) such that ν(K) ≅ D^{2} × S^{1}. Assume that K represents a torsion element of order n in H_{1}(Y, ℤ). Let 𝔰 be a Spin^{c} structure on Y. Then we have

∑_{k=0}^{pn−1} ŜW_{Y_{p/q}}(𝔰 ⊗ L_{k})
= p ∑_{k=0}^{n−1} ŜW_{Y} (𝔰 ⊗ L_{k},) + q ∑_{k∈ℤ} SW_{Y_{0}}(𝔰 ⊗ L_{k})
+s(p, q, n, u).

Proof. Following from Proposition 6.2, we know that

∑_{k=0}^{pn−1} ξ_{Y_{p/q}}(𝔰 ⊗ L_{k}, g_{Y_{p/q}}) − p ∑_{k=0}^{n−1} ξ_{Y} (𝔰 ⊗ L_{k}, g_{Y}) − ∑_{i=1}^{p} SF_{[θ_{0},θ_{i}]}(/ ∂^{V}) (33)

depends only on p, q, n and u = u(𝔰) ∈ {0, 2/n, ⋯, 2(n − 1)/n}, which is denoted by s(p, q, n, u). By subtracting (33) from the surgery formula for the Seiberg-Witten invariants in Proposition 6.1, we obtain the proof of this theorem.

Now we can establish the equivalence between the modified version of the Seiberg-Witten invariant ŜW and the Casson-Walker invariant for rational homology 3-spheres.

Theorem 6.4. For any rational homology 3-sphere Y, we have

∑_{𝔰∈Spin^{c}(Y)} ŜW_{Y} (𝔰) = \frac{1}{2}|H_{1}(Y, ℤ)|λ(Y)

where λ(Y) is the Casson-Walker invariant.

Proof. We first derive the surgery formula for the invariant ∑_{𝔰∈Spin^{c}(Y)} ŜW_{Y} (𝔰) from Theorem 6.3 and the Seiberg-Witten invariant for Y_{0} (a rational homology S^{1} × S^{2}, i.e., b_{1}(Y_{0}) = 1) (see [8] [3]):

∑_{𝔰∈Spin^{c}(Y_{p/q})} ŜW_{Y_{p/q}}(𝔰)
= p ∑_{𝔰∈Spin^{c}(Y)} ŜW_{Y} (𝔰) + q ∑_{i=0}^{∞} a_{j}j^{2} + |H_{1}(Y, ℤ)|s(p, q, n)

(34)

where s(p, q, n) = ∑_{u} s(p, q, n, u)/n and a_{j} is the coefficient of the symmetrized Alexander polynomial of Y_{0},

A(t) = |Torsion(H_{1}(Y_{0}, ℤ))| + ∑_{j=1}^{∞} a_{j}(t^{j} + t^{−j})

normalized such that

A(1) = |Torsion(H_{1}(Y_{0}, ℤ))|.

Set λ̄(Y) = \frac{1}{2}|H_{1}(Y, ℤ)|λ(Y) as the normalized Casson-Walker invariant.

Then the surgery formula in [11] for λ̄(Y) can be expressed as (cf. [10]):

\begin{aligned}λ̄(Y_{p/q}) & = & pλ̄(Y) + q ∑_{i=0}^{∞} a_{j}j^{2} \\ & & +|H_{1}(Y, ℤ)|(\frac{q(n^{2} − 1)}{12n^{2}} − \frac{ps(p, q)}{2}). \end{aligned}  (35)

Here s(p, q) is the Dedekind sum of relatively prime integers p and q (cf. [11]). Comparing (34) and (35), we only need to show that

s(p, q, n) = \frac{q(n^{2} − 1)}{12n^{2}} − \frac{ps(p, q)}{2}.  (36)

Since s(p, q, n) is independent of the manifold Y, we can choose some examples that can be computed explicitly, and use them to identify the coefficient s(p, q, n). The Lens space L(p, q) can be obtained by a p/qsurgery on an unknot in S^{3}. The calculation of Nicolaescu [9] for L(p, q) gives us that

∑_{𝔰∈Spin^{c}(L(p,q))} ŜW_{L(p,q)}(𝔰) = −\frac{ps(p, q)}{2}.

This implies that (36) holds for n = 1. Now we can prove (36) by induction on n. This is exactly the same argument as in the proof of Theorem 7.5 in [10] on the equivalence of their theta invariant and the Casson-Walker invariant. The example is the Seifert manifold M(n, 1; −n, 1; q, −p), obtained by p/q surgery on a knot of order n in L(n, 1)#\overline{L(n, 1)}. By Kirby calculus it is possible to show that M(n, 1; −n, 1; q, −p) can be obtained as (−n)-surgery on a knot in the Lens space L(pn − q, q), and can be obtained as a sequence of surgeries on knots of order less than n, see the proof of Theorem 7.5 in [10] for details.

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Matilde Marcolli Max–Planck–Institut für Mathematik, Vivatsgasse 7, D-53111 Bonn, Germany. marcolli@mpim-bonn.mpg.de

Bai-Ling Wang Department of Pure Mathematics, University of Adelaide, Adelaide SA 5005, Australia. bwang@maths.adelaide.edu.au