Variants of Equivariant Seiberg-Witten Floer Homology

Matilde Marcolli, Bai-Ling Wang

Abstract

For a rational homology 3-sphere Y with a Spin^{c} structure 𝔰, we show that simple algebraic manipulations of our construction of equivariant Seiberg-Witten Floer homology in [5] lead to a collection of variants HF_{∗,U(1)}^{SW,−}(Y, 𝔰), HF_{∗,U(1)}^{SW,∞}(Y, 𝔰) HF_{∗,U(1)}^{SW,+}(Y, 𝔰), ĤF_{∗}^{SW} (Y, 𝔰) and HF_{red,∗}^{SW}(Y, 𝔰) which are topological invariants. We establish a long exact sequence relating HF_{∗,U(1)}^{SW,±}(Y, 𝔰) and HF_{∗,U(1)}^{SW,∞}(Y, 𝔰). We show they satisfy a duality under orientation reversal, and we explain their relation to the equivariant Seiberg-Witten Floer (co)homologies introduced in [5]. We conjecture the equivalence of these versions of equivariant Seiberg-Witten Floer homology with the Heegaard Floer invariants introduced by Ozsváth and Szabó. Key words: rational homology 3-spheres, equivariant Seiberg-Witten Floer homology, Spin^{c} structures, topological invariants. Mathematics Subject Classification. Primary: 57R58. Secondary: 57R57, 58J10.

1 Introduction

For any rational homology 3-sphere Y with a Spin^{c} structure 𝔰, we constructed in [5] an equivariant Seiberg-Witten Floer homology HF_{∗,U(1)}^{SW}(Y, 𝔰), which is a topological invariant. In this paper, we will generalize this construction to provide a collection of equivariant Seiberg-Witten Floer homologies HF_{∗,U(1)}^{SW,−}(Y, 𝔰), HF_{∗,U(1)}^{SW,∞}(Y, 𝔰), HF_{∗,U(1)}^{SW,+}(Y, 𝔰), ĤF_{∗}^{SW} (Y, 𝔰) and HF_{red,∗}^{SW}(Y, 𝔰), all of which are topological invariants, such that HF_{∗,U(1)}^{SW,+}(Y, 𝔰) is isomorphic to the equivariant Seiberg-Witten Floer homology HF_{∗,U(1)}^{SW}(Y, 𝔰) constructed in [5]. The construction utilizes the U(1)-invariant forms on U(1)-manifolds twisted with coefficients in the Laurent polynomial algebra over integers.

In analogy to Austin and Braam’s construction of equivariant instanton Floer homology in [1], the equivariant Seiberg-Witten Floer homology HF_{∗,U(1)}^{SW}(Y, 𝔰) is the homology of the complex (CF_{∗,U(1)}^{SW}(Y, 𝔰), D), where CF_{∗,U(1)}^{SW}(Y, 𝔰) is generated by equivariant de Rham forms over all U(1)-orbits of the solutions of 3-dimensional Seiberg-Witten equations on (Y, 𝔰) modulo based gauge transformations (Cf.[5]). More specifically,

CF_{∗,U(1)}^{SW}(Y, 𝔰) = ⨁_{a∈M_{Y}^{∗} (𝔰)}ℤ[Ω] ⊗ (ℤη_{a} ⊕ ℤ1_{a}) ⊕ ℤ[Ω] ⊗ ℤ1_{θ},  (1)

where M_{Y} (𝔰) = M_{Y}^{∗} (𝔰) ∪ {θ} is the equivalence classes of solutions to the Seiberg-Witten equations for a good pair of metric and perturbations, consists of the irreducible monopoles M_{Y}^{∗} (𝔰) and the unique reducible monopole θ. We used the notation η_{a} to denote a 1-form on O_{a} ≅ S^{1}, such that the cohomology class [η_{a}] is an integral generator of H^{1}(O_{a}). Similarly, we denote by 1_{a} the 0-form given by the constant function.

Each generator is endowed with a grading such that, for any k ≥ 0,

gr(Ω^{k} ⊗ η_{a}) = 2k + gr(a), gr(Ω^{k} ⊗ 1_{a}) = 2k + gr(a) + 1, and gr(Ω^{k} ⊗ 1_{θ}) = 2k,  (2)

where gr : M_{Y}^{∗} (𝔰) → ℤ is the relative grading with respect to the reducible monopole θ. This corresponds to grading equivariant de Rham forms on each orbit O_{a} by codimension (Cf.[5] §5 for details).

The differential operator D can be expressed explicitly in components as the form:

D(Ω^{k} ⊗ η_{a}) = ∑_{{b∈M^{∗}_{Y} (𝔰) \atop gr(a)−gr(b)=1}}n_{ab}Ω^{k} ⊗ η_{b} + ∑_{{c∈M^{∗}_{Y} (𝔰) \atop gr(a)−gr(c)=2}} m_{ac}Ω^{k} ⊗ 1_{c} − Ω^{k−1} ⊗ 1_{a}
+n_{aθ}Ω^{k} ⊗ 1_{θ}(if gr (a) =1);
D(Ω^{k} ⊗ 1_{a}) = − ∑_{{b∈M^{∗}_{Y} (𝔰) \atop gr(a)−gr(b)=1}}n_{ab}Ω^{k} ⊗ 1_{b};
D(Ω^{k} ⊗ 1_{θ}) = ∑_{{d∈M^{∗}_{Y} (𝔰) \atop gr(d)=−2}} n_{θd}Ω^{k} ⊗ 1_{d}.

(3)

where n_{ab}, n_{aθ} and n_{θd} is the counting of flowlines from a to b (if gr(a) − gr(b) = 1), from a to θ (if gr(a) = 1) and from θ to d (if gr(d) = −2), and m_{ac} (if gr(a) − gr(c) = 2) is described as a relative Euler number associated to the 2-dimensional moduli space of flowlines from a to c (Cf. Lemma 5.7 of [5]). In the next section, we shall briefly review the construction and various relations among the coefficients, as established in [5]. These identities ensure that D^{2} = 0. Notice that, in the complex CF_{∗,U(1)}^{SW}(Y, 𝔰) and in the expression of the differential operator, only terms with non-negative powers of Ω are considered. We modify the construction as follows.

Definition 1.1. Let CF_{∗,U(1)}^{SW,∞}(Y, 𝔰) be the graded complex generated by

{Ω^{k} ⊗ η_{a}, Ω^{k} ⊗ 1_{a}, Ω^{k} ⊗ 1_{θ} : a ∈ M_{Y}^{∗} (𝔰), k ∈ ℤ.}

with the grading gr and the differential operator D given by (2) and (3) respectively. Let CF_{∗,U(1)}^{SW,−}(Y, 𝔰) be the subcomplex of CF_{∗,U(1)}^{SW,∞}(Y, 𝔰), generated by those generators with negative power of Ω. The quotient complex is denoted by CF_{∗,U(1)}^{SW,+}(Y, 𝔰). Their homologies are denoted by HF_{∗,U(1)}^{SW,∞}(Y, 𝔰), HF_{∗,U(1)}^{SW,−}(Y, 𝔰) and HF_{∗,U(1)}^{SW,+}(Y, 𝔰) respectively.

The main results in this paper relate these homologies to the equivariant Seiberg-Witten-Floer homology HF_{∗,U(1)}^{SW}(Y, 𝔰) and cohomology HF_{U(1)}^{SW,∗}(Y, 𝔰) constructed in [5] and establish some of their main properties.

Theorem 1.2. For any rational homology 3-sphere Y with a Spin^{c} structure 𝔰 ∈ Spin^{c}(Y), these homologies satisfy the following properties:

1. HF_{∗,U(1)}^{SW,∞}(Y, 𝔰) ≅ ℤ[Ω, Ω^{−1}].

2. HF_{∗,U(1)}^{SW,+}(Y, 𝔰) ≅ HF_{∗,U(1)}^{SW}(Y, 𝔰) where HF_{∗,U(1)}^{SW}(Y, 𝔰) is the equivariant Seiberg-Witten Floer homology for (Y, 𝔰) constructed in [5].

3. HF_{∗,U(1)}^{SW,−}(Y, 𝔰) ≅ HF_{U(1)}^{SW,∗}(−Y, 𝔰) where HF_{U(1)}^{SW,∗}(−Y, 𝔰) is the equivariant Seiberg-Witten Floer cohomology for (−Y, 𝔰) constructed in [5].

4. There exists a long exact sequence

\xymatrix{⋯ → HF_{∗,U(1)}^{SW,−}(Y, 𝔰) \ar[r]^{l_{∗}} & HF_{∗,U(1)}^{SW,∞}(Y, 𝔰) \ar[r]^{π_{∗}} & HF_{∗,U(1)}^{SW,+}(Y, 𝔰) \ar[r]^{δ_{∗}} & HF_{∗−1,U(1)}^{SW,−}(Y, 𝔰) → ⋯} (4)

relating these homologies. Moreover, HF_{∗,U(1)}^{SW,−}(Y, 𝔰), HF_{∗,U(1)}^{SW,∞}(Y, 𝔰), HF_{∗,U(1)}^{SW,+}(Y, 𝔰) and HF_{red,∗}^{SW}(Y, 𝔰) = Coker(π_{∗}) ≅ Ker(l_{∗−1}) are all topological invariants of (Y, 𝔰).

5. There is a u-action on HF_{∗,U(1)}^{SW,−}(Y, 𝔰), HF_{∗,U(1)}^{SW,∞}(Y, 𝔰) and HF_{∗,U(1)}^{SW,+}(Y, 𝔰) respectively which decreases the degree by two, and is related to the cutting down moduli spaces of flowlines by a geometric representative of a degree 2 characteristic form. The long exact sequence (4) is a long exact sequence of ℤ[u]-modules.

6. There is a homology group ĤF_{∗}^{SW} (Y, 𝔰), which is also a topological invariant of (Y, 𝔰), such that the following sequence is exact:

\xymatrix{⋯ → ĤF_{∗}^{SW} (Y, 𝔰) \ar[r] & HF_{∗,U(1)}^{SW,+}(Y, 𝔰) \ar[r]^{u} & HF_{∗−2,U(1)}^{SW,+}(Y, 𝔰) \ar[r] & ĤF_{∗−1}^{SW}(Y, 𝔰) → ⋯}  (5)

and that ĤF^{SW} (Y, 𝔰) is non-trivial if and only if HF_{∗,U(1)}^{SW,+}(Y, 𝔰) is non-trivial.

The u-action in the main theorem is induced from a u-action on the chain complex

u : CF_{∗,U(1)}^{SW,∞} → CF_{∗,U(1)}^{SW,∞},

which decreases the degree by 2. We will show that this u-action is homotopic to the obvious Ω^{−1} action on the chain complex CF_{∗,U(1)}^{SW,∞}. Thus, the induced u-action on HF_{∗,U(1)}^{SW,±}(Y, 𝔰) endows them with ℤ[u]-module structures.

Let ĈF_{∗}^{SW} (Y, 𝔰) be the subcomplex of CF_{∗,U(1)}^{SW,+}(Y, 𝔰) such that the following sequence is a short exact sequence of chain complexes:

\xymatrix{0 → ĈF_{∗}^{SW} (Y, 𝔰) \ar[r] & CF_{∗,U(1)}^{SW,+}(Y, 𝔰) \ar[r]^{Ω^{−1}} & CF_{∗,U(1)}^{SW,+}(Y, 𝔰) → 0}

We can define ĤF_{∗}^{SW} (Y, 𝔰) to be the homology of ĈF_{∗}^{SW} (Y, 𝔰).

In recent work [7] [8], Ozsváth and Szabó introduced Heegaard Floer invariants HF_{∗}^{±}(Y, 𝔰), HF_{∗}^{∞}(Y, 𝔰), ĤF_{∗}(Y, 𝔰), and HF_{red,∗}(Y, 𝔰), with exact sequences relating them. In view of their construction, the result of Theorem 1.2, together with the identification of our HF_{∗,U(1)}^{SW,∞}(Y, 𝔰) and the HF_{∗}^{∞}(Y, 𝔰) of Ozsváth and Szabó, suggest the following conjecture.

Conjecture 1.3. For any rational homology 3-sphere Y with a Spin^{c} structure 𝔰 ∈ Spin^{c}(Y), there are isomorphisms

\begin{matrix}HF_{∗,U(1)}^{SW,+}(Y, 𝔰) ≅ HF_{∗}^{+}(Y, 𝔰), & HF_{∗,U(1)}^{SW,−}(Y, 𝔰) ≅ HF_{∗}^{−}(Y, 𝔰); \\ ĤF_{∗}^{SW} (Y, 𝔰) ≅ ĤF_{∗}(Y, 𝔰), & HF_{red,∗}^{SW}(Y, 𝔰) ≅ HF_{red,∗}(Y, 𝔰). \end{matrix}

Acknowledgments This research was supported in part by the Humboldt Foundation’s Sofja Kovalevskaya Award.

2 Review of equivariant Seiberg-Witten Floer homology

In this section, we recall some of basic ingredients in the definition of the equivariant Seiebrg-Witten Floer homology from [5] (See [5] for all the details).

Let (Y, 𝔰) be a rational homology 3-sphere Y with a Spin^{c} structure 𝔰 ∈ Spin^{c}(Y). For a good pair of metric and perturbation (a co-closed imaginary-valued 1-form ν ) on Y, the 3-dimensional Seiebrg-Witten equations on (Y, 𝔰) (Cf. [2] [3] [4] [5]):

\{ \begin{matrix}∗F_{A} = σ(ψ, ψ) + ν \\ / ∂_{A}ψ = 0, \end{matrix}  (6)

for a pair of Spin^{c} connection A and a spinor ψ, have only finitely many irreducible solutions (modulo the gauge transformations), denoted by M_{Y}^{∗} (𝔰) the set of equivalence classes of irreducible solutions to (6), and θ is the unique reducible solution (modulo the gauge transformations). Write M_{Y} (𝔰) = M_{Y}^{∗} (𝔰) ∪ {θ}.

Gauge classes of finite energy solutions to the 4-dimensional Seiebrg-Witten equations, perturbed as in [2] [3] [5], can be regarded as moduli spaces of flowlines of the Chern-Simons-Dirac functional on the gauge equivalence classes of Spin^{c} connections and spinors for (Y, 𝔰). These can be partitioned into moduli spaces of flowlines between pairs of critical points from M_{Y} (𝔰). Each is a smooth oriented manifold which can be compactfied to a smooth manifold with corners by adding broken flowlines that split through intermediate critical points.

The spectral flow of the Hessian operator of the Chern-Simons-Dirac functional defines a relative grading on M_{Y} (𝔰):

gr(·, ·) : M_{Y} (𝔰) × M_{Y} (𝔰) → ℤ.

In particular, using the unique reducible point θ in M_{Y} (𝔰), there is a ℤ-lifting of the relative grading given by gr(a) = gr(a, θ).

Let a be an irreducible monopole in M_{Y} (𝔰), then for any b ≠ a in M_{Y} (𝔰), the moduli space of flowlines from a to b, denoted by M(a, b) has dimension gr(a) − gr(b) > 0 (if non-empty). The moduli space of flowlines from θ to d ∈ M_{Y}^{∗} (𝔰), denoted by M(θ, d) has dimension −gr(d) − 1 > 0 (if nonempty). Note that all these moduli spaces of flowlines admit an ℝ-action given by the ℝ-translation of flowlines: the corresponding quotient spaces are denoted by M̂(a, b) and M̂(θ, d), respectively.

For any irreducible critical points a and c in M_{Y} (𝔰) with gr(a) − gr(c) = 2, we can construct a canonical complex line bundle over M(a, c) and a canonical section as follows (see section 5.3 in [5]). Choose a base point (y_{0}, t_{0}) on Y × ℝ, and a complex line ℓ_{y_{0}} in the fiber W_{y_{0}} of the spinor bundle W over y_{0} ∈ Y. We choose ℓ_{y_{0}} so that it does not contain the spinor part ψ of any irreducible critical point. Since there are only finitely many critical points we can guarantee such choice exists. Denote the based moduli space of M(a, c) by M(O_{a}, O_{c}) as in [5], where O_{a} and O_{c} are the U(1)-orbits of monopoles on the based configuration space. We consider the line bundle

L_{ac} = M(O_{a}, O_{c}) ×_{U(1)} (W_{y_{0}}/ℓ_{y_{0}}) → M(a, c)  (7)

with a section given by

s([A, Ψ]) = ([A, Ψ], Ψ(y_{0}, t_{0})).  (8)

For a generic choice of (y_{0}, t_{0}) and ℓ_{y_{0}}, the section s of (8) has no zeroes on the boundary strata of the compactification of M(a, c). This determines a trivialization of L_{ac} away from a compact set in M(a, c). The line bundle L_{ac} over M(a, c), with the trivialization φ specified above, has a well-defined relative Euler class (Cf. Lemma 5.7 in [5]). Definition 2.1. 1. For any two irreducible critical points a and b in M_{Y} (𝔰) with gr(a) − gr(b) = 1,

we define n_{ab} := #(M̂(a, b)), the number of flowlines in M(a, b) counting with orientations. Similarly, for any a ∈ M_{Y} (𝔰) with gr(a) = 1 and any d ∈ M_{Y} (𝔰) with gr(d) = −2, we define n_{aθ} := #(M̂(a, θ)) and n_{θd} := #(M̂(θ, b)), respectively.

2. For any two irreducible critical points a and c in M_{Y} (𝔰) with gr(a) − gr(c) = 1, we define m_{ac} to be the relative Euler number of the canonical line bundle L_{ac} (7) with the canonical trivialization given by (8).

The following proposition states various relations satisfied by the integers defined in Definition 2.1, whose proof can be found in Remark 5.8 of [5]. Proposition 2.2. 1. For any irreducible critical point a in M_{Y}^{∗} (𝔰) and any critical point c in M_{Y} (𝔰)

with gr(a) − gr(c) = 2, we have the following identity:

∑_{{b∈M^{∗}_{Y} (𝔰) \atop gr(a)−gr(b)=1}} n_{ab}n_{bc} = 0.

2. Let a and d be two irreducible critical points with gr(a) − gr(d) = 3. Assume that all the critical points c with gr(a) > gr(c) > gr(d) are irreducible. Then we have the identity

∑_{c_{1}:gr(a)−gr(c_{1})=1} n_{a,c_{1}}m_{c_{1},d} − ∑_{c_{2}:gr(c_{2})−gr(d)=1} m_{a,c_{2}}n_{c_{2},d} = 0.

When gr(a) = 1 and gr(d) = −2, we have the identity

∑_{c_{1}:gr(c_{1})=0} n_{a,c_{1}}m_{c_{1},d} + n_{aθ}n_{θd} − ∑_{c_{2}:gr(c_{2})=−1} m_{a,c_{2}}n_{c_{2},d} = 0.

With the help of this Proposition, we can check that the equivariant Seiberg-Witten-Floer complex CF_{∗,U(1)}^{SW}(Y, 𝔰) as given in (1) with the grading and the differential operator given by (2) and (3) is well-defined, and we denote its homology by HF_{∗,U(1)}^{SW}(Y, 𝔰). The equivariant Seiberg-Witten-Floer cohomology, denoted by HF_{U(1)}^{SW,∗}(Y, 𝔰), is the homology of the dual complex Hom(CF_{∗,U(1)}^{SW}(Y, 𝔰), ℤ). The main result in [5] shows that the equivariant Seiberg-Witten Floer homology HF_{∗,U(1)}^{SW}(Y, 𝔰) and cohomology HF_{U(1)}^{SW,∗}(Y, 𝔰) are topological invariants of (Y, 𝔰).

3 Variants of equivariant Seiberg-Witten Floer homology

As mentioned in the introduction, we will generalize the construction of the equivariant Seiberg-Witten Floer homology in several ways.

First, we denote by CF_{∗,U(1)}^{SW,∞}(Y, 𝔰) the graded complex generated by

{Ω^{k} ⊗ η_{a}, Ω^{k} ⊗ 1_{a}, Ω^{k} ⊗ 1_{θ} : a ∈ M_{Y}^{∗} (𝔰), k ∈ ℤ.}

More precisely, for any irreducible critical orbits O_{a}, we set

\begin{aligned}C_{∗,U(1)}^{∞}(O_{a}) & = & ℤ[Ω, Ω^{−1}] ⊗ Ω_{0}^{∗}(O_{a}) \\ & := & ⨁_{k∈ℤ}(ℤΩ^{k} ⊗ η_{a} + ℤΩ^{k} ⊗ 1_{a}) \end{aligned}

with the grading gr(Ω^{k} ⊗ η_{a}) = 2k + gr(a) and gr(Ω^{k} ⊗ 1_{a}) = 2k + gr(a) + 1, and we set

C_{∗,U(1)}^{∞}(θ) = ⨁_{k∈ℤ}ℤ.Ω^{k} ⊗ 1_{θ}

with the grading gr(Ω^{k} ⊗ 1_{θ}) = 2k.

We then consider

CF_{∗,U(1)}^{SW,∞}(Y, 𝔰) = ⨁_{a∈M_{Y} (𝔰)}ℤ[Ω, Ω^{−1}] ⊗ Ω_{0}^{∗−dim(O_{a})}(O_{a}),  (9)

with the grading and the differential operator given by (2) and (3) respectively. That is, CF_{∗,U(1)}^{SW,∞}(Y, 𝔰) is given by

\begin{aligned}& ⨁_{a∈M_{Y} (𝔰)}C_{∗,U(1)}^{∞}(O_{a}) \\ = & ⨁_{a∈M_{Y}^{∗} (𝔰)}C_{∗,U(1)}^{∞}(O_{a}) ⊕ C_{∗,U(1)}^{∞}(θ). \end{aligned}

Theorem 3.1. Define HF_{∗,U(1)}^{SW,∞}(Y, 𝔰) to be the homology of (CF_{∗,U(1)}^{SW,∞}(Y, 𝔰), D). Then we have

HF_{∗,U(1)}^{SW,∞}(Y, 𝔰) ≅ ℤ[Ω, Ω^{−1}].

Proof. Consider the filtration of CF_{∗,U(1)}^{SW,∞}(Y, 𝔰) according to the grading of the critical points

F_{n}C_{∗,U(1)}^{∞} := ⨁_{gr(a)≤n} C_{∗,U(1)}^{∞}(O_{a})

the corresponding spectral sequence E_{kl}^{r}. The filtration is exhaustive, that is,

CF_{∗,U(1)}^{SW,∞}(Y, 𝔰) = ⋃_{n} F_{n}C_{∗,U(1)}^{∞},

and

⋯ ⊂ F_{n−1}C_{∗,U(1)}^{∞} ⊂ F_{n}C_{∗,U(1)}^{∞} ⊂ F_{n+1}C_{∗,U(1)}^{∞} ⊂ ⋯ ⊂ CF_{∗,U(1)}^{SW,∞}(Y, 𝔰).

Moreover, by the compactness of the moduli space of critical orbits, the set of indices gr(a) is bounded from above and below, hence the filtration is bounded. Thus, the spectral sequence converges to HF_{∗,U(1)}^{SW,∞}(Y, 𝔰).

We compute the E^{0}-term:

E_{kl}^{0} = F_{k}C_{k+l,U(1)}^{∞}/F_{k−1}C_{k+l,U(1)}^{∞}
= ⨁_{a∈M_{Y} (𝔰):gr(a)=i≤k}C_{k+l−i,U(1)}^{∞}(O_{a})/  ⨁_{a∈M_{Y} (𝔰):gr(a)=i≤k−1}C_{k+l−i,U(1)}^{∞}(O_{a})
= ⨁_{a∈M_{Y} (𝔰):gr(a)=k}C_{l,U(1)}^{∞}(O_{a}).

For k ≠ 0 this complex is just the direct sum of the separate complexes (C_{∗,U(1)}^{∞}(O_{a}), ∂_{U(1)}) on each orbit O_{a} with gr(a) = k:

⋯ → Ω ⊗ 1_{a} →^{0} Ω ⊗ η_{a} →^{−1} 1 ⊗ 1_{a} →^{0} 1 ⊗ η_{a} →^{−1} Ω^{−1} ⊗ 1_{a} → ⋯  (10)

In the case k = 0 we have

E_{0,l}^{0} = C_{l,U(1)}^{∞}(θ) ⊕ ⨁_{a∈M_{Y}^{∗} (𝔰):gr(a)=0}C_{l,U(1)}^{∞}(O_{a}),

which again is a direct sum of the complexes (C_{∗,U(1)}^{∞}(O_{a}), ∂_{U(1)}), here ∂_{U(1)} is the equivariant de Rham differential, and of the complex with generators Ω^{r} ⊗ 1_{θ}⟩ in degree l = 2r and trivial differentials.

We then compute the E_{pq}^{1} term directly: we have

E_{kl}^{1} = H_{k+l}(E_{k,∗}^{0}) = \{ \begin{matrix}ℤ.Ω^{r} ⊗ 1_{θ} & k = 0, l = 2r \\ 0 & k ≠ 0, \end{matrix}

since each complex (10) is acyclic. Thus, the only non-trivial E^{1}-terms are of the form E_{0l}^{1} = ℤ.Ω^{r} ⊗ 1_{θ}, l = 2r, with trivial differentials, so that the spectral sequence collapses and we obtain the result.

3.1 Long exact sequence

Definition 3.2. Let CF_{∗,U(1)}^{SW,−}(Y, 𝔰) be the subcomplex of CF_{∗,U(1)}^{SW,∞}(Y, 𝔰), generated by

{Ω^{k} ⊗ η_{a}, Ω^{k} ⊗ 1_{a}, Ω^{k} ⊗ 1_{θ} : a ∈ M_{Y}^{∗} (𝔰), k ∈ ℤ and k < 0},

whose homology groups are denoted by HF_{∗,U(1)}^{SW,−}(Y, 𝔰). The quotient complex is denoted by CF_{∗,U(1)}^{SW,+}(Y, 𝔰), with the homology groups denoted by HF_{∗,U(1)}^{SW,+}(Y, 𝔰).

Theorem 3.3. 1. HF_{∗,U(1)}^{SW,+}(Y, 𝔰) ≅ HF_{∗,U(1)}^{SW}(Y, 𝔰), where HF_{∗,U(1)}^{SW}(Y, 𝔰) is the equivariant Seiberg-Witten-Floer homology defined in [5].

2. There is an exact sequence of ℤ-modules which relates these variants of equivariant Seiberg-Witten-Floer homologies:

\xymatrix{⋯ → HF_{∗,U(1)}^{SW,−}(Y, 𝔰) \ar[r]^{l_{∗}} & HF_{∗,U(1)}^{SW,∞}(Y, 𝔰) \ar[r]^{π_{∗}} & HF_{∗,U(1)}^{SW,+}(Y, 𝔰) \ar[r]^{δ_{∗}} & HF_{∗−1,U(1)}^{SW,−}(Y, 𝔰) → ⋯}

Proof. It is easy to see that CF_{∗,U(1)}^{SW,+}(Y, 𝔰) = CF_{∗,U(1)}^{SW}(Y, 𝔰), with the same grading and differentials, hence HF_{∗,U(1)}^{SW,+}(Y, 𝔰) ≅ HF_{∗,U(1)}^{SW}(Y, 𝔰). The long exact sequence in homology is induced by the short exact sequence of chain complexes:

0 → CF_{∗,U(1)}^{SW,−}(Y, 𝔰) → CF_{∗,U(1)}^{SW,∞}(Y, 𝔰) → CF_{∗,U(1)}^{SW,+}(Y, 𝔰) → 0.

From the above long exact sequence, we can define

\begin{aligned}HF_{red,∗}^{SW}(Y, 𝔰) & = & Coker(π_{∗}) ≅ HF_{∗,U(1)}^{SW,+}(Y, 𝔰)/Ker(δ_{∗}) \\ & ≅ & Im(δ_{∗}) ≅ Ker(l_{∗−1}). \end{aligned}  (11)

3.2 The spectral sequence for HF_{∗,U(1)}^{SW,+}(Y, 𝔰)

We consider again the filtration by index of critical orbits,

F_{n}C_{∗,U(1)}^{+} := ⨁_{gr(a)≤n} C_{∗,U(1)}^{+}(O_{a}),

for

C_{∗,U(1)}^{+}(O_{a}) = ℤ[Ω] ⊗ Ω_{0}^{∗−dim(O_{a})}(O_{a}).

We have

\begin{matrix}E_{kl}^{0} = & F_{k}C_{k+l,U(1)}^{+}/F_{k−1}C_{k+l,U(1)}^{+} \\ = & ⨁_{gr(a)=k} C_{l,U(1)}^{+}(O_{a}). \end{matrix}

This is a direct sum of the complexes

⋯ →^{−1} Ω ⊗ 1_{a} →^{0} Ω ⊗ η_{a} →^{−1} 1 ⊗ 1_{a} →^{0} 1 ⊗ η_{a} → 0,  (12)

over each orbit O_{a} ≅ S^{1} and, in the case k = 0, the complex with generators Ω^{r} ⊗ 1_{θ} in degree l = 2r, and trivial differentials.

Thus, we obtain that E_{pq}^{1} = H_{p+q}(E_{p∗}^{0}) is of the form

E_{pq}^{1} = \{ \begin{matrix}0 & q > 0 \\ ℤ.1 ⊗ η_{a} & q = 0, gr(a) = k \end{matrix}

for k ≠ 0, and

E_{0q}^{1} = \{ \begin{matrix}ℤ.Ω^{r} ⊗ 1_{θ} & q = 2r > 0 \\ ℤ.1 ⊗ η_{a} ⊕ ℤ.1 ⊗ 1_{θ} & q = 0, gr(a) = 0. \end{matrix}

The differential d^{1} : E_{p,q}^{1} → E_{p−1,q}^{1} is of the form

\begin{aligned}d^{1}(1 ⊗ η_{a}) = & n_{ab}1 ⊗ η_{b} \\ & +n_{aθ}1 ⊗ 1_{θ} (if gr(a) = 1) \end{aligned}

Thus, we obtain

E_{pq}^{2} = \{ \begin{matrix}HF_{p}^{SW} (Y, 𝔰) & p ≠ 0, q = 0 \\ Ker(Δ_{1}) & p = 1, q = 0 \\ HF_{0}^{SW} (Y, 𝔰) ⊕ T_{0} & p = 0, q = 0 \\ ℤ.Ω^{r} ⊗ 1_{θ} & p = 0, q = 2r > 0. \end{matrix}

Here HF_{∗}^{SW} (Y, 𝔰) denotes the non-equivariant (metric and perturbation dependent) Seiberg–Witten Floer homology. This is the homology of the complex with generators 1 ⊗ η_{a} in degree gr(a) and boundary coefficients n_{ab} for gr(a) − gr(b) = 1. We also denoted by Δ_{1} the map

Δ_{1} : HF_{1}^{SW} (Y, 𝔰) → ℤ.1 ⊗ 1_{θ},
Δ_{1}(∑ x_{a}1 ⊗ η_{a}) = ∑ x_{a}n_{aθ}1 ⊗ 1_{θ},

where the coefficients x_{a} satisfy ∑ x_{a}n_{ab} = 0. Finally, the term T_{0} denotes the term

T_{0} = ℤ.1 ⊗ 1_{θ}/Im(Δ_{1}).

Notice then that the boundary d^{2} : E_{p,q}^{2} → E_{p−2,q+1}^{2} is trivial, hence the E_{p,q}^{3} terms are disposed as in the diagram:

\xymatrix{⋯ & 0 & 0 & 0 & 0 & ℤ.Ω^{2} ⊗ 1_{θ} & 0 & 0 & ⋯ \\ ⋯ & 0 & 0 & 0 & 0 & 0 & 0 & 0 & ⋯ \\ ⋯ & 0 & 0 & 0 & 0 & ℤ.Ω ⊗ 1_{θ} & 0 & 0 & ⋯ \\ ⋯ & 0 & 0 & 0 & 0 & 0 & 0 & 0 & ⋯ \\ ⋯ & HF_{4}^{SW} & HF_{3}^{SW} \ar[uurrr]^{d^{3}} & HF_{2}^{SW} & Ker(Δ_{1}) & HF_{0}^{SW} ⊕ T_{0} & HF_{−1}^{SW} & HF_{−2}^{SW} & ⋯}

The differential d^{3} : E_{p,q}^{3} → E_{p−3,q+2}^{3} is given by the expression

d^{3}([∑ x_{a}1 ⊗ η_{a}]) = ∑ x_{a}m_{ac}n_{cθ}Ω ⊗ 1_{θ},  (13)

for gr(a) − gr(c) = 2. The expression is obtained by considering the unique choice of a representative of the class [∑ x_{a}1 ⊗ η_{a}] in E_{p,q}^{3} whose boundary (3) defines a class in E_{p−3,q+2}^{3}.

The differential d^{4} : E_{p,q}^{4} → E_{p−4,q+3}^{4} is again trivial, and we obtain the E_{pq}^{5} of the form

\xymatrix{⋯ & 0 & 0 & 0 & 0 & ℤ.Ω^{2} ⊗ 1_{θ} & 0 & 0 & ⋯ \\ ⋯ & 0 & 0 & 0 & 0 & 0 & 0 & 0 & ⋯ \\ ⋯ & 0 & 0 & 0 & 0 & T_{1} & 0 & 0 & ⋯ \\ ⋯ & 0 & 0 & 0 & 0 & 0 & 0 & 0 & ⋯ \\ HF_{5}^{SW} \ar[uuuurrrrr]^{d^{5}} & HF_{4}^{SW} & Ker(Δ_{3}) & HF_{2}^{SW} & Ker(Δ_{1}) & HF_{0}^{SW} ⊕ T_{0} & HF_{−1}^{SW} & HF_{−2}^{SW} & ⋯}

where again we denote by T_{1} the term

T_{1} = ℤ.Ω ⊗ 1_{θ}/Im(Δ_{3}).

Thus, by iterating the process, we observe that all the differentials d^{2k} : E_{p,q}^{2k} → E_{p−2k,q+2k+1}^{2k} are trivial and the differentials d^{2k+1} : E_{p,q}^{2k+1} → E_{p−2k−1,q+2k}^{2k+1} consists of one map for p = 2k + 1, q = 0:

Δ_{2k+1} : HF_{2k+1}^{SW} → ℤ.Ω^{k} ⊗ 1_{θ},

induced by

Δ_{2k+1}(∑ x_{a}1 ⊗ η_{a}) = ∑ x_{a}m_{aa_{2k−1}}m_{a_{2k−1}a_{2k−3}} ⋯ m_{a_{3}a_{1}}n_{a_{1}θ}Ω^{k} ⊗ 1_{θ}.

Here we have gr(a) = 2k+1 and gr(a_{r}) = r. Notice that these maps agree with the morphism Δ_{∗}, which is obtained in [5] as the connecting homomorphism in the long exact sequence relating equivariant and non-equivariant Seiberg–Witten Floer homologies.

We thus obtain the following structure theorem for equivariant Seiberg–Witten Floer homology.

Theorem 3.4. The equivariant Seiberg–Witten Floer homology HF_{∗,U(1)}^{SW,+}(Y, 𝔰) has the form

HF_{∗,U(1)}^{SW,+}(Y, 𝔰) = \{ \begin{matrix}Ker(Δ_{2k+1}) & ∗ = 2k + 1 > 0 \\ HF_{2k}^{SW} (Y, 𝔰) ⊕ T_{k} & ∗ = 2k ≥ 0 \\ HF_{∗}^{SW} (Y, 𝔰) & ∗ < 0 \end{matrix}

where T_{k} is the term

T_{k} = ℤ.Ω^{k} ⊗ 1_{θ}/Im(Δ_{2k+1}).

This result refines the long exact sequence obtained in [5]:

\xymatrix{HF_{∗,U(1)}^{SW}(Y, 𝔰) \ar[r]^{i_{∗}} & HF_{∗}^{SW} (Y, 𝔰, g, ν) \ar@{-->}[dl]^{Δ_{∗}} \\ ℤ[Ω] \ar[u]^{j_{∗}}}

Similar results can be obtained for HF_{∗,U(1)}^{SW,−}(Y, 𝔰).

3.3 Topological invariance

Note that the definitions of these homologies depend on the Seiberg-Witten equations, which use the metric and perturbation on (Y, 𝔰). By the result of [5], we know that HF_{∗,U(1)}^{SW,+}(Y, 𝔰) ≅ HF_{∗,U(1)}^{SW}(Y, 𝔰) is a topological invariant of (Y, 𝔰), we first recall this topological invariance as stated in Theorem 6.1 [5].

Theorem 3.5. (Theorem 6.1 [5]) Let (Y, 𝔰) be a rational homology sphere with a Spin^{c} structure. Suppose given two metrics g_{0} and g_{1} on Y and perturbations ν_{0} and ν_{1} such that Ker(/ ∂_{ν_{0}}^{g_{0}}) = Ker(/ ∂_{ν_{1}}^{g_{1}}) = 0, so that the corresponding monopole moduli spaces M_{Y} (𝔰, g_{0}, ν_{0}) and M_{Y} (𝔰, g_{1}, ν_{1}) consist of finitely many isolated points. Then there exists an isomorphism between the equivariant Seiberg-Witten Floer homologies HF_{∗,U(1)}^{SW}(Y, 𝔰, g_{0}, ν_{0}) and HF_{∗,U(1)}^{SW}(Y, 𝔰, g_{1}, ν_{1}), with a degree shift given by the spectral flow of the Dirac operator / ∂_{ν_{t}}^{g_{t}} along a path of metrics and perturbations connecting (g_{0}, ν_{0}) and (g_{1}, ν_{1}). That is, if the complex spectral flow along the path (g_{t}, ν_{t}) is denoted by SF_{ℂ}(/ ∂_{ν_{t}}^{g_{t}}), then for any k ∈ ℤ,

HF_{k,U(1)}^{SW}(Y, 𝔰, g_{0}, ν_{0}) ≅ HF_{k+2SF_{ℂ}(/ ∂_{ν_{t}}^{g_{t}}),U(1)}^{SW}(Y, 𝔰, g_{1}, ν_{1}).

From Theorem 3.1, we know that

HF_{∗,U(1)}^{SW,∞}(Y, 𝔰) ≅ ℤ[Ω, Ω^{−1}]

is independent of (Y, 𝔰), up to a degree shift as given in Theorem 3.5. Thus, applying the five lemma to the long exact sequence in Theorem 3.3, we obtain that HF_{∗,U(1)}^{SW,−}(Y, 𝔰) and HF_{red,∗}^{SW}(Y, 𝔰) are also topological invariants of (Y, 𝔰).

Theorem 3.6. HF_{∗,U(1)}^{SW,−}(Y, 𝔰) and HF_{red,∗}^{SW}(Y, 𝔰) are topological invariants of (Y, 𝔰), in the sense that, given any two metrics g_{0} and g_{1} on Y and perturbations ν_{0} and ν_{1}, with Ker(/ ∂_{ν_{0}}^{g_{0}}) = Ker(/ ∂_{ν_{1}}^{g_{1}}) = 0, there exist isomorphisms

\begin{matrix}HF_{k,U(1)}^{SW,−}(Y, 𝔰, g_{0}, ν_{0}) ≅ HF_{k+2SF_{ℂ}(/ ∂_{ν_{t}}^{g_{t}}),U(1)}^{SW,−}(Y, 𝔰, g_{1}, ν_{1}) \\ HF_{red,k}^{SW}(Y, 𝔰, g_{0}, ν_{0}) ≅ HF_{red,k+2SF_{ℂ}(/ ∂_{ν_{t}}^{g_{t}})}^{SW}(Y, 𝔰, g_{1}, ν_{1}). \end{matrix}

Here SF_{ℂ}(/ ∂_{ν_{t}}^{g_{t}}) denotes the complex spectral flow of the Dirac operator / ∂_{ν_{t}}^{g_{t}} along the path (g_{t}, ν_{t}).

4 Properties of equivariant Seiberg-Witten Floer homologies

In this section, we briefly discuss some of the algebraic structures and properties of the equivariant Seiberg-Witten Floer homologies defined in the previous section.

Note that for any irreducible critical points a and b in M_{Y}^{∗} (𝔰), the associated integer m_{ac} is the counting of points in the geometric representative of the relative first Chern class of the canonical line bundle (7) over M(a, c), we can apply this fact to define a u-action on the chain complex CF_{∗,U(1)}^{SW,∞}(Y, 𝔰)

u : CF_{∗,U(1)}^{SW,∞}(Y, 𝔰) ⟶ CF_{∗,U(1)}^{SW,∞}(Y, 𝔰)

which decreases the grading by two. The action is given in terms of its actions on generators as follows:

u(Ω^{n} ⊗ η_{a}) = ∑_{{c∈M^{∗}(Y,𝔰) \atop gr(a)−gr(c)=2}}m_{ac}Ω^{n} ⊗ η_{c}.

u(Ω^{n} ⊗ 1_{a}) = \{ \begin{matrix}∑_{{c∈M^{∗}(Y,𝔰) \atop gr(a)−gr(c)=2}}m_{ac}Ω^{n} ⊗ 1_{c} & if gr(a) ≠ 1 \\ ∑_{{c∈M^{∗}(Y,𝔰) \atop gr(c)=−1}} m_{ac}Ω^{n} ⊗ 1_{c} + n_{aθ}Ω^{n} ⊗ 1_{θ} & if gr(a) = 1 \end{matrix}
u(Ω^{n} ⊗ 1_{θ}) = ∑_{{d∈M^{∗}_{Y} (𝔰) \atop gr(d)=−2}} n_{θd}Ω^{n} × η_{d} + Ω^{n−1} ⊗ 1_{θ}.

(14)

Proposition 4.1. The u-action defined (14) on the chain complex CF_{∗,U(1)}^{SW,∞}(Y, 𝔰) is homotopic to the Ω^{−1}-action acting on CF_{∗,U(1)}^{SW,∞}(Y, 𝔰). The induced actions on CF_{∗,U(1)}^{SW,±}(Y, 𝔰) define ℤ[u]-module structures on HF_{∗,U(1)}^{SW,^{±}}(Y, 𝔰).

Proof. Define H : CF_{∗,U(1)}^{SW,∞}(Y, 𝔰) ⟶ CF_{∗,U(1)}^{SW,∞}(Y, 𝔰) by its actions on the generators as follows:

H(Ω^{n} ⊗ η_{a}) = 0,
H(Ω^{n} ⊗ 1_{a}) = Ω^{n} ⊗ η_{a},
H(Ω^{n} ⊗ 1_{θ}) = 0.

Then it is a direct calculation to show that we have:

(u − Ω^{−1})(Ω^{k} ⊗ η_{a}) = m_{ac}Ω^{k} ⊗ η_{c} − Ω^{k−1} ⊗ η_{a} = (DH + HD)(Ω^{k} ⊗ η_{a})

(u − Ω^{−1})(Ω^{k} ⊗ 1_{a}) = m_{ac}Ω^{k} ⊗ 1_{c} − Ω^{n−1} ⊗ 1_{a} (+n_{aθ}Ω^{n} ⊗ 1_{θ} if gr(a) = 1) = (DH + HD)(Ω^{k} ⊗ 1_{a}),

(u − Ω^{−1})(Ω^{k} ⊗ 1_{θ}) = n_{θd}Ω^{n} ⊗ η_{d} = (DH + HD)(Ω^{k} ⊗ 1_{θ}).

Thus the claim follows using the chain homotopy u − Ω^{−1} = D ∘ H + H ∘ D.

Thus, on the homological level, we can identify the u-action with the induced Ω^{−1} action on various homologies. In particular, we see that there is a subcomplex ĈF_{∗}^{SW} (Y, 𝔰) of CF_{∗,U(1)}^{SW,+}(Y, 𝔰) such that the following short exact sequence of chain complexes holds:

\xymatrix{0 → ĈF_{∗}^{SW} (Y, 𝔰) \ar[r] & CF_{∗,U(1)}^{SW,+}(Y, 𝔰) \ar[r]^{Ω^{−1}} & CF_{∗,U(1)}^{SW,+}(Y, 𝔰) → 0.}  (15)

Proposition 4.2. Let ĤF_{∗}^{SW} (Y, 𝔰) be the homology of ĈF_{∗}^{SW} (Y, 𝔰), then ĤF_{∗}^{SW} (Y, 𝔰) is also a topological invariant of (Y, 𝔰), and it is determined by the following long exact sequence

\xymatrix{⋯ → ĤF_{∗}^{SW} (Y, 𝔰) \ar[r] & HF_{∗,U(1)}^{SW,+}(Y, 𝔰) \ar[r]^{u} & HF_{∗−2,U(1)}^{SW,+}(Y, 𝔰) \ar[r] & ĤF_{∗−1}^{SW}(Y, 𝔰) → · · · .}

Moreover, ĤF^{SW} (Y, 𝔰) is non-trivial if and only if HF_{∗,U(1)}^{SW,+}(Y, 𝔰) is non-trivial.

Proof. The long exact sequence follows from the short exact sequence of chain complexes (15) and Proposition 4.1. This long exact sequence implies that ĤF_{∗}^{SW} (Y, 𝔰) is also a topological invariant of (Y, 𝔰).

Note that, from the compactness of M_{Y} (𝔰), we see that each element in HF_{∗,U(1)}^{SW,+}(Y, 𝔰) can be annihilated by a sufficiently large power of Ω^{−1}. Hence, u is an isomorphism on HF_{∗,U(1)}^{SW,+}(Y, 𝔰) if and only if HF_{∗,U(1)}^{SW,+}(Y, 𝔰) is trivial. Then the last claim follows from this observation and the long exact sequence.

If we think of the set of Spin^{c} structures on Y as the set of equivalence classes of nowhere vanishing vector fields on Y (Cf.[9]), then there is a natural bijection between Spin^{c}(Y) and Spin^{c}(−Y) where −Y is the same Y with the opposite orientation. Theorem 4.3. Let (Y, 𝔰) be a rational homology 3-sphere with a Spin^{c} structure 𝔰, and (−Y, 𝔰) denote Y with the opposite orientation and the corresponding Spin^{c} structure. Then there is a natural isomorphism

HF_{U(1)}^{SW,∗}(Y, 𝔰) ≅ HF_{∗,U(1)}^{SW,−}(−Y, 𝔰)

where HF_{U(1)}^{SW,∗}(Y, 𝔰) is the equivariant Seiberg-Witten-Floer cohomology defined in [5].

Proof. Note that HF_{U(1)}^{SW,∗}(Y, 𝔰) is the homology of the dual complex Hom(CF_{∗,U(1)}^{SW,+}(Y, 𝔰), ℤ). We start to construct a natural pairing

⟨·, ·⟩ :  CF_{∗,U(1)}^{SW,∞}(Y, 𝔰) × CF_{∗,U(1)}^{SW,∞}(−Y, 𝔰) ⟶ ℤ  (16)

which satisfies

⟨D_{Y} (ξ_{1}), ξ_{2}⟩ = ⟨ξ_{1}, D_{−Y} (ξ_{2})⟩,  ⟨Ω^{−1}(ξ_{1}), ξ_{2}⟩ = ⟨ξ_{1}, Ω^{−1}(ξ_{2})⟩.  (17)

for any element ξ_{1} ∈ CF_{∗,U(1)}^{SW,∞}(Y, 𝔰) and any element ξ_{2} ∈ CF_{∗,U(1)}^{SW,∞}(−Y, 𝔰).

Then we will show that the above pairing is non-degenerate when restricted to CF_{∗,U(1)}^{SW,+}(Y, 𝔰) × CF_{∗,U(1)}^{SW,−}(−Y, 𝔰).

From the nature of the Seiberg-Witten equations, we see that there is an identification

M_{Y} (𝔰) → M_{−Y} (𝔰)

for a good pair of metric and perturbation on (Y, 𝔰) and the corresponding metric and perturbation on (−Y, 𝔰). Then the relative gradings with respect to the unique reducible monopole in M_{Y} (𝔰) and M_{−Y} (𝔰) respectively, satisfies

gr_{−Y} (a^{−}) = −gr_{Y} (a) − 1,

where a^{−} is the element in M_{−Y}^{∗} (𝔰) corresponding to a ∈ M_{Y}^{∗} (𝔰), we assume that gr_{Y} (θ) = gr_{−Y} (θ^{−}). Moreover, there is an natural identification between the moduli spaces of flowlines for (Y, 𝔰) and (−Y, 𝔰), that is,

M_{Y ×ℝ}(a, b) ≅ M_{−Y ×ℝ}(b^{−}, a^{−}).

Now we define the pairing on CF_{∗,U(1)}^{SW,∞}(Y, 𝔰) × CF_{∗,U(1)}^{SW,∞}(−Y, 𝔰) such that the following pairings are the only non-trivial pairings:

⟨Ω^{n} ⊗ η_{a}, Ω^{−n−1} ⊗ 1_{a^{−}}⟩ = 1
⟨Ω^{n} ⊗ 1_{a}, Ω^{−n−1} ⊗ η_{a^{−}}⟩ = 1
⟨Ω^{n} ⊗ 1_{θ}, Ω^{−n−1} ⊗ 1_{θ^{−}}⟩ = 1.

It is a direct calculation to show that this pairing satisfies the relation (17) and the restriction of this pairing to CF_{∗,U(1)}^{SW,+}(Y, 𝔰) × CF_{∗,U(1)}^{SW,−}(−Y, 𝔰) is non-degenerate. Then the claim follows from the definition.

Let ĤF^{SW,∗}(Y, 𝔰) and HF_{±,U(1)}^{SW,∗}(Y, 𝔰) denote the homology groups of the dual complexes Hom(ĈF_{∗}^{SW} (Y, 𝔰), ℤ) and Hom(CF_{∗,U(1)}^{SW,±}(Y, 𝔰), ℤ) of ĈF_{∗}^{SW} (Y, 𝔰) and CF_{∗,U(1)}^{SW,±}(Y, 𝔰) respectively. From the proof the above Theorem 4.3, we actually establish the following duality between these homologies.

Theorem 4.4. For any rational homology 3-sphere Y with a spinc structure 𝔰, there exist natural isomorphisms

ĤF^{SW,∗}(Y, 𝔰) ≅ ĤF_{∗}^{SW} (−Y, 𝔰),  HF_{±,U(1)}^{SW,∗}(Y, 𝔰) ≅ HF_{∗,U(1)}^{SW,∓}(−Y, 𝔰).  (18)

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