Equivariant Seiberg-Witten-Floer cohomology

We develop an equivariant version of Seiberg-Witten-Floer cohomology for finite group actions on rational homology $3$-spheres. Our construction is based on an equivariant version of the Seiberg-Witten-Floer stable homotopy type, as constructed by Manolescu. We use these equivariant cohomology groups to define a series of $d$-invariants $d_{G,c}(Y,\mathfrak{s})$ which are indexed by the group cohomology of $G$. These invariants satisfy a Froyshov-type inequality under equivariant cobordisms. Lastly we consider a variety of applications of these $d$-invariants: concordance invariants of knots via branched covers, obstructions to extending group actions over bounding $4$-manifolds, Nielsen realisation problems for $4$-manifolds with boundary and obstructions to equivariant embeddings of $3$-manifolds in $4$-manifolds.


Introduction
In this paper we develop an equivariant version of Seiberg-Witten-Floer cohomology for rational homology 3-spheres equipped with the action of a finite group. Our approach is modelled on the construction of a Seiberg-Witten-Floer stable homotopy type due to Manolescu [48] which we now briefly recall. Let Y be a rational homology 3-sphere and s a spin c -structure on Y . Given a metric g on Y , the construction of [48] yields an S 1 -equivariant stable homotopy type SW F (Y, s, g). The Seiberg-Witten-Floer cohomology of (Y, s) is then given (up to a degree shift) by the S 1 -equivariant cohomology of SW F (Y, s, g): HSW * (Y, s) = H * +2n(Y,s,g) S 1 (SW F (Y, s, g)), where n(Y, s, g) is a rational number given by a certain combination of eta invariants. The stable homotopy type SW F (Y, s, g) depends on the choice of metric, but only up to a suspension. Given two metrics g 0 , g 1 one obtains a canonical homotopy equivalence where SF ({D s }) denotes the spectral flow for the family of Dirac operators {D s } determined by a path of metrics {g s } from g 0 to g 1 . The rational numbers n(Y, s, g) are defined in such a way that they split the spectral flow in the sense that (1.2) SF ({D s }) = n(Y, s, g 1 ) − n(Y, s, g 0 ).
Our primary motivation for considering the equivariant d-invariants is that they are necessary for the formulation of our equivariant generalisation of Frøyshov's inequality described below.
Knot concordance invariants. Let K be a knot in S 3 and let Y = Σ 2 (K) be the double cover of S 3 branched over K. Then Y has an action of G = Z 2 generated by the covering involution. Further, Y has a spin c -structure t 0 uniquely determined by the condition that it arises from a spin structure. Set F = Z 2 . Then H * G ∼ = F[Q], where deg(Q) = 1. For each j ≥ 0, we define an invariant δ j (K) ∈ Z by setting δ j (K) = 4δ Z2,Q j (Σ 2 (K), t 0 ).
Let σ(K) and g 4 (K) denote the signature and smooth 4-genus of K.
One can obtain even more knot concordance invariants by considering higher order cyclic branched covers, see Remark 6.7.
1.2. Applications. We outline here some of the applications of equivariant Seiberg-Witten-Floer cohomology. These are considered in more detail in Section 7.
1.2.1. Non-extendable actions (Section 7.2). Let Y be a rational homology 3-sphere equipped with an orientation preserving action of G and let W be a smooth 4manifold which bounds Y . The equivariant d-invariants give obstructions to extending the action of G over W . Example 1.9. The Brieskorn homology sphere Y = Σ(p, q, r) where p, q, r are pairwise coprime is the branched cyclic p-fold cover of the torus knot T q,r . Let τ : Y → Y be a generator of the Z p -action determined by this covering. For certain values of p, q, r it can be shown that Y bounds a contractible 4-manifold, for example Σ (2,3,13) bounds a contractible 4-manifold [2]. It can be shown that τ is smoothly isotopic to the identity, hence it follows that τ can be extended as a diffeomorphism over any 4-manifold bounded by Y . On the other hand we show in Proposition 7.2 that if p is prime then δ Zp,1 (Y, s) = −λ(Y ) is minus the Casson invariant of Y (where s is the unique spin c -structure on Y ), which is non-zero. We futher show that the non-vanishing of δ Zp,1 (Y, s) implies that τ can not be extended as a smooth Z p -action to any contractible 4-manifold bounded by Y . This partially recovers the non-extendability results of Anvari-Hambleton [6,7] for Brieskorn homology 3-spheres bounded by contractible 4-manifolds.
On the other hand, our non-extendability result also holds in situations not covered by Anvari-Hambleton. Suppose now that Y = Σ(p, q, r) bounds a rational homology 4-ball W . For example, Fintushel-Stern showed that Σ(2, 3, 7) bounds a rational homology 4-ball, although it does not bound an integral homology 4-ball [25]. More examples can be found in [4,59]. We show in Section 7.2 that the non-vanishing of δ Zp,1 (Σ(p, q, r)), where p is prime implies that the Z p -action can not be extended to any rational homology 4-ball W bounded by Y , provided that p does not divide the order of H 2 (W ; Z).

1.2.2.
Realisation problems (Section 7.3). Let W be a smooth 4-manifold with boundary an integral homology sphere Y . Suppose that a finite group G acts on H 2 (W ; Z) preserving the intersection form. We say that the action of G on H 2 (W ; Z) can be realised by diffeomorphisms if there is a smooth orientation preserving action of G on W inducing the given action on H 2 (W ; Z). The equivariant d-invariants give obstructions to realising such actions by diffeomorphism. This extends the non-realisation results of [10], [11] for closed 4-manifolds to the case of 4-manifolds with non-empty boundary. Example 1.10. Suppose that b 1 (W ) = 0, that H 2 (W ; Z) has no 2-torsion and even intersection form. Suppose that Y is an L-space. Suppose that an action of G = Z p on H 2 (W ; Z) is given, where p is prime and that the subspace of H 2 (W ; R) fixed by G is negative definite. If σ(W )/8 < −δ(Y, s) (where s is the unique spin c structure on Y ) then the action of Z p on H 2 (W ; Z) is not realisable by a smooth Z p -action on W . Note that we are not making any assumptions about the action of Z p on the boundary. 1.2.3. Equivariant embeddings of 3-manifolds in 4-manifolds (Section 7.4). Let Y be a rational homology 3-sphere equipped with an orientation preserving action of G. By an equivariant embedding of Y into a 4-manifold X, we mean an embedding Y → X such that the action of G on Y extends over X.
Example 1.11. Let Y = Σ(2, 2s − 1, 2s + 1) where s is odd, equipped with the involution τ obtained from viewing Y as the branched double cover Σ 2 (T 2s−1,2s+1 ). Then Y embeds in S 4 [13,Theorem 2.13]. On the other hand, δ j (Y, s) = 0 for some j. We will show that the non-vanishing of this invariant implies that Y can not be equivariantly embedded in S 4 .
It is known that every 3-manifold Y embeds in the connected sum # n (S 2 × S 2 ) of n copies of S 2 × S 2 for some sufficiently large n [1, Theorem 2.1]. Aceto-Golla-Larson define the embedding number ε(Y ) of Y to be the smallest n for which Y embeds in # n (S 2 × S 2 ). Here we consider an equivariant version of the embedding number. To obtain interesting results we need to make an assumption on the kinds of group actions allowed. Definition 1.12. Let G = Z p = τ where p is a prime number. We say that a smooth, orientation preserving action of G on X = # n (S 2 × S 2 ) is admissible if H 2 (X; Z) τ = 0, where H 2 (X; Z) τ = {x ∈ H 2 (X; Z) | τ (x) = x}. We define the equivariant embedding number ε(Y, τ ) of (Y, τ ) to be the smallest n for which Y embeds equivariantly in # n (S 2 ×S 2 ) for some admissible Z p -action on # n (S 2 ×S 2 ), if such an embedding exists. We set ε(Y, τ ) = ∞ if there is no such embedding. Example 1.13. Let Y = Σ(2, 3, 6n + 1) = Σ 2 (T 3,6n+1 ) and equip Y with the covering involution τ . We show that 2n ≤ ε(Σ(2, 3, 6n + 1), τ ) ≤ 12n.
1.3. Comparison with other works. In [29], the authors introduce equivariant versions of several types of Floer homology, mostly focusing on the case that the group is Z 2 . In particular they define a Z 2 -equivariant version of HF − , which is a module over H * S 1 ×Z2 (pt; Z 2 ) = Z 2 [U, Q]. This construction shares many similarities with the equivariant Seiberg-Witten-Floer cohomology constructed in this paper, such as a localisation isomorphism and a spectral sequence relating the equivariant and ordinary Floer homologies. In fact, it seems reasonable to conjecture that our constructions are isomorphic.
In [5], the authors consider a Z 2 -equivariant Heegaard-Floer homology HF B − (K) for a branched double cover Y = Σ 2 (K) of a knot K, constructed in a manner similar to involutive Heegaard-Floer homology [30] except that the involution arises from the covering involution on Y . These groups are modules over the ring Z 2 [U, Q]/(Q 2 ). From this group they obtain knot concordance invariants δ(K), δ(K). A similar approach was taken in [21], where the authors obtain ιcomplexes [31,Definition 8.1] , we suspect that the group HF B − (K) may be isomorphic to the Zequivariant Seiberg-Witten-Floer homology of Σ 2 (K).
In [44,Remark 3.1], the authors define equivariant Seiberg-Witten-Floer homology in the special case that G acts freely on Y . Their construction coincides with ours in such cases.
1.4. Structure of the paper. In Section 2 we recall the construction of Seiberg-Witten-Foer spectra using finite dimensional approximation and the Conley index. In Section 3 we extend this construction to the G-equivariant setting, arriving at the construction of the G-equivariant Seiberg-Witten-Floer cohomology in §3.4. In the remainder of Section 3 we introduce the equivariant d-invariants and establish their basic properties. Section 4 is concerned with the behaviour of equivariant Seiberg-Witten-Floer cohomology and the d-invariants under equivariant cobordism. In Section 5 we specialise to the case that G is a cyclic group of prime order. In Section 6 we consider the case of branched double covers of knots with their natural involution to obtain knot concordance invariants. Finally in Section 7 we carry out some explicit computations of d-invariants and consider various applications.
Let g be a Riemannian metric on Y and let s be a spin c -structure with associated spinor bundle S. Let ρ : T Y → End(S) denote Clifford multiplication, satisfying The spinor bundle S is equipped with a Hermitian metric , which we take to be anti-linear in the first variable. Let su(S) be the Lie algebra bundle of trace-free skew-adjoint endomorphisms of S and sl(S) the Lie algebra bundle of trace-free endomorphisms of S. Then ρ induces an isomorphism ρ : T Y → su(S) which extends by complexification to an isomorphism ρ : T Y C → sl(S) satisfying ρ(v) = −ρ(v) * . Using the metric g to identify T Y and T * Y we will also view ρ as a map ρ : T * Y → su(S). We extend ρ to 2-forms by It follows that ρ(λ) = −ρ( * λ) for any 2-form λ. Define a Hermitian inner product on su(S) by a, b = 1 2 tr(a * b). Then for any tangent vectors u, v, we have ρ(u), ρ(v) = g(u, v). Define a map Then it follows that In particular, τ (φ, φ) is imaginary and τ (cφ, cφ) = |c| 2 τ (φ, φ). We also have the identity for all spinors φ and vectors v. Let L = det(S) be the determinant line bundle of S. Let Γ(L) denote the space of U (1)-connections on L, which is an affine space over iΩ 1 (Y ). We will write such a connection as 2A. Then 2A determines a Spin c (3)-connection on S whose u(1) part is A and whose spin(3) part is the Levi-Civita connection. Abusing terminology, we will refer to A as a spin c -connection.
Given a spin c -connection A, we let D A denote the associated Dirac operator on S. Fix a reference spin c -connection A 0 . Then we may write A = A 0 + a for some a ∈ iΩ 1 (Y ). It follows that Since b 1 (Y ) = 0, it follows that L admits a flat connection. We will assume that A 0 defines a flat connection on L.
We define the configuration space of Y to be C(Y ) depends on g and s but we omit this from the notation. C(Y ) is an affine space modelled on iΩ 1 (Y ) ⊕ Γ(S). In particular, the tangent space This defines a (constant) Riemannian metric on C(Y ). We will need to work with Sobolev completions. Given a flat reference spin c -connection A 0 , Sobolev norms are defined using A 0 and g. Fix an integer k ≥ 4. Later we will work with the L 2 k+1 -completion of C(Y ) and L 2 k+2 -gauge transformations. Having fixed a reference connection A 0 , we identify C(Y ) with iΩ 1 (Y ) ⊕ Γ(S). Thus an element (A, φ) ∈ C(Y ) will be identified with (a, φ) ∈ iΩ 1 (Y ) ⊕ Γ(S), where A = A 0 + a. To simplify notation, we will write D a in place of D A0+a .
The Chern-Simons-Dirac functional L : It follows that L is gauge invariant and we can regard L as a function on the quotient space C(Y )/G. The goal of Seiberg-Witten-Floer theory is to construct some sensible notion of Morse homology of L on C(Y )/G. Consider the formal L 2 -gradient of L. By this, we mean the function grad(L) : for all (a, φ) ∈ C(Y ) and all (a ′ , φ ′ ) ∈ iΩ 1 (Y ) ⊕ Γ(S). A short calculation gives These are the 3-dimensional Seiberg-Witten equations.
A trajectory for the downwards gradient flow is a differentiable map x : .
A key observation is that such trajectories can be re-interpreted as solutions of the 4-dimensional Seiberg-Witten equations on the cylinder X = R × Y . Definition 2.1. A Seiberg-Witten trajectory x(t) = (a(t), φ(t)) is said to be of finite type if both L(x(t)) and ||φ(t)|| C 0 are bounded functions of t.

2.2.
Restriction to the global Coulomb slice. Define the global Coulomb slice (with respect to A 0 ) to be the subspace Given (a, φ) ∈ C(Y ), there exists an element of V which is gauge equivalent to (a, φ), namely where d * (a − df ) = 0, so ∆f = d * (a). If we impose the condition Y f dvol Y = 0, then there is a unique solution to these equations given by f = Gd * a, where G is the Green's operator for the Laplacian ∆ = dd * on functions.
We have a globally defined map Π : Restricting to the global Coulomb slice V uses up all of the gauge symmetry except for the S 1 subgroup of constant gauge transformations. Instead of working on C(Y ) with full gauge symmetry, we work on V with S 1 symmetry.
As b 1 (Y ) = 0, every map u : Y → S 1 can be written as u = e f for some f : Y → iR. Moreover, f is unique up to addition of an integer multiple of 2πi. We define G 0 to be the subgroup of gauge transformations of the form u = e f for some f : Y → iR with Y f dvol = 0. It is easy to see that G = G 0 × S 1 .
We have that G 0 acts freely on C(Y ) and the quotient space can be identified with V . This determines a metricg on V as follows. Take the restriction of the L 2 -metric on C(Y ) to the subbundle of the tangent bundle orthogonal to the gauge orbits. This construction is G 0 -invariant and descends to a metricg on V .
The Chern-Simons-Dirac functional L is gauge invariant, hence the gradient grad(L) is orthogonal to the gauge orbits. It follows that the projection of grad(L) to V coincides with taking the gradient of L| V with respect tog. So the trajectories of grad(L) on C(Y ) project to the trajectories of L| V , where the gradient of L| V is taken using the metricg. Thus the trajectories on V have the form for a function f : Y → iR. The function f is uniquely determined by the conditions that Y f dvol Y = 0 and that * da is the linear part and is given by the non-linear terms. Let χ denote the gradient of L| V with respect tog. Then χ = l + c extends to a map The map l is a linear Fredholm operator. Using Sobolev multiplication and an estimate on the unique solution to The flow lines of χ on V will be called Seiberg-Witten trajectories in the Coulomb gauge. We say that such a trajectory x(t) = (a(t), φ(t)) is of finite type if L(x(t)) and ||φ(t)|| C 0 are bounded independent of t. Clearly the finite type Seiberg-Witten trajectories in the Coulomb gauge are precisely the projection to V of the Seiberg-Witten trajectories in C(Y ) of finite type.
2.3. Finite dimensional approximation. Let V µ λ denote the direct sum of all eigenspaces of l in the range (λ, µ] and letp µ λ be the L 2 -orthogonal projection from V to V µ λ . Note that V µ λ is a finite dimensional subspace of V . For technical reasons we replace the projectionsp µ λ with smoothed out versions where ρ : R → R is smooth, non-negative, non-zero precisely on (0, 1) and R ρ(θ)dθ = 1. This is to make p µ λ vary continuously with µ and λ. The reason for doing this is to show that the Conley index is independent of the choices of µ, λ, up to a suspension. This is achieved by continuously increasing or decreasing µ, λ to get a continuous family of flows and using homotopy invariance of the Conley index under continuous deformation of the flow.
Consider the gradient flow equation This result will allow us to construct the Seiberg-Witten-Floer homotopy type of (Y, s) using Conley indices.
An isolated invariant set is a subset S ⊆ M such that S = Inv(N, ϕ) for some isolating neighbourhood. Note that S must be compact since it is a closed subset of N and N is required to be compact.
is not in N , then there exists τ with 0 ≤ τ < t with ϕ τ (x) ∈ L. • L is positively invariant in N , that is, if x ∈ L and t > 0 and such that ϕ s (x) ∈ N for all 0 ≤ s ≤ t, then ϕ s (x) ∈ L for all 0 ≤ s ≤ t. The Conley index is independent of the choice of index pair (N, L) in a strong way. Namely for any two pairs (N 1 , L 1 ), (N 2 , L 2 ), there is a canonical homotopy equivalence N 1 /L 1 ∼ = N 2 /L 2 . The composition of two such canonical homotopy equivalences N 1 /L 1 ∼ = N 2 /L 2 and N 2 /L 2 ∼ = N 3 /L 3 coincides up to homotopy with the canonical homotopy equivalence N 1 /L 1 ∼ = N 3 /L 3 (one says that the collection of Conley indices N/L forms a connected simple system). By abuse of terminology, if (N, L) is an index pair for S we say that I = N/L is "the" Conley index of S.
The negative gradient of f using the Euclidean metric is It follows that the downwards gradient flow is given by ϕ t (x) = (e t x 1 , . . . , e t x p , e −t x p+1 , . . . , e −t x n ).
Let S = {0} be the critical point. This is an isolated invariant set. In fact, the only invariant point of ϕ is the origin, so we could take N = D p × D n−p as an isolating neighbourhood (where D j is the closed j-dimensional unit disc). Then L = S p−1 × D n−p is an exit set for N . It is easy to see that (N, L) satisfies the condition for an index pair for S. The Conley index is I(S) = D p × D n−p /(S p−1 × D n−p ), which is homotopy equivalent to S p , a p-dimensional sphere. It can be shown that this is well-defined, up to G-equivariant homotopy equivalence. Moreover I G (S) has the based homotopy type of a finite G-CW complex.
Example 2.6. Consider again the example of the Morse function on R n given by Now suppose that G is a compact Lie group which acts linearly on R n preserving f . Note that f defines an O(n − p, p)-structure on R n and the fact that G preserves f just means that the action of G on R n factors through a homomorphism G → O(n − p, p). As G is compact, we may as well assume (after a linear change of coordinates) that G maps to the maximal compact subgroup O(n − p) × O(p). So we can decompose R n as where V + , V − are real orthogonal representations of G of dimensions n − p and p respectively. Once again, take S = {0} as our isolated invariant set. As our Conley index, we can take N = D(V − ) × D(V + ) and L = S(V − ) × D(V + ), hence Proof. First note that A H is compact because A is compact. Moreover Let M be a topological space and let P, Q ⊆ M be subspaces. Give Q the induced topology. Then Applying this to P = A, Q = M H , we get Then since S ⊆ int M (A) by the assumption that A is an isolating neighbourhood, it follows that We verify that L H is an exit set for N H . Let x ∈ N H and suppose ϕ t (x) / ∈ N H for some t > 0. Then it follows that Hence L H is an exit set for N H .
We check that L H is positively invariant in N H . Suppose x ∈ L H and there exists a t > 0 for which ϕ s (x) ∈ N H for all s ∈ [0, t]. Then since L is positively invariant in N , it follows that ϕ s (x) ∈ L for all s ∈ [0, t]. Hence ϕ s (x) ∈ L ∩ M H = L H for all s ∈ [0, t].
We have verified that (N H , L H ) is an index pair for S H . Moreover it is straightforward to check that (N/L) H = N H /L H .

Equivariant Spanier-Whitehead category.
In this section we recall the construction of the category C from [48], which is an S 1 -equivariant version of the Spanier-Whitehead category. In Section 3.3 we will modify this construction to accommodate a finite group action on Y .
We work with pointed topological spaces with a basepoint preserving action of S 1 . The objects of C are triples (X, m, n), where X is a pointed topological space with S 1 -action, and m, n ∈ Z. 1 We further require that X has the S 1 -homotopy type of an S 1 -CW complex, which holds for Conley indices on manifolds. The set of morphisms between two objects (X, m, n) to (X ′ , m ′ , n ′ ) will be denoted by {(X, m, n), (X ′ , m ′ , n ′ )} S 1 and is defined to be where [ , ] S 1 denotes the set of S 1 -equivariant homotopy classes and the colimit is taken over all k, l such that k ≥ m ′ − m and l ≥ n ′ − n. The maps that define the colimit are given by suspensions where we smash on the left and for any topological space Z, we let Z + denote the one-point compactification with its obvious basepoint.
Any pointed space X with S 1 -action defines an object of C, namely (X, 0, 0). We often simply write this as X. For any finite dimensional representation E of S 1 , we let Σ E denote the reduced suspension operation This operation extends to C by taking Σ E (X, m, n) = (Σ E X, m, n). We are mainly interested in the case that E is a real vector space with trivial S 1 -action, or E is a complex vector space with S 1 acting by scalar multiplication. If E is a real vector space with trivial action, then one finds that The isomorphism depends on a choice of isomorphism E ∼ = R dim R (E) . Up to homotopy there are two choices since GL(E, R) has two components. If E is a complex vector space and S 1 acts by scalar multiplication, then Σ E (X, m, n) ∼ = (X, m, n − dim C (E)).
The isomorphism is unique up to homotopy as GL(E, C) is connected. We can define desuspension by a real vector space E with trivial S 1 -action as follows: Then Σ −E Σ E Z ∼ = Z by an isomorphism which is canonical up to homotopy. We can define desuspension by a complex vector space E with S 1 acting by scalar multiplication by: Then Σ −E Σ E Z ∼ = Z by an isomorphism which is canonical up to homotopy. 1 In [48] n is allowed to take on rational values. This is needed to construct a Seiberg-Witten-Floer spectrum which does not depend on the choice of metric. For our purposes it suffices to consider only integral values of n.
For Z = (X, m, n) ∈ C, we define the reduced equivariant cohomology of Z to be The cohomology is well defined as a consequence of the Thom isomorphism.
2.6. Seiberg-Witten-Floer cohomology. Consider as before a rational homology 3-sphere Y and a spin c -structure s. Let R > 0, λ, µ be as in Proposition 2.2. We want to take the Conley index of the set of all critical points in B(R) and flow lines between them which lie in B(R) for all time for the approximate Seiberg-Witten flow l + p µ λ c. The problem is that there could be trajectories that go to infinity in a finite amount of time. Hence we do not have a flow {ϕ t } in the sense of a 1-parameter group of diffeomorphisms. To get around this issue, let u µ λ be a compactly supported smooth cutoff function which is identically 1 on B(3R). For consistency purposes we assume that u µ where ρ is smooth, compactly supported and ρ(t) = 1 for t < 3.
For each µ, λ, the vector field u µ λ (l + p µ λ c) is compactly supported, so it generates a well-defined flow ϕ µ λ,t on V µ λ . Since u µ λ = 1 on B(2R), Proposition 2.2 still applies to the trajectories of u µ λ (l + p µ λ c). It follows that where S µ λ is the set of critical points and flow lines between critical points for the approximate Seiberg-Witten flow l + p µ λ c which lie in B(R). Therefore S µ λ is an isolated invariant set. Moreover S 1 preserves the approximate flow, hence we may take the S 1 -equivariant Conley index This is an S 1 -equivariant homotopy type. However it is not quite an invariant of (Y, s) because it depends on the choice of metric g as well as the values of λ, µ, R. Note that it is independent of the choice of u µ λ because of the assumption that u µ λ = 1 on B(3R). To get a genuine invariant we must understand how I µ λ changes as we vary these parameters.
We use the following invariance property of the Conley index: suppose we have a family {ϕ t (s)} of flows depending continuously on s ∈ [0, 1]. Suppose that a fixed compact set A is an isolating neighbourhood for all s ∈ [0, 1] and let S(s) = Inv(A, ϕ t (s)). Then I(S 0 , ϕ t (0)) ∼ = I(S 1 , ϕ t (1)) by a canonical homotopy equivalence.
Consider increasing µ to µ ′ . The finite energy trajectories of l + p µ But ϕ µ is easily seen to be homotopic to the product of the flow of u µ The Conley index of a product of flows is just the smash product of Conley indices. Combined with Example 2.6, we see that where W − is the part of W spanned by negative eigenvalues of l. But W is contained in the positive eigenvalues of l, so W − = 0 and hence . Now consider decreasing λ to λ ′ . An identical argument to the one above gives does not depend on the values of µ, λ (provided µ, −λ are sufficiently large).
We have established that the homotopy type of SW F (Y, s, g) does not depend on the choices of µ and λ, or more precisely, any two choices of µ, λ are related by a canonical homotopy equivalence. One also checks that it does not depend on the choice of R. So up to homotopy, SW F (Y, s, g) depends only on Y, s and g.
Next we consider varying the metric g. Consider a smooth homotopy g s , s ∈ [0, 1] joining two metrics g 0 , g 1 , which is constant near s = 0. Assuming that the g s are all sufficiently close to each other in a suitable topology, we can arrange that there exists R, µ * , λ * such that Proposition 2.2 is true for all s ∈ [0, 1] and all µ, λ with µ ≥ µ * , λ ≤ λ * . This suffices, as compactness of [0, 1] implies that any smooth path g s can be broken up into finitely many sub-paths over which this assumption holds.
We assume that there exists some λ < λ * and µ > µ * such that λ, µ are not eigenvalues of l s for any s ∈ [0, 1]. This property will hold for all sufficiently small paths. The spaces (V µ λ ) s then form a smooth vector bundle over [0, 1]. We can trivialise this vector bundle and identify all these spaces with a single V µ λ . Further, we assume that B(R) s1 ⊂ B(2R) s2 for each s 1 , s 2 ∈ [0, 1]. Here we think of the balls as subsets of the same space V µ λ . Once again, this property will hold for all small enough paths. Then is a compact isolating neighbourhood for S µ λ in any metric g s with the flow (ϕ µ λ ) s . The Conley index will be independent of s and hence The reason is that some eigenvalues in (λ, µ) may change sign. On the other hand, any eigenvalue greater than µ or less than λ can not change sign, by our assumption that µ, λ are not eigenvalues of l s for any s ∈ [0, 1]. Hence the difference between (V 0 λ ) 0 and (V 0 λ ) 1 is given in terms of the spectral flow of the family of operators {l s }, s ∈ [0, 1]. The operator l can be split into real and complex components. The real part has no spectral flow, so we only need to consider the complex part, which is the Dirac operator D s . The spectral flow can be expressed using the Atiyah-Patodi-Singer (APS) index theorem on the cylinder X = [0, 1] × Y , see [8,9]. Let g be the metric on X given by g s in the vertical direction and (ds) 2 in the horizontal direction. Let S s denote the spinor bundle associated to (s, g s ). The bundles S s can all be identified with S = S 0 , but with varying Clifford multiplication. The spin c structure s lifts to a spin c structure on X. Let S ± denote the spinor bundles of this spin c structure. Then S ± can be identified with the pullback of S to X. Suppose for each s we have chosen a flat reference connection A s . Since we have identified S s with S for all s, we get an induced identification of L s = det(S s ) with L = L 0 . Then A s = A 0 + iα s for some closed real 1-form α s . The path of spin c connections {A s } fit together to form a spin c connection A on the determinant line L pulled back to X. Let D be the Dirac operator determined by g and A. Then D(ψ) = ∂ s ψ + D s ψ. After a possible reparametrisation we can assume that (g s , A s ) is constant near the boundary. Applying the APS index theorem to the Dirac operator D on the cylinder [0, 1] × Y , one can write the spectral flow SF ({D s }) as Let η sign (g s ) denote the eta invariant of the signature operator on Y defined by g s . Then from the APS index theorem for the signature operator together with the fact that the signature operator has no spectral flow, we find Combining Equations (2.1) and (2.6), we see that We will show that n(Y, s, g) is a rational number. Let (W, s W ) be a spin c 4-manifold bounding (Y, s). This always exists because Ω spin c 3 = 0. Extend the Dirac operator D on Y to a Dirac operator D in the same way as we did for the cylinder [0, 1] × Y . The APS index theorem for the Dirac operator and signature operator on W combined give This shows that n(Y, s, g) is a rational number since ind AP S ( D) is an integer and δ(W, s) is a rational number.
Definition 2.9. The Seiberg-Witten-Floer cohomology of (Y, s, g) is defined as where j ∈ Q and as usual the coefficient group F has been omitted from the notation.
Below we will show that HSW * (Y, s) is independent of the choice of metric g (and other auxiliary choices), hence it is a well defined topological invariant of the pair (Y, s).
Notice that because of the grading shift by 2n(Y, s, g) the cohomology groups HSW * (Y, s) are concentrated in rational degrees. It was shown by Lidman-Manolescu [45] that HSW * (Y, s) is isomorphic to the Seiberg-Witten monopole Floer cohomology as defined by Kronheimer-Mrowka [38]. Together with the equivalence of monopole Floer homology and Heegaard Floer homology due to the work of Kutluhan-Lee-Taubes [39,40,41,42,43], Colin-Ghiggini-Honda [16,17,18] and Taubes [60], we have isomorphisms * denotes the minus version of Heegaard Floer homology and HF * + denotes the plus version of Heegaard Floer cohomology (taken with respect to the same coefficient group F). Here we use a grading convention for HF − such that HF − (S 3 ) starts in degree 0. Through the work of [20,32,57], the isomorphism is known to preserve the absolute gradings. Using co-Borel, Tate or non-equivariant cohomologies gives similar isomorphisms to the other versions of Heegaard-Floer homology, see [45,Corollary 1.2.4] for the precise statement.
If we have two metrics g 0 , g 1 , then the spectral flow of a path joining them satisfies On the other hand, from the definition of spectral flow we have Hence the Seiberg-Witten-Floer cohomology H j+2n(Y,s,g) S 1 (SW F (Y, s, g)) is independent of the metric. The above isomorphism is canonical in the sense that it does not depend on the choice of path from g 0 to g 1 . This follows from the fact that the space of all metrics on Y is contractible, so any two paths with the same endpoints are homotopic.
induces a non-equivariant duality between X H and (X ′ ) H , in the sense of nonequivariant Spanier-Whitehead duality.
Consider the Conley index I µ λ associated to (Y, s, g) for suitably chosen R, µ, λ. One finds that reversing orientation of Y has the effect of reversing the Chern-Simons-Dirac flow. From [19], it follows that I µ λ (Y ) and I λ µ (Y ) are V µ λ -dual, so there exists a duality map is the dimension of the kernel of D. Desuspending, we obtain a duality map We also have (2.4) n(Y, s, g) + n(Y , s, g) = −k(D).

Fixed points.
Definition 2.11. Let s ≥ 0 be an integer. We say that a finite pointed • The action of S 1 is free on the complement X − X S 1 .
where c is the non-linear part of the Seiberg-Witten flow. Thus the restriction of the approximate , hence S 1 acts freely on N − N S 1 and therefore also on (I µ λ ) − (I µ λ ) S 1 . This proves the result.
Using the identities we see that: • If X is of type SWF at level s, then R + ∧ X is of type SWF at level s + 1, • If X is of type SWF at level s, then C + ∧ X is of type SWF at level s. Now let Z = (X, m, n) belong to the equivariant Spanier-Whitehead category C. We say that Z is of type SWF at level s if X is of type SWF of level s + m. The above remarks shows that this is a well-defined notion.
We have shown that the Let X be a space of type SWF at level s. Let ι : X S 1 → X denote the inclusion of the fixed point set. Using the localisation theorem in equivariant cohomology [22,III (3.8)], it follows that the pullback map ι * : is not identically zero. Therefore, we may define the d-invariant d(X) of X by Note that d(X) could potentially depend on the choice of coefficient group, so we may write the invariant as d(X; F) if we wish to indicate the dependence on F. We also define the δ-invariant of X by δ(X) = d(X)/2. Using Equation (2.5) and the Thom isomorphism, one finds: For notational convenience we also define δ(Y, s) = d(Y, s)/2.

Equivariant Seiberg-Witten-Floer cohomology
3.1. Assumption on G and F. Throughout this paper we will assume that one of the two following conditions hold: (1) G is an arbitrary finite group and F = Z/2Z, or (2) F is an arbitrary field and the order of G is odd.
Condition (1) ensures that we do not need to concern ourselves with questions of orientability. Condition (2) ensures that any S 1 -central extension G acts orientation preservingly on all of its finite dimensional representations. Hence under either condition, the Thom isomorphism holds without requiring local coefficients: Here X is any G-space and V is any finite dimensional representation of G.

Lifting G-actions.
Recall that Y denotes a rational homology 3-sphere. Suppose that a finite group G acts on Y by orientation preserving diffeomorphisms and suppose G preserves the isomorphism class of a spin c -structure s. We will construct a G-equivariant version of the Seiberg-Witten-Floer cohomology of (Y, s). Choose a G-invariant metric g on Y and a reference spin c -connection A 0 such that the connection on the determinant line L is flat. Let g ∈ G and choose a liftĝ : S → S of g to the spinor bundle S, which is possible since G preserves the isomorphism class of s. Thenĝ −1 A 0ĝ = A 0 + a for some a ∈ iΩ 1 (Y ). Since A 0 and g −1 A 0ĝ are flat, we must have da = 0. Moreover, b 1 (Y ) = 0 implies that a = df for some f : Y → iR. Setting g = e −fĝ , it follows that g is a lift of g which preserves A 0 . Any other lift of g that preserves A 0 is of the form c g with c ∈ U (1) a constant. Let G s denote the set of all possible lifts of elements of G which preserve A 0 . Then G s is a group and we have a central extension Now we carry out the construction of the Conley index of a finite-dimensional approximation of the Chern-Simons-Dirac flow G s -equivariantly, instead of just S 1 -equivariantly.
3.3. G s -equivariant Spanier-Whitehead category. In this section G denotes any S 1 central extension of G. We will construct a category C( G), the G-equivariant version of C.
Recall from Section 2.5 that the category C was constructed so that there exists a desuspension functor Σ −V for any real vector space V with trivial S 1 -action or any complex vector space where S 1 acts by scalar multiplication. We now construct a category C( G) in which we can desuspend by real representations of G, where S 1 acts trivially, and by complex representations, where S 1 acts by scalar multiplication. We are lead to consider the following two types of finite dimensional representations of G: • Type (1): V is a real representation of G and S 1 acts trivially.
• Type (2): V is a complex representation G and S 1 acts on V by scalar multiplication.
Type (1) representations correspond canonically to real representations of G.
Type (2) representations correspond to projective unitary representations of G such that the pullback to G of the central extension S 1 → U (n) → P U (n) gives an extension isomorphic to G. If G is split, then type (2) representations are in bijection with complex representations of G. However, the bijection depends on a choice of splitting of G.
To define stable homotopy groups we need to consider suspensions with explicitly chosen representations. In other words, we need to work at the level of representations and not just isomorphism classes. Let V 1 , . . . , V p be a complete set of irreducible representations of type (1), and W 1 , . . . , W q a complete set of irreducible representations of type (2). Any representation of type (1) is isomorphic to a direct sum of copies of V 1 , . . . , V p and likewise any representation of type (2) is a direct sum of copies of W 1 , . . . , W q .
Similarly, if n = (n 1 , . . . , n q ) ∈ Z q satisfies n ≥ 0, then we set The category C( G) has as objects triples (X, m, n) where • X is a pointed topological space with a basepoint preserving G-action and the homotopy type of a G-CW complex.
Let (X, m, n), (X ′ , m ′ , n ′ ) be two objects of C ( G). The set of morphisms from The colimit is taken over all k ∈ N p , l ∈ N q such that k ≥ m ′ − m and l ≥ n ′ − n. The maps that define the colimit are given by suspensions where we smash on the left.
Let Y be any pointed G-space. We obtain a functor Y ∧ : C( G) → C( G) which is defined on objects by Y ∧ (X, m, n) = (Y ∧ X, m, n) and on morphisms in the evident way. In particular, if V is any finite dimensional representation of G, we define the reduced suspension We define desuspension by a representation V of type (1) as follows: where the isomorphism is canonical up to homotopy. For any representation W of type (2)  where |m| and |n| are defined as follows: The cohomology is well defined as a consequence of the Thom isomorphism.

G-equivariant
Seiberg-Witten-Floer cohomology. Let Y be a rational homology 3-sphere and G a finite group acting on Y preserving the isomorphism class of a spin c -structure s. Let G s be the S 1 -central extension of G obtained by lifting G to the spinor bundle corresponding to s. We repeat the construction of the Conley index I µ λ (g) from Section 2.6, except that now we carry out the construction G s -equivariantly. Restricting to the subgroup S 1 ⊆ G s , I µ λ (g) agrees with the S 1equivariant Conley index as previously constructed.
We need to understand how I µ λ (g) depends on µ, λ, the choice of G-invariant metric g and the constant R. As in the S 1 case, first consider variations of µ, λ. Carrying out a similar argument but G s -equivariantly, we see that I µ λ (g) simply changes by suspension. Analogous to the non-equivariant case we define is defined as before, but now carries a G s -action. Note that V 0 λ (g) is the sum of a representation of type (1) and a representation of type (2), so the desuspension Σ −V 0 λ (g) is defined. Then up to canonical isomorphisms SW F (Y, s, g) depends only on the triple (Y, s, g).
We consider the dependence of SW F (Y, s, g) on the metric g. The argument is much the same as before except done G s -equivariantly. Let g 0 , g 1 be two Ginvariant metrics. The space of such metrics is contractible, so we may choose a path {g s } from g 0 to g 1 . Then as in the non-equivariant case, the signature operator has no spectral flow and we have (SW F (Y, s, g)).
By the argument above, the HSW * G (Y, s) depends only on (Y, s) and the Gaction.
For a group K we write H * K for H * K (pt). Since HSW * (Y, s) is defined using equivariant cohomology, it is a graded module over the ring H * Restricting from G s to S 1 , we obtain forgetful maps compatible with the module structures.
Observe that since S 1 is the identity component of G s , the action of G s on HSW * (Y, s) descends to an action of G. So we may regard HSW * (Y, s) as a G-module.
Theorem 3.2. There is a spectral sequence E p,q r abutting to HSW * G (Y, s) whose second page is given by Proof. For a G s -space M , let M Gs denote the Borel model for the G s -action and M S 1 the Borel model for the S 1 -action obtained by restriction. The composition M Gs → BG s → BG is a fibration with fibre M S 1 . Applying the Leray-Serre spectral sequence, we get a spectral sequence which abuts to H * Gs (M ) and has E p,q 2 = H p (BG; H q S 1 (M )). More generally if M is the formal desuspension of a G sspace, then via an application of the Thom isomorphism a similar spectral sequence exists. Applying this to HSW * G (Y, s) gives the theorem. Definition 3.3. Let Y be a rational homology 3-sphere and s a spin c -structure. We say that Y is an L-space (with respect to s and F) if the action of U on HSW * (Y, s) is injective. Equivalently HSW * (Y, s) is a free F[U ]-module of rank 1. . We stress that these isomorphisms depend on the choice of splitting.
Theorem 3.5. Suppose that G s is a split extension. If Y is an L-space (with respect to s and F), then the spectral sequence given in Theorem 3.2 degenerates at E 2 . Moreover we have Proof. If Y is an L-space (with respect to s and F) then where θ has degree d(Y, s). We claim that G acts trivially on HSW * (Y, s). This can be seen as follows. First, since HSW * (Y, s) is up to a degree shift the S 1equivariant cohomology of the Conley index I = I µ λ , it suffices to prove the result for I. Let ι : I S 1 → I be the inclusion of the S 1 fixed point set. Since Y is an L-space, U acts injectively on HSW * (Y, s). Together with the localisation theorem in equivariant cohomology this implies that ι * is injective. Hence it suffices to show that G acts trivially on H * S 1 (I S 1 ). But I S 1 has the homotopy type of a sphere, so if ν is a generator of H * S 1 (I S 1 ) and g ∈ G, then g * (ν) = ±ν according to whether or not g acts orientation preservingly. Our assumptions on G and F (see Section 3.1) ensures that g * (ν) = ν for all g ∈ G. This proves the claim.
Letting E p,q r denote the spectral sequence for HSW * G (Y, s), it follows easily that 3.5. Spaces of type G-SWF. We introduce a G-equivariant analogue of spaces of type SWF. We then define a G-equivariant analogue of the d-invariant.
Let G be an extension of G by S 1 . If G acts on a space X, then we get an induced action of G = G/S 1 on the fixed point set X S 1 . We write G = S 1 × G for the trivial extension of G.
Definition 3.6. Let s ≥ 0 be an integer. We say that a finite pointed G-CW complex X is of type G-SWF at level s if where V is a real representation of G of dimension s. • The action of S 1 is free on the complement X − X S 1 .
More generally, let V be a finite dimensional representation which is the direct sum of representations of type (1) and (2).
Assume that G is split and choose a splitting G ∼ = G. Let X be a space of type G-SWF at level s. Let ι : X S 1 → X denote the inclusion of the fixed point set. Recall that H * . The localisation theorem in equivariant cohomology implies that τ where deg(τ ) = s. Therefore it also follows that Then for each c ∈ H * G , it follows that there exists an x ∈ H * G (X) for which ι * (x) = c U k τ , for some k ≥ 0. Set Λ G (X) = H * G (X S 1 ). Then Λ G (X) is a free H * G [U ]module of rank 1 and ι : X S 1 → X induces a map This is the filtration induced by the fibration Let τ denote the generator of Λ G (X). Then for j ≥ 0 we have obvious identifications τ. Now let c be a non-zero element in H * G of degree |c| = deg(c). By the discussion above we know that c U k τ is in the image of ι * for some k ≥ 0. Hence we may define: Note that d G,ac (X) = d G,c (X) for any a ∈ F * . In concrete terms, the condition that ι * (x) ∈ F |c| and ι * (x) = c U k τ (mod F |c|+1 ) means that ι * (x) is of the form Remark 3.8. Let X be a space of type G-SWF. The definition of d G,c (X) does not depend on a choice of splitting of S 1 → G → G. Indeed, two splittings differ by a homomorphism φ : The change of splitting acts on H * G [U ] by sending U to U + α. Then since (U + α) k = U k + · · · , where · · · denotes terms involving lower powers of U , it follows that d G,c (X) does not depend on the choice of splitting of G. Proof. Let s be the level of X. First consider the case that c 1 , c 2 are homogeneous, that is, for some |c 1 |, |c 2 |. Then by Definition 3.7, we have that there exist x 1 ∈ H dG,c 1 (X)+|c1| G (X) and where + · · · denotes terms that are in the next stage of the filtration and k i = (d G,ci (X) − s)/2 for i = 1, 2. Note that if c 1 or c 2 are zero then we take x 1 or x 2 to be zero. If |c 1 | = |c 2 |, then by Definition 3.7, we have d G,c1+c2 (X) = max{d G,c1 (X), d G,c2 (X)}. Now suppose that |c 1 | = |c 2 |. Let k = max{k 1 , k 2 } and set and hence, from the definition of d G,c1+c2 (X), we have d G,c1+c2 (X) ≤ 2k + s = max{2k 1 + s, 2k 2 + s} = max{d G,c1 (X), d G,c2 (X)}.
Recall that the ordinary (non-equivariant) d-invariant of X, d(X), is defined by d(X) = min{j |∃x ∈ H j S 1 (X) ι * (x) = 0}. It is not hard to see that d(X) = d {e},1 (X), where {e} denotes the trivial group and 1 is the generator of H 0 (pt). Proposition 3.10. Let X be a space of type G-SWF for the trivial extension. Then Proof. By the definition of d G,1 (X), there exists x ∈ H dG,1(X) G (X) such that ι * (x) = U k τ + · · · , where k = (d G,1 (X) − s)/2 and s is the level of X. Let y ∈ H dG,1(X) S 1 (X) be the image of x under the map induced by S 1 → G. Then it follows that ι * (y) = U k τ ∈ H dG,1(X) S 1 (X S 1 ). In particular, ι * (y) = 0 and hence d G,1 (X) ≥ d(X) by the definition of d(X).
Let S 1 act trivially on R and act by scalar multiplication on C. Let V be a real representation of G. Then V R = R ⊗ R V and V C = C ⊗ R V may be regarded as representations of G = S 1 × G, where S 1 acts on the first factor and G on the second. Proposition 3.11. Let X be a space of type G-SWF for the trivial extension and let V be a finite dimensional representation of G of type (1) or (2), as in Section 3.3. Then for any c ∈ H * G , we have Proof. This result follows easily from the Thom isomorphism, together with the fact that in the type (2) case, the G-equivariant Euler class of V has the form This definition is well-defined by Proposition 3.11. We also define a corresponding δ-invariant by setting δ G,c (Z) = d G,c (Z)/2. Definition 3.12. Let X, Y be spaces of type G-SWF for the trivial extension of G, where X has level s and Y has level t. Let f : X → Y be an S 1 × G-equivariant map. Consider the restriction f S 1 : X S 1 → Y S 1 of f to the fixed point set. Note that H * G (X S 1 ) is a free H * G -module starting in degree s. Let τ X S 1 denote a generator. Then τ X S 1 is unique up to an element of F * . Similarly H * G (Y S 1 ) is a free H * G -module starting in degree t and we let τ Y S 1 denote a generator. Then there exists a uniquely determined µ ∈ H t−s G such that , then deg(f S 1 ) changes by an element of F * , hence deg(f S 1 ) is well-defined up to multiplication by elements of F * . If t < s, then deg(f S 1 ) = 0.
Note that suspension does not change the degree of f S 1 . Hence we can more generally speak of the degree of f S 1 when f is a stable map between spectra of type G-SWF.
Proof. We prove the result when c ∈ H |c| G is homogeneous. The general case follows easily from this. The inclusion of the fixed points sets gives a commutative diagram.
Consider the induced commutative diagram in equivariant cohomology: Then by commutativity of the diagram, we have It follows that 3.6. Alternative characterisation of d G,c . In this section we will give an alternative characterisation of d G,c which does not directly refer to ι * and is sometimes more convenient for computations. Let X be a space of type G-SWF for the trivial extension G. Set Λ * G = H * G (X S 1 ). The inclusion of the fixed points ι : Similarly there is a filtration on H * G (X) which comes from the spectral sequence for equivariant cohomology. We will denote this filtration by F j . Then ι * (F j ) ⊆ F j because the inclusion ι induces a map between spectral sequences.
Let c ∈ H * G be a non-zero element of degree |c|. Recall that the invariant d G,c (X) is defined by The localisation theorem in equivariant cohomology implies that upon localising with respect to U , ι * becomes an isomorphism: In particular, there exists an element θ ∈ H 2k+deg(τ ) G (X) such that ι * (θ) = U l τ for some l ≥ 0. Fix a choice of such a θ. The localisation isomorphism implies that ι * (x) = 0 if and only if U k x = 0 for some k ≥ 0.
Proposition 3.14. Let c ∈ H * G be a non-zero element of degree |c|. Then Then we need to show that d G,c (X) = a G,c (X). Suppose x ∈ H aG,c(X)+|c| G (X) satisfies U n x = cU k θ (mod F |c|+1 ) for some n, k ≥ 0. Then Since U is injective on Λ G we must have k + l ≥ n and we can cancel U n from both sides to get ι * (x) = cU k+l−n τ (mod F |c|+1 ).
Next recall that ι * is an isomorphism after localising with respect to U . Hence if ι * (y 1 ) = ι * (y 2 ), then U n y 1 = U n y 2 for some n ≥ 0 and we have From the definition of a G,c (X), it follows that a G,c

3.7.
Equivariant d-invariants for rational homology 3-spheres. We return to the setting that Y is a rational homology 3-sphere, G is a finite group acting on Y preserving the isomorphism class of a spin c -structure s. Choose a G-invariant metric g and let G s be the S 1 -central extension of G obtained by lifting G to the spinor bundle corresponding to s. Now suppose that G s is a trivial extension, hence G s ∼ = G. From the construction of the Conley index, one finds that SW F (Y, s, g) is of type G-SWF at level 0. We only define the invariants d G,c (Y, s) in the case that G s is a trivial extension. This is because the definition of d G,c (Y, s) uses localisation by U , but U ∈ H * S 1 does not necessarily extend to a class in H * Gs , unless G s is a trivial extension. The inclusion ι : (V 0 λ (R)) + → I µ λ of the S 1 -fixed points of the Conley index desuspends to a map ι : . This is a free H * G [U ]module and we let τ denote a generator. As in Section 3.5 we filter Λ s). The construction of ι * and Λ * G (Y, s) depend on the choice of metric g, but the construction for any two metrics are related by a canonical homomorphism. The d-invariants of (Y, s) are given by Recall that d(Y , s) = −d(Y, s). On the other hand, the behaviour of the invariants d G,c (Y, s) under orientation reversal is not so straightforward. For example, it follows from Proposition 3.10 that In particular, , we also get that We will show in Theorem 4.4 that the invariants d G,c satisfy a stronger positivity condition. Proof. If Y is an L-space (with respect to s and F) then where θ has degree d(Y, s). From Theorem 3.5, there exists a class θ ∈ HSW * G (Y, s) which maps to θ under the forgetful map HSW * G (Y, s) → HSW * (Y, s) and we have that HSW * G (Y, s) is a free H * G [U ]-module generated by θ. We must also have that ι * ( θ) = U k τ (mod F 1 ) for some k ≥ 0, where τ is a generator of Λ G (Y, s).

Behaviour under cobordisms
We show that equivariant cobordisms of rational homology 3-spheres induce maps on equivariant Seiberg-Witten-Floer cohomology. We follow the construction of Manolescu [48], incorporating the corrections due to Khandhawit [36]. Since our construction is a straightforward extension of that of Manolescu and Khandhawit, differing only in the replacement of S 1 by the larger group G s , we will be brief.

4.1.
Finite dimensional approximation. Let W be a compact, oriented smooth 4-manifold with boundary Y = ∂W a disjoint union of rational homology spheres Y = ∪ j Y j . Assume further that b 1 (W ) = 0 and that W is connected. If s is a spin c -structure on W , then the restriction of s to Y determines a spin c -structure s| Y on Y . Since the boundary of W is a union of rational homology 3-spheres, we have H 2 (W, ∂W ; R) ∼ = H 2 (W ; R) and by Poincaré-Lefschetz duality we obtain a non-degenerate intersection form on H 2 (W ; R). Given a metric g on W which is isometric to a product metric in a collar neighbourhood of ∂W , we let H + (W ) denote the space of self-dual L 2 -harmonic 2-forms on the cylindrical end manifold W obtained from W by attaching half-infinite cylinders [0, ∞) × Y to W . It follows from [8,Proposition 4.9] that the natural map H + (W ) → H 2 (W ; R) is injective and identifies H + (W ) with a maximal positive definite subspace of H 2 (W ; R).
Suppose now that G acts smoothly and orientation preservingly on W and that this action sends each connected component of ∂W to itself. Hence by restriction G acts on each Y i by orientation preserving diffeomorphisms. Assume further that G preserves the isomorphism class of a spin c -structure s on W . Set s i = s| Yi . Then the action of G on Y i preserves s i . Similar to Section 3.2 we obtain an S 1 -extension G s of G. Restricting to Y i , we obtain an isomorphism of extensions G s ∼ = G si . Hence if G s is split, then it follows that each of the extensions G si is also split. Moreover a splitting of G s determines corresponding splittings of each G si .
Choose a G-invariant metric g on W which is isometric to a product (−ǫ, 0] × Y in some equivariant collar neighbourhood of Y (see [33,Theorem 3.5] for existence of equivariant collar neighbourhoods). To see that such a metric exists, first choose a G-invariant metric g Y on Y . Then choose an arbitrary metric g ′ on W which equals (dt) 2 + g Y in some equivariant collar neighbourhood (−ǫ, 0] × Y . Then let g be obtained from g ′ by averaging over G. Let S ± denote the spinor bundles on W corresponding to s. We note here that under these assumptions G preserves the subspace H + (W ) of H 2 (W ; R) defined by g. Let Ω 1 g (W ) denote the space of 1-forms on W in double Coulomb gauge with respect to Y [36, Definition 1]. This space is easily seen to be preserved by the action of G on 1-forms. The double Coulomb gauge condition ensures that if a ∈ Ω 1 g (W ) and φ ∈ Γ(S + ), then (a, φ)| Yj lies in the global Coulomb slice corresponding to Y j . Let us temporarily assume that Y = ∂W is connected. Let A be a spin c -connection on W such that in a collar neighbourhood of Y it equals the pullback of A 0 . Using the same argument as in Section 3.2, we can assume that A is G s -invariant. Then using A as a reference connection, we obtain a map which may be thought of as the Seiberg-Witten equations on W together with boundary conditions: where p µ is the orthogonal projection from V to V µ −∞ . Taking a finite dimensional approximation as described in [48], [36], one obtains a map and L 0 is a Fredholm linear operator defined in [48,Section 9]. Since SW F (Y, s, g) = Σ −V 0 λ I µ λ , we can re-write the map Ψ µ,λ,U,U ′ as Taking the smash product with Ker(L 0 ) and using (4.1), we see that Ψ µ,λ,U,U ′ is stably equivalent to a map The real part of L 0 has zero kernel and cokernel isomorphic to H + (W ). The complex part of L 0 can be identified with the Dirac operator D A with Atiyah-Patodi-Singer (APS) boundary conditions. Thus where Ker AP S (D + A ), Coker AP S (D + A ) denote the kernel and cokernel of D + A with APS boundary conditions. Hence we obtain a G s -equivariant map Note that f is only a map in the stable sense. That is, f is a morphism in the category C(G s ).
Recall that the S 1 -fixed point set of I µ λ is V 0 λ (R) + . The inclusion (V 0 λ (R)) → I µ λ of the S 1 -fixed points desuspends to a map ι : S 0 → SW F (Y, s, g). By restricting to S 1 -fixed points we obtain a commutative diagram Using that the Seiberg-Witten equations reduce to linear equations on the S 1 -fixed point set, one finds that f So far we have restricted to the case that the boundary ∂W is connected. More generally if ∂W = ∪ j Y j is a union of rational homology 3-spheres then much the same construction applies. The Conley index I µ λ is now given by the smash product of the Conley indices of each component, hence f is now a map of the form We still have that the degree of f S 1 is e(H + (W )).
(2) If ∂W = Y 1 ∪ Y 2 has two connected components, then Proof. We will give the proof in the case that W is connected. The general case follows easily from this by applying the theorem to each component of W . To simplify notation we will write s instead of s| Y and write g instead of g| Y . In case (1), ∂W = Y is connected.
As in Section 4.1, choosing suitable metrics and reference connections we obtain a stable map such that the degree of f S 1 is e. Applying Proposition 3.13 to f , we obtain which simplifies to Combined with Equation (2.3) we get δ(W, s) ≤ δ c (Y, s). Next recall from Section 2.7 the duality map where k(D) = dim C (Ker(D)). By the definition of equivariant duality, we have that is a non-equivariant duality. It follows that ε S 1 has degree 1. Taking the map f , suspending by SW F (Y , s, g) and composing with ε, we obtain a stable map such that the degree of h S 1 is e. Applying Proposition 3.13 to h we obtain . Using Equations (2.3) and (2.4), we obtain δ(W, s) ≤ −δ G,ce (Y , s), or equivalently The proof of case (2) is similar. We start with the map s, g). Suspending by SW F (Y 1 , s, g) and applying the duality map corresponding to Y 1 we obtain a map h : is the dimension of the kernel of the Dirac operator on Y 1 . Applying Proposition 3.13 to this map and simplifying, we obtain the inequality δ G,ce (Y 1 , s| Y1 ) + δ(W, s) ≤ δ G,c (Y 2 , s| Y2 ). Definition 4.2. Let (Y 1 , s 1 ), (Y 2 , s 2 ) be rational homology 3-spheres equipped with spin c -structures. Suppose that G acts orientation preservingly on Y 1 , Y 2 and preserves the spin c -structures s 1 , s 2 . A G-equivariant rational homology cobordism from (Y 1 , s 1 ) to (Y 2 , s 2 ) is a rational homology cobordism W from Y 1 to Y 2 such that the G-action and spin c -structure s 1 ∪ s 2 on ∂W extend over W . We say that (Y 1 , s 1 ), (Y 2 , s 2 ) are G-equivariantly rational homology cobordant if there exists a G-equivariant rational homology cobordism from (Y 1 , s 1 ) to (Y 2 , s 2 ).
Similarly we define the notion of a G-equivariant integral homology cobordism and say that two integral homology 3-spheres Y 1 , Y 2 on which G acts are G-equivariantly integral homology cobordant if there is a G-equivariant integral homology cobordism from Y 1 to Y 2 . Note that since Y 1 , Y 2 are integral homology 3-spheres, they have unique spin c -structures which are automatically G-invariant and any G-equivariant integral homology cobordism from Y 1 to Y 2 has a unique spin c -structure which restricts on the boundary to the unique spin c -structures on Y 1 , Y 2 .
Theorem 4.4. Let Y be a rational homology 3-sphere, G a finite group acting on Y preserving orientation and the isomorphism class of a spin c -structure s and suppose that G s is a trivial extension. Then for any c 1 , Proof. The proof is similar to that of Theorem 4.1. Let W = [0, 1]×Y be the trivial cobordism from Y to itself. Choosing suitable metrics and reference connections we obtain a stable map s, g). Note that H + (W ) = {0} and hence e(H + (W )) = 1. Applying Proposition 3.13 to this map we see that for any c 1 , c 2 ∈ H * G with c 1 c 2 = 0, we have s, g)). From the definition of the δ-invariant it is clear that s, g)) ≤ δ G,c1 (SW F (Y , s, g))+δ G,c2 (SW F (Y, s, g)) and hence Hence we obtain 0 ≤ δ G,c1 (Y , s) + δ G,c2 (Y, s).

4.3.
Induced cobordism maps. In this section we show that equivariant cobordisms induce maps on equivariant Seiberg-Witten-Floer cohomology.
Theorem 4.5. Let W be a smooth, compact, oriented 4-manifold with boundary and with b 1 (W ) = 0. Suppose that G acts smoothly on W preserving the orientation and a spin c -structure s. Suppose that ∂W = Y 1 ∪ Y 2 where Y 1 , Y 2 are rational homology 3-spheres and set s i = s| Yi . Suppose G sends Y i to itself. Then there is a morphism of graded H * Gs -modules such that the following diagram commutes where the vertical arrows are the forgetful maps to non-equivariant Seiberg-Witten-Floer cohomology and SW (W, s) is the morphism of Seiberg-Witten-Floer cohomology groups induced by (W, s).
Proof. We give the proof in the case W is connected. The general case follows by a similar argument. As in the proof of Theorem 4.1, choosing suitable metrics and reference connections, we obtain a stable map where k(D 1 ) is the dimension of the kernel of the Dirac operator on Y 1 . The induced map in equivariant cohomology takes the form s 1 , g 1 )).
Using the Thom isomorphism, this is equivalent to s 1 , g 1 )).

Then since HSW
, we see that h * is equivalent to a map Since this is a map of equivariant cohomologies induced by an equivariant map of spaces, it follows that SW G (W, s) is a morphism of graded H * Gs -modules. Restricting to the subgroup S 1 → G s , we obtain the commutative diagram in the statement of the theorem. acts smoothly and orientation preservingly on a rational homology 3-sphere Y , preserving a spin c -structure s. The action of G is equivalent to giving an orientation preserving diffeomorphism τ : Y → Y such that τ p = id and τ * (s) = s. Choose a lift τ ′ ∈ G s of τ . Then (τ ′ ) p = ζ for some ζ ∈ S 1 . Replacing τ ′ by τ = ζ −1/p τ ′ , where ζ 1/p is a p-th root of ζ, we see that τ p = id. Hence G s is a trivial extension.

δ-invariants.
Definition 5.1. If p = 2, then for any integer j ≥ 0, we define d j (Y, s, τ, 2) = d Z2,Q j (Y, s). If p is odd, then for any integer j ≥ 0, we define d j (Y, s, τ, p) = d Zp,S j (Y, s). We also set δ j (Y, s, τ, p) = d j (Y, s, τ, p)/2. When p and τ are understood we will omit them from the notation and simply write d j (Y, s) and δ j (Y, s).
In the case p is odd, one may also consider the invariants d Zp,RS j (Y, s). For simplicity we will not consider these invariants.
Theorem 5.2. We have the following properties: Proof. (1) is a restatement of Proposition 3.10. (2) follows from Proposition 3.9 taking c 1 = Q j and c 2 = Q in the case p = 2 and c 1 = S j , c 2 = S in the case p is odd. (4) is a special case of Theorem 4.4. For (3), first note that the difference δ j (Y, s) − δ j+1 (Y, s) is always an integer because δ G,c (Y, s) + n(Y, s, g) ∈ Z for any metric g. From (2) and (4) and the fact that n(Y, s, g) + n(Y , s, g) ∈ Z, it follows that δ j (Y, s)+δ j (Y , s) is a non-negative, decreasing, integer-valued function. Hence the value of δ j (Y, s) + δ j (Y , s) must eventually be constant. Using (2) again, it follows that δ j (Y, s) and δ j (Y , s) are eventually constant. (5) is a restatement of Proposition 3.16.
Next, we specialise Theorem 4.1 to the case G = Z p . Theorem 5.3. Let W be a smooth, compact, oriented 4-manifold with boundary and with b 1 (W ) = 0. Suppose that τ : W → W is an orientation preserving diffeomorphism of order p and s a spin c -structure preserved by τ . Suppose each component of ∂W is a rational homology 3-sphere and that τ sends each component of ∂W to itself. Suppose that the subspace of H 2 (W ; R) fixed by τ is negative definite. Then for all j ≥ 0 we have (2) If ∂W = Y 1 ∪ Y 2 has two connected components, then Proof. Let H + (W ) denote a τ -invariant maximal positive definite subspace of H 2 (W ; R) (which always exits because G = τ is finite) and let e denote the image of the Euler class of H + (W ) in H * Zp . To deduce the result from Theorem 4.1, we just need to check that eQ j = 0 for all j ≥ 0 if p = 2 and eS j = 0 for all j ≥ 0 if p is odd.
In the case p = 2, e is the top Stiefel-Whitney class of H + (W ), which is easily seen to be Q b+(W ) because our assumption that the subspace of H 2 (W ; R) fixed by τ is negative definite implies that τ acts as −1 on H + (W ). Then clearly eQ j = 0 for all j ≥ 0. Now suppose p is odd. Let L i be the complex 1-dimensional representation on which τ acts as multiplication by ζ i , ζ = e 2πi/p . Any finite dimensional real representation of G is the direct sum of a trivial representation and copies of the underlying real representations of the L i for 1 ≤ i ≤ p − 1. The hypothesis that the subspace of H 2 (W ; R) fixed by τ is negative definite means that as a representation of G, H + (W ) contains no trivial summand. Hence H + (W ) admits a complex structure such that from which it is clear that eS j = 0 for all j ≥ 0.
Remark 5.4. Suppose that p is odd. Then as in the proof of Theorem 5.3, H + (V ) admits a complex structure. So if p is odd and the assumptions of Theorem 5.3 hold, then b + (W ) must be even.
To keep notation simple, we will henceforth set b ′ ± (W ) = b ± (W ) if p = 2 and b ′ ± (W ) = b ± (W )/2 if p is odd. Then (1) and (2) of Theorem 5.3 can be written more uniformly as: ). Corollary 5.5. Let W be a smooth, compact, oriented 4-manifold with boundary and with b 1 (W ) = 0. Suppose that τ : W → W is an orientation preserving diffeomorphism of order p and s a spin c -structure preserved by τ . Suppose that Y = ∂W is a rational homology 3-sphere. Suppose that the subspace of H 2 (W ; R) fixed by τ is zero. Then Proof. It suffices to prove (1) since (2)

5.2.
Some algebraic results. In this section we collect some algebraic results which will be useful for computing δ invariants. Let Y be a rational homology 3-sphere, τ : Y → Y an orientation preserving diffeomorphism of prime order p and s a spin c -structure preserved by τ . Take G = Z p = τ and F = Z p . Let {E p,q r , d r } r≥2 denote the spectral sequence relating equivariant and non-equivariant Seiberg-Witten-Floer cohomologies. So E p,q 2 = H p (Z p , HSW q (Y, s)) where Z p acts on HSW q (Y, s) via the action induced by τ . To simplify notation we will write H q for HSW q (Y, s) and d for d(Y, s). So E p,q 2 = H p (Z p , H q ). For fixed q, H q is a finite dimensional representation of Z p over F. Moreover, for all sufficiently large k, we have Recall that H * G is isomorphic to F[Q] for p = 2 and to F[R, S]/(R 2 ) for odd p. In the case p = 2 we will set S = Q 2 , so in all cases S ∈ H 2 G .
Lemma 5.6. If V is a finite dimensional representation of Z p over F = Z p , then S : is surjective for all i ≥ 0 and an isomorphism for all Proof. Since Z p acts freely on S 1 , it follows from [12, page 114] that there is an element ν ∈ H 2 (Z p ; Z) (independent of V ) such that the cup product ν : is an isomorphism for i > 0 and surjective for i = 0. Since V is a representation of Z p over F, the same statement holds if we replace ν by its image in H 2 (Z p ; F), which must have the form aS for some a ∈ F. Moreover a = 0 follows by considering the case that V = Z p is the trivial representation. Hence the cup product S : is an isomorphism for i > 0 and surjective for i = 0. We have by induction that and H 1 (Z p ; V ) can both be expressed as certain subquotients of V , it follows that Lemma 5.7. For each r ≥ 2, the map S : E p,q r → E p+2,q r is surjective for all p ≥ 0 and an isomorphism for all p ≥ r − 1.
is surjective for all p and an isomorphism for all p ≥ 1, by Lemma 5.6. This proves the case r = 2. Now we proceed by induction. Let r > 2 and suppose that S : E p,q r−1 → E p+2,q r−1 is surjective for all p ≥ 0 and an isomorphism for all p ≥ r − 1. Let x ∈ E p+2,q r . Then x = [y] for some y ∈ E p+2,q r−1 with d r−1 (y) = 0. By the inductive hypothesis y = Sz for some z ∈ E p,q r−1 . Then Lemma 5.8. For each r ≥ 2, the image of the differential d r is contained in (E * , * r ) red .
Proof. From (5.1) we have that there exists a k 0 such that H d+2k = F and H d+2k+1 = 0 for all k ≥ k 0 . Hence the action of τ is trivial in these degrees and we have for all k ≥ k 0 . Since SW F (Y, s, g) is a space of type Z p -SWF, the localisation theorem in equivariant cohomology implies that there exists a k 1 ≥ k 0 such that the generator x ∈ E 0,d+2k1 2 = F satisfies d r (x) = 0 for all r ≥ 2. Then if y ∈ E p,q 2 with q ≥ d + 2k 1 , it follows that y is of the form y = cU a x for some a ≥ 0, where c ∈ H p G . Hence d r (y) = 0 for all r ≥ 2. Now let y ∈ E p,q r where p, q are arbitrary. Then there exists some a ≥ 0 such that q + 2a ≥ d + 2k 1 , hence U a d r (y) = d r (U a y) = 0. Therefore d r (y) ∈ (E * , * r ) red .
Recall that H ∞ is a free F[U ]-module of rank 1 with generator in degree d.
Proof. The classes U j+mr θ, 0 ≤ j < m r+1 − m r form a basis for S r /S r+1 . Choose a lift x r ∈ E 0,d+2mr r of U mr θ ∈ S r . Then d r (U j x r ) = 0 for 0 ≤ j < m r+1 − m r , for if d r (U j x r ) = 0 for some 0 ≤ j < m r+1 − m r , then we would have U j+mr θ ∈ S r+1 . Observe that d r (U j x r ) ∈ E r, * r . By Lemma 5.7 and the definition of M r , we see that d r (U j x r ) can be identified with a non-zero element of M r . Moreover, we have that d r (U j x r ) ∈ (M r ) red , by Lemma 5.8. Now since the d r (U j x r ) are non-zero and have distinct degrees, they span a subspace of (M r ) red of dimension m r+1 − m r . Furthermore, this subspace lies in the image of Proposition 5.10. Suppose that τ acts trivially on HSW * (Y, s). Then Proof. Recall that d = d(Y, s). Hence δ(Y, s) = d/2. From the definition of the invariant δ 1 (Y, s), it follows that for all sufficiently large r, we have By Lemma 5.9, for each r ≥ 2, we have and summing from 2 to r − 1, we get However since τ acts trivially on HSW * (Y, s), we have that E p, * 2 = HSW * (Y, s) for all p ≥ 0. Hence m 2 = 0, M 2 = HSW * (Y, s) and (M 2 ) red = HSW red (Y, s). Taking r sufficiently large, we have 6. Branched double covers of knots 6.1. Concordance invariants. Let K ⊂ S 3 be a knot in S 3 . Let Y = Σ 2 (K) be the branched double cover of S 3 , branched over K. Let π : Y → S 3 denote the covering map. One finds that b 1 (Y ) = 0. Manolescu and Owens [50] define a knot invariant where t 0 is the spin c -structure induced from the unique spin structure on Σ 2 (K) (see [50,Section 2] for an explanation of this). It is shown in [50] that δ(K) is always integer-valued and descends to a surjective group homomorphism δ : C → Z, where C is the smooth concordance group of knots in S 3 . The covering involution on Y determines an action of G = Z 2 on Y preserving t 0 (by uniqueness of the underlying spin structure). Hence we may define the following knot invariants, for each j ≥ 0: Since d j (Σ 2 (K), t 0 ) − d(Σ 2 (K), t 0 ) ∈ 2Z, it follows that δ j (K) − δ(K) ∈ 4Z. Then, since δ(K) is integer-valued, it follows that the δ j (K) are also integer-valued and moreover we have δ j (K) = δ(K) (mod 4). Proposition 6.1. For each j ≥ 0, δ j (K) depends only on the concordance class of K, hence δ j descends to a concordance invariant δ j : C → Z.
Proof. For an oriented knot K, recall that −K denotes the knot obtained by reversing orientation on S 3 and K. It follows that Σ 2 (−K) = Σ 2 (K). A concordance of oriented knots K 1 , K 2 is a smooth embedding of Σ = [0, 1] × S 1 in [0, 1] × S 3 having boundary −K 1 ∪K 2 . Taking the double cover of [0, 1]×S 3 branched along Σ gives a Z 2 -equivariant cobordism W from Σ 2 (K 1 ) to Σ(K 2 ). From the calculations in [35,Section 3], one sees that W is a rational homology cobordism. We claim that W is spin. To see this, choose a smoothly embedded surface Σ in D 4 whose boundary is [34] we see that W ′ is spin. Since W is embedded in W ′ , it follows that W ′ is spin as well. Any spin structure t on W will restrict on each component of the boundary to the unique spin structure on the branched double cover Σ 2 (K i ). The result now follows by applying Corollary 4.3 to (W, t).
Proof. From Proposition 6.2 we have that δ j (K) = σ ′ (K) for j ≥ g 4 (K) + σ ′ (K) and δ j (−K) = −σ ′ (K) for j ≥ g 4 (K) − σ ′ (K). Hence j + (K) ≤ g 4 (K) + σ ′ (K) and Remark 6.7. In this section we have used branched double covers Σ 2 (K) of knots equipped with their natural Z 2 -action to obtain a sequence of concordance invariants. Similarly for any odd prime p we may consider the cyclic branched cover Σ p (K) with its natural Z p -action. Once again there is a canonical spin c -structure t 0 [28] and so we may define a sequence of invariants depending on a prime p and an integer j ≥ 0. By similar arguments to the p = 2 case one finds that these are integer-valued knot concordance invariants of K.

Computations and applications
7.1. Brieskorn homology spheres. Let p, q, r be pairwise coprime positive integers and let Y = Σ(p, q, r) be the corresponding Brieskorn integral homology 3-sphere. Then Y has a unique spin c -structure and so when speaking of the Floer homology of Y we omit the mention of the spin c -structure.
Recall that Σ(p, q, r) can be realised as the p-fold cyclic cover of S 3 branched along the torus knot T q,r . In particular, this construction defines an action of Z p on Y . Let τ : Y → Y denote the generator of this action. Recall that Σ(p, q, r) is obtained by taking the link of the singularity {(x, y, z) ∈ C 3 | x p + y q + z r = 0}. Then τ is given by (x, y, z) → (e 2πi/p x, y, z). This map is isotopic to the identity through the homotopy (x, y, z) → (e 2πiqrt x, e 2πiprt y, e 2πipqt z), t ∈ [0, (qr) * ], where 0 < (qr) * < p denotes the multiplicative inverse of qr mod p. It follows that τ acts trivially on HF + (Y ).
Henceforth we will assume that p is a prime number. Set F = Z p and recall that deg(S) = 2 if p is odd. Let s denote the unique spin c -structure on Y . As in Section 5, we let δ j (Y, s, τ, p) denote δ Zp,Q p (Y, s) for p = 2 or δ Zp,S p (Y, s) for odd p. We will further abbreviate this to δ j (Y, τ ). When p = 2, δ j (Y, τ ) = δ j (T q,r )/4, where δ j (K) denotes the knot concordance invariant introduced in Section 6.1. More generally, δ j (Y, τ ) = δ (p),j (T q,r )/4, where δ (p),j (K) is the knot concordance invariant defined in Remark 6.7. Proposition 7.2. Let p, q, r be positive, pairwise coprime integers and assume that p is prime. Then δ j (Σ(p, q, r), τ ) = −λ(Σ(p, q, r)) for all j ≥ 0, where λ(Σ(p, q, r)) is the Casson invariant of Σ(p, q, r). Furthermore, we have that σ j/p (T q,r ), for all j ≥ 0 where σ α (K) is the Tristram-Levine signature of K.
Proof. Recall that Y = Σ(p, q, r) is the boundary of a negative definite plumbing [52] whose plumbing graph has only one bad vertex in the terminology of [53].
From [54,Theorem 1.3] we have that χ(HF + red (Y )) is related to the Casson invariant λ(Y ) via the formula Moreover, from [26], [14], we have that λ(Σ(p, q, r)) = 1 8 where M (p, q, r) is the Milnor fibre M (p, q, r) = {(x, y, z) ∈ C 3 | x p + y q + z r = δ} ∩ D 6 (where δ is a sufficiently small non-zero complex number). Recall that M (p, q, r) is a compact smooth 4-manifold with boundary diffeomorphic to Σ(p, q, r). Moreover, M (p, q, r) has the homotopy type of a wedge of 2-spheres, so b 1 (M (p, q, r)) = 0. Further, M (p, q, r) is a p-fold cyclic cover of D 4 branched along a surface bounding T q,r . Hence the action of Z p = τ on Y extends to M (p, q, r). From [28, Lemma 2.1] it follows that there is a Z p invariant spin structure t 0 on M (p, q, r). Since M (p, q, r) is a cyclic p-fold cover of D 4 it follows that the subspace of H 2 (M (p, q, r); R) fixed by τ is zero. Hence Corollary 5.5 may be applied, giving Since τ acts trivially on HF + (Y ), Proposition 5.10 implies that Hence δ 0 (Y, τ ) ≤ −λ(Y ). On the other hand, δ 0 (Y, τ ) ≥ δ j (Y, τ ) ≥ −λ(Y ) for any j ≥ 0. Hence δ j (Y, τ ) = −λ(Y ) for all j ≥ 0. Therefore we also have for all j ≥ 0.
The above result shows that the values of δ (p),j (T q,r ) do not depend on j. In contrast, the values of δ (p),j (−T q,r ) usually do depend on j, as the following propositions illustrate.
Proof. The case n = 1 is already covered in Example 7.1, so we assume n ≥ 2. Set Y a,b = Σ 2 (T a,b ) = Σ(2, a, b) and let τ be the covering involution. So δ j (−T 3,6n−1 ) = 4δ j (Y 3,6n−1 ). From the computations in [54, Section 8] we find that d(Y 3,6n−1 ) = −2, SW F * red (Y 3,6n−1 ) = (F −2 ) n−1 , where the subscript indicates degree. To simplify notation we let where the bi-degree is given as follows. θ and all elements of V have bi-degree (0, −2), U has bidegree (0, 2) and Q has bidegree (1, 0). Then E p,q 2 = 0 for q < −2. It follows that all the differentials in the spectral sequence are zero on θ and on V , since d r sends E p,−2 r to E p+r,−1−r r and −1 − r < −2 for r ≥ 2. Hence d r is zero on all of E r and E * , * ∞ ∼ = E * , * 2 . Let F j denote the filtration on HSW * Z2 (Y 3,6n−1 ) corresponding to the spectral sequence, so F j /F j+1 ∼ = E j, * ∞ . In particular, F 1 /F 2 ∼ = F[U ]θ ⊕ V . Choose lifts of θ and V to F 1 . We lift U j θ by taking the lift of θ and applying U j . Hence we obtain a short exact sequence of F[U, Q]-modules Next, for each j ≥ 0, Q induces an isomorphism Q : F j /F j+1 → F j+1 /F j+2 hence by applying Q repeatedly to F[U ]θ ⊕ V , we obtain a splitting of the filtration {F j } as F[Q]-modules. The splittings give an isomorphism of F[Q]-modules . However, this is not necessarily an isomorphism of F[U, Q]-modules. Under this isomorphism, U corresponds to an endomorphism of the form where U j : HSW * (Y 3,6n−1 ) → HSW * +j (Y 3,6n−1 ) and U 2 = U . Since HSW * (Y 3,6n−1 ) is concentrated in even degrees we have that U j = 0 for odd j. Moreover, our construction is such that U j θ = 0 for j = 2. It follows that U j = 0 for j < 0, as V is concentrated in a single degree. So we get U = U + Q 2 U 0 for some U 0 : V → HSW 0 (Y 3,6n−1 ).
To simplify notation set d j = d j (Y 3,6n−1 ). Using Proposition 3.14 we obtain the following characterisation of d j .
Similarly, from Proposition 7.4, we see that σ ′ (T 3,6n+1 ) = 4n and j − (T 3,6n+1 ) = 2n. So we obtain an estimate g 4 (T 3,6n+1 ) ≥ 6n. Once again, this estimate is sharp since g 4 (T 3,6n+1 ) = (3 − 1)(6n + 1 − 1)/2 = 6n. 7.2. Non-extendable actions. Let Y be a rational homology 3-sphere equipped with an orientation preserving action of G. Let W be a smooth 4-manifold with boundary Y . In this section we are concerned with the question of whether the Gaction can be extended to W . In particular we are interested in finding obstructions to such an extension. this means t is fixed by all of G. From here, the rest of the proof is the same as for Proposition 7.6. Example 7.9. Let Y = Σ(p, q, r) where p, q, r are relatively prime and assume that p is prime. Let Z p act on Y as described in Section 7.1. Recall from Proposition 7.2 that δ Zp,1 (Y, s) = −λ(Σ(p, q, r)). As in Example 7.7, λ(Σ(p, q, r)) < 0 and hence δ Zp,1 (Y, s) > 0. Therefore by Proposition 7.8, if W is a compact, oriented, smooth rational homology 4-ball bounded by Y and if p does not divide the order of H 2 (W ; Z), then the action of G does not extend over W . Thus we have obtained a partial extension of the results of Anvari-Hambleton to the case of rational homology 4-balls.
Fintushel-Stern showed that Σ(2, 3, 7) bounds a rational homology 4-ball, although it does not bound an integral 4-ball [25]. Akbulut-Larson showed that Σ(2, 4n + 1, 12n + 5) and Σ(3, 3n + 1, 12n + 5) for n odd bound rational 4-balls but not integral 4-balls [4]. More examples, Σ(2, 4n+3, 12n+7) and Σ(3, 3n+2, 12n+7) for even n were constructed by Şavk [59]. Taking p = 2 or 3, the above Brieskorn spheres admit Z p -actions with non-zero delta invariants, as in Example 7.7. Hence the Z p -action does not extend to any oriented rational homology 4-ball W with boundary Y , provided the order of H 2 (W ; Z) is coprime to p. However, it does not seem straightforward to determine whether the above examples are bounded by rational 4-balls satisfying this coprimality condition. Proposition 7.10. Let Y be an integral homology 3-sphere and s the unique spin cstructure on Y . Let G act orientation preservingly on Y and suppose that the extension G s is trivial. Suppose that Y is the boundary of a smooth, compact, oriented 4-manifold with b 1 (W ) = 0 and suppose that H 2 (W ; Z) has no 2-torsion.
(1) If H 2 (W ; R) is positive definite and δ G,1 (Y, s) > 0, then the G-action on Y can not be extended to a smooth G-action on W acting trivially on H 2 (W ; Z). (2) If H 2 (W ; R) is negative definite and δ G,c (Y, s) < 0 for some c ∈ H * G , then the G-action on Y can not be extended to a smooth G-action on W acting trivially on H 2 (W ; Z).
Proof. Suppose the G-action on Y extends to a smooth G-action on W acting trivially on H 2 (W ; Z). Since H 2 (W ; Z) has no 2-torsion, a spin c -structure t on W is determined uniquely by c 1 (t). Since G acts trivially on H 2 (W ; Z), it follows that G preserves every spin c -structure. Furthermore t| Y = s for any spin c -structure on W by uniqueness of t.
If H 2 (W ; R) is negative definite, then Theorem 4.1 may be applied to any spin cstructure t on W , giving δ(W, t) ≤ δ G,c (Y, s) for all t and all c ∈ H * G . Since Y is an integral homology sphere, the intersection form on the H 2 (W ; Z)/torsion is unimodular. By the main theorem of [23], there exists a spin c -structure t such that δ(W, t) ≥ 0. Hence δ G,c (Y, s) ≥ 0. The proof in the case that H 2 (W ; R) is positive definite is similarly obtained. Example 7.11. Consider again Y = Σ(p, q, r) with the same Z p -action. Recall from Proposition 7.2 that δ Zp,1 (Y, s) = −λ(Σ(p, q, r)). As in Example 7.9, δ Zp,1 (Y, s) > 0. So by Proposition 7.10, the action of Z p on Y can not be extended to any smooth, compact, oriented 4-manifold W such that b 1 (W ) = 0, H 2 (W ; Z) has no 2-torsion and with Z p acting trivially on H 2 (W ; Z). 7.3. Realisation problems. In this section we are concerned with the following realisation problem. Let W be a smooth 4-manifold with boundary an integral homology sphere Y . Suppose that a finite group G acts on H 2 (W ; Z) preserving the intersection form. We say that the action of G on H 2 (W ; Z) can be realised by diffeomorphisms if there is a smooth orientation preserving action of G on W inducing the given action on H 2 (W ; Z).
For simplicity we will assume that G = Z p for a prime p so that all extensions G s are trivial.
Proposition 7.12. Let W be a smooth, compact, oriented 4-manifold with b 1 (W ) = 0 and with boundary Y = ∂W an L-space integral homology sphere. Suppose that an action of G = Z p on H 2 (W ; Z) is given and suppose that H 2 (W ; Z) has no 2-torsion. Suppose that the subspace of H 2 (W ; R) fixed by G is negative definite. If the action of G on H 2 (W ; Z) can be realised by diffeomorphisms, then for every spin c -structure s on W for which c 1 (s) is invariant.
Proof. This is essentially a special case of Theorem 5.3. Note that since H 2 (W ; Z) is assumed to have no 2-torsion, any spin c -structure s for which c 1 (s) is invariant is preserved by G. So if G is realisable by diffeomorphisms, then Theorem 5.3 gives δ(W, s) ≤ δ G,1 (Y, s| Y ). But we have assumed that Y is an L-space, so δ G,1 (Y, s| Y ) = δ(Y, s| Y ).
Example 7.13. We consider a specialisation of Proposition 7.12 as follows. Take G = Z p . Assume Y is an L-space integral homology 3-sphere and let s be the unique spin c -structure. Suppose that W is a smooth, compact, oriented 4-manifold with b 1 (W ) = 0 and with boundary Y . Suppose that the intersection form on H 2 (W ; Z) is even and that H 2 (W ; Z) has no 2-torsion. Then W is spin and it has a unique spin structure t. By uniqueness, the restriction of t to the boundary equals s. Suppose that an action of G = Z p on H 2 (W ; Z) is given and that the subspace of H 2 (W ; R) fixed by G is negative definite. Then applying Proposition 7.12 to (W, t), we find that δ(W, t) = −σ(W )/8 ≤ δ(Y, s). Therefore, if σ(W )/8 < −δ(Y, s) then the action of Z p on H 2 (W ; Z) is not realisable by a smooth Z p -action on W .
For example if W = K3#W 0 is the connected sum of a K3 surface with W 0 , the negative definite plumbing of the E 8 graph, then ∂W = Y = Σ(2, 3, 5) is the Poincaré homology 3-sphere which is an L-space. Then W satisfies all the above conditions and σ(W )/8 = −3 < δ(Y, s) = −1. Hence for any prime p, any Z p -action on H 2 (W ; Z) such that the invariant subspace of H 2 (W ; R) is negative definite can not be realised by a smooth Z p -action on W .
12. Then we obtain an equivariant splitting X = X + ∪ Y X − . Since Y is an integral homology sphere, the intersection forms on X ± are unimodular. They are also even, since X is spin. Moreover the Rochlin invariant of Y is 1. So the intersection forms of X ± must contain at least one E 8 or −E 8 summand. Proposition 7.2 implies that δ j (Y ) = 1 for all j ≥ 0 and Proposition 7.4 implies that δ j (Y ) = 0 for j = 0, 1 and δ j (Y ) = −1 for j ≥ 2. Since n ≤ 12, Corollary 5.5 applied to X ± then implies that the intersection form of X + must be of the form αH ⊕ (−E 8 ) for some α ≥ 2 (where H is the hyperbolic lattice) and similarly the intersection form of X − must be of the form α ′ H ⊕ (E 8 ) for some α ′ ≥ 2. The intersection form of X is then (α + α ′ + 8)H and so n = α + α ′ + 8 ≥ 2 + 2 + 8 = 12. This proves that ε(Σ (2, 3, 7), τ ) = 12.