Equivariant Lagrangian Floer homology via cotangent bundles of EGN$EG_N$

We provide a construction of equivariant Lagrangian Floer homology HFG(L0,L1)$HF_G(L_0, L_1)$ , for a compact Lie group G$G$ acting on a symplectic manifold M$M$ in a Hamiltonian fashion, and a pair of G$G$ ‐Lagrangian submanifolds L0,L1⊂M$L_0, L_1 \subset M$ . We do so by using symplectic homotopy quotients involving cotangent bundles of an approximation of EG$EG$ . Our construction relies on Wehrheim and Woodward's theory of quilts, and the telescope construction. We show that these groups are independent of the auxiliary choices involved in their construction, and are H∗(BG)$H^*(BG)$ ‐bimodules. In the case when L0=L1$L_0 = L_1$ , we show that their chain complex CFG(L0,L1)$CF_G(L_0, L_1)$ is homotopy equivalent to the equivariant Morse complex of L0$L_0$ . Furthermore, if zero is a regular value of the moment map μ$\mu$ and if G$G$ acts freely on μ−1(0)$\mu ^{-1}(0)$ , we construct two ‘Kirwan morphisms’ from CFG(L0,L1)$CF_G(L_0, L_1)$ to CF(L0/G,L1/G)$CF(L_0/G, L_1/G)$ (respectively, from CF(L0/G,L1/G)$CF(L_0/G, L_1/G)$ to CFG(L0,L1)$CF_G(L_0, L_1)$ ). Our construction applies to the exact and monotone settings, as well as in the setting of the extended moduli space of flat SU(2)$SU(2)$ ‐connections of a Riemann surface, considered in Manolescu and Woodward's work. Applied to the latter setting, our construction provides an equivariant symplectic side for the Atiyah–Floer conjecture.


Introduction
Lagrangian Floer homology is a group HF (M ; L 0 , L 1 ) 1 associated with a pair of Lagrangians L 0 , L 1 in a symplectic manifold M , provided these satisfy some assumptions. Depending on those assumptions, these groups are more or less complicated to define. A particularly difficult setting for defining these groups is when M and/or L 0 , L 1 have singularities. In practice, many interesting singular symplectic manifolds and Lagrangians arise as a symplectic reduction: if M is a Hamiltonian G-manifold for some compact Lie group, and L 0 , L 1 are G-Lagrangians, then unless the action is nice enough, M/ /G and L 0 /G, L 1 /G might be singular.
For instance, this is the setting of the Atiyah-Floer conjecture in its initial formulation [Ati88]: given an integral homology 3-sphere Y , its instanton homology I * (Y ) should be isomorphic to a Lagrangian Floer homology ]. Yet, Jeffrey [Jef94] and Huebschmann [Hue95] observed that the moduli space M (Σ) can be realized as the symplectic reduction of a smooth SU(2)-Hamiltonian space: (an open subset N of) the so-called extended moduli space. Moreover it contains smooth SU(2)-Lagrangians L 0 , L 1 such that L (H 0 ) = L 0 /G and L (H 1 ) = L 1 /G. Manolescu and Woodward [MW12] defined symplectic instanton homology HSI(Y ) as a (non-equivariant) Lagrangian Floer homology in N , and suggested that a good candidate for the symplectic side of the Atiyah-Floer conjecture would be an equivariant version HF SU (2) (N ; L 0 , L 1 ) of HSI(Y ) (as a substitute for HF (M (Σ); L (H 0 ), L (H 1 ))).
Several versions of equivariant Floer homologies appeared in the litterature, in different settings: • In [AB95], Austin and Braam defined an equivariant version of Morse homology, as a mixture between Morse and deRham homology, for an equivariant Morse-Bott function. • In [Vit99], Viterbo suggested the definition of an equivariant version of symplectic homology, for the reparametrization U(1)-action. This was implemented in [BO13], via a Borel construction. • In [MiR99,MiR03], [CGS00], Mundet i Riera and independently Cieliebak, Gaio and Salamon introduced the symplectic vortex equation, which among other things, furnishes a possible way of approaching Atiyah-Floer type 1 It is usually denoted HF (L 0 , L 1 ) when the symplectic manifold is fixed. As this will not be the case in this paper, we prefer to keep M in the notation.
problems. It was used to build homology group associated with a Hamiltonian G-manifold, that could be called an equivariant Floer homology.
In the closed string case (no Lagrangians), Frauenfelder [Fra04] defined "moment Floer homology". For a pair of Lagrangians, Woodward defined "quasimap Floer homology" [Woo11]. This equation is particularly wellsuited for relating equivariant invariants to invariants of the symplectic quotient [TX17,Woo15a,Woo15b,Woo15c,NWZ14] , in analogy with the Kirwan map [Kir84]. Their construction involves the theory of (∞, 1) categories. • In [KLZ19], in the case of a single Lagrangian L 0 = L 1 , Kim, Lau and Zheng define its equivariant Lagrangian Floer homology using a Borel construction similar (but slightly different, see Remark 4.37) to the one in this paper. • In gauge theory, Kronheimer and Mrowka [KM07] define several versions of Monopole homology, corresponding to U(1)-equivariant theories. For instanton homology, a first construction was given in Austin-Braam [AB96], and later generalized by Miller [Mil19]. Daemi and Scaduto also defined a version for knots [DS20], related to singular instanton homology. In this paper, we present a construction for equivariant Lagrangian Floer homology that applies to Manolescu and Woodward's setting (we do so in Section 9) and that is as simple as possible, both algebraically and analytically. Algebraically, it is based on the telescope construction, all we need is contained in Section 2, some of which might be new. Analytically, we work in a setting that allows us to use domain dependent perturbations, so transversality is standard [FHS95]. In particular we don't have to achieve equivariant transversality, a usually delicate problem.
Our personal motivation for this construction is to recast the Donaldson polynomials as an extended Field theory, where equivariant Floer homology groups would play the role of 3-morphism spaces in the target 3-category, and generalized Donaldson polynomials for 4-manifolds with corners would take their values in these groups. We refer to [Caz19] for more details.
1.1. Outline of the construction. It is known since Floer that Morse homology corresponds to Lagrangian Floer homology in a cotangent bundle. This identification provides a dictionnary between these two theories. Our strategy is to first reformulate the definition of equivariant homology in a Morse-theoretical way, and then use this dictionnary to translate this construction to that Lagrangian setting.
Let X be a closed smooth manifold acted on by a compact Lie group G. By definition, equivariant homology is the homology of the homotopy quotient using Morse-theoretic pushforwards of (1.6) id X × G i N : Now if X is a G-manifold, then M = T * X is a Hamiltonian G-manifold, and its zero section L 0 = L 1 = 0 X ⊂ T * X is a G-Lagrangian. In analogy with the isomorphism HF (T * X; 0 X , 0 X ) HM (X), one can define the equivariant Lagrangian Floer homology of (T * X; 0 X , 0 X ) by: It turns out that T * (X × G EG N ) = (T * X × T * EG N )/ /G, under which: (1.8) assuming (M ; L 0 , L 1 ) satisfy standard assumptions so that Floer homologies are well-defined. Constructing the maps from N to N + 1 will involve quilts that generalize the Morse-theoretic pushforwards.
1.2. Statement of results. Throughout this paper we work with Z 2 -coefficients to avoid sign discussions (but we believe that everything should work with Z coefficients as well, provided Lagrangians are endowed with equivariant relative Pinstructures). The following statement summarizes Propositions 4.36, 4.42,5.1, 5.2, 5.3 and 7.3. With CF N the Floer chain complex in (M × T * EG N )/ /G and α N : CF N → CF N +1 chain morphisms induced by the inclusions i N : EG N → EG N +1 . Its homotopy type is independent on the auxiliary choices involved in the constructions (Hamiltonian perturbations, almost complex structures). Its associated homology group If furthermore, in the sense of Definition 4.39, M admits a Z n Maslov G-cover L G of its Lagrangian grassmannian bundle, and L 0 and L 1 admit a ( L G , G)-grading. Then CF G (M ; L 0 , L 1 ) has an absolute grading over Z n .
In the case when the Lagrangians coincide, we prove the following, using the Piunikhin-Salamon-Schwarz construction [PSS96]: Theorem B. Let L ⊂ M be satisfying either Assumption 4.17 or 4.20, with L 0 = L 1 = L. Then CF G (M ; L, L)) is homotopy equivalent to CM G (L), and the resulting homology groups are isomorphic as H * (BG)-bimodules. In particular, if X is a smooth, compact G-manifold, then CM G (X) CF G (T * X; 0 X , 0 X ).
Theorem C. If the action of G on M is regular (in the sense of Definition 4.4, so that the symplectic quotient M/ /G is smooth), then the equivariant Floer complex of (M ; L 0 , L 1 ) is related to the non-equivariant one of the quotients (M/ /G; L 0 /G, L 1 /G) by two morphisms: Finally, we apply our construction to Manolescu and Woodward's setting: Theorem D. [Theorem 9.1] The construction outlined in Section 1.1 can also be implemented in Manolescu and Woodward's setting, i.e. when M = N (Σ ) is the open subest of the extended moduli space involved in [MW12]. We call the corresponding group HSI G (Y ) equivariant symplectic instanton homology. It is a relatively Z 8 -graded H * (BSU (2))-module, and in the case when Y is a rational homology sphere, an absolute Z 8 -grading can be fixed canonically.
We believe HSI G (Y ) is independent on the Heegaard splitting of Y , and outline a proof of this in Remark 9.2. Furthermore, it provides an equivariant symplectic side for the Atiyah-Floer conjecture: we believe it should be isomorphic to a suitably defined SU (2)-equivariant version of instanton homology [Mil19].
Remark 1.1. In [AMM98], Alexeev, Malkin and Meinrenken introduced "quasi-Hamiltonian spaces". These are spaces with a 2-form acted on by a Lie group, with a moment map, similar to Hamiltonian spaces, except that their moment map takes its value in the group, rather than the dual of its Lie algebra. One key difference is that their 2-form is not closed: usually these are not symplectic manifolds, and to our best knowledge Floer homology has never been defined in their setting.
Nevertheless, these spaces admit quotients similar to symplectic reduction, which are honest symplectic manifolds. We expect that our construction, suitably adapted, will apply to this setting. Notice that this is relevant to the Atiyah-Floer conjecture: originally these spaces were introduced as particularly nice substitutes to the extended moduli space: if Σ is a genus g closed surface, then G 2g can be endowed with the structure of a quasi-Hamiltonian space, and its reduction is the Atiyah-Bott moduli space M (Σ). Therefore, equivariant Floer homology for these spaces would provide an alternative construction for equivariant symplectic instanton homology HSI G (Y ).
1.3. Organization of the paper. In Section 2, we start by setting the algebraic framework of telescopes that we will be using in our constructions.
In Section 3, we provide more details about the Morse theoretical construction of equivariant homology outlined above.
In Section 4, after reviewing some standard material about Hamiltonian actions and Lagrangian Floer homology, we set our working assumptions and construct CF G (M, L 0 , L 1 ).
In Section 5, we define continuation maps to prove independence of perturbations.
In Section 6, we compute the case L 0 = L 1 , using a PSS construction. In Section 7, we define the bimodule structure on HF G (M, L 0 , L 1 ).
In Section 8, we construct the Kirwan maps of Theorem C. Finally, in Section 9 we focus on Manolescu and Woodward's setting and define equivariant symplectic instanton homology of a 3-manifold.

Telescopes
As in symplectic homology [Vit99,BO13] or Wrapped Floer homology [AS10], we will define the equivariant chain complex as a homotopy colimit. The telescope construction is a nice model for that, we first review its construction at the level of spaces, as it sheds light to the chain level construction that we will be using to define the equivariant Morse and Floer complexes.
Definition 2.1 (Telescopes). Let (X N , a N ) N ≥0 be a sequence of spaces, with increment maps a N : The mapping telescope of this sequence is defined by with I t = [0, 1], the subscript is here to indicate the variable we will use for elements in I t .
It is endowed with a shift map a : Tel(X N , a N ) → Tel(X N , a N ) defined by a(x, t) = (a N (x), t). The shift is a homotopy equivalence, as it is connected to the identity through the path of maps (a u ) 0≤u≤1 defined by: be such sequences of spaces, consider a sequence of maps f N : X N → Y N that commute with the increments a N , b N up to homotopy, in the sense that there exists maps k N : In this setting one can define a map Proof that the map is well-defined. These two quantities agree when t = 1 2 , since Furthermore, Tel(f N , k N ) maps both (x, 1) and (a N (x), 0) to the same element (f N +1 a N (x), 0). Remark 2.3. One could have used other formulas here, for example by setting but in view of the next section, the one we use has the advantage of being a cellular map.
Definition 2.4 (Homotopies between Telescopes). Consider now two sequences of maps f 0 N , f 1 N : X N → Y N that are homotopic and that commute with the increments a N , b N up to homotopy: we are given two maps Notice that this is the same setting as in the previous paragraph, replacing X N by Z N = X N × I u . Therefore we get a map between telescopes: and since Tel(Z N ) = Tel(X N )×I u , one can think of Tel(f N ) as a homotopy between . Proposition 2.5 (Products). Let (X m , a m ) and (Y n , b n ) be two sequences of spaces, then the map if m > n is a homotopy equivalence.
Proof. Left to the reader (see Figure 1).

Figure 1.
Map from a product of telescopes to the telescope of products.

Chain complexes.
This subsection is the algebraic transcription of the previous one.
Definition 2.6 (Telescopes of Chain complexes). Let (C N , α N ) N ≥0 be a sequence of chain complexes over Z 2 , with increment chain morphisms One can think of C N as the cellular complex of X N , equipped with some cellular decomposition. The telescope of this sequence is defined by with q a formal variable of degree one. If σ is a cell of X N the C N summand in Tel(C N , α N ) corresponds to the cells σ × {0}, while the qC N summand corresponds to the cells σ × [0, 1]. It is endowed with the differential Sometimes we may just denote it Tel(C N ) when there is no ambiguity.
Definition 2.7 (Morphisms between Telescopes). Let (C N , α N ) N ≥0 and (D N , β N ) N ≥0 be two sequences as before, consider a sequence of chain morphisms In this setting one can define a map Tel(ϕ N , κ N ) : Tel(C N ) → Tel(D N ) by Remark 2.8. If a represents a cell in X N , then Tel(f N , k N ) will map it to (f N •a, 0). If b represents a cell in X N , then qb corresponds to the cell b × I t in Tel(X N ) which by (2.1) is mapped to the union of (f N • b) × I t and k N (b × I t ) × {0}.

2
Proposition 2.13 (Products). Let (C m , α m ) and (D n , β n ) be two sequences of chain complexes, then the map and (y, v) ∈ D n [q] and denoting is a morphism of chain complexes.

Equivariant Morse homology
We begin by a quick review of Morse homology to set our notations, and refer for example to [AD14] for more details.
3.1. Morse homology. Let X be a compact smooth manifold of dimension n, and f : M → R a Morse function. Each critical point x has a Morse index ind(x). Denote respectively the set of critical points and index k critical points by Crit(f ) and Crit k (f ).
Definition 3.1. A pseudo-gradient for f is a vector field v ∈ X(X) on X such that for all x ∈ X \ Crit(f ), d x f.v < 0; and such that in a Morse chart near a critical point, v is the negative gradient of f for the standard metric on R n . Denote by X(X, f ) ⊂ X(X) the space of pseudo-gradients for f . This is a convex (hence contractible) space.
Definition 3.2. Let x ∈ Crit k (f ) and v ∈ X(X, f ). Define the stable (resp. unstable) submanifold of x by: with φ v t the flow at time y of v. S x and U x are smooth (non-proper) submanifolds diffeomorphic respectively to R k and R n−k .
Definition 3.3. A pseudo-gradient v is Palais-Smale if for any pair x, y of critical points, S x intersects U y transversally.
where R acts on U x ∩ S y by the flow of v. It is well-known that ∂ 2 = 0 and that HM * (X, f ) = H * (X, Z 2 ). Intuitively, x ∈ Crit k (f ) corresponds to a k-chain obtained by triangulating U x .
3.2. Pushforwards on Morse homology. Let F : X → Y be a differentiable map between two smooth compact manifolds, then F induces a pushforward in homology F * : H * (X) → H * (Y ). We now recall a Morse theoretic construction of such a map, and refer for example to [KM07, Section 2.8] for more details.
Endow X and Y with two Morse functions f : X → R, g : Y → R, and pseudogradients ∇f, ∇g. Let x ∈ Crit k (f ) be a generator of CM k (X, f ) (say k ≥ 1 for the following discussion). Heuristically, x corresponds to the k-chain of its unstable manifold U x , therefore its image F * x should correspond to F (U x ), which is a priori unrelated to g. Apply the flow of ∇g to it: for t large enough, most points will fall down to local minimums, except those points lying in a stable manifold S y of a critical point y of index l ≥ 1. If k = l, then a small neighborhood of those points will concentrate to U y , which now corresponds to a generator of CM (Y, g).
Therefore the previous discussion motivates the following definition. Assume that f , g and two pseudo-gradients ∇f, ∇g are chosen so that, for any critical points x and y of f and g respectively, The graph In order to translate this construction to Floer theory, it is convenient to think of Γ(F ) ∩ U x × S y as a moduli space of grafted flow line from x to y: By this we mean a pair of flow lines (γ − , γ + ), with These are Morse counterparts of quilts, as we shall see. Denoting M(x, y) the moduli space of such grafted lines, the identification with Γ(F ) ∩ U x × S y is given by: 3.3. The equivariant Morse complex. Let X be a closed G-manifold, and EG N finite dimensional smooth approximations of EG, with inclusions and define the equivariant Morse complex as be its dual complex. From the facts that Morse homology corresponds to singular homology, and that Morse pushforwards induce usual pushforwards, one gets: Proposition 3.5. Denoting HM G * and CM * G the homology groups associated with the above complexes, one has

Equivariant Lagrangian Floer homology
We first recall some basic facts about Hamiltonian actions.

Hamiltonian actions.
Definition 4.1. (Hamiltonian manifold) Let G be a Lie group. A Hamiltonian G-manifold (M, ω, µ) is a symplectic manifold (M, ω) endowed with a G-action by symplectomorphisms, induced by a moment map µ : M → g * . The moment map is G-equivariant with respect to this action and the coadjoint representation on g * , and satisfies the following equation: for each η ∈ g, where X η stands for the vector field on M induced by the infinitesimal action, i.e.
In other words, X η is the symplectic gradient of the function µ, η .
Example 4.2 (Induced Hamiltonian action on a cotangent bundle). A smooth action of G on X lifts to a Hamiltonian action on T * X defined by

Definition 4.3 (Weinstein correspondence [Wei81]
). The data of both the action and the moment map can be conveniently packaged as a Lagrangian submanifold that we will refer to as the Weinstein correspondence, defined as: When M = T * X is a cotangent bundle and the action and the moment maps are the ones canonically induced from a smooth action on the base X, Λ G (M ) corresponds to the conormal bundle of the graph of the action Definition 4.4. (Symplectic quotient) If (M, ω, µ) is a Hamiltonian manifold, its symplectic quotient (or reduction) is defined as When 0 is a regular value for µ, and G acts freely and properly on µ −1 (0), M/ /G is also a symplectic manifold. In this case, we will say that the action is regular.
is a Lagrangian correspondence, where ι and π stand respectively for the inclusion and the projection.
which is a smooth symplectic manifold. Indeed, it follows from the fact that G acts freely on EG N that zero is a regular value of the moment map, and that G acts freely on its zero level in M × T N . Likewise, for i = 0, 1, Recall that the approximation of EG comes with inclusions i N : Even if M is compact, M N is never compact (as soon as EG N = G), and will not always be convex at infinity in general. The following will be important for ensuring compactness of the moduli spaces of pseudo-holomorphic curves in the monotone setting: Proposition 4.7. Let M be a G-Hamiltonian manifold with moment map µ M , and L ⊂ M a G-Lagrangian. Let E be a closed manifold on which G acts freely, and B = E/G.
Equip E with a G-invariant Riemannian metric, inducing an equivariant almost complex structure J E on T * E and an almost complex structure J B on T * B. Assume also that M is equipped with an equivariant almost complex structure J M .
Let Q = (M × T * E)/ /G, it inherits a symplectic structure ω Q and a compatible almost complex structure J Q . The quotient Q fibers over T * B, with fibers M , and this fibration restricts to a fibration of (L × 0 E )/G over 0 B , with fibers L.
The projection Q → T * B is (J Q , J B )-holomorphic, and for each β ∈ T * B, the inclusion ι β : M → Q to the fiber over β satisfies: From the orthogonal splitting defined fiberwise by orthogonal projection to (O ⊥ q ) * . Composing this projection with the quotient projection Notice that J E descends to an almost complex structure J B on T * B, and under the identification (T * E)/ /G T * B corresponds to the almost complex structure induced by the Riemannian metric on B. With respect to these almost complex structure, the projection T N → B N is pseudo-holomorphic.
(This map corresponds to the projection to the GIT quotient T * E → T * E/G C , by Kempf-Ness theorem and the fact that all the points of T * E are semi-stable.) One then gets a projection  π : Q → T * B whose fiber is diffeomorphic to M . Indeed, given a point β = [(q 0 , p 0 )] ∈ T * B, the fiber π −1 (β) can be parametrized by the map ι β : where we view µ(m) as an element of O * q . In order to show that (ι β ) * ω Q = ω M , let m ∈ M and v, w ∈ T m M , one has: and since both −d m µ.v, −d m µ.w lie on the fiber direction, which is Lagrangian for ω E , the second summand vanishes.
4.3. Exact setting. The simplest setting for defining Lagrangian Floer homology is the exact and convex at infinity one, which we recall quickly: Definition 4.9. An exact symplectic manifold (M, λ) is convex at infinity if it is isomorphic to the positive symplectization of a contact manifold outside a compact subset. A Example 4.10. A cotangent bundle T * X with its Liouville form λ = p dq is exact and convex at infinity. If Z ⊂ X is a smooth submanifold, then its conormal bundle N Z ⊂ T * X is exact (ι * λ = 0) and cylindrical at infinity.
To ensure that the symplectic homotopy quotients will have such structures, one can impose equivariant analogues as follows.
If G is connected, this is equivalent to saying that Remark 4.12. If G is compact, any exact G-Hamiltonian manifold can be made G-exact: one can make the primitive λ equivariant by averaging over G: with ϕ g : M → M the multiplication by g. The same is true for Lagrangians.
Recall the contact analogues of Hamiltonian actions and symplectic reductions. These appeared in [Alb89], see also [Gei08, sec. 7.7] and the references therein.
Definition 4.13. Let (X, ξ = ker λ) be a contact manifold, a contact G-Hamiltonian action is an action by contactomorphisms, with a moment map µ : X → g * such that Say that such an action is regular if 0 ∈ g * is a regular value of µ and G acts freely on µ −1 (0).
When this is the case, the contact reduction Proposition 4.14. Let (X, λ, µ) be a contact G-manifold. Its symplectization SX = (R × X, e t λ) is G-Hamiltonian with moment map Φ(t, x) = e t µ(x), and is G-exact. Furthermore, if the G-action on X is regular, the same is true for the G-action on SX, and (SX)/ /G is isomorphic to S(X/ /G) as an exact symplectic manifold. If Proposition 4.16. Let X be a G-manifold, then its cotangent bundle T X is Gexact and G-convex at infinity. If Z ⊂ X is a G-invariant submanifold, then its conormal bundle N Z ⊂ T * X is G-exact (the Liouville form restricts to zero on N Z ) In defining HF G (M ; L 0 , L 1 ) we will assume: where µ L stands for the Maslov index.
If L ⊂ M is κ-monotone and M is connected, then M is κ-monotone. The converse is true if L is simply connected.
This permits to control energy of pseuo-holomorphic curves to ensure compactness of the moduli spaces involved in the construction of Floer homology. Still, as Maslov index two discs can obstruct the differential to square to zero, one usually makes an extra assumption on minimal Maslov numbers. Assumption 4.20. The symplectic manifold M is compact κ-monotone, and the Lagrangians L 0 , L 1 are κ-monotone and such that N L0 , N L1 ∈ AZ, with A ≥ 3.
Except for compactness, this setting transports to symplectic homotopy quotients, as we shall see.  Proof. Let g C = g⊕ig, and J a G-invariant almost-complex structure on M compatible with ω. Since the action is regular, we get an injective map, with Z = µ −1 (0) Denote by g C the (trivial) sub-bundle of T M |Z corresponding to the image of this map. Notice that the sub-bundle g ⊂ g C defined as the image of Z × g corresponds to the foliation of Z by orbits.
Let V stand for the orthogonal of g C in T M |Z (with respect to either the symplectic structure or the Riemannian metric induced by J), and W = V ∩ T L.
Let now u and u be as in the statement. One has but µ(u * (g, g C )) = 0 as g is a constant sub-bundle of g C . Moreover, with π : Z → M/ /G the projection.
Recall from [MW12,Lemma 4.4] that (under some assumptions), the symplectic quotient of a κ-monotone symplectic manifold is again κ-monotone. Here is a relative version that follows from the previous lemma: Proposition 4.22. Let (M, ω) be a symplectic manifold, endowed with a regular Hamiltonian action of a compact Lie group G with moment µ : M → g * , and L ⊂ µ −1 (0) a G-Lagrangian. If L is κ-monotone, then L/G is also κ-monotone. Moreover, the minimal Maslov number N L/G is a multiple of N L .
Proof. Let u be a disc in (M/ /G, L/G) an u a lift to (M, L). The first statement follows from: The statement about minimal maslov numbers follows from the fact that any disc can be lifted, and from Lemma 4.21.
2 Then let (M ; L 0 , L 1 ) be a G-Hamiltonian manifold with a pair of G-Lagrangians satisfying Assumption 4.20. Assume that N is large enough, so that EG N and EG N +1 are highly connected. By , the map (γ 1 , . . . , γ k−1 ) → (γ 1 (δ 1 ), . . . , γ k−1 (δ k−1 )) gives a bijection between I H (L) and the intersection of L 0 :=L 0 × L 2 × · · · , and (4.48) Ht (L N 0 ) and L N 1 intersect transversely a Floer datum. More generally, in the situation of a cyclic generalized Lagrangian correspondence L as above, a quilted Floer datum is a pair of sequences     Z 2 x, with differential ∂ defined by Proof. The transversality statement is a standard argument [FHS95]. In the monotone case, let K ⊂ M N be a compact subset containing L N 0 , L N 1 , and such that H t = 0 and J t = J M N outside K. Consider a strip u in M(x, y; H t , J t ), and assume by contradiction that its image is not contained in K. Then outside K, composing with the projection M N → B N , which is pseudo-holomorphic by Proposition 4.7, one gets a pseudo-holomorphic curve in B N that should have no maximum, which is a contradiction.
Therefore curves are contained in K, and the monotonicity assumptions ensure energy bounds, therefore Gromov compactness applies to these moduli spaces, and the rest of the argument is standard.
. We quickly recall some notions from Wehrheim-Woodward's theory to introduce some notations, and refer to [WW15] for the precise definitions.
Definition 4.28. A quilted surface S consists in: • A collection P(S) of patches: these are Riemann surfaces with boundary and strip-like ends. • A collection S(S) of seams: these are pairwise disjoint identifications of boundary components of patches, satisfying a local real-analycity condition.
• A collection B(S) of true boundaries: these are the boundary components of patches not belonging to any seam.
If S is a quilted surface, let its total space be the Riemann surface obtained by glueing together all the patches along the seams. Its boundary is given by the union of components of B(S), and the seams become real analytic curves in |S|. We will often define a quilted surface by giving its total space and seams.
Example 4.29. Let δ 1 , . . . , δ k be positive numbers, (Quilted half-strip) Let Z ± (δ 1 , . . . , δ k ) be the quilted surface whose total space is R ± × [0, δ 1 + · · · + δ k ] and seams the horizontal lines (4.59) Its patches consist in half-strips (4.60) (Quilted half-cylinder) Let C ± (δ 1 , . . . , δ k ) be the quilted surface whose total space is R ± × R/(δ 1 + · · · + δ k )Z and seams the horizontal lines It has same patches as Z ± (δ 1 , . . . , δ k ) (except that P 1 and P k are seamed together). where P i is as in (4.60) and P i ∈ P(S), compatible with the seams (i.e. it induces a continuous map on total spaces), pseudo-holomorphic, proper on the total spaces, and mapping true boundaries to true boundaries. Likewise, a quilted cylindrical end on S is a quilted map (4.64) : C ± (δ 1 , . . . , δ k ) → S compatible with the seams, pseudo-holomorphic, and proper on total spaces. A quilted surface with strip-like and cylindrical ends is a quilted surface S together with a collection of quilted strip-like and cylindrical ends such that the complement of the images of the ends in the total space is compact. Furthermore, each end is labelled either as an incoming, outgoing, or a free end. We denote by   consists in maps u P : P → M P for each patch P of S satisfying the seam condition: (4.67) (u P (x), u P (x)) ∈ L σ , for x ∈ σ, and σ a seam between P and P , and the boundary condition: for x ∈ b, and b a true boundary of P .
To define the moduli space of quilts that will be involved in α N we use perturbations as in [Sei08, sec. (8e)], adapted to the quilted setting.  consist in quilted 1-forms K = (K P ) P such that • on a boundary b of a patch P (belonging or not belonging to a seam), • on a (quilted) incoming or outgoing end (or at least far enough in the end) K P is only t-dependent (i.e. independent on s). • on a free end, K vanishes.
• the following transversality condition is satisfied on a quilted incoming or outgoing end: suppose a quilted end is decorated by the cyclic generalized Lagrangian correspondence: Notice that at an incoming (resp. outgoing) end, a perturbation datum P is asymptotic to a Floer datum F in (resp. F out ). In this situation we will write P : F in → F out . Denote by Remark 4.34. In most moduli spaces we will be considering, it would be enough to consider 1-forms of the form H z dt, except in Section 7, where these would not satisfy the boundary assumptions, when the boundary is not horizontal.
Remark 4.35. We don't perturb on the free ends, the reason is that we will need our actual Lagrangians there to rule out strip breaking. Our generalized Lagrangian correspondences at the free ends will generally not intersect transversely, but only cleanly in the sense of Pózniak [Poź99].
Let Z be the quilted surface that will be involved in defining the map α N : its total space is Z \ {0, i}, with a vertical seam at s = 0. We view it as a quilted surface with one incoming end at s → −∞, one outgoing end at s → +∞, and two free quilted strip-like ends near 0 and i. The quilted surface Z has two patches Z − = {s + it : 0 ≤ t ≤ 1, s ≤ 0}, and (4.75)   (4.79) (du P − K P ) 0,1 = 0, with limits x (resp. y) at s → −∞ (resp. s → +∞), and with unprescribed limits at the free quilted strip-like ends (or equivalently of finite energy). The superscript 0, 1 stands for the anti-holomorphic part, with respect to the almost complex structure J P . Let L(x, y; P N ) i ⊂ L(x, y; P N ) denote the subspace of quilts of index i, corresponding to the index of the relevant Fredholm section ∂ P N cutting out L(x, y; P N ). Since we do not perturb at the free end, which is in clean intersection, one should use weighted Sobolev spaces there, as in [LL13, sec. 2.3]: we refer the reader to this for details.  of regular perturbation datum P N for which L(x, y; P N ) i is smooth and of dimension i. The zero-dimensional part L(x, y; P N ) 0 is compact and can be used to define a chain map α N : CF N → CF N +1 . and the compactification L(x, y; P N ) 1 of its one-dimensional part L(x, y; P N ) 1 can be used to show that it is actually a chain map. Therefore, one can apply the telescope construction of Section 2.2 and define Proof. The first part of the statement (smoothness and expected dimension) follows from standard transversality arguments [FHS95]. Compactness of L(x, y; P N ) 0 follows from Gromov compactness. About the compactification L(x, y; P N ) 1 , a priori the degenerations that one could observe are (see Figure 6): (1) sphere bubbling in the interior of the patches, or disc bubbling at the true boundary component, (2) quilted sphere bubbling at the vertical seam, (3) strip breaking at the free ends, (4) strip breaking at the incoming or outgoing end. We show that the three first cases are ruled out by our assumptions. It will follow that the fourth case is the only one that can actually happen, and therefore (4.82) ∂L(x, y; from which the identity ∂α N + α N ∂ = 0 follows.
The first kind of degenerations are clearly ruled out by our assumptions, for the same reasons as they are ruled out from the moduli space of the differential.
Assume we have a quilted sphere bubble: we can fold it to a disc One can find another disc d in Λ N whose boundary coincides with the one of u. Indeed, one can lift u to a disc Then by glueing u and d together along their boundary we get a sphere u cap in M − N × M N +1 . As u is homotopic to the internal connected sum d#u cap , from Formula (4.29) and µ Λ N (d) = 0 we get that µ Λ N (u) = 2c 1 (u cap ). Since u is a nonconstant pseudo-holomorphic disc, it has positive area, and u cap has same area. By monotonicity and the minimal Maslov number assumption, one must have µ Λ N (u) ≥ A ≥ 3, which forces the principal component to live in a moduli space of negative dimension, which should be empty by transversality.
In the exact case, u cap has zero area, therefore u is constant. Suppose now that one has a strip breaking at the free end: by folding it one can view it as a strip Lift it to a strip Now we claim we can find another strip that coincides withũ on ∂ 0 Z, is entirely contained in ∆ M × N Γ(ι N ) , and such that ∂ 1d is in the intersection ∆ Li × Γ(ι N ). Indeed, just take d 1 M = d 2 M = u 1 M , and for d N , d N +1 just homotope ∂ 0 u N , ∂ 0 u N +1 to the zero section: Now, just like we did to exclude the previous bubbling, glued andũ along their ∂ 0 Z boundary to get a discũ cap with boundary in L i × L i × 0 N × 0 N +1 . By construction, u,ũ andũ cap have same symplectic area and Maslov index.
In the monotone setting, u has index greater that A, which would force the principal component to live in a moduli space of negative dimension, empty for transversality reasons.
In the exact setting,ũ cap must have zero area, which contradicts u being nonconstant.
Remark 4.37. In [KLZ19], The authors define equivariant self-Floer homology of a Lagrangian using similar symplectic homotopy quotients: they consider M × G µ −1 T N (0), which fibers over B N with fibers M and therefore admits a symplectic structure, by Thurston's theorem. We believe this space is equivalent to ours, i.e. the symplectic structure can be chosen so that it is symplectomorphic to M N . However, the increment maps are constructed differently (they don't use quilts), and it is unclear to us whether these give equivalent constructions. 4.6. Gradings. Depending on the setting, the Floer complex may admit some grading. In the case when Lagrangians are oriented, comparing the orientation of the direct sum T x L 0 ⊕ T x L 1 with the one of T x L at a transverse intersection point x provides an absolute Z 2 grading on CF (M ; L 0 , L 1 ).
If M, L 0 and L 1 are simply connected (and hence orientable, but not necessarily oriented), then CF (M ; L 0 , L 1 ) can be endowed with a relative grading over Z A , with A ∈ Z as in Assumption 4.20, or A = +∞ in the exact setting. Relative means that for x, y generators of CF (M ; L 0 , L 1 ), one has a number I(x, y) ∈ Z A corresponding to the difference in degrees. This will be the case in Section 9 (Manolescu and Woodward's setting).
In the above two cases, it is clear that M N , L N 0 , L N 1 will satisfy similar assumptions, and hence CF (M N ; L N 0 , L N 1 ) and their telescope will inherit the same kind of absolute or relative gradings.
It is usually convenient to endow the Lagrangians with the structure of a grading [Kon95,Sei00], in order to get a refined absolute grading on the Floer complex: This definition is motivated by the following immediate result: Proposition 4.40. Assume that the action of G on M is regular. Then the datum of a n-fold Maslov G-covering L n G of M is equivalent to a n-fold Maslov covering L n of M/ /G. If such a datum is given, a (L n G , G)-grading on L is equivalent to a L n -grading on L/G.  If L ⊂ M is a Lagrangian transverse to the fibers at every point, then the section s L takes its values in L , and admits a unique lift contained in L 0 , which is defined to be its canonical grading.
If now X is acted on by G (inducing a Hamiltonian action on M ), then L G = c −1 (L G ) is a Maslov G-cover as defined previously. With

Continuation maps
In this section we aim to apply Corollary 2.12 to show that the homotopy type of the equivariant Floer complex is independent on the choice of Floer and perturbation data.
For each N , with i = 0, 1, let F i N be a regular Floer datum for (M N ; L N 0 , L N 1 ), and Let also P i N be regular perturbation datum for (Z; M N , L N ), defining increment morphisms: We are going to introduce perturbation data, as summarized in the following diagram: These will be used to define maps: Let Q N be a perturbation datum on (Z; M N ; L N 0 , L N 1 ) going from F 0 N to F 1 N . Use it to define a moduli space C(x, y; Q N ) of Q N -perturbed pseudo-holomorphic maps. For generic Q N the moduli space is smooth and of expected dimension. Its zero dimensional part C(x, y; Q N ) 0 defines a chain map In order to prove that ϕ N commutes with α N , β N up to homotopy we define a parametrized moduli space, involving a family {R L N } L∈R of perturbations on (Z, M N , L N ) such that (see Figure 8): • if L 0, then R L N corresponds to the superposition of Q N shifted by L in the region s < L/2 of Z and P 1 N in s > L/2. Notice that when L is negatively large enough, these coincide with F 1 N at s = L/2. • if L 0, then R L N corresponds to the superposition of P 0 N shifted by L in the region s < L/2 of Z and Q N +1 in s > L/2. Notice that when L is large enough, these coincide with F 0 N +1 at s = L/2.  Its zero-dimensional part can be used to define a map (5.10) and its one-dimensional part can be used to show that κ N gives a homotopy Therefore one gets a continuation morphism be the moduli space associated with S v N , and let (5.14) be the corresponding parametrized moduli space.
is smooth and of expected dimension.
Its zero-dimensional part C(x, y) 0 = v∈[0,1] C v (x, y) −1 defines φ N : C N → D N , and its one-dimensional part C(x, y) 1 = v∈[0,1] C v (x, y) 0 compactifies to a cobordism that gives the relation compactifies to a cobordism that gives the relation: Therefore, by Proposition 2.11, different paths of perturbations induce the same maps on telescopes, up to homotopy.
To show that furthermore these are homotopy equivalences, by Corollary 2.12, one has to check that if (ϕ N , κ N ) (resp. (φ N ,κ N )) as in Proposition 5.1 are from (C N , α N ) to (D N , β N ) (resp. from (D N , β N ) to (C N , α N )), then their composition  (φ N ,κ N )).
Let {U v N } v∈[0,+∞) be a family of perturbations on (Z, M N , L N 0 , L N 1 ) going from F 0 N to F 0 N , and such that : it is a superposition of Q N andQ N shifted by v, as in Figure 10.
These define moduli spaces D v (x, y; U v N ) and a corresponding parametrized moduli space whose zero dimensional component defines the map φ N , and one-dimensional part serves to prove the Relation (5.20).
To construct H N and prove the Relation (5.21) we define a two-parameter family of perturbations on (Z, M N , L N ) parametrized by a square, and obtained by patching together five families V 1 Figure 11): (1) V 1 N is parametrized by (v, δ 1 ) ∈ [0, +∞) × [K, +∞) and corresponds to the superposition of U v N and P 0 N spaced by δ 1 , (2) V 2 N is parametrized by (L, δ 2 ) ∈ R × [K, +∞) and corresponds to the superposition of Q N andRL N spaced by δ 2 , (3) V 3 N is parametrized by (L, δ 3 ) ∈ R × [K, +∞) and corresponds to the superposition of R L N andQ N +1 spaced by δ 3 , (4) V 4 N is parametrized by (v, δ 4 ) ∈ [0, +∞) × [K, +∞) and corresponds to the superposition of P 0 N and U v N +1 and spaced by δ 4 , These "fill in" the four 2-cells in Diagram (5.22) (after removing the arrows R L N andRL N ). Notice that these four families have some overlaps as indicated in Figure 11 : for example V 1 N and V 2 N coincide when v = δ 2 and δ 1 = −L. Therefore one can patch together their parameter spaces as in Figure 12 to obtain a square with a smaller square at its center removed. Let then V 5 N be any family that extends smoothly the boundary of this central square: use it to fill the middle square. We then get a family {V w N } w∈[0,1] 2 that permits to define moduli spaces E w (x, y; V w N ) and Use their zero dimensional part to define H N , and get (5.21) from its one dimensional part: each side of the square giving respectively the summands α N φ N , κ N φ N ,φ N +1 κ N , φ N +1 α N , and the right hand side dH N + H N d corresponds to strip breaking at interior points.

Self-Floer homology and Morse homology
In this section we prove Theorem B (except for the H * (BG)-bimodule part, which will be proved in Proposition 7.5). To do so, we use its well-known nonequivariant analogue, involving the Piunikhin-Salamon-Schwarz isomorphism, and show that these maps commute with the increments up to homotopy. Such isomorphisms first appeared in [PSS96] for Hamiltonian Floer homology, and were extended to the Lagrangian setting in [Alb08]. We briefly review their construction, and refer to the later reference for more details.
Let γ : R ≥0 → L, (6.8) such that: • u is a Q-holomorphic curve, with Q a regular perturbation datum on Z from F to (H = 0, J = J 0 ), with J 0 a constant almost complex structure. • γ is a flowline for v.
• Fix first a one parameter family of perturbations R L N , L < 0, on Z[ψ(L)] (i.e. Z with the vertical seam at s = ψ(L)) such that when L 0, R L N is as in Figure 15 • (u N , u N +1 ) is a R L N -holomorphic quilt, with the obvious boundary and seam conditions Proposition 6.1. For regular perturbations, X (x, y) is smooth of expected dimension, its zero dimensional part defines a map κ N , and its one-dimensional part permits to prove (6.13). Therefore one has a well-defined map between telescopes Likewise, one can define analogously maps SSP N commuting with α N , j N up to dκ N + κ N d, and a map These two maps are inverse of each other up to homotopy.
Proof. The proof is the same proof that P SS − SSP = Id + dH + Hd, upgraded to the telescopic setting, in an analogous way to what we did in Section 5.

Bimodule structure
Recall that equivariant cohomology H * G (X) has a H * (BG)-module structure. By combining the (pair of pants) product structure on Floer cohomology with the Lagrangian correspondence between M N and B N we define a chain level bimodule structure analogous to it. Let One can then define maps defined by counting quilts as in the left side of Figure 16.
Remark 7.1. By stretching the quilt as in the right side of Figure 16, one can show that L N (resp. R N ) is obtained by composing the morphism ) induced by P N 0 (resp. P N 1 ) with the pair of pants product Remark 7.2. In Morse homology, these quilts correspond to grafted trees as in Figure 17.
Furthermore, these maps commute with the increment maps, respectively up to homotopies dκ L N + κ L N d and dκ R N + κ R N d, and therefore induce maps between telescopes Proof. Let us first prove (7.10) (the proof of (7.11) is similar). This can be done, at first glance, by considering the deformation suggested in Figure 18: one defines a moduli space Care must be taken however, since when t = 0 one has a seam condition tangent to the boundary condition, so the puncture is not a strip like end. To remedy this we instead consider and at t = − we continue the moduli space by stretching away the puncture as in Figure 19, i.e. we glue to it a parametrized moduli space It compactifies to a moduli space A by adding strip breaking at the ends at finite L (giving some homotopy terms) and when L → +∞ by adding A ∞ consisting in the picture drawn in Figure 19, with possibly a quilted disc bubble attached to it. Likewise, at t = we do a similar stretching (see Figure 20) and get another parametrized moduli space    The fact that the left and right actions commute up to homotopy is a straightforward deformation argument suggested in Figure 22.  The proof that L N and R N commute with the increment maps up to homotopies is a similar deformation argument, suggested in Figure 23. Again there can be problems at two times: • Between steps (2) and (3) when the two seams come together: different strip-like ends have to come together, so the strip-like end structure wouldn't be fixed. Therefore, as in the previous argument, we first stretch the free ends away as in Figure 24. Then one can shrink the width between the two seams and replace the two seams by a single one decorated by the composition Λ B N • P N +1 1 , with Λ B N = N Γ(i N ) /G 2 . As this composition is embedded and equals to P N 1 • Λ N , one can continue the moduli space as drawn in the picture.
• Between steps (3) and (4), this is analogous to what happened in the proof of (7.10).
Along this process, several kinds of bubblings/breakings might a priori occur, in addition to the ones that we already ruled out. We now detail each of these: • At step (5) of Figure 24, some figure 8 bubbling can occur in the stripshrinking process, from Bottman's removal of singularity theorem [Bot20]. These are drawn in Figure 25 (either the left or right bubble, depending on whether one approaches step (5) from step (4) or step (6)). As explained in Figure 25, such bubbles, after lifting and unfolding, give rise to a quilted sphere (i.e. a disc in T N × T N +1 with boundary in N Γ(i N ) ), and a disc in M with boundary in L 1 . breaking, corresponding to step (3). After lifting and unfolding, one gets a quilted disc and a disc in M with boundary in L 1 . By folding the quilted disc along the seam between T N and T N +1 , one gets a disc in T N × T N +1 with one boundary in 0 N × 0 N +1 , and another boundary in N Γ(i N ) ). As in the proof of Proposition 4.36, one can cap the part of the boundary in N Γ(i N ) with a disc in N Γ(i N ) , and get a disc with the whole boundary in 0 N × 0 N +1 . For case (A), we cap the quilt with a strip in 0 N as shown in the picture, so that the true boundary is mapped to a constant value (this is possible since 0 N is simply connected). This gives a quilted sphere, which folds to a disc in T N × T N +1 with boundary in N Γ(i N ) .
For case (B), the quilted part is different, with one boundary in 0 N ×0 N +1 and another in N Γ(i N ) . We cap the part in N Γ(i N ) with a strip in N Γ(i N ) , such that the true boundary gets mapped in Γ(i N ) ⊂ 0 N × 0 N +1 . We therefore end up with a disc in T N × T N +1 with boundary in 0 N × 0 N +1 . The same arguments as in the proof of Proposition 4.36 permit to show that all these bubblings/breakings are excluded by our assumptions. This ends the proof.
Proposition 7.5. The isomorphisms P SS G and SSP G defined in Section 6 preserve the H * (BG)-bimodule structures.
Proof. This is a routine deformation argument, using the deformation of Figure 28   For N ≥ 1, one has a sequence of Lagrangian correspondences: . This sequence of correspondences induces morphisms  = 0, 1). Notice that by unfolding the patches in M × T N , one can alternatively view these quilts as "foams" as in the right side of Figure 29. As usual, one can show that these commute with the increment maps up to homotopies dκ N + κ N d and dκ N + κ N d, and therefore induce maps between telescopes: Remark 8.2. In the setting of Section 6, where G acts on a smooth manifold X, M = T * X and L 0 = L 1 = 0 X , if furthermore the action of G on X is free, then the action on M is regular. And under the PSS isomorphisms, the morphisms K, K correspond at the homology level to the Cartan isomorphisms H G (X) H(X/G). One can ask whether K, K induce isomorphisms more generally. Notice that this would not contradict the non-injectivity of the classical Kirwan maps (at least in the obvious way) since under the PSS isomorphisms K does not correspond to a Kirwan map.

Equivariant Manolescu-Woodward's Symplectic Instanton homology
We now apply our construction to the setting of Symplectic Instanton Homology defined by Manolescu and Woodward [MW12], which is a slightly more involved one from the monotone setting of Section 4.4. We start by quickly reviewing their construction, and refer to [MW12] for more details.
9.1. Manolescu-Woodward's Symplectic Instanton homology. Let Σ be a connected oriented surface of genus g with one boundary component, and let G = SU (2) throughout this section. Associated to it is the extended moduli space defined by Jeffrey [Jef94] (M, ω) = M g (Σ ). This space is a (singular, degenerated) symplectic manifold with a Hamiltonian G-action with moment map µ. The important features of it is that it is smooth and nondegenerated near µ −1 (0), and its symplectic quotient is identified with the (singular) Atiyah-Bott flat moduli space M (Σ) of Σ, the closed surface obtained by capping Σ with a disc. If Y = H 0 ∪ Σ H 1 is a closed oriented connected 3-manifold with Σ as a Heegaard splitting, then associated to the two handlebodies is a pair of smooth G-Lagrangians L 0 , L 1 ⊂ M g (Σ ).
The Atiyah-Floer conjecture states that the instanton homology of Y should correspond to HF (M (Σ); L 0 /G, L 1 /G), wich is ill-defined since M (Σ), L 0 /G and L 1 /G are singular. However, (after a cutting construction on M g (Σ ) that we will outline) Manolescu and Woodward succeeded in defining HF (M g (Σ ); L 0 , L 1 ) and suggested to define HF G (M g (Σ ); L 0 , L 1 ), as an equivariant symplectic side for the Atiyah-Floer conjecture.
9.2. Equivariant Symplectic Instanton homology. Our strategy will be to apply their construction to the symplectic homotopy quotients of M , rather than to M . This last moduli space still has a projection π c N : N c N → B N , with fibers N c , and a fiber-preserving action of G.
Let > 0, and suppose we are given φ : M ≤1/2 → M ≤1/2− as above, and such that it is the identity on W := M <1/2−2 , and G-equivariant. Such a diffeomorphism can be constructed using the gradient flow ofμ N with respect to a G-invariant metric. Define then Fix a ground almost complex structure J N on N N such that R N is an almost complex submanifold w.r.t. J N , and such that the projection π N is J N -holomorphic (w.r.t a fixed almost complex structure on B N ). Let J N be the set of almost complex structures on N N that are equal to J N outside a compact subset of W N .
To the handlebodies of the Heegaard splitting Y = H 0 ∪ Σ H 1 is associated a pair of G-Lagrangians L 0 , L 1 ⊂ µ −1 (0) ⊂ M . These are simply connected (and therefore monotone), spin, and their minimal Maslov number is equal to 2N M . So we get two Lagrangians in N c N disjoint from R N : Let then CF N = CF (N c N ; L N 0 , L N 1 ; R N ) stand for the Floer complex of L N 0 , L N 1 relative to R N , i.e. its Floer differential is defined by counting strips u with intersection number u.R N = 0, for a generic almost complex structure J in J N (see [MW12, Section 2.2] for more about relative Lagrangian Floer homology). By positivity of intersection, such curves are actually disjoint from R N . The constructions in the previous sections carry through this setting: Theorem 9.1. As defined above, CF N is a chain complex (i.e. ∂ 2 = 0). Moreover, the increment α N : CF N → CF N +1 defined as in Section 4.5 (and counting quilts not intersecting R N and R N +1 ) are chain morphisms. Let then CSI G (Y ) = Tel(CF N , α N ), and HSI G (Y ) its homology group. It is relatively Z 8 -graded, and in the case when Y is a rational homology sphere, an absolute Z 8 -grading can be fixed canonically. Furthermore, it has the structure of an H * (BG)-module, as defined in Section 7.
Proof. We need to make sure that moduli spaces of curves behave the same way as in the previous sections, i.e. no new bubbling or breaking arise.
Suppose b is a bubble/breaking arising in a moduli space relevant to the theorem (either a strip for ∂, a quilted strip for α N , or a quilted pair of pants for the module structure). If b has nonzero area for the monotone formω N , then it is ruled out for exactly the same reasons as previously (it would have an index too large, forcing the principal component to live in a moduli space of negative dimension). Therefore, assume that b has zeroω N -area, and is therefore contained in R N .
Since the Lagrangians L N 0 , L N 1 are disjoint from R N , this rules out any degeneration having boundary or seam conditions in these: disc bubbling, quilted strip breaking, or quilted sphere bubbling at the seam for the module structure (which is decorated by (∆ T N × L i )/G 2 ). One is left with sphere bubbling in M , and quilted sphere bubbling at the seam for the increment moduli spaces.
Assume first that b is a sphere bubbling. Since b is in R N and that the almost complex structure equals J N , its image by π c N in B N is a pseudo-holomorphic sphere, and therefore is constant. Therefore b is included in a fiber of π c N , and the same reasoning as in [MW12,Prop. 2.10] applies, which we briefly sketch for the reader's convenience. The bubble b must have intersection number with R N at most −2, this would force the principal component (which generically intersects R N transversely) to intersect R N at points where no bubbles are attached, and this is impossible for a limit of curves disjoint from R N .
Assume then that b is a quilted sphere bubble at the seam of the increment, i.e. = (d N , d N +1 ) consists in two discs in R N , R N +1 satisfying a seam condition in the correspondence Λ N . This projects to a quilted sphere in B N , B N +1 , i.e. a disc in (B N × B N +1 , N Γ(i N ) /G 2 ), which is constant. Therefore d N and d N +1 are contained in fibers of π c N , π c N +1 . These two fibers can both be identified to N c , and under this identification and possibly after multiplying by an element in G, d N and d N +1 glue together to a sphere in R, which must have intersection with R less than −2, which leads to the same contradiction as for sphere bubbling.
About gradings, Manolescu and Woodward showed [MW12, Corollary 3.6] that the minimal Chern number of N is a positive multiple of 4, which implies that the minimal Maslov numbers of L 0 and L 1 (and therefore of L N 0 and L N 1 ) are positive multiples of 8. If Y is a rational homology sphere, by [AM90, Prop. III.1.1.(c)] L 0 and L 1 intersect transversely at the trivial representation: this fact was used by Manolescu and Woodward to define an absolute grading on HSI(Y ). It follows that L N 0 and L N 1 intersect cleanly along a copy of 0 B N . One can then perturb this intersection by a Morse function of 0 B N , and take the Morse indices as an absolute grading for the corresponding intersection points.
Remark 9.2. By following the same proof as in [MW12, Section 6], one should be able to show that HSI G (Y ) is independent on the choice of Heegaard splitting Σ. This would amount to defining a quilted version of equivariant Lagrangian Floer homology, and proving an equivariant analogue of Wehrheim and Woodward's geometric composition theorem (which, for our construction, should be a straightforward consequence of the non-equivariant version). Of course, another way of proving independence would be through an equivariant version of the Atiyah-Floer conjecture: we expect HSI G (Y ) to be isomorphic to a version of equivariant instanton homology defined in [Mil19] (or something similar, Miller defined SO(3)-equivariant versions, while ours is SU(2)-equivariant), and [DM] prove topological invariance of it.