# Equivariant Lagrangian Floer homology via cotangent bundles of $EG_{N}$

Guillem Cazassus  Mathematical institute, University of Oxford, Oxford OX2 6GG
<EMAIL_ADDRESS>
###### Abstract.

We provide a construction of equivariant Lagrangian Floer homology
$HF_{G}(L_{0},L_{1})$, for a compact Lie group $G$ acting on a symplectic
manifold $M$ in a Hamiltonian fashion, and a pair of $G$-Lagrangian
submanifolds $L_{0},L_{1}\subset M$.

We do so by using symplectic homotopy quotients involving cotangent bundles of
an approximation of $EG$. Our construction relies on Wehrheim and Woodward’s
theory of quilts, and the telescope construction.

We show that these groups are independent in the auxilliary choices involved
in their construction, and are $H^{*}(BG)$-bimodules. In the case when
$L_{0}=L_{1}$, we show that their chain complex $CF_{G}(L_{0},L_{1})$ is
homotopy equivalent to the equivariant Morse complex of $L_{0}$.

Furthermore, if zero is a regular value of the moment map $\mu$ and if $G$
acts freely on $\mu^{-1}(0)$, we construct two “Kirwan morphisms” from
$CF_{G}(L_{0},L_{1})$ to $CF(L_{0}/G,L_{1}/G)$ (respectively from
$CF(L_{0}/G,L_{1}/G)$ to $CF_{G}(L_{0},L_{1})$).

Our construction applies to the exact and monotone settings, as well as in the
setting of the extended moduli space of flat $SU(2)$-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.

This work was funded by EPSRC grant reference EP/T012749/1.

###### Contents

  1. 1 Introduction
    1. 1.1 Outline of the construction
    2. 1.2 Statement of results
    3. 1.3 Organization of the paper
  2. 2 Telescopes
    1. 2.1 Spaces
    2. 2.2 Chain complexes
  3. 3 Equivariant Morse homology
    1. 3.1 Morse homology
    2. 3.2 Pushforwards on Morse homology
    3. 3.3 The equivariant Morse complex
  4. 4 Equivariant Lagrangian Floer homology
    1. 4.1 Hamiltonian actions
    2. 4.2 Symplectic homotopy quotients
    3. 4.3 Increment correspondences
    4. 4.4 Exact setting
    5. 4.5 Monotone setting
    6. 4.6 Construction
    7. 4.7 Gradings
  5. 5 Continuation maps
  6. 6 Self-Floer homology and Morse homology
    1. 6.1 PSS isomorphisms
  7. 7 Bimodule structure
  8. 8 Relations with the symplectic quotient (Kirwan maps)
  9. 9 Equivariant Manolescu-Woodward’s Symplectic Instanton homology
    1. 9.1 Manolescu-Woodward’s Symplectic Instanton homology
    2. 9.2 Equivariant Symplectic Instanton homology

##  1\. Introduction

Lagrangian Floer homology is a group $HF(M;L_{0},L_{1})$111It 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.
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

(1.1) $HF(\mathscr{M}(\Sigma);\mathscr{L}(H_{0}),\mathscr{L}(H_{1}))$

in the Atiyah-Bott moduli space $\mathscr{M}(\Sigma)$ of flat SU(2)-connexions
(its SU(2)-character variety) of a Heegaard splitting
$Y=H_{0}\cup_{\Sigma}H_{1}$. Unfortunately, this moduli space is singular (as
well as the Lagrangians $\mathscr{L}(H_{0}),\mathscr{L}(H_{1})$): this is the
main source of difficulties of this conjecture, its “symplectic side”
$HF(\mathscr{M}(\Sigma);\mathscr{L}(H_{0}),\mathscr{L}(H_{1}))$ is currently
undefined. Nevertheless, this conjecture has been studied extensively, and
established in some settings where all moduli spaces are smooth [DS94, Sal95,
Weh05b, Weh05a, SW08, Dun13, DF18, Xu20, DFL21].

Yet, Jeffrey [Jef94] and Huebschmann [Hue95] observed that the moduli space
$\mathscr{M}(\Sigma)$ can be realized as the symplectic reduction of a smooth
SU(2)-Hamiltonian space: (an open subset $\mathscr{N}$ of) the so-called
extended moduli space. Moreover it contains smooth SU(2)-Lagrangians
$L_{0},L_{1}$ such that $\mathscr{L}(H_{0})=L_{0}/G$ and
$\mathscr{L}(H_{1})=L_{1}/G$. Manolescu and Woodward [MW12] defined symplectic
instanton homology $HSI(Y)$ as a (non-equivariant) Lagrangian Floer homology
in $\mathscr{N}$, and suggested that a good candidate for the symplectic side
of the Atiyah-Floer conjecture would be an _equivariant_ version
$HF_{SU(2)}(\mathscr{N};L_{0},L_{1})$ of $HSI(Y)$ (as a substitute for
$HF(\mathscr{M}(\Sigma);\mathscr{L}(H_{0}),\mathscr{L}(H_{1}))$).

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 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 well-suited for relating equivariant invariants to
invariants of the symplectic quotient [TX17, Woo15a, Woo15b, Woo15c, NWZ14],
in analogy with the Kirwan map [Kir84].

  * •

In [SS10], Seidel and Smith defined a version of equivariant Lagrangian Floer
homology for symplectic involutions ($G=\mathbb{Z}_{2}$). Generalizations to
finite group actions appeared in [HLS16, HLS20a, BH21, CH17].

  * •

In [HLS20b], Hendricks, Lipshitz and Sarkar define an equivariant Lagrangian
Floer homology for Lie group actions. Their construction involves the theory
of $(\infty,1)$ 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 4.38) 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 another construction for equivariant Lagrangian
Floer homology for Hamiltonian action of a compact Lie group. Our construction
applies to Manolescu and Woodward’s setting (Section 9, this gives an
equivariant symplectic side for the Atiyah-Floer conjecture) and 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 need 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

(1.2) $H_{*}^{G}(X)=H_{*}(X\times_{G}EG),$

and using a finite dimensional smooth approximation $\\{EG_{N}\\}_{N}$ of

(1.3) $EG=\lim_{\overrightarrow{\ N\ }}{EG_{N}},$

with inclusions $i_{N}\colon EG_{N}\to EG_{N+1}$, one can write

(1.4) $H_{*}^{G}(X)=\lim_{\overrightarrow{\ N\ }}H_{*}(X\times_{G}EG_{N}).$

Now pick a Morse function $f_{N}$ on $X\times_{G}EG_{N}$ for each $N$, one can
now write it as a limit of Morse homologies:

(1.5) $H_{*}^{G}(X)=\lim_{\overrightarrow{\ N\
}}HM_{*}(X\times_{G}EG_{N},f_{N}),$

using Morse-theoretic pushforwards of

(1.6) $id_{X}\times_{G}i_{N}\colon X\times_{G}EG_{N}\to X\times_{G}EG_{N+1}.$

Now if $X$ is a $G$-manifold, then $M=T^{*}X$ is a Hamiltonian $G$-manifold,
and its zero section $L_{0}=L_{1}=0_{X}\subset T^{*}X$ is a $G$-Lagrangian. In
analogy with the isomorphism $HF(T^{*}X;0_{X},0_{X})\simeq HM(X)$, one can
define the equivariant Lagrangian Floer homology of $(T^{*}X;0_{X},0_{X})$ by:

(1.7) $\displaystyle HF^{G}(T^{*}X;0_{X},0_{X})$ $\displaystyle=HM_{*}^{G}(X)$
$\displaystyle=\lim_{\overrightarrow{\ N\
}}HF(T^{*}(X\times_{G}EG_{N});0_{X\times_{G}EG_{N}},0_{X\times_{G}EG_{N}}).$

It turns out that

(1.8) $\displaystyle T^{*}(X\times_{G}EG_{N})$ $\displaystyle=(T^{*}X\times
T^{*}EG_{N})/\\!\\!/G\text{, under which:}$ (1.9) $\displaystyle
0_{X\times_{G}EG_{N}}$ $\displaystyle=(0_{X}\times 0_{EG_{N}})/G.$

Therefore (1.7) provides a general definition for $HF^{G}(M;L_{0},L_{1})$ for
more general $G$-Hamiltonian triples $(M;L_{0},L_{1})$:

(1.10) $HF^{G}(M;L_{0},L_{1})=\lim_{\overrightarrow{\ N\ }}HF((M\times
T^{*}EG_{N})/\\!\\!/G;(L_{0}\times 0_{EG_{N}})/G,(L_{1}\times 0_{EG_{N}})/G),$

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 $\mathbb{Z}_{2}$-coefficients to avoid sign
discussions (but we believe that everything should work with $\mathbb{Z}$
coefficients as well, provided Lagrangians are endowed with equivariant
relative Pin-structures). The following statement summarizes Propositions
4.36, 4.43,5.1, 5.2, 5.3 and 7.3.

######  Theorem A.

The construction outlined in Section 1.1 can be implemented in the exact and
monotone setting (Assumptions 4.17, 4.20): at the chain level, it defines a
_telescope_

(1.11) $CF_{G}(M;L_{0},L_{1})=\mathrm{Tel}(CF_{N},\alpha_{N}),$

With $CF_{N}$ the Floer chain complex in $(M\times T^{*}EG_{N})/\\!\\!/G$ and
$\alpha_{N}\colon CF_{N}\to CF_{N+1}$ chain morphisms induced by the
inclusions $i_{N}\colon EG_{N}\to EG_{N+1}$. Its homotopy type is independent
on the auxiliary choices involved in the constructions (Hamiltonian
perturbations, almost complex structures). Its associated homology group
$HF_{G}(M;L_{0},L_{1})$ is an $H^{*}(BG)$-bimodule.

If furthermore, in the sense of Definition 4.40, $M$ admits a $\mathbb{Z}_{n}$
Maslov $G$-cover $\widehat{\mathcal{L}}_{G}$ of its Lagrangian grassmannian
bundle, and $L_{0}$ and $L_{1}$ admit a
$(\widehat{\mathcal{L}}_{G},G)$-grading. Then $CF_{G}(M;L_{0},L_{1})$ has an
absolute grading over $\mathbb{Z}_{n}$.

The next three theorems are likely to hold for the other constructions of
equivariant Lagrangian Floer homology, but are new to our best knowledge.

In the case when the Lagrangians coincide, we prove the following, using the
Piunikhin-Salamon-Schwarz construction [PSS96]:

######  Theorem B.

Let $L\subset M$ 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 $CM_{G}(X)\simeq
CF_{G}(T^{*}X;0_{X},0_{X})$.

When the symplectic quotient is nice enough, we define “Kirwan morphisms”
relating equivariant Floer homology with Floer homology of the quotient:

######  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:

(1.12) $\displaystyle K$ $\displaystyle\colon CF_{G}(M;L_{0},L_{1})\to
CF(M/\\!\\!/G;L_{0}/G,L_{1}/G)$ (1.13) $\displaystyle K^{\prime}$
$\displaystyle\colon CF(M/\\!\\!/G;L_{0}/G,L_{1}/G)\to CF_{G}(M;L_{0},L_{1})$

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
$M=\mathscr{N}(\Sigma^{\prime})$ 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
$\mathbb{Z}_{8}$-graded $H^{*}(BSU(2))$-module, and in the case when $Y$ is a
rational homology sphere, an absolute $\mathbb{Z}_{8}$-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].

###  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.

######  Acknowledgments.

I would like to thank Ciprian Manolescu and Chris Woodward for suggesting this
project many years ago, as well as Frédéric Bourgeois, Paolo Ghiggini, Dominic
Joyce, Paul Kirk, Artem Kotelskiy, Ciprian Manolescu, Eckhard Meinrenken, Mike
Miller-Eismeier, Alex Oancea, James Pascaleff, Alex Ritter, Matt Stoffregen,
Claude Viterbo, Chris Woodward, Guangbo Xu and Wai-Kit Yeung for helpful
conversations. I also thank Julio Sampietro for pointing out a minor mistake
in the definition of $\Lambda_{N}$ in a previous version.

##  2\. 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.

###  2.1. Spaces

######  Definition 2.1 (Telescopes).

Let $(X_{N},a_{N})_{N\geq 0}$ be a sequence of spaces, with _increment_ maps

$a_{N}\colon X_{N}\to X_{N+1}.$

The _mapping telescope_ of this sequence is defined by

$\mathrm{Tel}(X_{N},a_{N})=\left(\mathbin{\coprod}_{N}X_{N}\times
I_{t}\right)/(x,1)\sim(a_{N}(x),0)$

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
$a\colon\mathrm{Tel}(X_{N},a_{N})\to\mathrm{Tel}(X_{N},a_{N})$ 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 $(a^{u})_{0\leq u\leq 1}$ defined by:

$a^{u}(x,t)=\begin{cases}(x,t+u)&\text{ if }t\leq 1-u,\\\
(a_{N}(x),t+u-1)&\text{ if }t\geq 1-u.\end{cases}$

######  Definition 2.2 (Maps between Telescopes).

Let $(X_{N},a_{N})_{N\geq 0}$ and $(Y_{N},b_{N})_{N\geq 0}$ be such sequences
of spaces, consider a sequence of maps

$f_{N}\colon X_{N}\to Y_{N}$

that commute with the increments $a_{N},b_{N}$ up to homotopy, in the sense
that there exists maps $k_{N}\colon X_{N}\times I_{t}\to Y_{N+1}$ such that

$\displaystyle k_{N}(\cdot,0)$ $\displaystyle=b_{N}f_{N}$ $\displaystyle
k_{N}(\cdot,1)$ $\displaystyle=f_{N+1}a_{N}.$

In this setting one can define a map

$\mathrm{Tel}(f_{N},k_{N})\colon\mathrm{Tel}(X_{N})\to\mathrm{Tel}(Y_{N})$

by setting

(2.1) $\mathrm{Tel}(f_{N},k_{N})(x,t)=\begin{cases}(f_{N}(x),2t)&\text{ if
}0\leq t\leq\frac{1}{2},\\\ (k_{N}(x,2t-1),0)&\text{ if }\frac{1}{2}\leq t\leq
1.\end{cases}$

###### Proof that the map is well-defined.

These two quantities agree when $t=\frac{1}{2}$, since

$(f_{N}(x),1)=(b_{N}f_{N}(x),1)=(k_{N}(x,0),0).$

Furthermore, $\mathrm{Tel}(f_{N},k_{N})$ 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

$\mathrm{Tel}(f_{N},k_{N})(x,t)=(k_{N}(x,t),t).$

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

$f_{N}^{0},f_{N}^{1}\colon X_{N}\to Y_{N}$

that are homotopic and that commute with the increments $a_{N},b_{N}$ up to
homotopy: we are given two maps

$\displaystyle f_{N}$ $\displaystyle\colon X_{N}\times I_{u}\to Y_{N},$
$\displaystyle k_{N}$ $\displaystyle\colon X_{N}\times I_{t}\times I_{u}\to
Y_{N+1},$

satisfying:

$\displaystyle f_{N}(\cdot,0)$ $\displaystyle=f_{N}^{0},$ $\displaystyle
f_{N}(\cdot,1)$ $\displaystyle=f_{N}^{1},$ $\displaystyle k_{N}(x,0,u)$
$\displaystyle=b_{N}f_{N}(x,u),$ $\displaystyle k_{N}(x,1,u)$
$\displaystyle=f_{N+1}(a_{N}(x),u).$

Notice that this is the same setting as in the previous paragraph, replacing
$X_{N}$ by $Z_{N}=X_{N}\times I_{u}$. Therefore we get a map between
telescopes:

$\mathrm{Tel}(f_{N})\colon\mathrm{Tel}(Z_{N})\to\mathrm{Tel}(Y_{N})$

and since $\mathrm{Tel}(Z_{N})=\mathrm{Tel}(X_{N})\times I_{u}$, one can think
of $\mathrm{Tel}(f_{N})$ as a homotopy between

$\mathrm{Tel}(f_{N}^{0}),\mathrm{Tel}(f_{N}^{1})\colon\mathrm{Tel}(X_{N})\to\mathrm{Tel}(Y_{N}).$

######  Proposition 2.5 (Products).

Let $(X_{m},a_{m})$ and $(Y_{n},b_{n})$ be two sequences of spaces, then the
map

(2.2)
$\Phi\colon\mathrm{Tel}(X_{m},a_{m})\times\mathrm{Tel}(Y_{n},b_{n})\to\mathrm{Tel}(X_{m}\times
Y_{m},a_{m}\times b_{m})$

defined, with $(x,u)\in X_{m}\times[0,1)$ and $(y,v)\in Y_{n}\times[0,1)$ and
denoting

(2.3) $a^{k}=a\circ\cdots\circ a\colon X_{m}\to X_{m+k},$

by

(2.4) $\Phi((x,u),(y,v))=\begin{cases}((x,y),\max(u,v))&\text{if }m=n\\\
((a^{n-m}x,y),v)&\text{if }m<n\\\ ((x,b^{m-n}y),u)&\text{if }m>n\\\
\end{cases}$

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.

###  2.2. Chain complexes

This subsection is the algebraic transcription of the previous one.

######  Definition 2.6 (Telescopes of Chain complexes).

Let $(C_{N},\alpha_{N})_{N\geq 0}$ be a sequence of chain complexes over
$\mathbb{Z}_{2}$, with _increment_ chain morphisms

$\alpha_{N}\colon C_{N}\to C_{N+1}.$

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

$\mathrm{Tel}(C_{N},\alpha_{N})=\bigoplus_{N}C_{N}\oplus qC_{N}$

with $q$ a formal variable of degree one. If $\sigma$ is a cell of $X_{N}$ the
$C_{N}$ summand in $\mathrm{Tel}(C_{N},\alpha_{N})$ corresponds to the cells
$\sigma\times\\{0\\}$, while the $qC_{N}$ summand corresponds to the cells
$\sigma\times[0,1]$. It is endowed with the differential

$\delta(a+qb)=da+qdb+\alpha_{N}b-b$

Sometimes we may just denote it $\mathrm{Tel}(C_{N})$ when there is no
ambiguity.

######  Definition 2.7 (Morphisms between Telescopes).

Let $(C_{N},\alpha_{N})_{N\geq 0}$ and $(D_{N},\beta_{N})_{N\geq 0}$ be two
sequences as before, consider a sequence of chain morphisms

$\varphi_{N}\colon C_{N}\to D_{N}$

that commute with the increment $\alpha_{N},\beta_{N}$ up to a homotopy
$\kappa_{N}\colon C_{N}\to D_{N+1}$:

$\varphi_{N+1}\alpha_{N}-\beta_{N}\varphi_{N}=d\kappa_{N}+\kappa_{N}d$

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In this setting one can define a map
$\mathrm{Tel}(\varphi_{N},\kappa_{N})\colon\mathrm{Tel}(C_{N})\to\mathrm{Tel}(D_{N})$
by

$\mathrm{Tel}(\varphi_{N},\kappa_{N})(a+qb)=\varphi_{N}(a)+\kappa_{N}(b)+q\varphi_{N}(b).$

######  Remark 2.8.

If $a$ represents a cell in $X_{N}$, then $\mathrm{Tel}(f_{N},k_{N})$ will map
it to $(f_{N}\circ a,0)$. If $b$ represents a cell in $X_{N}$, then $qb$
corresponds to the cell $b\times I_{t}$ in $\mathrm{Tel}(X_{N})$ which by
(2.1) is mapped to the union of $(f_{N}\circ b)\times I_{t}$ and
$k_{N}(b\times I_{t})\times\left\\{0\right\\}$.

One has
$\delta\mathrm{Tel}(\varphi_{N},\kappa_{N})=\mathrm{Tel}(\varphi_{N},\kappa_{N})\delta$:

$\displaystyle\delta\mathrm{Tel}(\varphi_{N},\kappa_{N})(a+qb)$
$\displaystyle=\delta(\varphi_{N}(a)+\kappa_{N}(b)+q\varphi_{N}(b))$
$\displaystyle=d\varphi a+\left[d\kappa b+\beta\varphi b\right]+qd\varphi
b-\varphi b$ $\displaystyle=\varphi da+\left[\kappa db+\varphi\alpha
b\right]+q\varphi db-\varphi b$
$\displaystyle=\mathrm{Tel}(\varphi_{N},\kappa_{N})\delta(a+qb).$

######  Proposition 2.9 (Compositions of telescopes).

If $(\varphi_{N},\kappa_{N})$ goes from $(C_{N},\alpha_{N})$ to
$(D_{N},\beta_{N})$ and $(\chi_{N},\lambda_{N})$ goes from $(D_{N},\beta_{N})$
to $(E_{N},\gamma_{N})$, then $\chi_{N}\varphi_{N}$ commutes with
$\alpha_{N},\gamma_{N}$ up to the homotopy
$\chi_{N}\kappa_{N}+\lambda_{N}\varphi_{N}$.

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Define their composition from $(C_{N},\alpha_{N})$ to $(E_{N},\gamma_{N})$ by:

$(\chi_{N},\lambda_{N})\circ(\varphi_{N},\kappa_{N})=(\chi_{N}\varphi_{N},\chi_{N}\kappa_{N}+\lambda_{N}\varphi_{N})$

Telescopes agree with composition:

$\mathrm{Tel}\left((\chi_{N},\lambda_{N})\circ(\varphi_{N},\kappa_{N})\right)=\mathrm{Tel}(\chi_{N},\lambda_{N})\circ\mathrm{Tel}(\varphi_{N},\kappa_{N}).$

###### Proof.

(In computations we drop the subscripts $N$ for readability)

Commutativity up to homotopy:

$\displaystyle\chi\varphi\alpha-\gamma\chi\varphi$
$\displaystyle=\chi(\beta\varphi+d\kappa+\kappa d)-(\chi\beta+d\lambda+\lambda
d)\varphi$
$\displaystyle=d(\chi\kappa+\lambda\varphi)+(\chi\kappa+\lambda\varphi)d.$

Compatibility with telescopes:

$\displaystyle\mathrm{Tel}(\chi_{N},\lambda_{N})\circ\mathrm{Tel}(\varphi_{N},\kappa_{N})(a+qb)$
$\displaystyle=\mathrm{Tel}(\chi_{N},\lambda_{N})(\varphi a+\kappa b+q\varphi
b)$ $\displaystyle=\chi\varphi a+\chi\kappa b+\lambda\varphi b+q\chi\varphi b$
$\displaystyle=\mathrm{Tel}((\chi_{N},\lambda_{N})\circ(\varphi_{N},\kappa_{N}))(a+qb).$

∎

######  Definition 2.10 (Homotopies between Telescopes).

Consider now two sequences $(\varphi_{N}^{0},\kappa_{N}^{0})$ and
$(\varphi_{N}^{1},\kappa_{N}^{1})$ from $(C_{N},\alpha_{N})$ to
$(D_{N},\beta_{N})$ as before (i.e. for $u=0,1$,
$\varphi_{N+1}^{u}\alpha_{N}-\beta_{N}\varphi_{N}^{u}=d\kappa_{N}^{u}+\kappa_{N}^{u}d$).

We say that $(\varphi_{N}^{0},\kappa_{N}^{0})$ and
$(\varphi_{N}^{1},\kappa_{N}^{1})$ are _homotopic_ if there exists two
sequences of maps

$\displaystyle\phi_{N}$ $\displaystyle\colon C_{N}\to D_{N},$
$\displaystyle\kappa_{N}$ $\displaystyle\colon C_{N}\to D_{N+1},$

such that

$\displaystyle\varphi^{1}-\varphi^{0}$ $\displaystyle=d\phi+\phi d,$
$\displaystyle\kappa^{1}-\kappa^{0}+\phi\alpha+\beta\phi$
$\displaystyle=d\kappa+\kappa d.$

Let then
$\mathrm{Tel}(\phi,\kappa)\colon\mathrm{Tel}(C,\alpha)\to\mathrm{Tel}(D,\beta)$
be defined by

$\mathrm{Tel}(\phi,\kappa)(a+qb)=\phi a+\kappa b+q\phi b.$

######  Proposition 2.11.

The morphism $\mathrm{Tel}(\phi,\kappa)$ defined above gives a homotopy
between $\mathrm{Tel}(\varphi^{0},\kappa^{0})$ and
$\mathrm{Tel}(\varphi^{1},\kappa^{1})$, in the sense that

$\mathrm{Tel}(\varphi^{0},\kappa^{0})-\mathrm{Tel}(\varphi^{1},\kappa^{1})=\delta\mathrm{Tel}(\phi,\kappa)+\mathrm{Tel}(\phi,\kappa)\delta.$

###### Proof.

$\displaystyle(\delta\mathrm{Tel}(\phi,\kappa)+\mathrm{Tel}(\phi,\kappa)\delta)(a+qb)$
$\displaystyle=\delta(\phi a+\kappa b+q\phi
b)+\mathrm{Tel}(\phi,\kappa)(da+\alpha b+b+qdb)$ $\displaystyle=\left[d\phi
a+d\kappa b+\beta\phi b+\phi b+qd\phi b\right]+\left[\phi da+\phi\alpha b+\phi
b+\kappa db+q\phi db\right]$ $\displaystyle=(d\phi+\phi
d)a+(\phi\alpha+\beta\phi+d\kappa+\kappa d)b+q(\phi d+d\phi)b$
$\displaystyle=(\varphi^{1}-\varphi^{0})a+(\kappa^{1}-\kappa^{0})b+q(\varphi^{1}-\varphi^{0})b$
$\displaystyle=\left[\mathrm{Tel}(\varphi^{0},\kappa^{0})-\mathrm{Tel}(\varphi^{1},\kappa^{1})\right](a+qb).$

∎

Together with Proposition 2.9 this implies:

######  Corollary 2.12.

Let $(\varphi,\kappa)$ and $(\chi,\lambda)$ be going respectively from
$(C,\alpha)$ to $(D,\beta)$ and from $(D,\beta)$ to $(C,\alpha)$, and such
that $(\chi,\lambda)\circ(\varphi,\kappa)$ and
$(\varphi,\kappa)\circ(\chi,\lambda)$ are respectively homotopic to
$(id_{C},0)$ and $(id_{D},0)$. Then $\mathrm{Tel}(\varphi,\kappa)$ and
$\mathrm{Tel}(\chi,\lambda)$ are homotopy equivalences between
$\mathrm{Tel}(C,\alpha)$ and $\mathrm{Tel}(D,\beta)$.

$\Box$

######  Proposition 2.13 (Products).

Let $(C_{m},\alpha_{m})$ and $(D_{n},\beta_{n})$ be two sequences of chain
complexes, then the map

(2.5)
$\Phi\colon\mathrm{Tel}(C_{m},\alpha_{m})\otimes\mathrm{Tel}(D_{n},\beta_{n})\to\mathrm{Tel}(C_{m}\otimes
D_{m},\alpha_{m}\otimes\beta_{m})$

defined, with $(x+qx^{\prime})\in C_{m}[q]$ and $(y,v)\in D_{n}[q]$ and
denoting

(2.6) $\alpha^{k}=\alpha\circ\cdots\circ\alpha\colon C_{m}\to C_{m+k},$

by

(2.7) $\Phi((x+qx^{\prime})\otimes(y+qy^{\prime}))=\begin{cases}(x\otimes
y)+q(x\otimes y^{\prime}+x^{\prime}\otimes y)&\text{if }m=n\\\
\alpha^{n-m}x\otimes(y+qy^{\prime})&\text{if }m<n\\\
(x+qx^{\prime})\otimes\beta^{m-n}y&\text{if }m>n\\\ \end{cases}$

is a morphism of chain complexes.

###### Proof.

Compute first:

$\displaystyle(Id\otimes\delta+\delta\otimes
Id)((x+qx^{\prime})\otimes(y+qy^{\prime}))$ $\displaystyle=$
$\displaystyle(x+qx^{\prime})\otimes(\mathrm{d}y+q\mathrm{d}y^{\prime}+\beta
y^{\prime}+y^{\prime})+(\mathrm{d}x+q\mathrm{d}x^{\prime}+\alpha
x^{\prime}+x^{\prime})\otimes(y+qy^{\prime})$

Before applying $\Phi$, notice that this is a sum of terms of degree $(m,n)$,
except $(x+qx^{\prime})\otimes\beta y^{\prime}$ and $\alpha
x^{\prime}+\otimes(y+qy^{\prime})$, which are of degree $(m,n+1)$ and
$(m+1,n)$ respectively. We will treat the cases $m=n$ and $m<n$ separately
(the case $m>n$ is analogous)

Case $m=n$:

$\displaystyle\Phi\circ(Id\otimes\delta+\delta\otimes
Id)((x+qx^{\prime})\otimes(y+qy^{\prime}))$ $\displaystyle=$ $\displaystyle
x\otimes(\mathrm{d}y+\beta
y^{\prime}+y^{\prime})+q[x^{\prime}\otimes(\mathrm{d}y+\beta
y^{\prime}+y^{\prime})+x\otimes\mathrm{d}y^{\prime}]+\alpha x\otimes\beta
y^{\prime}$ $\displaystyle+$ $\displaystyle(\mathrm{d}x+\alpha
x^{\prime}+x^{\prime})\otimes y+q[(\mathrm{d}x+\alpha
x^{\prime}+x^{\prime})\otimes y^{\prime}+\mathrm{d}x^{\prime}\otimes y]+\alpha
x^{\prime}\otimes\beta y$ $\displaystyle=$ $\displaystyle
x\otimes(\mathrm{d}y+\beta
y^{\prime}+y^{\prime})+q[x^{\prime}\otimes(\mathrm{d}y+\beta
y^{\prime})+x\otimes\mathrm{d}y^{\prime}]+\alpha x\otimes\beta y^{\prime}$
$\displaystyle+$ $\displaystyle(\mathrm{d}x+\alpha
x^{\prime}+x^{\prime})\otimes y+q[(\mathrm{d}x+\alpha x^{\prime})\otimes
y^{\prime}+\mathrm{d}x^{\prime}\otimes y]+\alpha x^{\prime}\otimes\beta y$

which is equal to:

$\displaystyle\delta\Phi((x+qx^{\prime})\otimes(y+qy^{\prime}))$
$\displaystyle=$ $\displaystyle\delta((x\otimes y)+q(x\otimes
y^{\prime}+x^{\prime}\otimes y))$ $\displaystyle=$
$\displaystyle\mathrm{d}x\otimes y+x\otimes\mathrm{d}y+q(dx\otimes
y^{\prime}+x\otimes dy^{\prime}+dx^{\prime}\otimes y+x^{\prime}\otimes dy)$
$\displaystyle+$ $\displaystyle(\alpha\beta)(x\otimes
y^{\prime}+x^{\prime}\otimes y)+x\otimes y^{\prime}+x^{\prime}\otimes y$
$\displaystyle=$ $\displaystyle\mathrm{d}x\otimes
y+x\otimes\mathrm{d}y+q(dx\otimes y^{\prime}+x\otimes
dy^{\prime}+dx^{\prime}\otimes y+x^{\prime}\otimes dy)$ $\displaystyle+$
$\displaystyle\alpha x\otimes\beta y^{\prime}+\alpha x^{\prime}\otimes\beta
y+x\otimes y^{\prime}+x^{\prime}\otimes y$

Case $m<n$: Let $k=n-m$, notice that $\Phi(\alpha
x^{\prime}+\otimes(y+qy^{\prime}))=\alpha^{k}x^{\prime}+\otimes(y+qy^{\prime})$,
whether $k=1$ or $k>1$.

$\displaystyle\Phi\circ(Id\otimes\delta+\delta\otimes
Id)((x+qx^{\prime})\otimes(y+qy^{\prime}))$ $\displaystyle=$
$\displaystyle(\alpha^{k}x)\otimes(\mathrm{d}y+y^{\prime}+q\mathrm{d}y^{\prime})+(\alpha^{k+1}x)\otimes(\beta
y^{\prime})$ $\displaystyle+$
$\displaystyle(\alpha^{k}\mathrm{d}x+\alpha^{k}x^{\prime})\otimes(y+qy^{\prime})+(\alpha^{k}x^{\prime})\otimes(y+qy^{\prime})$
$\displaystyle=$
$\displaystyle(\alpha^{k}x)\otimes(\mathrm{d}y+y^{\prime}+q\mathrm{d}y^{\prime})+(\alpha^{k+1}x)\otimes(\beta
y^{\prime})$ $\displaystyle+$
$\displaystyle(\alpha^{k}\mathrm{d}x)\otimes(y+qy^{\prime})$

On the other hand,

$\displaystyle\delta\Phi((x+qx^{\prime})\otimes(y+qy^{\prime}))$
$\displaystyle=$
$\displaystyle\delta\left[(\alpha^{k}x)\otimes(y+qy^{\prime})\right]$
$\displaystyle=$ $\displaystyle\mathrm{d}(\alpha^{k}x\otimes
y)+\alpha\beta(\alpha^{k}x\otimes y^{\prime})+\alpha^{k}x\otimes
y^{\prime}+q\mathrm{d}(\alpha^{k}x\otimes y^{\prime})$ $\displaystyle=$
$\displaystyle\mathrm{d}\alpha^{k}x\otimes
y+\alpha^{k}x\otimes\mathrm{d}y+\alpha^{k+1}x\otimes\beta y^{\prime}$
$\displaystyle+$ $\displaystyle\alpha^{k}x\otimes
y^{\prime}+q\mathrm{d}\alpha^{k}x\otimes
y^{\prime}+q\alpha^{k}x\otimes\mathrm{d}y^{\prime}$

which, since $\alpha\mathrm{d}=\mathrm{d}\alpha$, are equal. ∎

######  Remark 2.14.

The morphism $\Phi$ is probably a homotopy equivalence as well, but we will
not need that.

##  3\. Equivariant Morse homology

We begin with 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 $f\colon
M\to\mathbb{R}$ a Morse function. Each critical point $x$ has a Morse index
$\mathrm{ind}(x)$. Denote respectively the set of critical points and index
$k$ critical points by $\mathrm{Crit}(f)$ and $\mathrm{Crit}_{k}(f)$.

######  Definition 3.1.

A _pseudo-gradient_ for $f$ is a vector field $v\in\mathfrak{X}(X)$ on $X$
such that for all $x\in X\setminus\mathrm{Crit}(f)$, $d_{x}f.v<0$; and such
that in a Morse chart near a critical point, $v$ is the negative gradient of
$f$ for the standard metric on $\mathbb{R}^{n}$. Denote by
$\mathfrak{X}(X,f)\subset\mathfrak{X}(X)$ the space of pseudo-gradients for
$f$. This is a convex (hence contractible) space.

######  Definition 3.2.

Let $x\in\mathrm{Crit}_{k}(f)$ and $v\in\mathfrak{X}(X,f)$. Define the
_stable_ (resp. _unstable_) submanifold of $x$ by:

(3.1) $\displaystyle S_{x}$ $\displaystyle=\left\\{y\in
X:\lim_{t\to+\infty}\phi_{t}^{v}(y)=x\right\\},$ (3.2) $\displaystyle U_{x}$
$\displaystyle=\left\\{y\in X:\lim_{t\to-\infty}\phi_{t}^{v}(y)=x\right\\}.$

with $\phi_{t}^{v}$ the flow at time $y$ of $v$. $S_{x}$ and $U_{x}$ are
smooth (non-proper) submanifolds diffeomorphic respectively to
$\mathbb{R}^{k}$ and $\mathbb{R}^{n-k}$.

######  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.

######  Definition 3.4.

Assume $v\in\mathfrak{X}(X,f)$ is Palais-Smale, define the _Morse complex_

$CM_{*}(X,f),\ \partial\colon CM_{*}(X,f)\to CM_{*-1}(X,f)$

by

(3.3) $\displaystyle CM_{k}(X,f)$
$\displaystyle=\bigoplus_{x\in\mathrm{Crit}_{k}(f)}\mathbb{Z}_{2}\,.x$ (3.4)
$\displaystyle\partial x$
$\displaystyle=\sum_{y\in\mathrm{Crit}_{k-1}(f)}\\#\big{(}(U_{x}\cap
S_{y})/\mathbb{R}\big{)}\,.y,$

where $\mathbb{R}$ acts on $U_{x}\cap S_{y}$ by the flow of $v$. It is well-
known that $\partial^{2}=0$ and that $HM_{*}(X,f)=H_{*}(X,\mathbb{Z}_{2})$.
Intuitively, $x\in\mathrm{Crit}_{k}(f)$ corresponds to a $k$-chain obtained by
triangulating $U_{x}$.

###  3.2. Pushforwards on Morse homology

Let $F\colon X\to Y$ be a differentiable map between two smooth compact
manifolds, then $F$ induces a pushforward in homology $F_{*}\colon H_{*}(X)\to
H_{*}(Y)$. 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 $f\colon X\to\mathbb{R}$, $g\colon
Y\to\mathbb{R}$, and pseudo-gradients $\nabla f,\nabla g$. Let
$x\in\mathrm{Crit}_{k}(f)$ be a generator of $CM_{k}(X,f)$ (say $k\geq 1$ 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 $\nabla
g$ 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 $l\geq 1$. 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 $\nabla f,\nabla g$ are chosen so that,
for any critical points $x$ and $y$ of $f$ and $g$ respectively, The graph
$\Gamma(F)$ intersects $U_{x}\times S_{y}$ transversely in $X\times Y$. Define
then

$\displaystyle F_{*}$ $\displaystyle\colon CM_{*}(X,f)\to CM_{*}(Y,g)\text{
by}$ (3.5) $\displaystyle F_{*}x$
$\displaystyle=\sum_{y\in\mathrm{Crit}_{\mathrm{ind}\,x}}\\#\left(\Gamma(F)\cap
U_{x}\times S_{y}\right)y.$

This map induces the actual pushforward in homology [KM07, prop. 2.8.2].

In order to translate this construction to Floer theory, it is convenient to
think of $\Gamma(F)\cap U_{x}\times S_{y}$ as a moduli space of _grafted flow
line_ from $x$ to $y$: By this we mean a pair of flow lines
$(\gamma_{-},\gamma_{+})$, with

$\displaystyle\gamma_{-}$ $\displaystyle\colon\mathbb{R}_{-}\to M,\
\gamma_{-}^{\prime}(t)=\nabla f(\gamma_{-}(t)),$ $\displaystyle\gamma_{+}$
$\displaystyle\colon\mathbb{R}_{+}\to N,\ \gamma_{+}^{\prime}(t)=\nabla
g(\gamma_{+}(t))$

such that

$\displaystyle\lim_{t\to-\infty}{\gamma_{-}(t)}=x,$
$\displaystyle\lim_{t\to+\infty}{\gamma_{+}(t)}=y,$ $\displaystyle
F\left(\gamma_{-}(0)\right)=\gamma_{+}(0).$

These are Morse counterparts of quilts, as we shall see. Denoting
$\mathcal{M}(x,y)$ the moduli space of such grafted lines, the identification
with $\Gamma(F)\cap U_{x}\times S_{y}$ is given by:

(3.6) $(\gamma_{-},\gamma_{+})\mapsto(\gamma_{-}(0),\gamma_{+}(0)).$

Figure 2. A grafted line.

###  3.3. The equivariant Morse complex

Let $X$ be a closed $G$-manifold, and $EG_{N}$ finite dimensional smooth
approximations of $EG$, with inclusions

(3.7) $i_{N}\colon EG_{N}\to EG_{N+1}.$

Fix Morse functions $f_{N}$ on $X_{N}=X\times_{G}EG_{N}$, let the increment
maps be

(3.8) $j_{N}=id_{X}\times_{G}i_{N}\colon X_{N}\to X_{N+1},$

and define the _equivariant Morse complex_ as

(3.9)
$CM^{G}_{*}(X,\\{f_{N}\\})=\mathrm{Tel}\Big{(}CM(X_{N},f_{N}),(j_{N})_{*}\Big{)}.$

Let also the _equivariant Morse cochain complex_ $CM_{G}^{*}(X,\\{f_{N}\\})$
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

(3.10) $\displaystyle HM^{G}_{*}(X,\\{f_{N}\\})$ $\displaystyle=H^{G}_{*}(X),$
(3.11) $\displaystyle HM_{G}^{*}(X,\\{f_{N}\\})$ $\displaystyle=H_{G}^{*}(X).$

##  4\. Equivariant Lagrangian Floer homology

We first recall some basic facts about Hamiltonian actions.

###  4.1. Hamiltonian actions

######  Definition 4.1.

(Hamiltonian manifold) Let $G$ be a Lie group. A Hamiltonian $G$-manifold
$(M,\omega,\mu)$ is a symplectic manifold $(M,\omega)$ endowed with a
$G$-action by symplectomorphisms, induced by a moment map $\mu\colon
M\to\mathfrak{g}^{*}$. The moment map is $G$-equivariant with respect to this
action and the coadjoint representation on $\mathfrak{g}^{*}$, and satisfies
the following equation:

(4.1) $\iota_{X_{\eta}}\omega=d\langle\mu,\eta\rangle,$

for each $\eta\in\mathfrak{g}$, where $X_{\eta}$ stands for the vector field
on $M$ induced by the infinitesimal action, i.e.

(4.2) $X_{\eta}(m)=\frac{d}{dt}_{|t=0}(e^{t\eta}m).$

In other words, $X_{\eta}$ is the symplectic gradient of the function
$\langle\mu,\eta\rangle$.

######  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

(4.3) $g\cdot(q,p)=(gq,p\circ(D_{q}L_{g})^{-1}),$

with $g\in G$, $(q,p)\in T^{*}X$. Its moment map $\mu\colon
T^{*}X\to\mathfrak{g}^{*}$ is defined by

(4.4) $\mu(q,p)\cdot\xi=p(X_{\xi}(q)).$

######  Definition 4.3 (Weinstein correspondence [Wei81]).

The data of both the action and the moment map can be conveniently packaged as
a Lagrangian submanifold

(4.5) $\Lambda_{G}(M)\subset T^{*}G\times M^{-}\times M,$

that we will refer to as the _Weinstein correspondence_ , defined as:

(4.6) $\Lambda_{G}(M)=\\{((q,p),m,m^{\prime}):m^{\prime}=q.m,\
R_{g^{-1}}^{*}p=\mu(m)\\}.$

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 $X$,
$\Lambda_{G}(M)$ corresponds to the conormal bundle of the graph of the action

(4.7) $\Gamma_{G}(X)\subset G\times X\times X,$

where one identifies $T^{*}X$ with $(T^{*}X)^{-}$ via $(q,p)\mapsto(q,-p)$.

######  Definition 4.4.

(Symplectic quotient) If $(M,\omega,\mu)$ is a Hamiltonian manifold, its
_symplectic quotient_ (or _reduction_) is defined as

(4.8) $M/\\!\\!/G=\mu^{-1}(0)/G.$

When $0$ is a regular value for $\mu$, and $G$ acts freely and properly on
$\mu^{-1}(0)$, $M/\\!\\!/G$ is also a symplectic manifold. In this case, we
will say that the action is _regular_.

######  Definition 4.5.

(Canonical Lagrangian correspondence between $M$ and $M/\\!\\!/G$) If the
action is regular in the sense of Definition 4.5, the image of the map

(4.9) $\iota\times\pi\colon\mu^{-1}(0)\to M^{-}\times M/\\!\\!/G$

is a Lagrangian correspondence, where $\iota$ and $\pi$ stand respectively for
the inclusion and the projection.

######  Definition 4.6.

($G$-Lagrangian) A $G$-Lagrangian of a Hamiltonian $G$-manifold $M$ is a
Lagrangian submanifold $L\subset M$ that is both contained in the zero level
$\mu^{-1}(0)$, and $G$-invariant. When the $G$-action is regular, the
$G$-Lagrangians of $M$ are in one-to-one correspondence with the Lagrangians
on $M/\\!\\!/G$ (though a Lagrangian in $M$ need not be a $G$-Lagrangian to
induce a Lagrangian on $M/\\!\\!/G$).

###  4.2. Symplectic homotopy quotients

Let now $M$ be a Hamiltonian $G$-manifold, with moment map $\mu_{M}$, and
$L_{0}$, $L_{1}$ a pair of $G$-Lagrangians. Let

(4.10) $\displaystyle T_{N}$ $\displaystyle=T^{*}EG_{N},\text{ and}$ (4.11)
$\displaystyle 0_{N}$ $\displaystyle\subset T_{N}$

stand for its zero section. For any $N$, let

(4.12) $M_{N}=(M\times T_{N})/\\!\\!/G,$

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 $M\times T_{N}$. Likewise, for
$i=0,1$,

(4.13) $L_{i}^{N}=L_{i}\times_{G}0_{N}$

is a smooth Lagrangian submanifold of $M_{N}$.

Even if $M$ is compact, $M_{N}$ is never compact (as soon as $EG_{N}\neq G$),
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 $\mu_{M}$, and $L\subset
M$ 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 $Q=(M\times T^{*}E)/\\!\\!/G$, it inherits a symplectic structure
$\omega_{Q}$ 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 $(L\times 0_{E})/G$ over $0_{B}$, with fibers $L$.

(4.14)
$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{\beta}}$$\textstyle{Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{L\
\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(L\times
0_{E})/G\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T^{*}B}$$\textstyle{0_{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

The projection $Q\to T^{*}B$ is $(J_{Q},J_{B})$-holomorphic, and for each
$\beta\in T^{*}B$, the inclusion $\iota_{\beta}\colon M\hookrightarrow Q$ to
the fiber over $\beta$ satisfies:

(4.15) $\iota_{\beta}^{*}\omega_{Q}=\omega_{M}.$

###### Proof.

For $q\in E$, let $O_{q}=T_{q}(G\cdot q)\subset E$ be the tangent space of the
$G$-orbit passing through $q$. With $\mu_{E}\colon T^{*}E\to\mathfrak{g}^{*}$
the moment map, one has

(4.16) $(\mu_{E})^{-1}(0)=\left\\{(q,p)\ |\ p_{|O_{q}}=0\right\\}.$

From the orthogonal splitting

(4.17) $T_{q}E=O_{q}\oplus O_{q}^{\perp},$

one gets a dual splitting

(4.18) $T_{q}^{*}E=O_{q}^{*}\oplus(O_{q}^{\perp})^{*},$

and with $(O^{\perp})^{*}=\bigcup_{q}(O_{q}^{\perp})^{*}$, one has
$(\mu_{E})^{-1}(0)=(O^{\perp})^{*}$, and a projection

(4.19) $T^{*}E\to(\mu_{E})^{-1}(0)$

defined fiberwise by orthogonal projection to $(O_{q}^{\perp})^{*}$. Composing
this projection with the quotient projection

(4.20) $(\mu_{T^{*}E})^{-1}(0)\to(T^{*}E)/\\!\\!/G\simeq T^{*}B,$

one obtains a projection $T^{*}E\to T^{*}B$.

Notice that $J_{E}$ descends to an almost complex structure $J_{B}$ on
$T^{*}B$, and under the identification $(T^{*}E)/\\!\\!/G\simeq T^{*}B$
corresponds to the almost complex structure induced by the Riemannian metric
on $B$. With respect to these almost complex structure, the projection
$T_{N}\to B_{N}$ is pseudo-holomorphic.

(This map corresponds to the projection to the GIT quotient $T^{*}E\to
T^{*}E/G^{\mathbb{C}}$, by Kempf-Ness theorem and the fact that all the points
of $T^{*}E$ are semi-stable.)

One then gets a projection

(4.21) $\mu_{diag}^{-1}(0)\hookrightarrow M\times T^{*}E\to T^{*}E\to T^{*}B$

which is $G$-invariant for the diagonal action, and therefore a projection

(4.22) $\pi\colon Q\to T^{*}B$

whose fiber is diffeomorphic to $M$. Indeed, given a point
$\beta=[(q_{0},p_{0})]\in T^{*}B$, the fiber $\pi^{-1}(\beta)$ can be
parametrized by the map $\iota_{\beta}\colon M\to Q=(M\times T^{*}E)/\\!\\!/G$
defined by

(4.23) $m\mapsto[m,(q_{0},p_{0}-\mu(m))],$

where we view $\mu(m)$ as an element of $O_{q}^{*}$. In order to show that
$(\iota_{\beta})^{*}\omega_{Q}=\omega_{M}$, let $m\in M$ and $v,w\in T_{m}M$,
one has:

(4.24)
$(\iota_{\beta})^{*}(\omega_{Q})_{m}(v,w)=\omega_{M}(v,w)+\omega_{E}(-d_{m}\mu.v,-d_{m}\mu.w)$

and since both $-d_{m}\mu.v,-d_{m}\mu.w$ lie on the fiber direction, which is
Lagrangian for $\omega_{E}$, the second summand vanishes.

∎

###  4.3. Increment correspondences

Recall that the approximation of $EG$ comes with inclusions $i_{N}\colon
EG_{N}\to EG_{N+1}$. These induce Lagrangian _increment correspondences_
$\Lambda_{N}\subset(M_{N})^{-}\times M_{N+1}$ defined as follows. Inside
$M^{-}\times T_{N}^{-}\times M\times T_{N+1}$, consider (modulo implicitly
exchanging the second and third factors)

(4.25) $\widehat{\Lambda}_{N}=\Delta_{M}\times N_{\Gamma(i_{N})},$

where $\Delta_{M}\subset M^{-}\times M$ is the diagonal, and
$N_{\Gamma(i_{N})}\subset(T_{N})^{-}\times T_{N+1}$ is given by:

(4.26) $N_{\Gamma(i_{N})}=\left\\{(q,p),(q^{\prime},p^{\prime})\ :\
q^{\prime}=i_{N}(q),\ p=p^{\prime}\circ\mathrm{d}i_{N}\right\\}.$

Notice that, under the identification of $T_{N}$ with $(T_{N})^{-}$ via
$(q,p)\mapsto(q,-p)$, $N_{\Gamma(i_{N})}$ corresponds to the conormal bundle
of the graph of $i_{N}$. The group $G\times G$ acts on $M^{-}\times
T_{N}^{-}\times M\times T_{N+1}$ by:

(4.27)
$(g_{1},g_{2}).(m,t,m^{\prime},t^{\prime})=(g_{1}m,g_{1}t,g_{2}m^{\prime},g_{2}t^{\prime})$

and with moment map $\Phi=(-\Phi_{1},\Phi_{2})$, with

(4.28) $\displaystyle\Phi_{1}(m,t,m^{\prime},t^{\prime})$
$\displaystyle=\mu_{M}(m)+\mu_{T_{N}}(t),$ (4.29)
$\displaystyle\Phi_{2}(m,t,m^{\prime},t^{\prime})$
$\displaystyle=\mu_{M}(m^{\prime})+\mu_{T_{N+1}}(t^{\prime}).$

Notice that $\widehat{\Lambda}_{N}$ is a $G$-Lagrangian for the diagonal
action of this $G\times G$-action (this mostly follows from the equivariance
of $i_{N}$), but not a $(G\times G)$-Lagrangian222as Julio Sampietro pointed
out to us..

However, it intersects $\Phi^{-1}(0)$ cleanly, let
$I_{N}=\widehat{\Lambda}_{N}\cap\Phi^{-1}(0)$ be this intersection. Indeed, it
follow from the fact that $\widehat{\Lambda}_{N}$ is a $G^{diag}$-Lagrangian
(in particular contained in $(-\Phi_{1}+\Phi_{2})^{-1}(0)$) that
$I_{N}=\widehat{\Lambda}_{N}\cap\Phi_{1}^{-1}(0)=\widehat{\Lambda}_{N}\cap\Phi_{2}^{-1}(0)$,
and one can see that these two intersections are transverse, which implies
clean intersection with $\Phi^{-1}(0)=\Phi_{1}^{-1}(0)\cap\Phi_{2}^{-1}(0)$.

It follows that $\widetilde{\Lambda}_{N}:=(G\times G).I_{N}$ (which also
equals $(G\times\\{0\\}).I_{N}=(\\{0\\}\times G).I_{N}$) is a $(G\times
G)$-Lagrangian, which gives our desired increment correspondence:

(4.30) $\Lambda_{N}=\widetilde{\Lambda}_{N}/(G\times G)\subset M_{N}^{-}\times
M_{N+1}.$

###  4.4. Exact setting

The simplest setting for defining Lagrangian Floer homology is the exact and
convex at infinity one, which we recall quickly:

######  Definition 4.8.

A symplectic manifold $(M,\omega)$ is _exact_ if $\omega=d\lambda$.

A Lagrangian $\iota\colon L\hookrightarrow M$ inside an exact symplectic
manifold $(M,\lambda)$ is _exact_ if $\iota^{*}\lambda=df$.

######  Definition 4.9.

An exact symplectic manifold $(M,\lambda)$ is _convex at infinity_ if it is
isomorphic to the positive symplectization of a contact manifold outside a
compact subset.

A Lagrangian $L$ inside a convex symplectic manifold $(M,\lambda)$ is
_cylindrical at infinity_ if it is of the form $\mathbb{R}_{+}\times\Lambda$
at infinity, where $\Lambda$ is a Legendrian.

######  Example 4.10.

A cotangent bundle $T^{*}X$ with its Liouville form $\lambda=p\,dq$ is exact
and convex at infinity. If $Z\subset X$ is a smooth submanifold, then its
conormal bundle $N_{Z}\subset T^{*}X$ is exact ($\iota^{*}\lambda=0$) and
cylindrical at infinity.

To ensure that the symplectic homotopy quotients will have such structures,
one can impose equivariant analogues as follows.

######  Definition 4.11.

A $G$-Hamiltonian manifold $(M,\omega)$ is _equivariantly exact_ (or _$G$
-exact_) if $\omega=d\lambda$, with $\lambda$ a $G$-invariant 1-form.

If $G$ is connected, this is equivalent to saying that

(4.31) $\forall\xi\in\mathfrak{g},\ \mathcal{L}_{X_{\xi}}\lambda=0,$

i.e. the form $\iota_{X_{\xi}}\lambda+\left\langle\mu,\xi\right\rangle$ is
closed.

A $G$-Lagrangian $\iota\colon L\hookrightarrow M$ in a $G$-exact symplectic
manifold $(M,\lambda)$ is _$G$ -exact_ if $\iota^{*}\lambda=df$, with $f$ a
$G$-invariant smooth function.

######  Remark 4.12.

If $G$ is compact, any exact $G$-Hamiltonian manifold can be made $G$-exact:
one can make the primitive $\lambda$ equivariant by averaging over $G$:

(4.32) $\frac{1}{\mathrm{Vol}G}\int_{g\in G}\varphi_{g}^{*}\lambda,$

with $\varphi_{g}\colon M\to M$ 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 $(X,\xi=\ker\lambda)$ be a contact manifold, a contact $G$-Hamiltonian
action is an action by contactomorphisms, with a moment map $\mu\colon
X\to\mathfrak{g}^{*}$ such that

(4.33) $\iota_{X_{\xi}}\lambda+\langle\mu,\xi\rangle=0,\
\forall\xi\in\mathfrak{g}.$

Say that such an action is _regular_ if $0\in\mathfrak{g}^{*}$ is a regular
value of $\mu$ and $G$ acts freely on $\mu^{-1}(0)$.

When this is the case, the _contact reduction_ $X/\\!\\!/G=\mu^{-1}(0)/G$
inherits a contact form.

A Legendrian $\Lambda\subset X$ is a $G$-Legendrian if it is $G$-invariant and
contained in $\mu^{-1}(0)$.

######  Proposition 4.14.

Let $(X,\lambda,\mu)$ be a contact $G$-manifold. Its symplectization
$SX=(\mathbb{R}\times X,e^{t}\lambda)$ is $G$-Hamiltonian with moment map
$\Phi(t,x)=e^{t}\mu(x)$, 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 $\Lambda\subset X$ is a $G$-Legendrian, then
$S\Lambda=\mathbb{R}\times\Lambda\subset SX$ is a $G$-exact Lagrangian,
$(S\Lambda)/\\!\\!/G\simeq S(\Lambda/\\!\\!/G)$.

######  Definition 4.15.

A $G$-exact symplectic manifold $(M,\lambda)$ is _$G$ -convex at infinity_ if
it is isomorphic (as $G$-exact Hamiltonian manifolds) to the positive
symplectization of a contact $G$-manifold outside a compact subset.

A $G$-Lagrangian $L$ inside a $G$-convex symplectic manifold $(M,\lambda)$ is
_$G$ -cylindrical at infinity_ if it is of the form
$\mathbb{R}_{+}\times\Lambda$ at infinity, where $\Lambda$ is a
$G$-Legendrian.

######  Proposition 4.16.

Let $X$ be a $G$-manifold, then its cotangent bundle $TX$ is $G$-exact and
$G$-convex at infinity. If $Z\subset X$ is a $G$-invariant submanifold, then
its conormal bundle $N_{Z}\subset T^{*}X$ is $G$-exact (the Liouville form
restricts to zero on $N_{Z}$)

In defining $HF^{G}(M;L_{0},L_{1})$ we will assume:

######  Assumption 4.17.

The symplectic manifold $(M,\omega=\mathrm{d}\lambda)$ is $G$-exact and
$G$-convex at infinity. The Lagrangians $L_{0},L_{1}\subset M$ are $G$-exact,
$G$-cylindrical at infinity, and disjoint outside a compact subset.

It then follows from the above discussion that for each $N$, $M_{N}$ is exact
and convex at infinity, $L_{0}^{N},L_{1}^{N}\subset M_{N}$ are exact,
cylindrical at infinity and disjoint outside a compact subset. Furthermore,
one can show that $\Lambda_{N}\subset(M_{N})^{-}\times M_{N+1}$ is exact and
cylindrical at infinity as well (but we will not use this).

###  4.5. Monotone setting

The monotone setting is another simple case where Lagrangian Floer homology is
easy to define, see for example [MW12, sec. 2.1].

If $L\subset M$ is a Lagrangian submanifold, let
$\mu_{L}\colon\pi_{2}(M,L)\to\mathbb{Z}$ denote the Maslov index. If $u$ is a
disc in $M$ with boundary in $L$, $v$ a sphere in $M$, and $u\\#v$ be the disc
corresponding to the connected sum at an interior point (see Figure 3), recall
that

(4.34) $\mu_{L}(u\\#v)=\mu_{L}(u)+2c_{1}(v).$

Figure 3. Interior connected sum $u\\#v$.

######  Definition 4.18.

Let $\kappa>0$, a symplectic manifold $(M,\omega)$ is _$\kappa$ -monotone_ if

(4.35) $[\omega]_{|\pi_{2}(M)}=\kappa\,c_{1}(TM).$

A Lagrangian submanifold $L\subset M$ is _$\kappa$ -monotone_ if

(4.36) $2[\omega]_{|\pi_{2}(M,L)}=\kappa\,\mu_{L},$

where $\mu_{L}$ stands for the Maslov index.

If $L\subset M$ is $\kappa$-monotone and $M$ is connected, then $M$ is
$\kappa$-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.

######  Definition 4.19.

The _minimal Chern number_ $N_{M}$ of $M$ is the positive generator of the
image of $c_{1}(TM)\colon\pi_{2}(M)\to\mathbb{Z}$.

The _minimal Maslov number_ $N_{L}$ of $L$ is the positive generator of the
image of $\mu_{L}\colon\pi_{2}(M,L)\to\mathbb{Z}$

If $M$ is connected, $N_{L}$ divides $2N_{M}$, and if $L$ is simply connected,
$N_{L}=2N_{M}$.

Floer homology is well-defined under the following:

######  Assumption 4.20.

The symplectic manifold $M$ is compact $\kappa$-monotone, and the Lagrangians
$L_{0},L_{1}$ are $\kappa$-monotone and such that $N_{L_{0}},N_{L_{1}}\in
A\mathbb{Z}$, with $A\geq 3$.

Except for compactness, this setting transports to symplectic homotopy
quotients, as we shall see.

######  Lemma 4.21.

Let $(M,\omega)$ be a symplectic manifold, endowed with a regular Hamiltonian
action of a compact Lie group $G$ with moment $\mu\colon
M\to\mathfrak{g}^{*}$, and $L\subset\mu^{-1}(0)$ a $G$-Lagrangian. Let
$u\colon(D^{2},\partial D^{2})\to(M/\\!\\!/G,L/G)$ be a disc, and
$\overline{u}\colon(D^{2},\partial D^{2})\to(\mu^{-1}(0),L)$ a lift of $u$
(which exists since $D^{2}$ is contractible). Then the Maslov indices are
equal:

(4.37) $\mu_{L/G}(u)=\mu_{L}(\overline{u}).$

###### Proof.

Let $\mathfrak{g}^{\mathbb{C}}=\mathfrak{g}\oplus i\mathfrak{g}$, and $J$ a
$G$-invariant almost-complex structure on $M$ compatible with $\omega$. Since
the action is regular, we get an injective map, with $Z=\mu^{-1}(0)$

(4.38) $\displaystyle Z\times\mathfrak{g}^{\mathbb{C}}$ $\displaystyle\to
TM_{|Z}$ (4.39) $\displaystyle(x,v+iw)$ $\displaystyle\mapsto
X_{v}(x)+J_{x}X_{w}.$

Denote by $\underline{\mathfrak{g}}^{\mathbb{C}}$ the (trivial) sub-bundle of
$TM_{|Z}$ corresponding to the image of this map. Notice that the sub-bundle
$\underline{\mathfrak{g}}\subset\underline{\mathfrak{g}}^{\mathbb{C}}$ defined
as the image of $Z\times\mathfrak{g}$ corresponds to the foliation of $Z$ by
orbits.

Let $V$ stand for the orthogonal of $\underline{\mathfrak{g}}^{\mathbb{C}}$ in
$TM_{|Z}$ (with respect to either the symplectic structure or the Riemannian
metric induced by $J$), and $W=V\cap TL$.

Let now $u$ and $\overline{u}$ be as in the statement. One has

(4.40)
$\mu_{L}(\overline{u})=\mu(\overline{u}^{*}(\underline{\mathfrak{g}},\underline{\mathfrak{g}}^{\mathbb{C}}))+\mu(\overline{u}^{*}(V,W)),$

but
$\mu(\overline{u}^{*}(\underline{\mathfrak{g}},\underline{\mathfrak{g}}^{\mathbb{C}}))=0$
as $\underline{\mathfrak{g}}$ is a constant sub-bundle of
$\underline{\mathfrak{g}}^{\mathbb{C}}$. Moreover,

(4.41) $\mu(\overline{u}^{*}(V,W))=\mu_{L/G}(u),$

since

(4.42) $\displaystyle V$ $\displaystyle\simeq\pi^{*}T(M/\\!\\!/G)\text{, and}$
(4.43) $\displaystyle W$ $\displaystyle\simeq\pi^{*}T(L/G),$

with $\pi\colon Z\to M/\\!\\!/G$ the projection. ∎

Recall from [MW12, Lemma 4.4] that (under some assumptions), the symplectic
quotient of a $\kappa$-monotone symplectic manifold is again
$\kappa$-monotone. Here is a relative version that follows from the previous
lemma:

######  Proposition 4.22.

Let $(M,\omega)$ be a symplectic manifold, endowed with a regular Hamiltonian
action of a compact Lie group $G$ with moment $\mu\colon
M\to\mathfrak{g}^{*}$, and $L\subset\mu^{-1}(0)$ a $G$-Lagrangian. If $L$ is
$\kappa$-monotone, then $L/G$ is also $\kappa$-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 $\overline{u}$ a lift to $(M,L)$.
The first statement follows from:

(4.44)
$\kappa\mu_{L/G}(u)=\kappa\mu_{L}(\overline{u})=2\omega(\overline{u})=2\omega_{M/\\!\\!/G}(u).$

The statement about minimal maslov numbers follows from the fact that any disc
can be lifted, and from Lemma 4.21. ∎

######  Lemma 4.23.

If $M$ is $\kappa$-monotone, then the diagonal $\Delta_{M}\subset M^{-}\times
M$ is $\kappa$-monotone. Furthermore, $N_{\Delta_{M}}=2N_{M}$.

###### Proof.

A disc

(4.45) $u=(u_{1},u_{2})\colon(D^{2},\partial D^{2})\to(M^{-}\times
M,\Delta_{M})$

gives rise to a sphere

(4.46) $v=u_{1}\cup_{\partial D^{2}}u_{2}\colon
S^{2}=(D^{2})^{-}\cup_{\partial D^{2}}D^{2}\to M,$

and conversely any sphere in $M$ gives a disc in $(M^{-}\times M,\Delta_{M})$.
The claim follows from $\mu_{\Delta_{M}}(u)=2c_{1}(v)$. Indeed, $u$ is
homotopic to $\alpha\\#\beta$ (see Figure 4), with

(4.47) $\displaystyle\alpha=(u_{1},u_{1})$ $\displaystyle\colon
D^{2}\to\Delta_{M}\text{, and}$ (4.48)
$\displaystyle\beta=(u_{1},u_{1})\cup_{\partial D^{2}}(u_{1},u_{2})$
$\displaystyle\colon(D^{2})^{-}\cup_{\partial D^{2}}D^{2}\to M.$

Then (4.34) gives
$\mu_{\Delta_{M}}(u)=\mu_{\Delta_{M}}(\alpha)+2c_{1}(\beta)$, but
$\mu_{\Delta_{M}}(\alpha)=0$ as $\alpha\subset\Delta_{M}$, and
$\beta=(u_{1}\cup_{\partial D^{2}}u_{1})\times u$ so
$c_{1}(\beta)=c_{1}(u_{1}\cup_{\partial D^{2}}u_{1})+c_{1}(u)$, but as
$u_{1}\cup_{\partial D^{2}}u_{1}$ is homotopic to a constant map,
$c_{1}(u_{1}\cup_{\partial D^{2}}u_{1})=0$.

Figure 4. $u$ is homotopic to $\alpha\\#\beta$.

∎

######  Lemma 4.24.

Let $Z\subset X$ with $\pi_{2}(X,Z)=0$, then any disc

(4.49) $u\colon(D^{2},\partial D^{2})\to(T^{*}X,N_{Z})$

is homotopic to a constant disc and therefore has Maslov index zero.

$\Box$

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 Lemma 4.24 it follows that $L_{i}\times 0_{N}\subset M\times T_{N}$ is
$\kappa$-monotone and $N_{L_{i}\times 0_{N}}=N_{L_{i}}$. By Lemmas 4.21 and
Proposition 4.22, $L_{i}^{N}$ is monotone and $N_{L_{i}^{N}}\in A\mathbb{Z}$.

###  4.6. Construction

Fix first a $G$-invariant almost complex structure on $M$ and a $G$-invariant
Riemannian metric on each $EG_{N}$, so to get almost complex structures
$J_{M_{N}},J_{B_{N}}$ on $M_{N}$ and $B_{N}=(T_{N})/\\!\\!/G$ respectively, as
in Section 4.2.

Let $\mathcal{J}(M_{N})$ stand for the space of compatible almost complex
structures on $M_{N}$ agreeing with our fixed $J_{M_{N}}$ outside the preimage
by $\pi_{N}$ of a compact subset of $B_{N}$ containing $0_{B_{N}}$, and

(4.50) $\mathcal{J}_{t}(M_{N})=C^{\infty}([0,1],\mathcal{J}(M_{N})).$

Let $\mathcal{H}(M_{N})$ stand for the space of compactly supported smooth
functions on $M_{N}$, and
$\mathcal{H}_{t}(M_{N})=C^{\infty}([0,1],\mathcal{H}(M_{N}))$. If
$H_{t}\in\mathcal{H}_{t}(M_{N})$, let $X_{H_{t}}$ stand for its symplectic
gradient, and $\phi_{H_{t}}^{t}$ its flow.

######  Definition 4.25 (Perturbed (generalized) intersection points).

Let $\mathcal{I}_{H_{t}}(L_{0}^{N},L_{1}^{N})$ stand for the set of
$H_{t}$-perturbed intersection points, i.e. Hamiltonian chords
$\gamma\colon[0,1]\to M_{N}$ of $X_{H_{t}}$ with $\gamma(0)\in L_{0}^{N}$ and
$\gamma(1)\in L_{1}^{N}$. These are in one to one correspondence with
$\phi_{H_{t}}^{1}(L_{0}^{N})\cap L_{1}^{N}$.

More generally, if $\underline{L}$ is a cyclic _generalized Lagrangian
correspondence_ from $M_{0}$ to itself, i.e. a sequence of Lagrangian
correspondences

(4.51) $\underline{L}=\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
9.79582pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{M_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
15.41388pt\raise 6.0361pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.74722pt\hbox{$\scriptstyle{L_{0}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 33.79582pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
33.79582pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{M_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
57.4826pt\raise 6.0361pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.74722pt\hbox{$\scriptstyle{L_{1}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 77.38747pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
77.38747pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
103.06175pt\raise 6.0861pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.69722pt\hbox{$\scriptstyle{L_{k-1}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 114.88747pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
114.88747pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{M_{k}=M_{0},}$}}}}}}}\ignorespaces}}}}\ignorespaces$

$\delta_{1},\ldots,\delta_{k-1}$ are positive integers, and
$\underline{H}=(H^{1}_{t},\ldots,H^{k-1}_{t})$ a sequence of Hamiltonians in
$M_{1},\ldots,M_{k-1}$ respectively, let
$\mathcal{I}_{\underline{H}}(\underline{L})$ be the set of _$\underline{H}$
-perturbed generalized intersection points_, i.e. Hamiltonian chords
$\gamma_{i}\colon[0,\delta_{i}]\to M_{i}$ such that for all $i=1,\ldots,k-2$,
with $\gamma_{k}:=\gamma_{1}$,

(4.52) $(\gamma_{i}(\delta_{i}),\gamma_{i+1}(0))\in L_{i},$

with $L_{i}^{\prime}=(id\times\phi_{H^{i}_{t}}^{\delta_{i}})(L_{i})$, the map
$(\gamma_{1},\ldots,\gamma_{k-1})\mapsto(\gamma_{1}(\delta_{1}),\ldots,\gamma_{k-1}(\delta_{k-1}))$
gives a bijection between $\mathcal{I}_{\underline{H}}(\underline{L})$ and the
intersection of

(4.53) $\displaystyle\left(\prod\underline{L}^{\prime}\right)_{0}:=$
$\displaystyle L_{0}^{\prime}\times L_{2}^{\prime}\times\cdots\text{, and}$
(4.54) $\displaystyle\left(\prod\underline{L}^{\prime}\right)_{1}:=$
$\displaystyle L_{1}^{\prime}\times L_{3}^{\prime}\times\cdots\text{ in}$
(4.55) $\displaystyle\prod\underline{M}:=$ $\displaystyle M_{0}^{-}\times
M_{1}\times\cdots\times M_{k-1}^{\pm}.$

######  Definition 4.26.

As in [Sei08], call a pair
$(H_{t},J_{t})\in\mathcal{H}_{t}(M_{N})\times\mathcal{J}_{t}(M_{N})$ such that
$\phi_{H_{t}}^{1}(L_{0}^{N})$ and $L_{1}^{N}$ intersect transversely a _Floer
datum_.

More generally, in the situation of a cyclic generalized Lagrangian
correspondence $\underline{L}$ as above, a _quilted Floer datum_ is a pair of
sequences

(4.56)
$(\underline{H}_{t},\underline{J}_{t})\in\mathcal{H}_{t}(\underline{M})\times\mathcal{J}_{t}(\underline{M}):=\prod_{i=1}^{k-1}\mathcal{H}_{t}(M_{i})\times\mathcal{J}_{t}(M_{i}),$

for which $\mathcal{I}_{\underline{H}}(\underline{L})$ is cut out
transversely, i.e. $\left(\prod\underline{L}^{\prime}\right)_{0}$ and
$\left(\prod\underline{L}^{\prime}\right)_{1}$ intersect transversely in
$\prod\underline{M}$.

Let $Z=\\{s+it:0\leq t\leq 1\\}\subset\mathbb{C}$ be the strip, and
$\partial_{0}Z=\\{t=0\\}$, $\partial_{1}Z=\\{t=1\\}$ its two boundaries. For
$J_{t}\in\mathcal{J}_{t}(M_{N})$, $H_{t}\in\mathcal{H}_{t}(M_{N})$ and
$x,y\in\mathcal{I}(L_{0},L_{1};H_{t})$, let
$\widetilde{\mathcal{M}}(x,y;H_{t},J_{t})$ stand for the moduli space of
perturbed $J_{t}$-holomorphic strips

(4.57) $u\colon Z\to M$

satisfying the _Floer equation_

(4.58) $\partial_{s}u+J_{t}(\partial_{t}u-X_{H_{t}})=0,$

the Larangian boundary conditions $u_{|\partial_{0}Z}\subset L_{0}^{N}$,
$u_{|\partial_{1}Z}\subset L_{1}^{N}$, and asymptotic to $x$ and $y$ when
$s\to-\infty$ and $s\to+\infty$ respectively. Let then
$\mathcal{M}(x,y;H_{t},J_{t})$ be its quotient by $\mathbb{R}$ (modulo
translations in the $s$-direction).

For $i\in\mathbb{Z}$, let $\widetilde{\mathcal{M}}(x,y;H_{t},J_{t})_{i}$ and
$\mathcal{M}(x,y;H_{t},J_{t})_{i}$ denote the subsets of curves with Maslov
index $I(u)=i+1$.

######  Proposition 4.27.

Assume that $(M;L_{0},L_{1})$ either satisfy the exact (4.17) or monotone
(4.20) assumptions. There exists a comeagre subset

(4.59)
$\mathcal{H}\mathcal{J}_{t}^{reg}(M_{N})\subset\mathcal{H}_{t}(M_{N})\times\mathcal{J}_{t}(M_{N})$

of regular peturbations such that, for
$\mathcal{F}=(H_{t},J_{t})\in\mathcal{H}\mathcal{J}_{t}^{reg}(M_{N})$,
$\phi_{H_{t}}^{1}(L_{0}^{N})$ intersects $L_{1}^{N}$ transversely;
$\widetilde{\mathcal{M}}(x,y;H_{t},J_{t})_{i}$ and
$\mathcal{M}(x,y;H_{t},J_{t})_{i}$ are smooth of dimension $i+1$ and i
respectively. When $i=0$, $\mathcal{M}(x,y;H_{t},J_{t})_{0}$ is a finite set.
In this case, define

(4.60)
$CF(M_{N};L_{0}^{N},L_{1}^{N};\mathcal{F})=\bigoplus_{x\in\mathcal{I}(L_{0},L_{1};H_{t})}\mathbb{Z}_{2}\,x,$

with differential $\partial$ defined by

(4.61) $\partial
x=\sum_{y\in\mathcal{I}(L_{0},L_{1};H_{t})}\\#\mathcal{M}(x,y;H_{t},J_{t})_{0}\,y$

Then $\partial^{2}=0$, therefore one can define
$HF(M_{N};L_{0}^{N},L_{1}^{N};\mathcal{F})$ as the homology group of this
chain complex.

###### Proof.

The transversality statement is a standard argument [FHS95]. In the monotone
case, let $K\subset M_{N}$ be a compact subset containing
$L_{0}^{N},L_{1}^{N}$, and such that $H_{t}=0$ and $J_{t}=J_{M_{N}}$ outside
$K$. Consider a strip $u$ in $\widetilde{\mathcal{M}}(x,y;H_{t},J_{t})$, and
assume by contradiction that its image is not contained in $K$. Then outside
$K$, composing with the projection $M_{N}\to B_{N}$, 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. ∎

#### The increment map $\alpha_{N}\colon CF_{N}\to CF_{N+1}$

For each $N$ fix a Floer datum $\mathcal{F}_{N}$ for
$(M_{N};L_{0}^{N},L_{1}^{N})$, and let

(4.62) $CF_{N}=CF(M_{N};L_{0}^{N},L_{1}^{N};\mathcal{F}_{N}).$

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_ $\underline{S}$ consists in:

  * •

A collection $\mathcal{P}(\underline{S})$ of _patches_ : these are Riemann
surfaces with boundary and strip-like ends.

  * •

A collection $\mathcal{S}(\underline{S})$ of _seams_ : these are pairwise
disjoint identifications of boundary components of patches, satisfying a local
real-analycity condition.

  * •

A collection $\mathcal{B}(\underline{S})$ of _true boundaries_ : these are the
boundary components of patches not belonging to any seam.

If $\underline{S}$ is a quilted surface, let its total space

(4.63)
$\left|\underline{S}\right|=\left(\coprod_{P\in\mathcal{P}(\underline{S})}P\right)/\sim$

be the Riemann surface obtained by glueing together all the patches along the
seams. Its boundary is given by the union of components of
$\mathcal{B}(\underline{S})$, and the seams become real analytic curves in
$\left|\underline{S}\right|$. We will often define a quilted surface by giving
its total space and seams.

######  Example 4.29.

Let $\delta_{1},\ldots,\delta_{k}$ be positive numbers,

(Quilted half-strip) Let $\underline{Z}_{\pm}(\delta_{1},\ldots,\delta_{k})$
be the quilted surface whose total space is
$\mathbb{R}_{\pm}\times[0,\delta_{1}+\cdots+\delta_{k}]$ and seams the
horizontal lines

(4.64)
$\mathbb{R}_{\pm}\times\\{\delta_{1}\\},\mathbb{R}_{\pm}\times\\{\delta_{1}+\delta_{2}\\},\ldots,\mathbb{R}_{\pm}\times\\{\delta_{1}+\cdots+\delta_{k-1}\\}.$

Its patches consist in half-strips

(4.65)
$P_{i}=\mathbb{R}_{\pm}\times[\delta_{1}+\cdots+\delta_{i-1},\delta_{1}+\cdots+\delta_{i}].$

(Quilted half-cylinder) Let
$\underline{C}_{\pm}(\delta_{1},\ldots,\delta_{k})$ be the quilted surface
whose total space is
$\mathbb{R}_{\pm}\times\big{(}\mathbb{R}/(\delta_{1}+\cdots+\delta_{k})\mathbb{Z}\big{)}$
and seams the horizontal lines

(4.66)
$\mathbb{R}_{\pm}\times\\{\delta_{1}\\},\mathbb{R}_{\pm}\times\\{\delta_{1}+\delta_{2}\\},\ldots,\mathbb{R}_{\pm}\times\\{\delta_{1}+\cdots+\delta_{k}\\}.$

It has same patches as $\underline{Z}_{\pm}(\delta_{1},\ldots,\delta_{k})$
(except that $P_{1}$ and $P_{k}$ are seamed together).

######  Definition 4.30.

A _quilted strip-like end_ on $\underline{S}$ is a quilted map

(4.67)
$\underline{\epsilon}\colon\underline{Z}_{\pm}(\delta_{1},\ldots,\delta_{k})\to\underline{S},$

i.e. a collection of maps $\epsilon_{1},\ldots,\epsilon_{k}$

(4.68) $\epsilon_{i}\colon P_{i}\to P_{i}^{\prime}$

where $P_{i}$ is as in (4.65) and
$P_{i}^{\prime}\in\mathcal{P}(\underline{S})$, 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 $\underline{S}$ is a quilted map

(4.69)
$\underline{\epsilon}\colon\underline{C}_{\pm}(\delta_{1},\ldots,\delta_{k})\to\underline{S}$

compatible with the seams, pseudo-holomorphic, and proper on total spaces.

A _quilted surface with strip-like and cylindrical ends_ is a quilted surface
$\underline{S}$ 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

(4.70) $\mathcal{E}(\underline{S})=\mathcal{E}_{\rm
in}(\underline{S})\cup\mathcal{E}_{\rm out}(\underline{S})\cup\mathcal{E}_{\rm
free}(\underline{S}).$

the set of quilted ends of $\underline{S}$.

######  Definition 4.31.

A _decoration_ $(\underline{M},\underline{L})$ of a quilted surface
$\underline{S}$ consists in

  * •

a symplectic manifold $M_{P}$ for each patch $P\in\mathcal{P}(\underline{S})$,

  * •

a Lagrangian correspondence $L_{\sigma}\subset(M_{P})^{-}\times
M_{P^{\prime}}$ for each seam $\sigma\in\mathcal{S}(\underline{S})$ between
two patches $P,P^{\prime}$,

  * •

a Lagrangian submanifold $L_{b}\subset M_{P}$ for each true boundary of $P$.

######  Definition 4.32.

Let $(\underline{S},\underline{M},\underline{L})$ be a decorated quilted
surface. A (topological) _quilt_

(4.71) $\underline{u}\colon\underline{S}\to(\underline{M},\underline{L})$

consists in maps $u_{P}\colon P\to M_{P}$ for each patch $P$ of
$\underline{S}$ satisfying the _seam condition_ :

(4.72) $(u_{P}(x),u_{P^{\prime}}(x))\in L_{\sigma},$

for $x\in\sigma$, and $\sigma$ a seam between $P$ and $P^{\prime}$, and the
_boundary condition_ :

(4.73) $u_{P}(x)\in L_{b},$

for $x\in b$, and $b$ a true boundary of $P$.

To define the moduli space of quilts that will be involved in $\alpha_{N}$ we
use perturbations as in [Sei08, sec. (8e)], adapted to the quilted setting.

######  Definition 4.33 (Perturbation datum).

Let $(\underline{S},\underline{M},\underline{L})$ be a decorated quilted
surface with strip like ends. Let

(4.74)
$\mathcal{K}(\underline{S},\underline{M},\underline{L})\subset\Omega^{1}(\underline{S};\mathcal{H}(\underline{M})):=\prod_{P\in\mathcal{P}(\underline{S})}\Omega^{1}(P;\mathcal{H}({M}_{P})),$

consist in quilted 1-forms $\underline{K}=(K_{P})_{P}$ such that

  * •

on a boundary $b$ of a patch $P$ (belonging or not belonging to a seam),

(4.75) $K_{P}(\xi)_{|L_{b}}=0\text{, }\forall\xi\in Tb.$

  * •

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, $\underline{K}$ 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:

(4.76) $\underline{L}=\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
9.79582pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{M_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
15.41388pt\raise 6.0361pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.74722pt\hbox{$\scriptstyle{L_{0}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 33.79582pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
33.79582pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{M_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
57.4826pt\raise 6.0361pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.74722pt\hbox{$\scriptstyle{L_{1}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 77.38747pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
77.38747pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
103.06175pt\raise 6.0861pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.69722pt\hbox{$\scriptstyle{L_{k-1}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 114.88747pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
114.88747pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{M_{k}=M_{0},}$}}}}}}}\ignorespaces}}}}\ignorespaces$

with $M_{0}$ a point if we are considering a strip-like end. By the two
conditions above, on the end, $\underline{K}$ is of the form
$K_{P_{i}}=H_{i}(t)\mathrm{d}t$ on the patch $P_{i}$ decorated by $M_{i}$.

Then, we want $\left(\prod\underline{L}^{\prime}\right)_{0}$ and
$\left(\prod\underline{L}^{\prime}\right)_{1}$ as in Definition 4.25 to
intersect transversely in $\prod\underline{M}$.

Let $\mathcal{J}(P,M_{P})=C^{\infty}(P,\mathcal{J}(M_{P}))$ stand for the
space of domain-dependent almost complex structures, and

(4.77)
$\mathcal{J}(\underline{S},\underline{M},\underline{L})\subset\prod_{P\in\mathcal{P}(\underline{S})}\mathcal{J}(P,M_{P})$

be those $\underline{J}=(J_{P})_{P}$ that are only $t$-dependent on quilted
ends.

The set of _perturbation data_ is denoted

(4.78)
$\mathcal{K}\mathcal{J}(\underline{S},\underline{M},\underline{L}):=\mathcal{K}(\underline{S},\underline{M},\underline{L})\times\mathcal{J}(\underline{S},\underline{M},\underline{L}).$

Notice that at an incoming (resp. outgoing) end, a perturbation datum
$\mathcal{P}$ is asymptotic to a Floer datum $\mathcal{F}_{in}$ (resp.
$\mathcal{F}_{out}$). In this situation we will write
$\mathcal{P}\colon\mathcal{F}_{in}\to\mathcal{F}_{out}$. Denote by

(4.79)
$\mathcal{K}\mathcal{J}(\underline{S},\underline{M},\underline{L},\mathcal{F}_{in},\mathcal{F}_{out})\subset\mathcal{K}\mathcal{J}(\underline{S},\underline{M},\underline{L})$

the subset of such perturbation datum.

######  Remark 4.34.

In most moduli spaces we will be considering, it would be enough to consider
1-forms of the form $H_{z}\mathrm{d}t$, 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 $\underline{Z}$ be the quilted surface that will be involved in defining
the map $\alpha_{N}$: its total space is $Z\setminus\\{0,i\\}$, with a
vertical seam at $s=0$. We view it as a quilted surface with one incoming end
at $s\to-\infty$, one outgoing end at $s\to+\infty$, and two free quilted
strip-like ends near $0$ and $i$. The quilted surface $\underline{Z}$ has two
patches

(4.80) $\displaystyle Z_{-}$ $\displaystyle=\\{s+it:0\leq t\leq 1,\ s\leq
0\\}\text{, and}$ (4.81) $\displaystyle Z_{+}$ $\displaystyle=\\{s+it:0\leq
t\leq 1,\ s\geq 0\\},$

one seam $\sigma=\\{s=0\\}$ and four true boundary components
$\partial_{i}Z_{\pm}=\\{\pm s\geq 0,\ t=i\\}$, $i=0,1$.

Let $(\underline{M}_{N},\underline{L}_{N})$ be the following decoration of
$\underline{Z}$: $Z_{-}$ and $Z_{+}$ are decorated respectively by $M_{N}$ and
$M_{N+1}$, $\sigma$ by $\Lambda_{N}$, $\partial_{i}Z_{-}$ by $L_{i}^{N}$ and
$\partial_{i}Z_{+}$ by $L_{i}^{N+1}$, see Figure 5.

Figure 5. The quilted surface $\underline{Z}$ and its decoration
$(\underline{M}_{N},\underline{L}_{N})$.

For an input $x\in\mathcal{I}(L_{0}^{N},L_{1}^{N},\mathcal{F}_{N})$, an output
$y\in\mathcal{I}(L_{0}^{N+1},L_{1}^{N+1},\mathcal{F}_{N+1})$ and a
perturbation datum

(4.82)
$\mathcal{P}_{N}=(\underline{K},\underline{J})\in\mathcal{K}\mathcal{J}(\underline{Z},\underline{M}_{N},\underline{L}_{N};\mathcal{F}_{N},\mathcal{F}_{N+1}),$

let the moduli space of perturbed quilted maps
$\mathcal{L}(x,y;\mathcal{P}_{N})$ consist in quilts

(4.83)
$\underline{u}\colon\underline{Z}\to(\underline{M}_{N},\underline{L}_{N})$

satisfying the perturbed Cauchy-Riemann equation on a patch $P$:

(4.84) $(\mathrm{d}u_{P}-K_{P})^{0,1}=0,$

with limits $x$ (resp. $y$) at $s\to-\infty$ (resp. $s\to+\infty$), 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
$\mathcal{L}(x,y;\mathcal{P}_{N})_{i}\subset\mathcal{L}(x,y;\mathcal{P}_{N})$
denote the subspace of quilts of index $i$, corresponding to the index of the
relevant Fredholm section $\overline{\partial}_{\mathcal{P}_{N}}$ cutting out
$\mathcal{L}(x,y;\mathcal{P}_{N})$. 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.

######  Proposition 4.36.

There exists a comeagre subset

(4.85)
$\mathcal{K}\mathcal{J}(\underline{Z},\underline{M}_{N},\underline{L}_{N};\mathcal{F}_{N},\mathcal{F}_{N+1})^{\rm
reg}\subset\mathcal{K}\mathcal{J}(\underline{Z},\underline{M}_{N},\underline{L}_{N};\mathcal{F}_{N},\mathcal{F}_{N+1})$

of regular perturbation datum $\mathcal{P}_{N}$ for which
$\mathcal{L}(x,y;\mathcal{P}_{N})_{i}$ is smooth and of dimension $i$. The
zero-dimensional part $\mathcal{L}(x,y;\mathcal{P}_{N})_{0}$ is compact and
can be used to define a chain map

$\alpha_{N}\colon CF_{N}\to CF_{N+1}.$

and the compactification $\overline{\mathcal{L}}(x,y;\mathcal{P}_{N})_{1}$ of
its one-dimensional part $\mathcal{L}(x,y;\mathcal{P}_{N})_{1}$ 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

(4.86)
$CF^{G}(M;L_{0},L_{1};\\{\mathcal{F}_{N},\mathcal{P}_{N}\\}_{N})=\mathrm{Tel}(CF_{N},\alpha_{N}).$

###### Proof.

The first part of the statement (smoothness and expected dimension) follows
from standard transversality arguments [FHS95]. Compactness of
$\mathcal{L}(x,y;\mathcal{P}_{N})_{0}$ follows from Gromov compactness.

About the compactification $\overline{\mathcal{L}}(x,y;\mathcal{P}_{N})_{1}$,
a priori the degenerations that one could observe are (see Figure 6):

  1. (1)

sphere bubbling in the interior of the patches, or disc bubbling at the true
boundary component,

  2. (2)

quilted sphere bubbling at the vertical seam,

  3. (3)

strip breaking at the free ends,

  4. (4)

strip breaking at the incoming or outgoing end.

Figure 6. A priori bubling in $\overline{\mathcal{L}}(x,y;\mathcal{P}_{N})$.

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

(4.87)
$\partial\overline{\mathcal{L}}(x,y;\mathcal{P}_{N})_{1}=\coprod_{z}\mathcal{M}(x,z;\mathcal{F}_{N})\times\mathcal{L}(z,y;\mathcal{P}_{N})\cup\coprod_{z}\mathcal{L}(x,z;\mathcal{P}_{N})\times\mathcal{M}(z,y;\mathcal{F}_{N+1}),$

from which the identity $\partial\alpha_{N}+\alpha_{N}\partial=0$ 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.

We will rule out (2) as a special case of (3). Suppose that one has a strip
breaking at the free end: by folding it one can view it as a strip

(4.88) $u\colon(Z;\partial_{0}Z,\partial_{1}Z)\to(M_{N}^{-}\times
M_{N+1};\Lambda_{N},L_{i}^{N}\times L_{i}^{N+1}).$

Lift it to a strip

(4.89) $\displaystyle\tilde{u}$
$\displaystyle=(u_{M}^{1},u_{N},u_{M}^{2},u_{N+1})\colon$ (4.90)
$\displaystyle(Z;\partial_{0}Z,\partial_{1}Z)$ $\displaystyle\to(M^{-}\times
T_{N}^{-}\times M\times T_{N+1};I_{N},L_{i}\times 0_{N}\times L_{i}\times
0_{N+1}),$

with limits $x=(x_{M},x_{N},x_{M},x_{N+1})$ and
$y=(y_{M},y_{N},y_{M},y_{N+1})$ at the ends. This is always possible: first
lift the $\partial_{0}Z$ boundary to $I_{N}$ (using the fibration
$I_{N}\to\Lambda_{N}$), then extend this lift to the strip $Z$, and the other
boundary $\partial_{1}Z$ will automatically be mapped to $L_{i}\times
0_{N}\times L_{i}\times 0_{N+1}$. Notice that $\Delta_{M}\times
N_{\Gamma(\iota_{N})}$ and $L_{i}\times 0_{N}\times L_{i}\times 0_{N+1}$
intersect cleanly along $\Delta_{L_{i}}\times\Gamma(\iota_{N})$.

Now we claim we can find another strip

(4.91) $\tilde{d}=(d_{M}^{1},d_{N},d_{M}^{2},d_{N+1})\colon Z\to M^{-}\times
T_{N}^{-}\times M\times T_{N+1}$

that coincides with $\tilde{u}$ on $\partial_{0}Z$, is entirely contained in
$\Delta_{M}\times N_{\Gamma(\iota_{N})}$, and such that
$\partial_{1}\tilde{d}$ is in the intersection
$\Delta_{L_{i}}\times\Gamma(\iota_{N})$. Indeed, take
$d_{M}^{1}=d_{M}^{2}=u_{M}^{1}$, and for $d_{N},d_{N+1}$ homotope
$\partial_{0}u_{N},\partial_{0}u_{N+1}$ to the zero section:

(4.92) $\displaystyle d_{N}(s,t)$ $\displaystyle=(1-t)u_{N}(s,0),$ (4.93)
$\displaystyle d_{N+1}(s,t)$ $\displaystyle=(1-t)u_{N+1}(s,0).$

Now, glue $\tilde{d}$ and $\tilde{u}$ along their $\partial_{0}Z$ boundary to
get a disc $\tilde{u}_{cap}$ with boundary in $L_{i}\times 0_{N}\times
L_{i}\times 0_{N+1}$. By construction, $u$, $\tilde{u}$ and $\tilde{u}_{cap}$
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, $\tilde{u}_{cap}$ must have zero area, which contradicts
$u$ being nonconstant.

Assume now we have a quilted sphere bubble of type (2): we can fold it to a
disc

(4.94) $u\colon(D^{2},\partial D^{2})\to(M_{N}^{-}\times
M_{N+1},\Lambda_{N}).$

Pick a path in $\Lambda_{N}$ that connects a boundary point of $u$ to a point
in $L_{i}^{N}\times L_{i}^{N+1}$. Such a path exists: for example one can
follow the seam of the principal component of the quilt the bubble is attached
to. This path allows one to view $u$ as a strip

(4.95) $(Z;\partial_{0}Z,\partial_{1}Z)\to(M_{N}^{-}\times
M_{N+1};\Lambda_{N},L_{i}^{N}\times L_{i}^{N+1})$

of the kind (3) with same area and index, which therefore can be ruled out for
the same reasons.

∎

######  Remark 4.37.

If $G$ is not simply connected, lifting discs of type (2) to discs with
boundary in $\widehat{\Lambda}_{N}$ (and not $\widetilde{\Lambda}_{N}$) might
not be possible, and for this reason $\Lambda_{N}$ might not be monotone.
Fortunately, the above trick using $L_{i}^{N}\times L_{i}^{N+1}$ allows us to
overcome this problem.

######  Remark 4.38.

In [KLZ19], The authors define equivariant self-Floer homology of a Lagrangian
using similar symplectic homotopy quotients: they consider
$M\times_{G}\mu_{T_{N}}^{-1}(0)$, 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.

###  4.7. 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 $T_{x}L_{0}\oplus T_{x}L_{1}$ with the one of $T_{x}L$ at a transverse
intersection point $x$ provides an absolute $\mathbb{Z}_{2}$ 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 $\mathbb{Z}_{A}$, with $A\in\mathbb{Z}$ as in
Assumption 4.20, or $A=+\infty$ in the exact setting. Relative means that for
$x,y$ generators of $CF(M;L_{0},L_{1})$, one has a number
$I(x,y)\in\mathbb{Z}_{A}$ 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 $M_{N},L_{0}^{N},L_{1}^{N}$ will
satisfy similar assumptions, and hence $CF(M_{N};L_{0}^{N},L_{1}^{N})$ 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:

######  Definition 4.39 ([Sei00]).

For $n\geq 2$, or $n=+\infty$, an $n$-fold Maslov covering of $M$ is a
$\mathbb{Z}_{n}$-covering $\mathcal{L}^{n}\to\mathcal{L}$ of the Lagrangian
grassmannian bundle $\mathcal{L}\to M$.

If $L\subset M$ is a Lagrangian, its tangent subspaces defines a section
$s_{L}\colon L\to\mathcal{L}$ of $\mathcal{L}_{|L}\to L$.

Assume $M$ is given an $n$-fold Maslov covering $\mathcal{L}^{n}$, a
$\mathcal{L}^{n}$-_grading_ on $L$ is a lift $\tilde{L}\colon
L\to\mathcal{L}^{n}$ of $s_{L}$.

If in addition $(M,\omega,\mu)$ is $G$-Hamiltonian and $L\subset M$ is a
$G$-Lagrangian, one can define equivariant analogues of Maslov coverings and
gradings:

######  Definition 4.40.

Assume $M$ is $G$-Hamiltonian and $L\subset M$ is a $G$-Lagrangian. Let
$\mathcal{L}_{G}\subset\mathcal{L}_{|\mu^{-1}(0)}\to\mu^{-1}(0)$ consist in
$G$-Lagrangian subspaces. An $n$-fold Maslov $G$-covering of $M$ is a
$\mathbb{Z}_{n}$-covering $\mathcal{L}^{n}_{G}\to\mathcal{L}_{G}$ of
$\mathcal{L}_{G}\to\mu^{-1}(0)$, with a lift to $\mathcal{L}^{n}_{G}$ of the
$G$-action on $\mathcal{L}_{G}$.

If $L\subset M$ is a $G$-Lagrangian, then $s_{L}$ as defined previously is a
section of ${\mathcal{L}_{G}}_{|L}$. A $(\mathcal{L}_{G}^{n},G)$-_grading_ on
$L$ is a $G$-equivariant lift $\tilde{L}\colon L\to\mathcal{L}^{n}_{G}$ of
$s_{L}$.

This definition is motivated by the following immediate result:

######  Proposition 4.41.

Assume that the action of $G$ on $M$ is regular. Then the datum of a $n$-fold
Maslov $G$-covering $\mathcal{L}_{G}^{n}$ of $M$ is equivalent to a $n$-fold
Maslov covering $\mathcal{L}^{n}$ of $M/\\!\\!/G$. If such a datum is given, a
$(\mathcal{L}_{G}^{n},G)$-_grading_ on $L$ is equivalent to a
$\mathcal{L}^{n}$-_grading_ on $L/G$.

######  Example 4.42.

Let $M=T^{*}X$ be a cotangent bundle. The bundle $\mathcal{L}\to M$ admits a
“section by Maslov cycles” $\mathfrak{m}$: for $m\in M$ let

(4.96) $\displaystyle\mathfrak{m}_{m}$ $\displaystyle=\\{l\in\mathcal{L}_{m}\
:\ l\text{ is not transverse to the fiber at }m\\},$ (4.97)
$\displaystyle\mathfrak{m}$ $\displaystyle=\bigcup_{m\in M}\mathfrak{m}_{m},$
(4.98) $\displaystyle\mathcal{L}^{\pitchfork}$
$\displaystyle=\mathcal{L}\setminus\mathfrak{m}.$

It follows that $\mathcal{L}$ admits a Maslov $\mathbb{Z}$-cover
$c\colon\widehat{\mathcal{L}}\to\mathcal{L}$. Fix a component
$\widehat{\mathcal{L}}^{\pitchfork}_{0}$ of
$\widehat{\mathcal{L}}^{\pitchfork}=c^{-1}(\mathcal{L}^{\pitchfork})$.

If $L\subset M$ is a Lagrangian transverse to the fibers at every point, then
the section $s_{L}$ takes its values in $\mathcal{L}^{\pitchfork}$, and admits
a unique lift contained in $\widehat{\mathcal{L}}^{\pitchfork}_{0}$, which is
defined to be its canonical grading.

If now $X$ is acted on by $G$ (inducing a Hamiltonian action on $M$), then
$\widehat{\mathcal{L}}_{G}=c^{-1}(\mathcal{L}_{G})$ is a Maslov $G$-cover as
defined previously. With

(4.99) $\displaystyle\mathcal{L}^{\pitchfork}_{G}$
$\displaystyle=\mathcal{L}_{G}\setminus\mathfrak{m},$ (4.100)
$\displaystyle\widehat{\mathcal{L}}^{\pitchfork}_{G}$
$\displaystyle=c^{-1}(\mathcal{L}^{\pitchfork}_{G}),$

if $L\subset M$ is a $G$-Lagrangian transverse to the fibers, then its
canonical grading takes its values in $\widehat{\mathcal{L}}^{\pitchfork}_{G}$
and defines a $G$-grading.

If $M$ has a Maslov cover, and $L_{0},L_{1}\subset M$ are $G$-graded, by
Example 4.42 $L_{0}\times 0_{N},L_{1}\times 0_{N}\subset M\times T_{N}$ are
$G$-graded, and by Proposition 4.41 $L_{0}^{N},L_{1}^{N}\subset M_{N}$ are
graded. Therefore:

######  Proposition 4.43.

If $L_{0},L_{1}\subset M$ are $G$-graded in addition of satisfying Assumptions
4.17, 4.20, then $CF_{G}(M;L_{0},L_{1})$ is absolutely
$\mathbb{Z}_{n}$-graded.

##  5\. 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 $\mathcal{F}^{i}_{N}$ be a regular Floer datum
for $(M_{N};L_{0}^{N},L_{1}^{N})$, and

(5.1) $\displaystyle C_{N}$
$\displaystyle=CF(M_{N};L_{0}^{N},L_{1}^{N};\mathcal{F}^{0}_{N}),$ (5.2)
$\displaystyle D_{N}$
$\displaystyle=CF(M_{N};L_{0}^{N},L_{1}^{N};\mathcal{F}^{1}_{N}).$

Let also $\mathcal{P}^{i}_{N}$ be regular perturbation datum for
$(\underline{Z};\underline{M}_{N},\underline{L}_{N})$, defining increment
morphisms:

(5.3) $\displaystyle\alpha_{N}\colon C_{N}\to C_{N+1}\text{, with }i=0,$ (5.4)
$\displaystyle\beta_{N}\colon D_{N}\to D_{N+1}\text{, with }i=1.$

We are going to introduce perturbation data, as summarized in the following
diagram:

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These will be used to define maps:

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Let $\mathcal{Q}_{N}$ be a perturbation datum on
$(Z;M_{N};L_{0}^{N},L_{1}^{N})$ going from $\mathcal{F}^{0}_{N}$ to
$\mathcal{F}^{1}_{N}$. Use it to define a moduli space
$\mathcal{C}(x,y;\mathcal{Q}_{N})$ of $\mathcal{Q}_{N}$-perturbed pseudo-
holomorphic maps. For generic $\mathcal{Q}_{N}$ the moduli space is smooth and
of expected dimension. Its zero dimensional part
$\mathcal{C}(x,y;\mathcal{Q}_{N})_{0}$ defines a chain map

(5.7) $\varphi_{N}\colon C_{N}\to D_{N}.$

Figure 7. The perturbation $\mathcal{Q}_{N}$ on
$(Z;M_{N};L_{0}^{N},L_{1}^{N})$.

In order to prove that $\varphi_{N}$ commutes with $\alpha_{N},\beta_{N}$ up
to homotopy we define a parametrized moduli space, involving a family
$\\{\mathcal{R}^{L}_{N}\\}_{L\in\mathbb{R}}$ of perturbations on
$(\underline{Z},\underline{M}_{N},\underline{L}_{N})$ such that (see Figure
8):

  * •

if $L\ll 0$, then $\mathcal{R}^{L}_{N}$ corresponds to the superposition of
$\mathcal{Q}_{N}$ shifted by $L$ in the region $s<L/2$ of $\underline{Z}$ and
$\mathcal{P}_{N}^{1}$ in $s>L/2$. Notice that when $L$ is negatively large
enough, these coincide with $\mathcal{F}_{N}^{1}$ at $s=L/2$.

  * •

if $L\gg 0$, then $\mathcal{R}^{L}_{N}$ corresponds to the superposition of
$\mathcal{P}_{N}^{0}$ shifted by $L$ in the region $s<L/2$ of $\underline{Z}$
and $\mathcal{Q}_{N+1}$ in $s>L/2$. Notice that when $L$ is large enough,
these coincide with $\mathcal{F}_{N+1}^{0}$ at $s=L/2$.

Figure 8. The perturbation $\mathcal{R}^{L}_{N}$ for large $\left|L\right|$.

For each $L$, let then $\mathcal{H}_{L}(x,y;\mathcal{R}^{L}_{N})$ be the
moduli space of $\mathcal{R}^{L}_{N}$-perturbed pseudo-holomorphic quilts
$\underline{u}\colon\underline{Z}\to(\underline{M}_{N},\underline{L}_{N})$,
and the corresponding parametrized moduli space:

(5.8)
$\mathcal{H}(x,y;\\{\mathcal{R}^{L}_{N}\\}_{L})=\bigcup_{L\in\mathbb{R}}\mathcal{H}_{L}(x,y;\mathcal{R}^{L}_{N}),$

######  Proposition 5.1.

For generic families $\\{\mathcal{R}^{L}_{N}\\}_{L}$,
$\mathcal{H}(x,y;\\{\mathcal{R}^{L}_{N}\\}_{L})$ is smooth and of expected
dimension (i.e. the index of the corresponding Fredholm operator).

Its zero-dimensional part

(5.9)
$\mathcal{H}(x,y;\\{\mathcal{R}^{L}_{N}\\}_{L})_{0}=\bigcup_{L\in\mathbb{R}}\mathcal{H}_{L}(x,y;\mathcal{R}^{L}_{N})_{-1}$

can be used to define a map

(5.10) $\kappa_{N}\colon C_{N}\to D_{N+1},$

and its one-dimensional part

(5.11)
$\mathcal{H}(x,y;\\{\mathcal{R}^{L}_{N}\\}_{L})_{1}=\bigcup_{L\in\mathbb{R}}\mathcal{H}_{L}(x,y;\mathcal{R}^{L}_{N})_{0}$

can be used to show that $\kappa_{N}$ gives a homotopy

(5.12)
$\varphi_{N+1}\alpha_{N}-\beta_{N}\varphi_{N}=\mathrm{d}\kappa_{N}+\kappa_{N}\mathrm{d}.$

Therefore one gets a continuation morphism

(5.13) $\mathrm{Tel}(\varphi_{N},\kappa_{N})\colon CF_{G}^{0}\to CF_{G}^{1},$

with $CF_{G}^{0}=\mathrm{Tel}(C_{N},\alpha_{N})$ and
$CF_{G}^{1}=\mathrm{Tel}(D_{N},\beta_{N})$.

###### Proof.

Smoothness of these moduli spaces follows from parametrized transversality
[MS12, Def. 3.1.6, Th. 3.1.6].

The boundary of the compactification of
$\mathcal{H}(x,y;\\{\mathcal{R}^{L}_{N}\\}_{L})_{1}$ consists in three pieces:

  * •

When $L\to-\infty$, the quilt breaks in a strip in $M_{N}$ perturbed by
$\mathcal{Q}_{N}$, and a quilt perturbed by $\mathcal{P}^{1}_{N}$, leading to
a contribution for $\beta_{N}\varphi_{N}$

  * •

When $L\to+\infty$, the quilt breaks in a strip in $M_{N+1}$ perturbed by
$\mathcal{Q}_{N+1}$, and a quilt perturbed by $\mathcal{P}^{0}_{N}$, leading
to a contribution for $\varphi_{N+1}\alpha_{N}$

  * •

At finite $L$, one can have strip breaking at either the incoming or outgoing
end, contributing to $\mathrm{d}\kappa_{N}+\kappa_{N}\mathrm{d}$.

Indeed, all other possible bubbling or strip breaking is excluded, as in the
proofs of Propositions 4.27, 4.36. ∎

Now suppose that two different choices
$(\mathcal{Q}_{N}^{0},\mathcal{R}_{N}^{L,0})$ and
$(\mathcal{Q}_{N}^{1},\mathcal{R}_{N}^{L,1})$ of regular perturbations were
made, leading to maps $(\varphi_{N}^{0},\kappa_{N}^{0})$ and
$(\varphi_{N}^{1},\kappa_{N}^{1})$. In order to construct homotopies
$(\phi_{N},\kappa_{N})$ as in Definition 2.10 we introduce two kinds of
families of perturbations $\mathcal{S}_{N}^{v}$, $\mathcal{T}_{N}^{L,v}$, with
$L\in\mathbb{R}$ and $v\in[0,1]$.

Let $\\{\mathcal{S}_{N}^{v}\\}_{v\in[0,1]}$ be a family of perturbations on
$(Z,M_{N},L_{0}^{N},L_{1}^{N})$ from $\mathcal{F}_{N}^{0}$ to
$\mathcal{F}_{N}^{1}$ such that, when $v=0$ or 1,
$\mathcal{S}_{N}^{v}=\mathcal{Q}_{N}^{v}$. Let
$\mathcal{C}^{v}(x,y,\mathcal{S}_{N}^{v})$ be the moduli space associated with
$\mathcal{S}_{N}^{v}$, and let

(5.14)
$\mathcal{C}(x,y;\\{\mathcal{S}_{N}^{v}\\})=\bigcup_{v\in[0,1]}\mathcal{C}^{v}(x,y;\mathcal{S}_{N}^{v})$

be the corresponding parametrized moduli space.

######  Proposition 5.2.

For generic families $\\{\mathcal{S}^{v}_{N}\\}_{v}$,
$\mathcal{C}(x,y;\\{\mathcal{S}_{N}^{v}\\})$ is smooth and of expected
dimension.

Its zero-dimensional part
$\mathcal{C}(x,y)_{0}=\bigcup_{v\in[0,1]}\mathcal{C}^{v}(x,y)_{-1}$ defines
$\phi_{N}\colon C_{N}\to D_{N}$, and its one-dimensional part
$\mathcal{C}(x,y)_{1}=\bigcup_{v\in[0,1]}\mathcal{C}^{v}(x,y)_{0}$
compactifies to a cobordism that gives the relation

(5.15)
$\varphi^{1}_{N}-\varphi^{0}_{N}=\mathrm{d}\phi_{N}+\phi_{N}\mathrm{d}.$

###### Proof.

The proof is analogous to the proof of Proposition 5.1 ∎

Let $\\{\mathcal{T}_{N}^{L,v}\\}_{(L,v)\in\mathbb{R}\times[0,1]}$ be a family
of perturbations on $(\underline{Z},\underline{M}_{N},\underline{L}_{N})$ such
that:

  * •

if $L\ll 0$, then $\mathcal{T}_{N}^{L,v}$ corresponds to the superposition of
$\mathcal{S}_{N}^{v}$ shifted by $L$ in the region $s<L/2$ of $\underline{Z}$
and $\mathcal{P}_{N}^{1}$ in $s>L/2$.

  * •

if $L\gg 0$, then $\mathcal{T}_{N}^{L,v}$ corresponds to the superposition of
$\mathcal{P}_{N}^{0}$ shifted by $L$ in the region $s<L/2$ of $\underline{Z}$
and $\mathcal{S}_{N+1}^{v}$ in $s>L/2$.

  * •

when $v=0$ or 1, $\mathcal{T}_{N}^{L,v}=\mathcal{R}_{N}^{L,v}$.

Let $\mathcal{K}_{L}^{v}(x,y;\\{\mathcal{T}_{N}^{L,v}\\})$ be the associated
moduli space of quilts, and

(5.16) $\mathcal{K}(x,y;\\{\mathcal{T}_{N}^{L,v}\\})=\bigcup_{v\in[0,1],\
L\in\mathbb{R}}\mathcal{K}_{L}^{v}(x,y;\mathcal{T}_{N}^{L,v}).$

######  Proposition 5.3.

For generic $\\{\mathcal{T}_{N}^{L,v}\\}$,
$\mathcal{K}(x,y;\\{\mathcal{T}_{N}^{L,v}\\})$ is smooth and of expected
dimension. Its zero-dimensional part
$\mathcal{K}(x,y)_{0}=\bigcup_{v\in[0,1],\
L\in\mathbb{R}}\mathcal{K}_{L}^{v}(x,y)_{-2}$ defines the map
$\kappa_{N}\colon C_{N}\to D_{N+1}$ and its one-dimensional part

(5.17) $\mathcal{K}(x,y)_{1}=\bigcup_{v\in[0,1],\
L\in\mathbb{R}}\mathcal{K}_{L}^{v}(x,y)_{-1}$

compactifies to a cobordism that gives the relation:

(5.18)
$\kappa^{1}_{N}-\kappa^{0}_{N}+\phi_{N+1}\alpha_{N}+\beta_{N}\phi_{N}=\mathrm{d}\kappa_{N}+\kappa_{N}\mathrm{d}.$

Therefore, by Proposition 2.11, different paths of perturbations induce the
same maps on telescopes, up to homotopy.

###### Proof.

The boundary of the compactification of $\mathcal{K}(x,y)_{1}$ consists in
five kind of degenerations (see Figure 9):

  1. (1)

at $v=0$, contribution for $\kappa_{N}^{0}$,

  2. (2)

at $v=1$, contribution for $\kappa_{N}^{1}$,

  3. (3)

at $L\to-\infty$, contribution for $\beta_{N}\phi_{N}$,

  4. (4)

at $L\to+\infty$, contribution for $\phi_{N+1}\alpha_{N}$,

  5. (5)

at finite $L$ anf $v\in(0,1)$, strip breaking at incoming and outgoing ends,
contributing to $\mathrm{d}\kappa_{N}+\kappa_{N}\mathrm{d}$.

∎

Figure 9. The moduli space $\mathcal{K}(x,y)_{1}$ and its compactification.

To show that furthermore these are homotopy equivalences, by Corollary 2.12,
one has to check that if $(\varphi_{N},\kappa_{N})$ (resp.
$(\tilde{\varphi}_{N},\tilde{\kappa}_{N})$) as in Proposition 5.1 are from
$(C_{N},\alpha_{N})$ to $(D_{N},\beta_{N})$ (resp. from $(D_{N},\beta_{N})$ to
$(C_{N},\alpha_{N})$), then their composition
$(\tilde{\varphi}_{N},\tilde{\kappa}_{N})\circ(\varphi_{N},\kappa_{N})$ is
homotopic to $(id_{C_{N}},0)$, i.e. there exists

(5.19) $(\phi_{N},H_{N})\colon(C_{N},\alpha_{N})\to(C_{N},\alpha_{N})$

such that:

(5.20) $\displaystyle\tilde{\varphi}_{N}\varphi_{N}-id$
$\displaystyle=\mathrm{d}\phi_{N}+\phi_{N}\mathrm{d},$ (5.21)
$\displaystyle\tilde{\kappa}_{N}\phi_{N}+\tilde{\varphi}_{N+1}\kappa_{N}+\phi_{N+1}\alpha_{N}+\alpha_{N}\phi_{N}$
$\displaystyle=\mathrm{d}H_{N}+H_{N}\mathrm{d}.$

Let $\mathcal{Q}_{N}$, $\mathcal{R}_{N}^{L}$ (resp. $\tilde{\mathcal{Q}}_{N}$,
$\tilde{\mathcal{R}}_{N}^{\tilde{L}}$) be the (families of) perturbations as
earlier involved in the definitions of $(\varphi_{N},\kappa_{N})$ (resp.
$(\tilde{\varphi}_{N},\tilde{\kappa}_{N})$).

Let $\\{\mathcal{U}_{N}^{v}\\}_{v\in[0,+\infty)}$ be a family of perturbations
on $(Z,M_{N},L_{0}^{N},L_{1}^{N})$ going from $\mathcal{F}_{N}^{0}$ to
$\mathcal{F}_{N}^{0}$, and such that :

  * •

when $v=0$, it is just $\mathcal{F}_{N}^{0}$ (unperturbed)

  * •

when $v\gg 0$, it is a superposition of $\mathcal{Q}_{N}$ and
$\tilde{\mathcal{Q}}_{N}$ shifted by $v$, as in Figure 10.

Figure 10. The perturbation $\mathcal{U}_{N}^{v}$ for large $v$.

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