# Coulomb branch algebras via symplectic cohomology

Eduardo González  ,  Cheuk Yu Mak  and  Dan Pomerleano

###### Abstract.

Let $(\bar{M},\omega)$ be a compact symplectic manifold with convex boundary
and $c_{1}(T\bar{M})=0$. Suppose that $(\bar{M},\omega)$ is equipped with a
convex Hamiltonian $G$-action for some connected, compact Lie group $G$. We
construct an action of the pure Coulomb branch of $G$ on the $G$-equivariant
symplectic cohomology of $\bar{M}.$ Building on work of Teleman [Teleman1], we
use this construction to characterize the Coulomb branches of Braverman-
Finkelberg-Nakajima [BFN] in terms of equivariant symplectic cohomology.

##  1\. Introduction

### Background

Fix a compact, connected Lie group $G$ and a quaternionic representation $E$
of $G$. Physicists associate a 3D N = 4 supersymmetric gauge theory to such a
pair. This gauge theory defines various moduli of vacua, one of which is
called the Coulomb branch, $\mathcal{M}(G,E)$. Physical considerations predict
that $\mathcal{M}(G,E)$ should have a number of remarkable properties, for
example it should be a (possibly singular) hyper-Kähler manifold with an
SU(2)-action. However, the physical definition of $\mathcal{M}(G,E)$ involves
quantum corrections which are difficult to interpret mathematically. In [BFN],
Braverman-Finkelberg-Nakajima proposed a definition of $\mathcal{M}(G,E)$ for
cotangent type representations $E$, i.e. those representations which are
isomorphic to $\mathbb{V}\oplus\mathbb{V}^{\vee}$ for some complex
representation $\mathbb{V}$.111Constructions of Coulomb branches for more
general representations have recently been proposed in [Raskin, Teleman3],
however the interaction of these constructions with symplectic topology is not
clear at present.  For these representations, they define $\mathcal{M}(G,E)$
to be the spectrum $\operatorname{Spec}(\mathcal{C}_{3}(G;E))$ of certain
Poisson algebras $\mathcal{C}_{3}(G;E).$ For our purposes, it will be useful
to note that the algebras $\mathcal{C}_{3}(G;E)$ naturally arise in a one-
parameter family $\mathcal{C}_{3}^{\circ}(G;E)$ over $\mathbb{C}[\mu]$ (of
which $\mathcal{C}_{3}(G;E)$ is the zero fiber) which incorporates the central
rescaling of the representation.

The primordial example of a Coulomb branch is the pure Coulomb branch,
$\mathcal{M}(G,0)$, which occurs when the “matter representation” $E$ is
trivial. $\mathcal{C}_{3}(G;0)$ is by definition the “semi-infinite” homology
of the based loop space of $G$, $\hat{H}_{*}^{G}(\Omega G)$, equipped with its
Pontryagin product. An important result ([BFM]) identifies $\mathcal{M}(G,0)$
with the universal centralizer of the Langlands dual group $G^{\vee}.$ The
variety $\mathcal{M}(G,0)$ is a affine group scheme over
$H^{*}(BG,\mathbb{C})$ which acts on all of the other Coulomb branches.
Moreover, each $\mathcal{M}(G,E)$ contains a dense free
$\mathcal{M}(G,0)$-orbit. When the matter representation is non-trivial, the
BFN construction extends and unifies many known constructions in geometric
representation theory — for example based maps from $\mathbb{C}P^{1}$ to a
flag variety or slices in affine Grassmannians.

### Pure Coulomb branches

If $(M,\omega)$ is a compact (monotone) symplectic manifold with a
Hamiltionian $G$ action, then $\mathcal{C}_{3}(G;0)$ acts on the equivariant
quantum cohomology, $QH_{G}^{*}(M)$ ([Teleman2, GMP22]). Our first result is
an analogue of this construction for compact symplectic manifolds
$(\bar{M},\omega)$ with convex boundary and vanishing first Chern class.
Recall that having convex boundary means that in a neighborhood of the
boundary, there is a primitive $\theta$ of $\omega$ such that the
$\omega$-dual of $\theta$ points outward along the boundary. A Hamiltonian
$G$-action on $(\bar{M},\omega)$ is convex if $\theta$ can be chosen
$G$-invariant and also to have nowhere dense spectrum. Given
$(\bar{M},\omega)$ with a (convex) Hamiltonian $G$-action, we can consider its
equivariant symplectic cohomology, $SH_{G}^{*}(\bar{M}),$ which is the direct
limit of equivariant Hamiltonian Floer groups of cylindrical Hamiltonians of
positive slope. We prove:

######  Theorem 1.1.

Let $(\bar{M},\omega)$ be a compact symplectic manifold with convex boundary
and $c_{1}(T\bar{M})=0$. Suppose further that $(\bar{M},\omega)$ is equipped
with a convex Hamiltonian $G$-action. Then there is an algebra homomorphism:

(1) $\displaystyle\mathcal{S}:\mathcal{C}_{3}(G;0)\to SH_{G}^{*}(\bar{M}).$

In fact, for technical reasons discussed below, we first construct a ring
homomorphism:

(2) $\displaystyle\mathcal{S}_{T}:\hat{H}_{*}^{T}(\Omega G)\to
SH_{T}^{*}(\bar{M}).$

We then show that this map is equivariant with respect to the natural Weyl
group actions on both sides. The map (1) is then obtained by passing to Weyl-
invariants.

The starting point for our work is Seidel’s observation that for closed
symplectic manifolds, a loop $\gamma_{t}$ of Hamiltonian symplectomorphisms
induces an action on Hamiltonian Floer theory. Seidel’s construction has been
extended to include (convex) $S^{1}$-actions on convex symplectic manifolds in
[Ritter, LJ21]. The new feature, compared to the closed case, is that a
Hamiltonian loop can modify the behavior of the Hamiltonian at infinity. This
means that given a Hamiltonian $H$ of some fixed slope, there may be no
continuation map between the Floer cohomology of the twisted Hamiltonian,
$\gamma_{t}^{*}H$, and the Floer cohomology $H$ due to the failure of the
maximum principle for interpolating solutions. However, under suitable
geometric assumptions, there is a well-defined continuation map to a
Hamiltonian $H^{\prime}$ of higher slope, giving rise to an action on
symplectic cohomology.

The map (2) is a “parameterized” or “family” version of Seidel’s construction
over cycles in $\Omega G.$ The idea is to represent cycles in
$\hat{H}_{*}^{T}(\Omega G)$ “geometrically” by suitably decorated equivariant
maps from a compact, smooth manifold into $\Omega G$ (this is an equivariant
version of “geometric homology” due to [BaumDouglas, Jak98]). One needs to
choose these cycles (as well as the auxiliary data needed to define Floer
cohomology) carefully to ensure that the maximum principle holds over their
associated families. A further complication is that the version of equivariant
Floer cohomology that we consider involves considering further families over
the Borel space $BT$, which is an infinite limit of finite-dimensional spaces.
Additional care is required to ensure that there is a continuation solution
from the twisted Hamiltonian to some Hamiltonian of finite slope (in other
words, that the slope needed to dominate the twisted Hamiltonian does not
“escape to infinity” over the finite dimensional approximations). These
problems were most easily overcome $T$-equivariantly as opposed to
$G$-equivariantly, which resulted in the slightly indirect construction
mentioned above.

######  Remark 1.2.

For the discussion which follows, it will be relevant to note that above,
$SH_{G}^{*}(\bar{M})$ was defined over the Novikov field $\Lambda$ with ground
field $\mathbb{C}$. However, when $(\bar{M},\omega,\theta)$ is a Liouville
domain (meaning $\theta$ can be extended over all of $M$), symplectic
cohomology can be also be defined over $\mathbb{C}$. The map (1) can also be
defined for this version of symplectic cohomology, which we denote by
$SH_{G}^{*}(\bar{M},\mathbb{C})$.

### Coulomb branches with matter

In [BFN], the Coulomb branch algebra is defined via a convolution diagram
involving a certain infinite rank vector bundle over the affine Grassmannian
of the complexification of $G$. In [Teleman1], Teleman gave an elegant and
direct construction of $\mathcal{C}_{3}^{\circ}(G;E)$ as the algebra of
functions on a scheme obtained by gluing two copies of
$\mathring{\mathcal{M}}(G,0)=\operatorname{Spec}(\mathcal{C}_{3}^{\circ}(G;0))$
by group multiplication with a certain easily described rational section of
the Toda integrable system, $\epsilon_{V}$. This construction characterizes
$\mathring{\mathcal{M}}(G,E):=\operatorname{Spec}(\mathcal{C}_{3}^{\circ}(G;E))$
as the universal equivariant, affine compactification of
$\mathring{\mathcal{M}}(G,0)$ such that $\epsilon_{V}$ extends to a section of
the Toda system (see Corollary 6.9). While Teleman’s construction is
algebraic, it was motivated by consideration of 2D boundary conditions for the
pure gauge theory and the gauge theory with matter $E$.

Our second result realizes a closed-string version of this story. Given a
Liouville domain with convex $G$-action, it is natural to search for actions
on the ordinary cohomology as opposed to the symplectic cohomology. There is
an acceleration map

(3) $\displaystyle\operatorname{ac}:H^{*}_{G}(\bar{M},\mathbb{C})\to
SH_{G}^{*}(\bar{M},\mathbb{C}).$

In cases where this map is injective, it makes sense to look for subalgebras
of $\mathcal{C}_{3}(G;0)$ which preserve its image. The example relevant to
Coulomb branches is the seemingly simple case where $\bar{M}=\bar{\mathbb{V}}$
is a unit ball inside of a complex vector space $\mathbb{V}$ equipped with a
unitary representation of $G$. In this case, we consider the $G\times S^{1}$
equivariant symplectic cohomology $SH^{*}_{G\times S^{1}}(\bar{\mathbb{V}})$,
where the additional $S^{1}$-factor corresponds to the diagonal rotation
action. The algebra $\mathcal{C}_{3}^{\circ}(G;0):=\hat{H}_{G}(\Omega G)[\mu]$
is naturally a subalgebra of $\hat{H}_{G\times S^{1}}(\Omega(G\times S^{1}))$
and so (1) therefore induces an action:

(4) $\displaystyle\mathcal{C}_{3}^{\circ}(G;0)\to SH_{G\times
S^{1}}(\bar{\mathbb{V}},\mathbb{C}).$

We give the following symplectic interpretation of the ingredients in
Teleman’s construction:

######  Theorem 1.3.

The following hold:

  1. (1)

There is an isomorphism $\Gamma(\mathcal{O}_{\epsilon_{\mathbb{V}}})\cong
SH^{*}_{G\times S^{1}}(\bar{\mathbb{V}},\mathbb{C}).$ The inclusion
$\epsilon_{\mathbb{V}}$ corresponds to the homomorphism (4).

  2. (2)

There is a commutative diagram:

(5) ${\mathcal{C}_{3}^{\circ}(G;E)}$${H^{*}_{G\times
S^{1}}(\mathbb{V},\mathbb{C})}$${\mathcal{C}_{3}(G;0)[\mu]}$${SH^{*}_{G\times
S^{1}}(\bar{\mathbb{V}},\mathbb{C}).}$$\scriptstyle{\mathcal{S}}$$\scriptstyle{i}$$\scriptstyle{\operatorname{ac}}$$\scriptstyle{\mathcal{S}}$

These calculations allows us to rephrase Teleman’s characterization of
$\mathring{\mathcal{M}}(G,E)$ geometrically as the universal equivariant,
affine compactification of the group scheme $\mathring{\mathcal{M}}(G,0)$
whose coordinate ring fits into (5) (see Corollary 6.13). This result should
be understood as a decategorification of the expectation that a suitable
equivariant “fully” wrapped Fukaya category of $\bar{\mathbb{V}}$ should
define a boundary condition for the pure gauge theory while a suitable
infinitesimally wrapped Fukaya category should define a boundary condition for
the gauge theory with matter. Boundary conditions for the gauge theory with
matter have been studied from the physics perspective [Hilburn1] and from an
algebro-geometric perspective [Hilburn2].

Comparison with [GMP22] suggests a natural (and potentially easy) extension of
our results here. Namely, [GMP22] also studies actions of the quantum Coulomb
branch algebra $\mathcal{C}_{3}(G;0)_{\hbar}:=\hat{H}_{*}^{S^{1}\times
G}(\Omega G)$ on loop-equivariant quantum cohomology. This suggests that the
module action induced by (1) should lift to an action of
$\mathcal{C}_{3}(G;0)_{\hbar}$ on the version of equivariant symplectic
cohomology which also incorporates loop equivariance,
$SH_{S^{1}_{\operatorname{rot}}\times G}^{*}(\bar{M})$. Section 7 of
[Teleman1] provides a similar characterization of the quantum Coulomb branch
algebras $\mathcal{C}_{3}(G;E)_{\hbar}$ as subalgebras of the pure quantum
Coulomb branch algebra $\mathcal{C}_{3}(G;0)_{\hbar}$. Somewhat more
speculatively, we also note that Teleman develops parallel results for
$K$-theoretic Coulomb branches. Correspondingly, we expect an analogue of
Theorem 1.1 for (a suitable equivariant version of) the $K$-theoretic
symplectic cohomology recently constructed by Large [Large].

### Acknowledgements

D.P. would like to thank Constantin Teleman for his generous and patient
explanations of [Teleman1]. The authors would also like to thank Justin
Hilburn for informative discussions about Coulomb branches. C.M. was partly
supported by the Simons Collaboration on Homological Mirror Symmetry, Award
#652236, the Royal Society University Research Fellowship, and the University
of Southampton FSS ECR Career Development Fund while working on this project.
D.P. was partly supported by the Simons Collaboration in Homological Mirror
Symmetry, Award # 652299 while working on this project.

##  2\. Floer theory over a family, pull-back and push-forward

In this section, we explain how to define the Floer cohomology for a
Hamiltonian fibre bundle with Liouvile fibres over a smooth finite dimensional
base. It is analogous to [Hutchings] and similar ideas have been used to
define equivariant Floer cohomology ([SS10], [BO17], [SeidelGM], [LJ20]).

Let $(\bar{M},\omega)$ be a compact symplectic manifold with a convex boundary
(i.e. $\bar{M}$ is a strong convex filling of its boundary). Let $\theta$ be a
primitive one form of $\omega$ near the boundary such that the $\omega$-dual
of $\theta$ points outwards along the boundary. We will denote by $M$ the
symplectic completion of $\bar{M}$. In other words,

$M=\bar{M}\cup_{\partial\bar{M}}[1,\infty)\times\partial\bar{M}$

where the Liouville one form on the cylindrical end
$[1,\infty)\times\partial\bar{M}$ is given by $r\alpha$ for $r\in[1,\infty)$
with $\alpha:=\theta|_{\partial\bar{M}}$. Throughout this text, we will assume
that $c_{1}(TM)=0$ so that we can put a $\mathbb{Z}$-grading on the Floer
complexes later on (see the paragraph after Definition 2.9).

The Reeb vector field $R_{\alpha}$ on $(\partial\bar{M},\alpha)$ is the unique
vector field characterized by $\alpha(R_{\alpha})=1$ and
$\iota_{R_{\alpha}}d\alpha=0$. The flow of the Reeb vector field is called
Reeb flow. The action spectrum of $(\partial\bar{M},\alpha)$, denoted by
$\operatorname{Spec}(\partial\bar{M},\alpha)$, consists of the period of the
orbits of the Reeb flow.

######  Definition 2.1.

A Hamiltonian function $H\in C^{\infty}(S^{1}\times M)$ is cylindrical if
there is a positive function
$\mathbf{s}:S^{1}\times\partial\bar{M}\to\mathbb{R}_{>0}$ that is invariant
under the Reeb flow and a function $c:S^{1}\to\mathbb{R}$ such that
$H(r,x)=r\mathbf{s}(x)+c$ in the complement of a compact set.

The function $\mathbf{s}$ is called the slope of $H$. The space of cylindrical
Hamiltonian is a vector space over $\mathbb{R}$ and hence convex. We define a
partial ordering on the space of cylindrical Hamiltonians by

(6) $\displaystyle H_{1}\leq_{\mathbf{s}}H_{2}\text{ if
}\mathbf{s}_{H_{1}}(x)\leq\mathbf{s}_{H_{2}}(x)\text{ for all }x$

where $\mathbf{s}_{H_{i}}$ is the slope of $H_{i}$ for $i=1,2$. We say that a
Hamiltonian function $H=(H_{t})_{t\in S^{1}}\in C^{\infty}(S^{1}\times M)$ is
mean-normalized if $\int_{\partial\bar{M}}H_{t}\alpha\wedge\omega^{n-1}=0$ for
all $t\in S^{1}$. Unless otherwise stated, all Hamiltonians in the rest of the
paper are mean-normalized.

######  Definition 2.2.

Let $G$ be a compact connected Lie group. A Hamiltonian $G$-action on
$(\bar{M},\omega,\theta)$ is called convex if

  * •

$\theta$ is $G$-invariant and

  * •

the spectrum
$\operatorname{Spec}(\partial\bar{M},\alpha:=\theta_{|\partial\bar{M}})$ is
nowhere dense.

The hypothesis that $\theta$ is $G$-invariant implies that the action on
$[1,\infty)\times\partial\bar{M}$ is the product of the trivial action on
$[1,\infty)$ and a $G$-action on $\partial\bar{M}$.

######  Lemma 2.3.

Suppose that $(\bar{M},\omega,\theta)$ is equipped with an convex Hamiltonian
$G$-action.

Let $\gamma\in\Omega G\subset\Omega\operatorname{Ham}(M)$ and $K_{\gamma}$ be
a generating Hamiltonian function of $\gamma$. Then $K_{\gamma}$ is a
cylindrical Hamiltonian function. Conversely, if $H$ is a cylindrical
Hamiltonian function, then outside a compact set, the Hamiltonian flow of $H$
on the cylindrical end preserves $\theta$.

###### Proof.

Let $v\in\mathfrak{g}$ and $X_{v}$ be the induced Hamiltonian vector field.
Since $\theta$ is $G$-invariant, we have

(7) $\displaystyle
0=\mathcal{L}_{X_{v}}\theta=\iota_{X_{v}}\omega+d(r\alpha(X_{v}|_{r=1}))$

The last equality uses that $X_{v}$ is independent of $r$ on the cylindrical
end. Let

$\displaystyle\mathbf{s}_{v}:=-\alpha(X_{v}|_{r=1}):\partial\bar{M}\to\mathbb{R}.$

By (7), Hamiltonian vector field of the Hamiltonian function $r\mathbf{s}_{v}$
is $X_{v}$. Let $R_{\alpha}$ be the Reeb vector field. The function
$\mathbf{s}_{v}$, which is the slope of the Hamiltonian function
$r\mathbf{s}_{v}$, is invariant under Reeb flow because
$R_{\alpha}(\mathbf{s}_{v})=d\mathbf{s}_{v}(R_{\alpha})=\omega(X_{v},R_{\alpha})|_{r=1}=-dr(X_{v})=0$.
Now, let $(X_{t})_{t\in S^{1}}$ be the vector field which generates $\gamma$.
By the assumption that $\gamma\in\Omega G$, we know that for each $t\in
S^{1}$, $X_{t}=X_{v_{t}}$ for some $v_{t}\in\mathfrak{g}$. Therefore
$(K_{\gamma})_{t}(r,y)=r\mathbf{s}_{v_{t}}(y)$ up to a constant so
$K_{\gamma}$ is a cylindrical Hamiltonian function. The converse is proved in
[Ritter, Lemma C.3]. ∎

Let $B$ be a smooth finite dimensional closed manifold.

######  Definition 2.4.

We say that $p:P\to B$ is an admissible bundle over $B$ if it is a Hamiltonian
fibre bundle with fiber $(M,\omega,\theta)$ and the structure group lies in
$G$.

Since the $G$ action is convex, $p$ restricts to a subbundle
$\bar{p}:\bar{P}\to B$ with fibres $\bar{M}$. Moreover, there is a radial
coordinate function $r:P\setminus\bar{P}\to[1,\infty)$ such that for any fiber
$p^{-1}(b)$ , the symplectic form near the boundary equals to
$\omega_{b}=d(r\theta_{b})$.

Let $\eta:B\to\mathbb{R}$ be a Morse function, $g_{\eta}$ be a Riemannian
metric on $B$ satisfying the Morse-Smale condition and $\nabla$ be a
$G$-connection on $P$ such that $\nabla$ is flat near critical points of
$\eta$. The space of such connections $\nabla$ is non-empty and contractible.
The set of critical points of $\eta$ is denoted by
$\operatorname{critp}(\eta)$. A gradient trajectory $\tau$ of $\eta$ is a
solution to the negative gradient flow equation
$\frac{d\tau}{ds}=-\operatorname{grad}(\eta)(\tau(s))$. For every
$c\in\operatorname{critp}(\eta)$, we can use the flatness of $\nabla$ near $c$
to obtain a neighborhood $N_{c}$ of $c$ together with a decomposition

(8) $\displaystyle p^{-1}(N_{c})\simeq M\times N_{c}$

such that $\nabla$ is trivial with respect to the decomposition.

######  Definition 2.5.

We call a triple $(\eta,g_{\eta},\nabla)$ as above an admissible base triple.

A Hamiltonian function of $P$ is a function $H:S^{1}\times P\to\mathbb{R}$.
For $t\in S^{1}$ and $b\in B$, we denote its restriction to $p^{-1}(b)$ by
$H_{b}:S^{1}\times p^{-1}(b)\to\mathbb{R}$.

######  Definition 2.6.

A Hamiltonian function $H\in C^{\infty}(S^{1}\times P)$ is cylindrical if for
every $b\in B$, $H_{b}:S^{1}\times p^{-1}(b)\to\mathbb{R}$ is a cylindrical
Hamiltonian.

This is a well-defined notion because the structure group of $P$ is $G$ and
$H_{b}$ being cylindrical is a property that is invariant under $G$-action.

######  Definition 2.7.

A cylindrical Hamiltonian $H$ is compatible with $(\eta,g_{\eta},\nabla)$ if:

  * •

$H_{c}:=(H_{c,t})_{t\in S^{1}}$ is non-degenerate for every
$c\in\operatorname{critp}(\eta)$

  * •

locally near each $c\in\operatorname{critp}(\eta)$, with respect to the
decomposition (8), we have that $H_{b}$ is pulled back from the first factor.

  * •

for any gradient trajectory $\tau$ of $\eta$ and two real numbers $s_{1}\leq
s_{2}$, we have $H_{\tau(s_{1})}\geq_{\mathbf{s}}H_{\tau(s_{2})}$ with respect
to the parallel transport map along $\tau$ induced by $\nabla$ (recall
$\leq_{\mathbf{s}}$ from (6)).

  * •

for any gradient trajectory $\tau$ of $\eta$ between two critical points
$c_{0}$ and $c_{1}$, the parallel transport map does not map a Hamiltonian
orbit of $H_{c_{0}}$ to a Hamiltonian orbit of $H_{c_{1}}.$

A cylindrical Hamiltonian $H$ is compatible with $\eta$ if it is compatible
with an admissible base triple $(\eta,g_{\eta},\nabla)$ for some $g_{\eta}$
and $\nabla$. A cylindrical Hamiltonian $H$ is admissible if it is compatible
with some $\eta$.

######  Lemma 2.8.

Given any admissible base triple $(\eta,g_{\eta},\nabla)$ and any constant
$a\in\mathbb{R}_{>0}\setminus\operatorname{Spec}(\partial\bar{M},\alpha)$,
there is a cylindrical Hamiltonian $H$ that is compatible with
$(\eta,g_{\eta},\nabla)$ with slope $\mathbf{s}_{H}=a$.

###### Proof.

Let $a\in\mathbb{R}_{>0}\setminus\operatorname{Spec}(\partial\bar{M},\alpha)$.
For each critical point $c$ of $\eta$, we first choose a non-degenerate
cylindrical Hamiltonian function $H_{c}$ on $M$ with slope $a$. Using the
product decomposition (8), we can pull it back to a cylindrical Hamiltonian
over a neighborhood $\cup_{c\in\operatorname{critp}(\eta)}N_{c}$ of the
critical points of $\eta$ such that both the first and the second bullets are
satisfied. The fact that $H_{c}$ is non-degenerate means that all Hamiltonian
orbits are lying inside a compact set. Since there are only finitely many
critical points of $\eta$ (because $B$ is compact and $\eta$ is Morse), by a
generic perturbation of each individual $H_{c}$ over a compact set, we can
further assume that the fourth bullet is also satisfied.

We can then smoothly extend the cylindrical Hamiltonian to the rest of $P$
(i.e. over the complement of $\cup_{c\in\operatorname{critp}(\eta)}N_{c}$)
such that $H_{b}$ has slope $a$ for all $b\in B$. Such an extension is always
possible because the space of cylindrical Hamiltonian with a fixed slope is
convex. Since the slope is the same for all $H_{b}$, the third bullet is
automatically satisfied.

∎

A pair $\vec{x}=(c,x)$ consisting of a critical point $c$ of $\eta$ and a
$1$-periodic orbit $x$ of $H_{c}$ in $p^{-1}(c)$ is called a generator of
$(\eta,H)$. Let $\Lambda$ be the Novikov field

$\Lambda:=\left\\{\sum_{i=0}^{\infty}b_{i}q^{r_{i}}|r_{i}\in\mathbb{R},\lim_{i\to\infty}r_{i}=\infty,b_{i}\in\mathbb{C}\right\\}$

######  Definition 2.9.

The Floer cochain complex of $(\eta,H)$ is defined to be

(9) $\displaystyle CF^{*}(P,H,\eta,\Lambda):=\oplus\Lambda\cdot\vec{x}$

where the sum is taken over all generators $\vec{x}=(c,x)$ of $(\eta,H)$.

We often suppress the coefficient field and denote the group by
$CF^{*}(P,H,\eta)$. We can define a $\mathbb{Z}$-grading on $CF^{*}(P,H,\eta)$
as follows. Since we assume that $c_{1}(TM)=0$, we can and will fix once and
for all a homotopy class of trivialization of the canonical bundle of $M$. We
then choose a symplectomorphism identifying $M$ with a reference fibre of $P$.
The structure group of the bundle $P\to B$ is $G$, which is connected, so it
acts trivially on the homotopy class of the trivialization of the canonical
bundle. Therefore, by taking parallel transport from the reference fibre, we
get a well-defined homotopy class of trivialization of the canonical bundle on
all other fibres. We define the grading of $\vec{x}=(c,x)$ to be the sum of
the Morse grading of $c$ and the Floer grading of $x$, where the Floer grading
of $x$ is the Conley-Zehnder index of $x$ with respect to the homotopy class
of trivialization of the canonical bundle of the fibre $M$ over $c$. Finally,
the Novikov variable $q$ will be of degree zero.

######  Remark 2.10.

The $\mathbb{Z}$-grading on $CF^{*}(P,H,\eta)$ depends (only) on the homotopy
class of trivialization of the canonical bundle of $M$ as well as the
symplectomorphism identifying $M$ with the reference fibre of $P$.

Choose an $S^{1}$-dependent fiberwise almost complex structure $(J_{b})_{b\in
B}$ which is compatible with $(\omega_{b})_{b\in B}$ and is of contact type in
each fiber. Recall that $J_{b}$ is of contact type if $\theta_{b}\circ
J_{b}=dr$ outside a compact set of $p^{-1}(b)$. Define the moduli space
$\widetilde{\mathcal{M}}(\vec{x}_{0},\vec{x}_{1},(J_{b})_{b\in B})$ to be the
space of pairs $(\tau,u)$ where

  * •

$\tau:\mathbb{R}\to B$ is a gradient trajectory of $\eta$ such that
$\lim_{s\to-\infty}\tau(s)=c_{0}$ and $\lim_{s\to\infty}\tau(s)=c_{1}$

  * •

$u:\mathbb{R}\times S^{1}\to P$ is a smooth map which lifts $\tau$ and solves
the Floer equation

(13)
$\displaystyle\left\\{\begin{array}[]{ll}(du^{vert}-X_{H_{\tau(s)}}\otimes
dt)_{J_{\tau(s)}}^{0,1}=0.\\\ \lim_{s\to-\infty}u(s,t)=x_{0}(t)\\\
\lim_{s\to\infty}u(s,t)=x_{1}(t)\end{array}\right.$

where $du^{vert}$ denotes the projection of $du$ to the fibrewise direction
$T^{vert}P$ under the splitting $TP=T^{vert}P\oplus T^{hor}P$ by $\nabla$.

We let

$\mathcal{M}(\vec{x}_{0},\vec{x}_{1}):=\widetilde{\mathcal{M}}(\vec{x}_{0},\vec{x}_{1})/\mathbb{R}.$

By trivializing $P$ over $\tau$ using the connection $\nabla$, a solution
$(\tau,u)\in\mathcal{M}(\vec{x}_{0},\vec{x}_{1})$ can be recast as either:

  * •

Floer trajectories in the fiber over $c_{0}=c_{1}$ (when $\tau$ is constant)
or

  * •

continuation solutions for a Hamiltonian $(H_{s,t})_{(s,t)\in\mathbb{R}\times
S^{1}}$ with respect to an $\omega$-compatible almost complex structure of
contact type $(J_{s,t})_{(s,t)\in\mathbb{R}\times S^{1}}$ on $M$ (when $\tau$
is non constant).

The modui space $\mathcal{M}(\vec{x}_{0},\vec{x}_{1})$ is the zero locus of a
Fredholm section and has a well-defined virtual dimension given by

$\operatorname{vdim}\mathcal{M}(\vec{x}_{0},\vec{x}_{1})=|\vec{x}_{0}|-|\vec{x}_{1}|-1.$

For $(\tau,u)\in\mathcal{M}(\vec{x}_{0},\vec{x}_{1})$, its topological energy
is defined by

(14) $\displaystyle E(u):=\int_{\mathbb{R}\times
S^{1}}(du^{vert})^{*}\omega+\int_{0}^{1}(H_{c_{0}})_{t}(x_{0}(t))dt-\int_{0}^{1}(H_{c_{1}})_{t}(x_{1}(t))dt.$

The term $(du^{vert})^{*}\omega$ is defined to be the pull-back of the
fibrewise $2$-form along the composition $T(\mathbb{R}\times S^{1})\to TP\to
T^{vert}P$. In other words, if we trivialize $P$ along $\tau$ by $\nabla$ and
identify $u$ is as a map to a single fibre of $P$ then $E(u)$ is the
topological energy in the literature (cf. [AbSe, Equation (148)]) and the
first term of (14) is the area of $u$. The space
$\mathcal{M}(\vec{x}_{0},\vec{x}_{1})$ splits as a disjoint union

$\mathcal{M}(\vec{x}_{0},\vec{x}_{1}):=\cup_{A}\mathcal{M}(\vec{x}_{0},\vec{x}_{1};A).$

according to the relative homology class $A\in
H_{2}(P,\vec{x}_{0}\cup\vec{x}_{1})$ of a solution. The topological energy is
a locally constant function on $\mathcal{M}(\vec{x}_{0},\vec{x}_{1})$ which
depends only on $A$, so we can denote it by $E(A)$.

######  Proposition 2.11.

Let $H$ be a cylindrical Hamiltonian function compatible with $\eta$.

  * •

For a generic choice of compatible $J$ that is of contact type, the space
$\mathcal{M}(\vec{x}_{0},\vec{x}_{1})$ is a manifold of the expected dimension
for any pair of generators $(\vec{x}_{0},\vec{x}_{1})$.

  * •

Moreover, if $\operatorname{vdim}\mathcal{M}(\vec{x}_{0},\vec{x}_{1})\leq 1$,
then for any $E\in\mathbb{R}$, the moduli space

$\cup_{A,E(A)<E}\mathcal{M}(\vec{x}_{0},\vec{x}_{1};A)$

admits a Gromov compactification
$\cup_{A,E(A)<E}\overline{\mathcal{M}}(\vec{x}_{0},\vec{x}_{1};A)$ making it a
compact manifold with boundary.

###### Proof.

Since $M$ is a symplectic Calabi-Yau manifold with a convex end, the strategy
of proof for both transversality (see e.g. [MSJbook], [HoferSalamon]) and
compactness ([Viterbo], [Ritter]) is standard.

To run the transversality argument, we need to avoid solutions $(\tau,u)$ such
that $\tau$ is non-constant but $u$ is $s$-independent with respect to the
parallel transport map along $\tau$ induced by $\nabla$. These solutions are
ruled out exactly by the fourth bullet of Definition 2.7.

To run the argument for compactness, first note that when
$\operatorname{vdim}\mathcal{M}(\vec{x}_{0},\vec{x}_{1})\leq 1$, then by
genericity, we can assume that $u(s,t)$ misses all $J_{s,t}$-holomorphic
spheres for all $(s,t)\in\mathbb{R}\times S^{1}$ and
$u\in\mathcal{M}(\vec{x}_{0},\vec{x}_{1})$ because the the image of the
evaluation map of a $1$-pointed $J_{s,t}$-holomorphic sphere is of real
codimension $4$. In particular, we can avoid sphere bubblings. It remains to
show that we have a $C^{0}$ a priori estimate and a uniform geometric energy
bound of solutions $(\tau,u)$. The $C^{0}$ a priori estimate for $u$ is proved
in [Ritter, Theorem C.11]. More precisely, the fact that $H$ is a cylindrical
Hamiltonian implies that conditions (2) and (3) in [Ritter, Theorem C.11] are
satisfied (cf. [Ritter, Theorem C.6]). The condition that $(H_{z},J_{z})$
becomes independent of $s$ for large $|s|$ follows from the second bullet of
Definition 2.7. The condition $\partial_{s}H_{z}\leq 0$ in [Ritter, Theorem
C.11] translates to $\partial_{s}H_{\tau(s)}\leq 0$ in our setting, which is
not necessarily true on the nose because there is a constant term
$H_{\tau(s)}$. But the Floer equation does not depend on the constant term in
$H_{\tau(s)}$ so the Theorem remains valid as long as
$\partial_{s}\mathbf{s}_{H_{\tau(s)}}\leq 0$, which is precisely the third
bullet of Definition 2.7.

To obtain the uniform geometric energy bound on $u$, we suppose that $u$ lies
over a given gradient trajectory $\tau$ and trivialize $P$ along this
trajectory. We can then apply the standard energy estimate to give an upper
bound to the geometric energy in terms of the topological energy and the
Hamiltonian $H$. This energy estimate may vary as we vary $\tau$, but the
moduli of gradient trajectories can be compactified by including broken
trajectories and so this gives a uniform upper bound for the geometric energy
of $u$. ∎

Choose a generic $J$ and define a differential on $CF(P,H,\eta)$ by

(15)
$\displaystyle\partial_{CF}(\vec{x}_{1}):=\sum_{\vec{x}_{0},|\vec{x}_{0}|=|\vec{x}_{1}|+1}\sum_{(\tau,u)\in\mathcal{M}_{0}(\vec{x}_{0},\vec{x}_{1})}s(u)q^{E(u)}\vec{x}_{0}$

where $s(u)\in\\{-1,1\\}$ is the sign of $(\tau,u)$. The sum is well-defined
because of the compactness of the moduli (Proposition 2.20). For a discussion
of how the signs are determined, see [Hutchings, Section 3 and 6].

######  Lemma 2.12.

We have that

$\partial_{CF}^{2}=0$

Moreover, $HF(P,H,\eta):=H(CF(P,H,\eta),\partial_{CF})$ is independent of the
choice of $J$.

###  2.1. Continuation and invariance

We want to discuss the natural maps arising from varying $H$ and
$(\eta,g_{\eta},\nabla)$.

Let $H^{\prime}$ be another cylindrical Hamiltonian on $P$ compatible with
$(\eta,g_{\eta},\nabla)$ and such that the slope
$\mathbf{s}_{H^{\prime}_{b}}\geq\mathbf{s}_{H_{b}}$ for all $b\in B$. In this
case, we can define the continuation map

(16) $\displaystyle HF(P,H,\eta)\to HF(P,H^{\prime},\eta)$

as follows. A monotone homotopy from $H$ to $H^{\prime}$ is a one-parameter
family of cylindrical Hamiltonian $(H_{s})_{s\in\mathbb{R}}$ such that
$H_{s}=H^{\prime}$ for $s\ll 0$, $H_{s}=H$ for $s\gg 0$ and for every $b\in
B$, we have $\partial_{s}\mathbf{s}_{(H_{s})_{b}}\leq 0$. We say that a
monotone homotopy is compatible with $(\eta,g_{\eta},\nabla)$ if
$\partial_{s}\mathbf{s}_{(H_{s})_{\tau(s)}}\leq 0$ along any gradient
trajectory $\tau:\mathbb{R}\to B$ of $\eta$ (again, with respect to the
parallel transport map along $\tau$ induced by $\nabla$). Note that for a
monotone increasing smooth function $\rho:\mathbb{R}\to[0,1]$ such that
$\rho(s)=0$ for $s\ll 0$ and $\rho(s)=1$ for $s\gg 0$, the family

$\displaystyle H_{s}=\rho(s)H+(1-\rho(s))H^{\prime},\quad(\text{for
s}\in\mathbb{R})$

is a monotone homotopy that is compatible with $(\eta,g_{\eta},\nabla)$ if
both $H$ and $H^{\prime}$ are compatible with $(\eta,g_{\eta},\nabla)$.

Let $\vec{x}=(c,x)$ and $\vec{x}^{\prime}=(c^{\prime},x^{\prime})$ be a
generator of $CF(P,H,\eta)$ and $CF(P,H^{\prime},\eta)$ respectively. Choose a
one-parameter family of $S^{1}$-dependent fiberwise compatible almost complex
structures $(J_{s})_{s\in\mathbb{R}}$ that are of contact type. Define the
moduli space $\mathcal{M}(\vec{x}^{\prime},\vec{x},(H_{s})_{s\in\mathbb{R}})$
to be the space of pairs $(\tau,u)$ where

  * •

$\tau:\mathbb{R}\to B$ is a gradient trajectory of $\eta$ such that
$\lim_{s\to-\infty}\tau(s)=c^{\prime}$ and $\lim_{s\to\infty}\tau(s)=c$

  * •

$u:\mathbb{R}\times S^{1}\to P$ is a smooth map which lifts $\tau$ and solves
the Floer equation

(20)
$\displaystyle\left\\{\begin{array}[]{ll}(du^{vert}-X_{(H_{s})_{\tau(s)}}\otimes
dt)_{(J_{s})_{\tau(s)}}^{0,1}=0.\\\ \lim_{s\to-\infty}u(s,t)=x^{\prime}(t)\\\
\lim_{s\to\infty}u(s,t)=x(t)\end{array}\right.$

The virtual dimension of
$\mathcal{M}(\vec{x}^{\prime},\vec{x},(H_{s})_{s\in\mathbb{R}})$ is
$|\vec{x}^{\prime}|-|\vec{x}|$. Similarly,
$\mathcal{M}(\vec{x}^{\prime},\vec{x},(H_{s})_{s\in\mathbb{R}})$ splits as a
disjoint union

$\mathcal{M}(\vec{x}^{\prime},\vec{x},(H_{s})_{s\in\mathbb{R}}):=\cup_{A\in
H_{2}(P,\vec{x}^{\prime}\cup\vec{x})}\mathcal{M}(\vec{x}^{\prime},\vec{x},(H_{s})_{s\in\mathbb{R}};A)$

and we can define the topological energy using the same formula (14) with
$x_{0}$ and $x_{1}$ being replaced with $x^{\prime}$ and $x$, respectively.

######  Proposition 2.13.

Let $(H_{s})_{s\in\mathbb{R}}$ be a monotone homotopy from $H$ to $H^{\prime}$
that is compatible with $(\eta,g_{\eta},\nabla)$.

  * •

For a generic choice of $(J_{s})_{s\in\mathbb{R}}$, the space
$\mathcal{M}(\vec{x}^{\prime},\vec{x},(H_{s})_{s\in\mathbb{R}})$ is a manifold
of the expected dimension for any pair of generators
$(\vec{x},\vec{x}^{\prime})$.

  * •

Moreover, if its virtual dimension is $\leq 1$, then for any $E\in\mathbb{R}$,
the union of those components with $E(A)<E$ admits a Gromov compactification

$\cup_{A,E(A)<E}\overline{\mathcal{M}}(\vec{x}^{\prime},\vec{x},(H_{s})_{s\in\mathbb{R}};A)$

making it a compact manifold with boundary.

The proof is similar to Proposition 2.20. The compatibility with
$(\eta,g_{\eta},\nabla)$ provides us a $C^{0}$ a priori estimate of the
solutions as above. Therefore, we can define a linear map

(23)
$\displaystyle\begin{array}[]{ll}\kappa_{(H_{s})_{s\in\mathbb{R}}}:CF(P,H,\eta)\to
CF(P,H^{\prime},\eta)\\\
\kappa_{(H_{s})_{s\in\mathbb{R}}}(\vec{x})=\sum_{\vec{x}^{\prime},|\vec{x}^{\prime}|=|\vec{x}|}\sum_{(\tau,u)\in\mathcal{M}(\vec{x}^{\prime},\vec{x},(H_{s})_{s\in\mathbb{R}})}s(\tau,u)q^{E(u)}\vec{x}^{\prime}\end{array}$

where $s(u)\in\\{-1,1\\}$ is the sign of $(\tau,u)$.

######  Lemma 2.14.

The linear map $\kappa_{(H_{s})_{s\in\mathbb{R}}}$ is chain map. Moreover, the
induced map on cohomology is independent of the choice of the
$(\eta,g_{\eta},\nabla)$-compatible monotone homotopy and the family of
cylindrical almost complex structure.

###### Proof.

The fact that $\kappa_{(H_{s})_{s\in\mathbb{R}}}$ is a chain map follows from
looking at the boundary of the one dimensional moduli of
$\overline{\mathcal{M}}(\vec{x}^{\prime},\vec{x},(H_{s})_{s\in\mathbb{R}})$.
The induced map on cohomology being independent of $J$ can be proved by the
standard homotopy argument [MSJbook]. To run the argument for the proof of the
independence of choice of monotone homotopy, we need to show that the space of
monotone homotopies that are compatible with $(\eta,g_{\eta},\nabla)$ is path
connected. Indeed, the space is convex so it is path connected. ∎

It is more tricky to vary $\eta$. Let $H$ and $H^{\prime}$ be cylindrical
Hamiltonians that are compatible with $(\eta,g_{\eta},\nabla)$ and
$(\eta^{\prime},g_{\eta}^{\prime},\nabla^{\prime})$ respectively. Let
$(\eta_{s})_{s\in\mathbb{R}}:B\to\mathbb{R}$ be a family of functions such
that $\eta_{s}=\eta^{\prime}$ for $s\ll 0$ and $\eta_{s}=\eta$ for $s\gg 0$,
$(\nabla_{s})_{s\in\mathbb{R}}$ be a family of connections for $P\to B$ such
that $\nabla_{s}=\nabla^{\prime}$ for $s\ll 0$ and $\nabla_{s}=\nabla$ for
$s\gg 0$, and $(H_{s})_{s\in\mathbb{R}}:P\to\mathbb{R}$ be a family of
cylindrical Hamiltonians such that $H_{s}=H^{\prime}$ for $s\ll 0$ and
$H_{s}=H$ for $s\gg 0$. We can use $\eta_{s}$ to define a chain map between
the Morse cochain of $\eta$ and $\eta^{\prime}$ by counting solutions
$\tau:\mathbb{R}\to B$ of

$\frac{d}{ds}\tau(s)=-\operatorname{grad}(\eta_{s},g_{\eta_{s}})|_{\tau(s)}.$

Suppose that $(H_{s})_{s\in\mathbb{R}}$ is a family of cylindrical Hamiltonian
on $P$ which is compatible with
$((\eta_{s})_{s\in\mathbb{R}},(g_{\eta_{s}})_{s\in\mathbb{R}},(\nabla_{s})_{s\in\mathbb{R}})$
in the sense that

(24) $\displaystyle\partial_{s}\mathbf{s}_{H_{\tau(s)}}\leq 0\text{ along any
gradient trajectory $\tau$ of $\eta_{s},g_{\eta_{s}}$ with respect to
$(\nabla_{s})_{s\in\mathbb{R}}$.}$

Then, by choosing a generic family of $S^{1}$-dependent cylindrical almost
complex structure of contact type $(J_{s})_{s\in\mathbb{R}}$ on $P$, we can
define a chain map (the continuation map)

(27)
$\displaystyle\begin{array}[]{ll}\kappa_{(H_{s})_{s\in\mathbb{R}},(\eta_{s})_{s\in\mathbb{R}}}:CF(P,H,\eta)\to
CF(P,H^{\prime},\eta^{\prime})\\\
\kappa_{(H_{s})_{s\in\mathbb{R}},(\eta_{s})_{s\in\mathbb{R}}}(\vec{x})=\sum_{\vec{x}^{\prime},|\vec{x}^{\prime}|=|\vec{x}|}\sum_{(\tau,u)\in\mathcal{M}(\vec{x}^{\prime},\vec{x},(H_{s})_{s\in\mathbb{R}},(\eta_{s})_{s\in\mathbb{R}})}s(u)q^{E(u)}\vec{x}^{\prime}\end{array}$

where
$\mathcal{M}(\vec{x}^{\prime},\vec{x},(H_{s})_{s\in\mathbb{R}},(\eta_{s})_{s\in\mathbb{R}})$
is the moduli of rigid solutions $(\tau,u)$ such that $\tau$ is a gradient
trajectory of $(\eta_{s},g_{\eta_{s}})_{s\in\mathbb{R}}$ and $u$ is a solution
of (20) covering $\tau$. The term $E(u)$ and $s(u)\in\\{-1,-1\\}$ are again
the topological energy (14) and the sign of $u$, respectively. When
$(\eta_{s},g_{\eta_{s}},\nabla_{s})_{s\in\mathbb{R}}$ are independent of $s$,
it recovers $\kappa_{(H_{s})_{s\in\mathbb{R}}}$.

In general, the condition (24) is implicit and hard to verify so we introduce
the following defintion.

######  Definition 2.15.

The relation $\leq_{P}$ on the space of cylindrical Hamiltonians on $P$ is
defined by

(28) $\displaystyle H^{\prime}\geq_{P}H\text{ if there is
}a\notin\operatorname{Spec}(\partial\bar{M},\alpha)\text{ such that
}\min\mathbf{s}_{H^{\prime}}\geq a\geq\max\mathbf{s}_{H}$

To be crystal clear, the definitions are
$\min\mathbf{s}_{H^{\prime}}:=\min_{b\in
B}\min_{\partial\bar{p}^{-1}(b)}\mathbf{s}_{H^{\prime}_{b}}$ and
$\max\mathbf{s}_{H}:=\max_{b\in
B}\max_{\partial\bar{p}^{-1}(b)}\mathbf{s}_{H_{b}}$. Since
$\operatorname{Spec}(\partial\bar{M},\alpha)$ is nowhere dense, we can rewrite
the condition (28) as $\min\mathbf{s}_{H^{\prime}}>\max\mathbf{s}_{H}$ or
$\min\mathbf{s}_{H^{\prime}}=\max\mathbf{s}_{H}\notin\operatorname{Spec}(\partial\bar{M},\alpha)$.
Note that this relation is transitive but not reflexive (cf. (31) and
Corollary 2.17).

######  Lemma 2.16.

Let $H$ and $H^{\prime}$ be cylindrical Hamiltonians that are compatible with
$(\eta,g_{\eta},\nabla)$ and
$(\eta^{\prime},g_{\eta}^{\prime},\nabla^{\prime})$ respectively. Suppose that
$H^{\prime}\geq_{P}H$. Then there exist
$(\eta_{s},g_{\eta_{s}},\nabla_{s})_{s\in\mathbb{R}}$ and
$(H_{s})_{s\in\mathbb{R}}$ as above such that $(H_{s})_{s\in\mathbb{R}}$ is
compatible with $(\eta_{s},g_{\eta_{s}},\nabla_{s})_{s\in\mathbb{R}}$.
Moreover, the induced map on cohomology

(29)
$\displaystyle\kappa_{(H_{s})_{s\in\mathbb{R}},(\eta_{s})_{s\in\mathbb{R}}}:HF(P,H,\eta)\to
HF(P,H^{\prime},\eta^{\prime})$

is independent of the choice of
$(\eta_{s},g_{\eta_{s}},\nabla_{s})_{s\in\mathbb{R}}$,
$(H_{s})_{s\in\mathbb{R}}$ and $(J_{s})_{s\in\mathbb{R}}$.

###### Proof.

For fixed $(\eta_{s},g_{\eta_{s}},\nabla_{s})_{s\in\mathbb{R}}$, the space of
$(H_{s})_{s\in\mathbb{R}}$ from $H$ to $H^{\prime}$ that is compatible with
$(\eta_{s},g_{\eta_{s}},\nabla_{s})_{s\in\mathbb{R}}$ is convex so by a
homotopy argument as in the proof of Lemma 2.14, we conclude that it is
independent of the choice of $(H_{s})_{s\in\mathbb{R}}$ and
$(J_{s})_{s\in\mathbb{R}}$.

To argue that it is independent of the choice of
$(\eta_{s},g_{\eta_{s}},\nabla_{s})_{s\in\mathbb{R}}$, we first consider the
case that the slopes are constant functions that are independent of $b\in B$.
In other words, we have two real numbers $\mathbf{s}^{\prime},\mathbf{s}$ such
that $\mathbf{s}_{H^{\prime}_{b}}=\mathbf{s}^{\prime}$ and
$\mathbf{s}_{H_{b}}=\mathbf{s}$ for all $b\in B$. Moreover, we have
$\mathbf{s}^{\prime}\geq\mathbf{s}$ and
$\mathbf{s}^{\prime},\mathbf{s}\notin\operatorname{Spec}(\partial\bar{M},\alpha)$.
In this case, we can choose a monotone decreasing function
$f_{\mathbf{s}}:\mathbb{R}\to[\mathbf{s},\mathbf{s}^{\prime}]$ such that
$f_{\mathbf{s}}(s)=\mathbf{s}^{\prime}$ for $s\ll 0$,
$f_{\mathbf{s}}(s)=\mathbf{s}$ for $s\gg 0$ and a family
$(H_{s})_{s\in\mathbb{R}}$ from $H$ to $H^{\prime}$ such that
$\mathbf{s}_{(H_{s})_{b}}=f_{\mathbf{s}}(s)$ for all $b\in B$. This choice of
$(H_{s})_{s\in\mathbb{R}}$ is compatible with any family
$(\eta_{s},g_{\eta_{s}},\nabla_{s})_{s\in\mathbb{R}}$ from
$(\eta,g_{\eta},\nabla)$ to
$(\eta^{\prime},g_{\eta}^{\prime},\nabla^{\prime})$. Therefore, we can apply a
homotopy argument with fixed $(H_{s})_{s\in\mathbb{R}}$ and varying
$(\eta_{s},g_{\eta_{s}},\nabla_{s})_{s\in\mathbb{R}}$ to conclude that (29) is
independent of $(\eta_{s},g_{\eta_{s}},\nabla_{s})_{s\in\mathbb{R}}$ when
$\mathbf{s}_{H^{\prime}_{b}}=\mathbf{s}^{\prime}$ and
$\mathbf{s}_{H_{b}}=\mathbf{s}$ for all $b\in B$.

Now, we consider the general case. Let
$\check{\mathbf{s}}\in\mathbb{R}_{>0}\notin\operatorname{Spec}(\partial\bar{M},\alpha)$
be such that
$\min\mathbf{s}_{H^{\prime}}\geq\check{\mathbf{s}}\geq\max\mathbf{s}_{H}$. Let
$\check{H}$ and $\check{H}^{\prime}$ be a cylindrical Hamiltonian such that
$\mathbf{s}_{\check{H}_{b}}=\check{\mathbf{s}}=\mathbf{s}_{\check{H}^{\prime}_{b}}$
for all $b\in B$ and is compatible with $(\eta,g_{\eta},\nabla)$ and
$(\eta^{\prime},g_{\eta}^{\prime},\nabla^{\prime})$, respectively.

We are going to show that (29) equals to the composition

(30) $\displaystyle HF(P,H,\eta)\to HF(P,\check{H},\eta)\to
HF(P,\check{H}^{\prime},\eta^{\prime})\to HF(P,H^{\prime},\eta^{\prime})$

where the first and last maps are independent of choices by Lemma 2.14 and the
second map is independent of choices by above.

First of all, by gluing the auxiliary data defining these three maps, we can
show the existence of $(H_{s})_{s\in\mathbb{R}}$ that is compatible with
$(\eta_{s},g_{\eta_{s}},\nabla_{s})_{s\in\mathbb{R}}$.

Conversely, let $(\eta_{s},g_{\eta_{s}},\nabla_{s})_{s\in\mathbb{R}}$ and
$(H_{s})_{s\in\mathbb{R}}$ be a choice of data used to define (29). By the
independent of choice of $(H_{s})_{s\in\mathbb{R}}$, we can assume that it
satisfies the following property: there exists $R_{1}>R_{2}>0$ such that

  * •

$\eta_{s}=\eta^{\prime}$ and $\nabla_{s}=\nabla^{\prime}$ for $s<-R_{2}$, and
$\eta_{s}=\eta$ and $\nabla_{s}=\nabla$ for $s>R_{2}$

  * •

$H_{-R_{1}}=\check{H}^{\prime}$, $H_{R_{1}}=\check{H}$

  * •

for $s\in[-R_{1},R_{1}]$, $\mathbf{s}_{(H_{s})_{b}}$ is a constant independent
of $b\in B$ and $\partial_{s}\mathbf{s}_{H_{s}}\leq 0$

We have shown that the map $HF(P,\check{H},\eta)\to
HF(P,\check{H}^{\prime},\eta^{\prime})$ does not depend on auxiiliary choices.
In particular, we can use
$(\eta_{s},g_{\eta_{s}},\nabla_{s})_{s\in\mathbb{R}}$ to define it. Therefore,
by choosing $R_{1}\gg R_{2}\gg 0$, we can make
$(\eta_{s},g_{\eta_{s}},\nabla_{s})_{s\in\mathbb{R}}$ and
$(H_{s})_{s\in\mathbb{R}}$ coincide with the glued data coming the composition
of the three maps in (30), where the firs map corresponds to $s<-R_{1}$, the
second map corresponds to $-R_{1}<s<R_{1}$ and the third map corresponds to
$s>R_{1}$. It shows that (29) equals to (30).

∎

Even though we denote the Floer cohomology as $HF(P,H,\eta)$, it also depends
on $(g_{\eta},\nabla)$. For the collection of data
$\\{(H,\eta,g_{\eta},\nabla)\\}$ such that $H$ is cylindrical and compatible
with $(\eta,g_{\eta},\nabla)$, we define a reflexive and transitive relation
on it by

(31)
$\displaystyle(H^{\prime},\eta^{\prime},g_{\eta}^{\prime},\nabla^{\prime})\geq_{P}(H,\eta,g_{\eta},\nabla)\text{
if }H^{\prime}\geq_{P}H\text{ or the quadruples are identical}$

Clearly, there is an upper bound for any two elements so it forms a directed
set.

######  Corollary 2.17.

The collection of cohomology groups

$\\{HF(P,H,\eta):H\text{ is cylindrical and compatible with
}(\eta,g_{\eta},\nabla)\\}$

together with the continuation maps forms a direct system with respect to
$\geq_{P}$.

In particular, when $\mathbf{s}_{H_{b}}$ is a constant independent of $b\in
B$, then $HF(P,H,\eta)$ is independent of the choice of $\eta$ such that $H$
is compatible with $\eta$.

As a consequence of Corollary 2.17, for each
$a\in\mathbb{R}_{>0}\setminus\operatorname{Spec}(\partial\bar{M},\alpha)$, the
Floer cohomology

$HF(P,a):=HF(P,H,\eta)$

is independent of admissible $H$ up to natural isomorphisms.

###### Sketch of proof of Corollary 2.17.

For $i=0,1,2$, let $\mathbf{H}_{i}:=HF(P,H_{i},\eta_{i})$ be elements in the
collection such that $H_{0}\leq_{P}H_{1}\leq_{P}H_{2}$. Let $\kappa_{ij}$ be
the continuation map from $\mathbf{H}_{i}$ to $\mathbf{H}_{j}$ for $i\leq j$.
By gluing the auxiliary data defining $\kappa_{01}$ and $\kappa_{12}$, we can
express the composition $\kappa_{12}\circ\kappa_{01}$ as a map given by
counting rigid solutions with respect to the glued auxiliary data. Since the
map $\mathbf{H}_{0}\to\mathbf{H}_{2}$ is independent of choices (Lemma 2.16),
it shows that $\kappa_{01}\circ\kappa_{12}=\kappa_{02}$.

∎

###  2.2. Product structure

Let $SH^{*}(P):=\varinjlim_{a}HF(P,a)$. We are going to define a product
structure on $SH^{*}(P)$. Roughly speaking, it comes from counting Floer pair
of pants lying over a Morse gradient tree with two inputs and one output. To
ensure that the perturbation term in the Floer equation is smooth, we need to
pay special attention at the trivalent point of the Morse gradient tree. The
details are given in the section.

Let $T_{2,1}$ be the unique trivalent Stasheff tree with three external edges,
two of which are incoming and one which is outgoing. Each of the two incoming
edges $\vec{e}_{1},\vec{e}_{2}$ is identified with $[0,\infty)$ and the
outgoing edge is identified with $(-\infty,0]$.

The classical approach to achieving transversality for these operations
involves equipping each edge with a different Morse function. Rather than
doing this, we achieve transversality of gradient flow solutions for maps from
$T_{2,1}$ by perturbing the gradient flow equation (for a single Morse
function). The main definition is the following [Abouzaidtop, Definition 2.6]:

######  Definition 2.18.

A gradient flow perturbation datum on $T_{2,1}$ is a choice, for each edge
$e\in E(T_{2,1})$ of a smoothly varying family of vector fields,

$X_{e}:e\to C^{\infty}(TB)$

vanishes away from a bounded subset of $e.$

Given a gradient flow perturbation datum as above, for each edge $e\in
E(T_{2,1})$ and any map $\tau:e\to B$, one can ask for $\tau$ to solve the
perturbed gradient flow equation (for $\eta$ with respect to $X_{e}$):

(32)
$\displaystyle\frac{d}{ds}\tau(s)=(-\operatorname{grad}(\eta)+X_{e}(s))|_{\tau(s)}\text{
for all }s\in e.$

######  Definition 2.19.

Let $\eta:B\to\mathbb{R}$, be a Morse function and fix gradient flow
perturbation data perturbation data $\\{X_{e}\\}_{i=0,1,2}$ as above. Suppose
that $c_{0},c_{1},c_{2}$ lie in $\operatorname{critp}(\eta)$. With respect to
this data, let

(33) $\mathcal{M}(c_{0},c_{1},c_{2})$

denote the moduli space of continuous maps $\tau:T_{2,1}\to B$ whose
restriction to each edge is a solution to (32).

Note that near the vertex, the perturbation data can be arbitrary. It is not
difficult to show that for a generic choice of perturbation data, our moduli
spaces are cut out transversally Somewhat informally, this corresponds to the
fact that infinitesimally, solutions to the perturbed gradient flow equations
correspond to intersections of the unstable and stable manifolds under
perturbations by the diffeomorphisms $\phi_{e}$ given by integrating the
vector fields $X_{e}$. As the vector fields $X_{e}$ can be chosen arbitrarily,
these diffeomorphisms are essentially arbitrary (for a complete proof, see
[Abouzaidtop, Section 7]). Futhermore, when our data is chosen generically,
the zero dimensional components of the moduli spaces above induce maps between
orientation lines as before, hence an operation on Morse complexes.

Let $\Sigma=\mathbb{P}^{1}\setminus\\{0,1,2\\}$. For $i=1,2$, let
$\epsilon_{i}:[0,\infty)\times S^{1}\to\Sigma$ be a holomorphic embedding such
that $\lim_{s\to\infty}\epsilon_{i}(s,t)=i\in\mathbb{P}^{1}$. Also let
$\epsilon_{0}:(-\infty,0]\times S^{1}\to\Sigma$ be a holomorphic embedding
such that $\lim_{s\to-\infty}\epsilon_{i}(s,t)=0\in\mathbb{P}^{1}$. In other
words, $\epsilon_{i}$ is a positive cylindrical end and $\epsilon_{0}$ a
negative cylindrical end. We require that the images of $\epsilon_{i}$ are
pairwise disjoint. Let $\pi_{\Sigma}\to T_{2,1}$ be the continuous map such
that $\pi_{\Sigma}(\epsilon_{i}(s,t))=e_{i}(s)$ for all $i=0,1,2$, and it maps
the complement of the cylindrical ends of $\Sigma$ to the trivalent vertex of
$T_{2,1}$. Let
$a_{0},a_{1},a_{2}\in\mathbb{R}_{>0}\setminus\operatorname{Spec}(\partial\bar{M},\alpha)$
be such that $a_{0}>a_{1}+a_{2}$. Let $(\eta,g_{\eta},\nabla)$ be an
admissible base triple as before, and $H_{i}$ be a cylindrical Hamiltonian of
$P$ that is compatible with $(\eta,g_{\eta},\nabla)$ and has a constant slope
$a_{i}$ (cf. Lemma 2.8). We choose a perturbation data which consists of a
domain and time dependent cylindrical Hamiltonian $H=(H_{z})_{z\in\Sigma}$ of
$P$ (i.e. $H_{z}$ is a $S^{1}$-dependent cylindrical Hamiltonian of $P$) and
$\beta\in\Omega^{1}(\Sigma)$ such that

  * •

if $z\notin\epsilon_{i}([1,\infty)\times S^{1})$ for all $i$, then $H_{z}$ is
time independent and $(H_{z})_{b}$ is $G$-invariant for all $b\in B$

  * •

for each $i=0,1,2$, there is $R\gg 0$ such that $H_{\epsilon_{i}(s,t)}=H_{i}$
when $|s|>R$

  * •

for all $z\in\Sigma$, $\mathbf{s}(z):=\mathbf{s}_{H_{z}}$ is a constant
function on $\partial\bar{M}$

  * •

for each $i=0,1,2$, we have
$\partial_{s}\mathbf{s}_{H_{\epsilon_{i}(s,t)}}\leq 0$ over all $(s,t)$

  * •

$\beta=dt$ over the cylindrical ends, and

  * •

$d(\mathbf{s}(z)\beta)\leq 0$

The condition that $a_{0}>a_{1}+a_{2}$ guarantees the existence of $H$ and
$\beta$ satisfying all the conditions. We can view
$H\otimes\beta\in\Omega^{1}(\Sigma,C^{\infty}(S^{1}\times P))$ and it will
give us a perturbation term in the upcoming Floer equation we consider.

We also choose a domain and time dependent contact type fibrewise compatible
almost complex structure $J=(J_{z})_{z\in\Sigma}$ of $P$ such that

  * •

for $z\notin\epsilon_{i}([1,\infty)\times S^{1})$ for all $i$, $(J_{z})_{b}$
is $G$-invariant for all $b\in B$

  * •

for each $i=0,1,2$, there is $R\gg 0$ such that $J_{\epsilon_{i}(s,t)}=J_{i}$
when $|s|>R$, where $J_{i}$ is an almost complex structure defining
$CF(P,H_{i},\eta)$

For $i=0,1,2$, let $\vec{x}_{i}:=(c_{i},x_{i})$ a generator in
$CF(P,\mathbf{s}_{i})$. We define
$\mathcal{M}(\vec{x}_{0},\vec{x}_{1},\vec{x}_{2})$ to be the moduli space of
pairs $(\tau,u)$ such that $\tau\in\mathcal{M}(c_{0},c_{1},c_{2})$ and
$u:\Sigma\to P$ satisfies the following statements

  * •

$u(z)\in P_{\tau(\pi_{\Sigma}(z))}$

  * •

$(du^{vert}-X_{(H_{z})_{\tau(\pi_{\Sigma}(z))}}\otimes\beta)^{0,1}_{(J_{s})_{\tau(\pi_{\Sigma}(z))}}=0$

  * •

$\lim_{|s|\to\infty}u(\epsilon_{i}(s,t))=x_{i}(t)$,

In the second bullet, $du^{vert}$ is defined with respect to the connection on
$P\to B$ along $\tau$. Over the vertex $0\in T_{2,1}$, in a prori, there is an
ambiguity of the identification of $P_{\tau(0)}$ with $M$ up to an element in
$G$. However, $H_{z}$ and $J_{z}$ are chosen to be $G$-invariant for
$z\in\pi_{\Sigma}^{-1}(0)$ so the Floer equation is independent of this
ambiguity. Moreover, the perturbation term and almost complex structure in the
equation depend smoothly on $z$. The virtual dimension of
$\mathcal{M}(\vec{x}_{0},\vec{x}_{1},\vec{x}_{2})$ is
$|\vec{x}_{0}|-|\vec{x}_{1}|-|\vec{x}_{2}|$.

For $A\in H_{2}(P,\vec{x}_{0}\cup\vec{x}_{1}\cup\vec{x}_{2})$, we can define
$\mathcal{M}(\vec{x}_{0},\vec{x}_{1},\vec{x}_{2};A)$ to be the subset of
$\mathcal{M}(\vec{x}_{0},\vec{x}_{1},\vec{x}_{2})$ such that
$u_{*}[\Sigma]=A$. The topological energy of $u$ is defined to be (cf. [AbSe,
Equation (148)])

$\displaystyle E(u)$
$\displaystyle=\int_{\Sigma}(du^{vert})^{*}\omega-\int_{\Sigma}d(u^{*}H\otimes\beta)$
$\displaystyle=\int_{\Sigma}(du^{vert})^{*}\omega+\int_{0}^{1}(H_{c_{0}})_{t}(x_{0}(t))dt-\sum_{i=1}^{2}\int_{0}^{1}(H_{c_{i}})_{t}(x_{i}(t))dt$

which only depends on the class $A$.

######  Proposition 2.20.

The following statements hold:

  * •

For a generic choice of compatible $J$ that is of contact type, the space
$\mathcal{M}(\vec{x}_{0},\vec{x}_{1},\vec{x}_{2})$ is a manifold of the
expected dimension for any triple of generators
$(\vec{x}_{0},\vec{x}_{1},\vec{x}_{2})$.

  * •

Moreover, if the virtual dimension is $\leq 1$, then for any $E\in\mathbb{R}$,
the moduli space
$\cup_{A,E(A)<E}\mathcal{M}(\vec{x}_{0},\vec{x}_{1},\vec{x}_{2};A)$ admits a
Gromov compactification

$\cup_{A,E(A)<E}\overline{\mathcal{M}}(\vec{x}_{0},\vec{x}_{1},\vec{x}_{2};A)$

making it a compact manifold with boundary.

###### Proof.

The main point is again to employ a maximum principle to give a priori $C^{0}$
estimates of the solutions. The maximum principle we need is in [Ritter,
Remark C.10]. The $H$ in [Ritter, Remark C.10] corresponds to
$\frac{(H_{z})_{\tau(\pi_{\Sigma}(z))}}{\mathbf{s}(z)}$, which equals to $r$
(up to adding a constant) over the cylindrical end, and the $\beta$ in
[Ritter, Remark C.10] corresponds to our $\mathbf{s}(z)\beta$. Since $H_{z}$
is time independent away from the cylindrical end,
$\partial_{s}\mathbf{s}_{H_{\epsilon_{i}(s,t)}}\leq 0$ over all $(s,t)$ and
$d(\mathbf{s}(z)\beta)\leq 0$, [Ritter, Remark C.10] applies. ∎

For any pair of generators, define a product

(34)
$\displaystyle\vec{x}_{1}\cdot\vec{x}_{2}:=\sum_{\vec{x}_{0},|\vec{x}_{0}|=|\vec{x}_{1}|+|\vec{x}_{2}|}\sum_{(\tau,u)\in\mathcal{M}(\vec{x}_{0},\vec{x}_{1},\vec{x}_{2})}s(u)q^{E(u)}\vec{x}_{0}$

and extend it multi-linearly to the Floer complexes.

######  Proposition 2.21.

Let $a_{0}>a_{1}+a_{2}$.

  * •

The product structure $CF(P,H_{1},\eta)\times CF(P,H_{2},\eta)\to
CF(P,H_{0},\eta)$ descends to a product structure on cohomology.

  * •

The cohomological level product structure is independent of the auxiliary
choices made in the construction, namely, cylindrical ends,
$H=(H_{z})_{z\in\Sigma}$, $J=(J_{z})_{z\in\Sigma}$, and the perturbation
vector field $X_{e}$ on $T_{2,1}$.

  * •

Moreover, it is compatible with continuation maps so it induces a
(graded-)commutative and associative product structure on $SH^{*}(P)$.

###### Sketch of Proof.

Denote the product structure by

$\mu^{2}:CF(P,H_{1},\eta)\times CF(P,H_{2},\eta)\to CF(P,H_{0},\eta).$

By considering the boundary of $1$-dimensional strata of
$\overline{\mathcal{M}}(\vec{x}_{0},\vec{x}_{1},\vec{x}_{2})$, we get

$\mu^{2}(d(\vec{x}_{1}),\vec{x}_{2})+\mu^{2}(\vec{x}_{1},d(\vec{x}_{2}))=d\mu^{2}(\vec{x}_{1},\vec{x}_{2})$

so $\mu^{2}$ descends to cohomology $[\mu^{2}]:HF(P,H_{1},\eta)\times
HF(P,H_{2},\eta)\to HF(P,H_{0},\eta)$.

To see that $[\mu^{2}]$ is independent of choice, we apply a cobordism
argument. In other words, we choose a one-parameter family of auxiliary data
connecting two the ends (it is possible because the space of auxiliary data is
weakly contractible) and form the corresponding parametrized moduli space.
Counting the rigid elements in the parametrized moduli space gives us a chain
homotopy we need. It is important that the one-parameter family of auxiliary
data is chosen such that maximum principle applies as in the proof of
Proposition 2.20.

Once we know that it is independent of auxiliary choices, together with the
standard gluing argument we can prove the (graded-)commutativity and
associativity because they correspond to interpolating two different choices.

The compatibility with continuation map means that

$\kappa_{a_{0},a_{0}^{\prime}}([\mu_{a_{0},a_{1},a_{2}}](\kappa_{a_{1}^{\prime},a_{1}}(\vec{x}_{1}),\kappa_{a_{2}^{\prime},a_{2}}(\vec{x}_{2})))=[\mu_{a_{0}^{\prime},a_{1}^{\prime},a_{2}^{\prime}}](\vec{x}_{1},\vec{x}_{2})$

where $\kappa_{j,k}:HF(P,j)\to HF(P,k)$ is the coninuation map when $j\leq k$
and $[\mu_{l,j,k}]:HF(P,j)\times HF(P.k)\to HF(P,l)$ is the product structure
when $l>j+k$. This compatibility follows from the gluing argument and
independence of auxiliary choices. ∎

The product operation can be defined not only for cylindrical Hamiltonians
with constant slopes, it can also be defined as long as the maximum principle
can be achieved. The analogue of Lemma 2.16 for the product operation is as
follow.

######  Lemma 2.22.

For $i=0,1,2$, let $H_{i}$ be a cylindrical Hamiltonians that is compatible
with $\eta_{i}$. Suppose that
$\max{\mathbf{s}_{H_{1}}}+\max{\mathbf{s}_{H_{2}}}\leq\min\mathbf{s}_{H_{0}}$.
After making an appropriate auxiliary choice, we can define a product
operation

$CF(P,H_{1},\eta_{1})\times CF(P,H_{2},\eta_{2})\to CF(P,H_{0},\eta_{0})$

Moreover, the induced map on cohomology

(35) $\displaystyle HF(P,H_{1},\eta_{1})\times HF(P,H_{2},\eta_{2})\to
HF(P,H_{0},\eta_{0})$

is independent of the auxiliary choice in the construction.

###  2.3. Pull-back

We want to discuss the functorality between two admissible bundles with the
same fibre $(M,\omega,\theta)$ but different bases $B$ and $B^{\prime}$ (both
$B$ and $B^{\prime}$ are smooth closed manifolds). Let $h:B^{\prime}\to B$ be
a smooth map and $p^{\prime}:P^{\prime}\to B^{\prime}$ be the fiber bundle
obtained by pulling back $p$ along $h$. Let $\tilde{h}:P^{\prime}\to P$ be the
induced map covering $h$. Let $\eta$ and $\eta^{\prime}$ be Morse functions on
$B$ and $B^{\prime}$ respectively and assume that $H$ and $H^{\prime}$ are
Hamiltonians on $P$ and $P^{\prime}$ that are compatible with
$(\eta,g_{\eta},\nabla)$ and
$(\eta^{\prime},g_{\eta}^{\prime},\nabla^{\prime})$ respectively.

Let $(H_{s})_{s\in\mathbb{R}_{\geq 0}}$ and
$(H^{\prime}_{s})_{s\in\mathbb{R}_{\leq 0}}$ be two families of cylindrical
Hamiltonians, on $P$ and $P^{\prime}$ respectively, such that $H_{s}=H$ for
$s\gg 0$, $H^{\prime}_{s}=H^{\prime}$ for $s\ll 0$ and
$\tilde{h}^{*}H_{s}=\tilde{h}^{*}H_{0}=H^{\prime}_{0}=H^{\prime}_{s}$ for $s$
in an open neighborhood of $0$. Choose one-parameter families of
$S^{1}$-dependent fiberwise compatible almost complex structures
$(J_{s})_{s\in\mathbb{R}_{\geq 0}}$ and
$(J^{\prime}_{s})_{s\in\mathbb{R}_{\leq 0}}$ that are of contact type on $P$
and $P^{\prime}$, respectively. They are chosen such that $J_{s}$ is
independent of $s$ for $s\gg 0$, $J^{\prime}_{s}$ is independent of $s$ for
$s\ll 0$, and
$\tilde{h}^{*}J_{s}=\tilde{h}^{*}J_{0}=J^{\prime}_{0}=J^{\prime}_{s}$ for $s$
in an open neighborhood of $0$. Note that $\tilde{h}^{*}J_{s}$ is well-defined
because $J_{s}$ is fiberwise and the restriction of $\tilde{h}$ to fibres are
isomorphisms.

Given a generator $\vec{x}=(c,x)$ of $H$ and
$\vec{x}^{\prime}=(c^{\prime},x^{\prime})$ of $H^{\prime}$, we can consider
the moduli space
$\mathcal{M}_{pb}(\vec{x}^{\prime},\vec{x},(H^{\prime}_{s})_{s\in\mathbb{R}_{\leq
0}},(H_{s})_{s\in\mathbb{R}_{\geq 0}})$ which consists of
$(\tau^{-},\tau^{+},u^{-},u^{+})$ such that

  * •

$\tau^{-}:(-\infty,0]\to B^{\prime}$ is a gradient trajectory of
$\eta^{\prime}$ with $\lim_{s}\tau^{-}(s)=c^{\prime}$,

  * •

$\tau^{+}:[0,\infty)\to B$ is a gradient trajectory of $\eta$ with
$\lim_{s}\tau^{+}(s)=c$,

  * •

$\tau^{+}(0)=h(\tau^{-}(0))$,

  * •

$u^{-}:(-\infty,0]\times S^{1}\to P^{\prime}$ is covering $\tau^{-}$ and
satisfies the Floer equation

$(du^{vert}-X_{(H^{\prime}_{s})_{\tau^{-}(s)}}\otimes
dt)^{0,1}_{(J^{\prime}_{s})_{\tau^{-}(s)}}=0$

with $\lim_{s\to-\infty}u^{-}(s,t)=x^{\prime}(t)$,

  * •

$u^{+}:[0,\infty)\times S^{1}\to P$ is covering $\tau^{+}$ and satisfies the
Floer equation

$(du^{vert}-X_{(H_{s})_{\tau^{+}(s)}}\otimes
dt)^{0,1}_{(J_{s})_{\tau^{+}(s)}}=0$

with $\lim_{s\to\infty}u^{+}(s,t)=x(t)$,

  * •

$u^{+}(0,t)=\tilde{h}(u^{-}(0,t))$,

Notice that $\tau(s):=(\tau^{-}(-s),\tau^{+}(s))$ for $s\in\mathbb{R}_{\geq
0}$ is a gradient trajectory of the function $B^{\prime}\times B\to\mathbb{R}$
given by $(b^{\prime},b)\mapsto\eta(b)-\eta^{\prime}(b^{\prime})$ starting at
a point on $\operatorname{graph}(h)\subset B^{\prime}\times B$. Similarly, let
$\pi_{P}:P^{\prime}\times P\to P$ and $\pi_{P^{\prime}}:P^{\prime}\times P\to
P^{\prime}$ be the projection to the factors, then
$u(s,t):=(u^{-}(-s,t),u^{+}(s,t))\in P^{\prime}\times P$ for
$(s,t)\in\mathbb{R}_{\geq 0}\times S^{1}$ is a solution to the Floer equation

(36)
$\displaystyle(du^{vert}-X_{\pi_{P}^{*}(H_{s})_{\tau^{+}(s)}-\pi_{P^{\prime}}^{*}(H^{\prime}_{-s})_{\tau^{-}(-s)}}\otimes
dt)^{0,1}_{-(J_{-s})_{\tau^{-}(-s)}\oplus(J_{s})_{\tau^{+}(s)}}=0$

covering $\tau$ with fiberwise Lagrangian boundary condition
$\operatorname{graph}(\tilde{h})\subset P^{\prime}\times P$ over $\tau(0)$
along $\\{0\\}\times S^{1}\subset\mathbb{R}_{\geq 0}\times S^{1}$.

From this perspective (so-called ‘folding’ in [WWQuilt]), we can apply the
standard Fredholm theory package to study
$\mathcal{M}_{pb}(\vec{x}^{\prime},\vec{x},(H^{\prime}_{s})_{s\in\mathbb{R}_{\leq
0}},(H_{s})_{s\in\mathbb{R}_{\geq 0}})$.

######  Lemma 2.23.

For generic choice of $(J_{s})_{s\in\mathbb{R}_{\geq 0}}$ and
$(J^{\prime}_{s})_{s\in\mathbb{R}_{\leq 0}}$ as above, every element in the
moduli space
$\mathcal{M}_{pb}(\vec{x}^{\prime},\vec{x},(H^{\prime}_{s})_{s\in\mathbb{R}_{\leq
0}},(H_{s})_{s\in\mathbb{R}_{\geq 0}})$ is regular so it is a manifold of the
expected dimension.

###### Proof.

By our assumptions on $H_{s}$ and $H^{\prime}_{s}$, when $s$ is close to $0$
and $(p,p^{\prime})\in\operatorname{graph}(\tilde{h})$, we have
$H_{s,t}(p)-H_{-s,t}^{\prime}(p^{\prime})=0$ because
$\tilde{h}^{*}H_{s}=\tilde{h}^{*}H_{0}=H^{\prime}_{0}=H^{\prime}_{s}$ for $s$
in an open neighborhood of $0$. In other words, the Hamiltonian perturbation
term

$X_{\pi_{P}^{*}H_{s,t}-\pi_{P^{\prime}}^{*}H_{-s,t}^{\prime}}dt|_{T^{vert}\operatorname{graph}(\tilde{h})}$

is zero along the fiberwise Lagrangian boundary condition
$\operatorname{graph}(\tilde{h})$ so it falls into the standard Floer theory
package (see [SeidelBook, (8.6)], [WWQuilt, Section 5], [Cazassus, (4.70)]).
Therefore, the standard transversality proof applies. ∎

Next, we want to impose conditions on $(H_{s})_{s\in\mathbb{R}_{\geq 0}}$ and
$(H^{\prime}_{s})_{s\in\mathbb{R}_{\leq 0}}$ to ensure compactness of moduli.
As before, we define the topological energy $E(u^{-},u^{+})$ of
$(\tau^{-},\tau^{+},u^{-},u^{+})$ to be

(37) $\displaystyle E(u^{-},u^{+})$ $\displaystyle=E(u^{-})+E(u^{+})$
$\displaystyle E(u^{-})$ $\displaystyle=\int_{(-\infty,0]\times
S^{1}}((du^{-})^{vert})^{*}\omega+\int_{0}^{1}(H_{c^{\prime}})_{t}(x^{\prime}(t))dt$
$\displaystyle E(u^{+})$ $\displaystyle=\int_{[0,\infty)\times
S^{1}}((du^{+})^{vert})^{*}\omega-\int_{0}^{1}(H_{c})_{t}(x(t))dt$

It only depends on the relative homology class of its folding
$u:[0,\infty)\times S^{1}\to P^{\prime}\times P$, relative to the fibrewise
Lagrangian boundary condition and the asymptotic orbit (cf. (36)).

######  Lemma 2.24.

Suppose that $\partial_{s}\mathbf{s}_{(H_{s})_{\tau^{+}(s)}}\leq 0$ and
$\partial_{s}\mathbf{s}_{(H^{\prime}_{s})_{\tau^{-}(s)}}\leq 0$ for all $s$,
and for all gradient trajectory $\tau^{+}$ and $\tau^{-}$ of $\eta^{+}$ and
$\eta^{-}$, respectively. If the virtual dimension of the modui space
$\mathcal{M}_{pb}(\vec{x}^{\prime},\vec{x},(H^{\prime}_{s})_{s\in\mathbb{R}_{\leq
0}},(H_{s})_{s\in\mathbb{R}_{\geq 0}})$ is $\leq 1$, then for any
$E\in\mathbb{R}$, the subspace of solutions with topological energy at most
$E$ admits a Gromov compactification to a compact manifold with boundary.

###### Proof.

It suffices to prove an a priori $C^{0}$ bound for the solutions. By
trivializing $P^{\prime}$ over $\tau^{-}$ using $\nabla^{\prime}$,
trivializing $P$ over $\tau^{+}$ using $\nabla$ and identifying fibres of
$P^{\prime}$ to fibres of $P$ using $\tilde{h}$, we can regard $u^{-}$ and
$u^{+}$ as smooth maps to a fibre $(M,\omega,\theta)$, say the fibre
$P^{\prime}_{\tau^{-}(0)}\simeq P_{\tau^{+}(0)}$. The map
$u^{-}\\#u^{+}:\mathbb{R}\times S^{1}\to M$ defined by

(40)
$\displaystyle(u^{-}\\#u^{+})(s,t):=\left\\{\begin{array}[]{ll}u^{-}(s,t)&\text{
for }s\leq 0\\\ u^{+}(s,t)&\text{ for }s\geq 0\end{array}\right.$

is piecewise smooth and hence Lipschitz. Therefore, the first weak derivative
exist [EvansBook, 5.8.2b].

Recall that
$\tilde{h}^{*}H_{s}=\tilde{h}^{*}H_{0}=H^{\prime}_{0}=H^{\prime}_{s}$ and
$\tilde{h}^{*}J_{s}=\tilde{h}^{*}J_{0}=J^{\prime}_{0}=J^{\prime}_{s}$ for $s$
in an open neighborhood of $0$. Therefore, $u^{-}\\#u^{+}$ satisfies a Floer
equation with smooth auxiliary data (both $H$ and $J$). The existence of the
first weak derivative of $u^{-}\\#u^{+}$ allows us to apply elliptic
bootstrapping [MSJbook, Theorem B.4.1], so it is actually smooth. The required
$C^{0}$ estimate now follows from maximum principle [Ritter, Theorem C.11]
(cf. proof of Proposition 2.20). ∎

The virtual dimension of
$\mathcal{M}_{pb}(\vec{x}^{\prime},\vec{x},(H^{\prime}_{s})_{s\in\mathbb{R}_{\leq
0}},(H_{s})_{s\in\mathbb{R}_{\geq 0}})$ is $|\vec{x}^{\prime}|-|\vec{x}|$. To
see this, since we are using cohomological convention, the virtual dimension
of the moduli of pairs of gradient trajectory $(\tau^{-},\tau^{+})$ with
$h(\tau^{-}(0))=\tau^{+}(0)$ is precisely $|c^{\prime}|-|c|$, where $|\cdot|$
is the Morse index. Therefore, by trivializing over $\tau^{-}$ and $\tau^{+}$,
we see that the virtual dimension of $(u^{-},u^{+})$ for a given
$(\tau^{-},\tau^{+})$ is $|x^{\prime}|-|x|$. Therefore, the virtual dimension
is $|c^{\prime}|-|c|+|x^{\prime}|-|x|=|\vec{x}^{\prime}|-|\vec{x}|$.

######  Lemma 2.25.

Suppose that $\partial_{s}\mathbf{s}_{(H_{s})_{\tau^{+}(s)}}\leq 0$ and
$\partial_{s}\mathbf{s}_{(H^{\prime}_{s})_{\tau^{-}(s)}}\leq 0$ for all $s$,
and for all gradient trajectory $\tau^{+}$ and $\tau^{-}$ of $\eta^{+}$ and
$\eta^{-}$, respectively. The pull-back defined by

$\displaystyle CF^{*}(P,H,\eta)$ $\displaystyle\to
CF^{*}(P^{\prime},H^{\prime},\eta^{\prime})$ (41) $\displaystyle\vec{x}$
$\displaystyle\mapsto\sum_{\vec{x}^{\prime},|\vec{x}^{\prime}|=|\vec{x}|}\sum_{(u^{-},u^{+})\in\mathcal{M}_{pb}(\vec{x}^{\prime},\vec{x},(H^{\prime}_{s})_{s\in\mathbb{R}_{\leq
0}},(H_{s})_{s\in\mathbb{R}_{\geq
0}})}s(u^{-},u^{+})q^{E(u^{-},u^{+})}\vec{x}^{\prime}$

is a degree preserving chain map, where $s(u^{-},u^{+})\in\\{-1,1\\}$ is the
sign.

######  Lemma 2.26.

Suppose that $H^{\prime}\geq_{P^{\prime}}\tilde{h}^{*}H$. Then there exist
$(H^{\prime}_{s})_{s\in\mathbb{R}_{\leq 0}}$ and
$(H_{s})_{s\in\mathbb{R}_{\geq 0}}$ as above such that the assumptionn of
Lemma 2.25 is satisfied (and hence the pull-back map (41) is well-defined).
Moreover, the induced map on cohomology is independent of the choice of
$(H^{\prime}_{s})_{s\in\mathbb{R}_{\leq 0}}$, $(H_{s})_{s\in\mathbb{R}_{\geq
0}}$, $(J_{s})_{s\in\mathbb{R}_{\geq 0}}$ and
$(J^{\prime}_{s})_{s\in\mathbb{R}_{\leq 0}}$.

###### Proof.

The arguments are in parallel with the proof of Lemma 2.16 so we will omit. ∎

The following is the analogue of Corollary 2.17.

######  Corollary 2.27.

Suppose that $h^{\prime}:B^{\prime\prime}\to B^{\prime}$ is another smooth map
between smooth compact manifolds, and $P^{\prime\prime}\to B^{\prime\prime}$
is the pull-back of $P^{\prime}\to B^{\prime}$ along $h^{\prime}$. Let
$\tilde{h}^{\prime}:P^{\prime\prime}\to P^{\prime}$ be the induced map
covering $h^{\prime}$. Suppose that
$H^{\prime\prime}\geq_{P^{\prime\prime}}(\tilde{h}^{\prime})^{*}H^{\prime}$
and $H^{\prime}\geq_{P^{\prime}}\tilde{h}^{*}H$. Then the composition of the
pull-back maps $HF^{*}(P,H,\eta)\to
HF^{*}(P^{\prime},H^{\prime},\eta^{\prime})$ and
$HF^{*}(P^{\prime},H^{\prime},\eta^{\prime})\to
HF^{*}(P^{\prime\prime},H^{\prime\prime},\eta^{\prime\prime})$ is the pull-
back map $HF^{*}(P,H,\eta)\to
HF^{*}(P^{\prime\prime},H^{\prime\prime},\eta^{\prime\prime})$.

######  Example 2.28.

Let $h:B^{\prime}\to B$ (and hence $\tilde{h}:P^{\prime}\to P$) be an
embedding. Let $(\eta,g_{\eta},\nabla)$ be chosen such that there is no
gradient trajectory $\tau$ of $\eta$ connecting two critical points with
$\lim_{s\to-\infty}\tau(s)\in B^{\prime}$ and $\lim_{s\to\infty}\tau(s)\notin
B^{\prime}$, and every gradient trajectory of $\eta$ connecting two critical
points in $B^{\prime}$ is completely lying inside $B^{\prime}$. Let $H$ be a
cylindrical Hamiltonian that is compatible with $(\eta,g_{\eta},\nabla)$.
Suppose that $(\eta^{\prime},g_{\eta}^{\prime},\nabla^{\prime})$ and
$H^{\prime}$ are the restriction of $(\eta,g_{\eta},\nabla)$ and $H$ to
$P^{\prime}$ and $B^{\prime}$. Then
$CM(B^{\prime},\eta^{\prime},g_{\eta}^{\prime})$ is naturally a quotient
complex of $CM(B,\eta,g_{\eta})$ and the induced map on cohomology coincide
with the Morse theoretic pull-back. Similarly,
$CF(P^{\prime},H^{\prime},\eta^{\prime})$ is naturally a quotient complex of
$CF(P,H,\eta)$ and the induced map on cohomology coincide with the Floer
theoretic pull-back introduced above.

######  Lemma 2.29.

Let $E_{1}\subset E_{2}\subset\dots$ be a sequence of smooth compact
manifolds. Let $P_{n}:=E_{n}\times M$ and view the projection to the first
factor as an admissible bundle over $E_{n}$. Then
$\varprojlim_{n}HF^{*}(P_{n},a)=\varprojlim_{n}(H^{*}(E_{n})\otimes
HF^{*}(M,a))=(\varprojlim_{n}H^{*}(E_{n}))\otimes HF^{*}(M,a)$.

###### Proof.

Let $H_{M}$ be a non-degenerate cylindrical Hamiltonian of $M$ with constant
slope $a$. Let $\pi_{M}:P_{n}\to M$ be the projection so $\pi_{M}^{*}H_{M}$ is
a cylindrical Hamiltonian of $P_{n}$. Choose a Morse function
$\eta:E_{n}\to\mathbb{R}$ and take $\nabla$ to be the trivial connection. Let
$H$ be a $C^{2}$ small perturbation of $\pi_{M}^{*}H_{M}$ in a compact set so
that it is a still cylindrical Hamiltonian of $P_{n}$ with constant slope $a$
and it is compatible with $(\eta,g_{\eta},\nabla)$ such that $H_{c}=H_{M}$ for
every critical point $c$ of $\eta$.

Suppose that $(\tau,u)$ is a solution contributing to the differential. Then
$\pi_{M}\circ u$ satisfies the Floer equation

$(d(\pi_{M}\circ u)-X_{H_{\tau(s)}}\otimes dt)^{0,1}_{J_{\tau(s)}}=0$

When the perturbation is sufficiently small (so that $H_{\tau(s)}$ is very
close to $H_{M}$ for all $s$), the virtual dimension of $\pi_{M}\circ u$ must
be at least $0$, and when it is $0$, the input and the output are the same
orbit. This identifies $CF^{*}(P_{n},H,\eta)$ with $CM^{*}(E_{n},\eta)\otimes
CF^{*}(M,H_{M})$ as a cochain complex so $HF^{*}(P_{n},a)=H^{*}(E_{n})\otimes
HF^{*}(M,a)$.

When we consider $P_{n}\subset P_{n+1}$, we can choose $\eta$ on $E_{n+1}$
such that the critical points of $\eta$ outside $E_{n}$ forms a subcomplex,
and any gradient trajectory connecting two critical points in $E_{n}$ is
contained in $E_{n}$. Then the pull-back map $CF(P_{n+1},H,\eta)\to
CF(P_{n},H|_{P_{n}},\eta|_{E_{n}})$ can be identified with restricting to the
quotient complex obtained by killing the generators outside $P_{n}$. When the
perturbation is sufficiently small, it can in turn be identified with the
natural map $CM^{*}(E_{n+1},\eta)\otimes CF^{*}(M,H_{M})\to
CM^{*}(E_{n},\eta|_{E_{n}})\otimes CF^{*}(M,H_{M})$ so the result follows. ∎

######  Lemma 2.30.

The pull-back map $SH^{*}(P)\to SH^{*}(P^{\prime})$ is an algebra map.

###### Sketch of proof.

The proof of Lemma 2.30 combines the ideas in Section 2.2 and this section.
The moduli space we need to construct consists of solutions $(\tau,u)$ of a
coupled equation. The domain of $\tau$ is $T_{2,1}$ and the target is $B\sqcup
B^{\prime}$. There is $r\in\mathbb{R}$ such that $\tau(\\{s\leq r\\})\subset
B^{\prime}$ and $\tau(\\{s\geq r\\})\subset B$, where $s:T_{2,1}\to\mathbb{R}$
is the coordinate function on the edges of $T_{2,1}$. The map $\tau$ is a
negative ($X_{e}$-perturbed) gradient trajectory over any edge and it
satisfies the matching condition
$h(\tau_{B^{\prime}}(\\{s=r\\}))=\tau_{B}(\\{s=r\\})$, where
$\tau_{B^{\prime}}:=\tau|_{\\{s\leq r\\}}$ and $\tau_{B}:=\tau|_{\\{s\geq
r\\}}$. On the other hand, $u:\Sigma=\mathbb{P}^{1}\setminus\\{0,1,2\\}\to
P^{\prime}\sqcup P$ is a lift of $\tau$ that satisfies an appropriate Floer
equation. We can compactify this moduli space. If we consider the boundary of
the compactification of the one dimensional moduli space, then the limit when
$r$ goes to $-\infty$ corresponds to taking the product in $SH^{*}(P)$ and
then pulling back to $SH^{*}(P^{\prime})$. Similarly, the limit when $r$ goes
to $\infty$ corresponds to pulling back two classes from $SH^{*}(P)$ to
$SH^{*}(P^{\prime})$ and then taking the product in $SH^{*}(P^{\prime})$.

∎

######  Lemma 2.31.

Suppose that $h:B^{\prime}\to B$ is a fibre bundle with fibres $F$ such that
$H^{0}(F;\mathbb{C})=\mathbb{C}$, and $H^{*}(F;\mathbb{C})=0$ otherwise. Then
the pull-back map induces an isomorphism $HF^{*}(P,a)\simeq
HF^{*}(P^{\prime},a)$.

###### Proof.

We pick an admissible base triple $(\eta,g_{\eta},\nabla)$ for $P\to B$. Let
$g^{\prime}$ be a Riemannian metric on $B^{\prime}$ such that $h$ is a
Riemannian submersion. Let $N$ be a neighborhood of the critical points of
$\eta$ such that $\nabla$ is flat over $N$. So the bundle $P^{\prime}$ over
$h^{-1}(N)$ can be identified with $P^{\prime}|_{h^{-1}(N)}=M\times
h^{-1}(N)=M\times F\times N$. Let $\eta^{\prime}:B^{\prime}\to\mathbb{R}$ be a
Morse function obtained by perturbing $h^{*}\eta$ inside $h^{-1}(N)$. We pick
$\eta^{\prime}$ such that $(\eta^{\prime},g^{\prime},h^{*}\nabla)$ is an
admissible base triple.

We define a descending filtration on the chain complex $CF^{*}(P,a)$ and
$CF^{*}(P^{\prime},a)$ by taking $F^{p}CF^{*}(P,a)$ and
$F^{p}CF^{*}(P^{\prime},a)$ to be the subcomplex generated by $\vec{x}=(c,x)$
such that $|c|\geq p$, and the subcomplex generated by
$\vec{x}^{\prime}=(c^{\prime},x^{\prime})$ such that $|h(c^{\prime})|\geq p$,
respectively. By our choice of admissible base triples, the pull-back map
respects the fibration (i.e. it maps $F^{p}CF^{*}(P,a)$ to
$F^{p}CF^{*}(P^{\prime},a)$). It induces a homomorphism of the corresponding
spectral sequences.

The $E_{1}$-page of $CF^{*}(P,a)$ is given by
$(\oplus_{c\in\operatorname{critp}(\eta)}HF^{*-|c|}(M,a),d_{1})$, where the
restriction of $d_{1}$ to the group $HF(M,a)$ over $c$ is the sum of the
continuation map over each rigid gradient trajectory with input $c$. On the
other hand, by Lemma 2.29, the $E_{1}$-page of $CF^{*}(P^{\prime},a)$ is given
by $(\oplus_{c\in\operatorname{critp}(\eta)}H^{*}(F)\otimes
HF^{*-|c|}(M,a),d_{1}^{\prime})$. By our assumption on $H^{*}(F;\mathbb{C})$,
we have $H^{*}(F)\otimes HF^{*-|c|}(M,a)=HF^{*-|c|}(M,a)$. The homomorphism
between the components of the $E_{1}$-pages can be identified with
$H^{*}(\text{point})\otimes HF^{*-|c|}(M,a)\to H^{*}(F)\otimes
HF^{*-|c|}(M,a)$ so it is an isomorphism. This finishes the proof.

∎

###  2.4. Push-forward

We now explain how to define the push-forward map in this setup by simply
reversing the $s$-direction. To clarify, we follow the notation set up in the
first paragraph of the previous section.

This time we let $(H_{s})_{s\in\mathbb{R}_{\leq 0}}$ and
$(H^{\prime}_{s})_{s\in\mathbb{R}_{\geq 0}}$ be two families of cylindrical
Hamiltonians, on $P$ and $P^{\prime}$ respectively, such that $H_{s}=H$ for
$s\ll 0$, $H^{\prime}_{s}=H^{\prime}$ for $s\gg 0$ and
$\tilde{h}^{*}H_{s}=\tilde{h}^{*}H_{0}=H^{\prime}_{0}=H^{\prime}_{s}$ for $s$
in an open neighborhood of $0$. Choose one-parameter families of
$S^{1}$-dependent fiberwise compatible almost complex structures
$(J_{s})_{s\in\mathbb{R}_{\leq 0}}$ and
$(J^{\prime}_{s})_{s\in\mathbb{R}_{\geq 0}}$ that are of contact type on $P$
and $P^{\prime}$, respectively. They are chosen such that $J_{s}$ is
independent of $s$ for $s\gg 0$, $J^{\prime}_{s}$ is independent of $s$ for
$s\ll 0$, and
$\tilde{h}^{*}J_{s}=\tilde{h}^{*}J_{0}=J^{\prime}_{0}=J^{\prime}_{s}$ for $s$
in an open neighborhood of $0$.

Given a generator $\vec{x}=(c,x)$ of $H$ and
$\vec{x}^{\prime}=(c^{\prime},x^{\prime})$ of $H^{\prime}$, we can consider
the moduli space
$\mathcal{M}_{pf}(\vec{x};\vec{x}^{\prime},(H_{s})_{s\in\mathbb{R}_{\leq
0}},(H^{\prime}_{s})_{s\in\mathbb{R}_{\geq 0}})$ which consists of
$(\tau^{-},\tau^{+},u^{-},u^{+})$ such that

  * •

$\tau^{-}:(-\infty,0]\to B$ is a gradient trajectory of $\eta$ with
$\lim_{s}\tau^{-}(s)=c$,

  * •

$\tau^{+}:[0,\infty)\to B^{\prime}$ is a gradient trajectory of
$\eta^{\prime}$ with $\lim_{s}\tau^{+}(s)=c^{\prime}$,

  * •

$h(\tau^{+}(0))=\tau^{-}(0)$,

  * •

$u^{-}:(-\infty,0]\times S^{1}\to P$ is covering $\tau^{-}$ and satisfies the
Floer equation

$(du^{vert}-X_{(H_{s})_{\tau^{-}(s)}}\otimes
dt)^{0,1}_{(J_{s})_{\tau^{-}(s)}}=0$

with $\lim_{s\to-\infty}u^{-}(s,t)=x(t)$,

  * •

$u^{+}:[0,\infty)\times S^{1}\to P^{\prime}$ is covering $\tau^{+}$ and
satisfies the Floer equation

$(du^{vert}-X_{(H^{\prime}_{s})_{\tau^{+}(s)}}\otimes
dt)^{0,1}_{(J^{\prime}_{s})_{\tau^{+}(s)}}=0$

with $\lim_{s\to\infty}u^{+}(s,t)=x^{\prime}(t)$,

  * •

$\tilde{h}(u^{+}(0,t))=u^{-}(0,t)$,

The analogue of Lemma 2.23 and 2.24 are true. Moreover, the virtual dimension
of the moduli space
$\mathcal{M}_{pf}(\vec{x};\vec{x}^{\prime},(H_{s})_{s\in\mathbb{R}_{\leq
0}},(H^{\prime}_{s})_{s\in\mathbb{R}_{\geq 0}})$ is
$|\vec{x}|-|\vec{x}^{\prime}|-\dim(B)+\dim(B^{\prime})$ because the virtual
dimension of $(\tau^{-},\tau^{+})$ is
$|c|-\dim(B)+|c^{\prime}|-\dim(B^{\prime})$ this time. It leads to the
following analogue of Lemma 2.25, 2.26 and Corollary 2.27.

######  Lemma 2.32.

Suppose that $\partial_{s}\mathbf{s}_{(H^{\prime}_{s})_{\tau^{+}(s)}}\leq 0$
and $\partial_{s}\mathbf{s}_{(H_{s})_{\tau^{-}(s)}}\leq 0$ for all $s$, and
for all gradient trajectory $\tau^{+}$ and $\tau^{-}$ of $\eta^{+}$ and
$\eta^{-}$, respectively. The push-forward defined by

$\displaystyle CF^{*}(P^{\prime},H^{\prime},\eta^{\prime})\to
CF^{*}(P,H,\eta)$ (42)
$\displaystyle\vec{x}^{\prime}\mapsto\sum_{\vec{x},|\vec{x}|=|\vec{x}^{\prime}|+\dim(B)-\dim(B^{\prime})}\sum_{(u^{-},u^{+})\in\mathcal{M}_{pf}(\vec{x},\vec{x}^{\prime},(H_{s})_{s\in\mathbb{R}_{\leq
0}},(H^{\prime}_{s})_{s\in\mathbb{R}_{\geq
0}})}s(u^{-},u^{+})q^{E(u^{-},u^{+})}\vec{x}$

is a chain map of degree $\dim(B)-\dim(B^{\prime})$, where $E(u^{-},u^{+})$ is
defined by (37) and $s(u^{-},u^{+})$ is the sign.

######  Lemma 2.33.

Suppose that $\tilde{h}^{*}H\geq_{P^{\prime}}H^{\prime}$. Then there exist
$(H_{s})_{s\in\mathbb{R}_{\leq 0}}$ and
$(H^{\prime}_{s})_{s\in\mathbb{R}_{\geq 0}}$ as above such that the assumption
of Lemma 2.32 is satisfied (and hence the pull-forward map (42) is well-
defined). Moreover, the induced map on cohomology is independent of the choice
of $(H_{s})_{s\in\mathbb{R}_{\leq 0}}$,
$(H^{\prime}_{s})_{s\in\mathbb{R}_{\geq 0}}$, $(J_{s})_{s\in\mathbb{R}_{\leq
0}}$ and $(J^{\prime}_{s})_{s\in\mathbb{R}_{\geq 0}}$.

######  Corollary 2.34.

Suppose that $h^{\prime}:B^{\prime\prime}\to B^{\prime}$ is another smooth map
between smooth compact manifolds, and $P^{\prime\prime}\to B^{\prime\prime}$
is the pull-back of $P^{\prime}\to B^{\prime}$ along $h^{\prime}$. Let
$\tilde{h}^{\prime}:P^{\prime\prime}\to P^{\prime}$ be the induced map
covering $h^{\prime}$. Suppose that
$(\tilde{h}^{\prime})^{*}H^{\prime}\geq_{P^{\prime\prime}}H^{\prime\prime}$
and $\tilde{h}^{*}H\geq_{P^{\prime}}H^{\prime}$. Then the composition of the
push-forward maps
$HF^{*}(P^{\prime\prime},H^{\prime\prime},\eta^{\prime\prime})\to
HF^{*}(P^{\prime},H^{\prime},\eta^{\prime})$ and
$HF^{*}(P^{\prime},H^{\prime},\eta^{\prime})\to HF^{*}(P,H,\eta)$ is the push-
forward map $HF^{*}(P^{\prime\prime},H^{\prime\prime},\eta^{\prime\prime})\to
HF^{*}(P,H,\eta)$.

The analog of Lemma 2.30 is the following.

######  Lemma 2.35.

Denote the pushfoward and pullback by $h_{*}:SH^{*}(P^{\prime})\to
SH^{*-\dim(B)}(P)$ and $h^{*}:SH^{*}(P)\to SH^{*}(P^{\prime})$, respectively.
Then we have
$h_{*}(\vec{x}^{\prime}h^{*}(\vec{x}))=h_{*}(\vec{x}^{\prime})\vec{x}$ for all
$\vec{x}^{\prime}\in SH^{*}(P^{\prime})$ and $\vec{x}\in SH^{*}(P)$.

###### Proof.

It is the same as Lemma 2.30 except that one of the two inputs and the output
of the elements in the moduli space are swapped. ∎

##  3\. Equivariant Floer theory

Let $K\subset G$ be a connected closed subgroup.222We mainly consider $K$ to
be a maximal torus $T$ or $K=G$ in the paper. Let $EK_{1}\subset
EK_{2}\subset\dots$ be a $K$-equivariant smooth finite dimensional
approximation of a classifying space $EK$. In particular, each $EK_{n}$ is a
compact smooth manifold, $\cup_{n}EK_{n}=EK$ and the connectivity of $EK_{n}$
goes to infinity as $n$ goes to infinity. Let $BK_{n}:=EK_{n}/K$. For a
manfiold $X$ with a $K$ action, we denote $(X\times EK_{n})/K$ by
$X_{borel,n}$ and $(X\times EK)/K$ by $X_{borel}$.

Let $P\to B$ be an admissible bundle as in the previous section (see
Definition 2.4). We say that a $K$-action on $P$ is compatible with the
admissible structure if it covers a $K$-action on $B$ and the induced map
$P_{borel,n}\to B_{borel,n}$ is an admissible bundle (so the structure group
lies in $G$) for all $n$. Notice that a choice of a symplectomorphism between
$M$ and a reference fibre of $P\to B$ induces a symplectomorphism between $M$
and a reference fibre of $P_{borel,n}\to B_{borel,n}$ that is well-defined up
to an element in $G$. As a result, it induces a well-defined homotopy class of
trivialization of the canonical bundle of the reference fibre so a choice of
$\mathbb{Z}$-grading on $CF(P)$ determines a choice of $\mathbb{Z}$-grading on
$CF(P_{borel,n})$. We assume for the rest of the paper that if $B$ carries a
$K$-action, then $B$ also admits a $K$-orientation.

######  Example 3.1.

If $P=M$ and $B$ is a point, then a $K$-action on $P$ is compatible with the
admissible structure if and only if $M_{borel,n}\to BK_{n}$ is admissible.
This will hold if and only if the $K$ action on $P(=M)$ factors through the
convex $G$ action on $M$.

######  Definition 3.2.

Let $a\in\mathbb{R}_{>0}\setminus\operatorname{Spec}(\partial\bar{M},\alpha)$.
Suppose that $P$ has a $K$-action that is compatible with the admissible
structure. Then we define

(43) $\displaystyle HF^{*}_{K}(P,a):=\varprojlim_{n}HF^{*}(P_{borel,n},a)$

where the inverse limit is defined with respect to pull-back induced by the
inclusion $P_{borel,n}\to P_{borel,n+1}$ for all $n$.

This definition is well-defined thanks to Corollary 2.17 and 2.27.

######  Lemma 3.3.

The collection
$\\{HF^{*}_{K}(P,a):a\in\mathbb{R}_{>0}\setminus\operatorname{Spec}(\partial\bar{M},\alpha)\\}$
forms a direct system.

###### Proof.

It is a consequence of Corollary 2.17 and 2.27. ∎

######  Definition 3.4.

The $K$-equivariant symplectic cohomology of $P$ is defined to be

$SH^{*}_{K}(P)=\varinjlim_{a}HF^{*}_{K}(P,a)$

Continuing with Example 3.1, if $P=M$, then $SH^{*}_{K}(M)$ is the
$K$-equivariant symplectic cohomology of $M$. It is the main object of
interest in Section 4.

For our later applications, we mainly consider the case $P=B\times M$ and $K$
is the diagonal action. However, some results we need use $P$ that are not of
product type (e.g. Proposition 3.5 and Lemma 3.7).

###  3.1. Useful properties

####  3.1.1. Free actions

######  Proposition 3.5.

Suppose that $K$ acts on $P$ in a way compatible with the admissible structure
such that its action on $B$ is free. Then for any
$a\in\mathbb{R}_{>0}\setminus\operatorname{Spec}(\partial\bar{M},\alpha)$,
there is an isomorphism

(44) $\displaystyle HF^{*}(P^{\prime},a)\simeq HF^{*}_{K}(P,a)$

Proposition 3.5 is a consequence of the following:

######  Proposition 3.6.

Let $P^{\prime}\to B^{\prime}$ be an admissible bundle. Let $B_{1}\subset
B_{2}\subset\dots$ be a sequence of closed smooth manifolds such that there is
a sequence $h_{n}:B_{n}\to B^{\prime}$ making $B_{n}$ a $EK_{n}$-bundle over
$B^{\prime}$ and $h_{n+1}|_{B_{n}}=h_{n}$. Let $P_{n}\to B_{n}$ be the
admissible bundle obtained by pulling back along $h_{n}$. Then for any
$a\in\mathbb{R}_{>0}\setminus\operatorname{Spec}(\partial\bar{M},\alpha)$, the
pull-back maps $HF^{*}(P^{\prime},a)\to HF^{*}(P_{n},a)$ induces an
isomorphism

(45) $\displaystyle HF^{*}(P^{\prime},a)\simeq\varprojlim_{n}HF^{*}(P_{n},a).$

###### Proof of Proposition 3.5 assuming Proposition 3.6.

For each $n$, $P_{borel,n}$ is the pull-back of $P^{\prime}\to B^{\prime}$
along the $EK_{n}$-bundle map $h_{n}:B_{borel,n}\to B^{\prime}$. The result
directly follows from Proposition 3.6. ∎

###### Proof of Proposition 3.6.

The strategy of proof is similar to Lemma 2.31. We can set up the admissible
base triples $(\eta_{n},g_{n},\nabla_{n})$ and
$(\eta^{\prime},g^{\prime},\nabla^{\prime})$ for $P_{n}\to B_{n}$ and
$P^{\prime}\to B^{\prime}$ as in the proof of Lemma 2.31 so we get a
homomorphism of the respective $E_{1}$-page

(46)
$\displaystyle\oplus_{c^{\prime}\in\operatorname{critp}(\eta^{\prime})}HF^{*-|c^{\prime}|}(M,a)\to\oplus_{c^{\prime}\in\operatorname{critp}(\eta^{\prime})}H^{*}(EK_{n})\otimes
HF^{*-|c^{\prime}|}(M,a)$

Recall from the paragraph after Defintion 2.9 that we have a
$\mathbb{Z}$-grading on the Floer complexes. With $a$ being fixed, there is
$N>0$ such that $HF^{k}(M,a)=0$ for all $|k|>N$ and all
$c^{\prime}\in\operatorname{critp}(\eta^{\prime})$. We can take $n$ large
enough such that $H^{k}(EK_{n})=0$ for $1\leq k\leq 2N+\dim(B^{\prime})$. It
ensures that as a spectral sequence, the RHS of (46) has a direct summand
given by $H^{0}(EK_{n})\otimes HF^{*-|c^{\prime}|}(M,a)$. The map (46) gives
an isomorphism between the LHS and this direct summand. When we take inverse
limit over $n$ (cf. Lemma 2.29), only this summand survives so it gives the
desired isomorphism (45).

∎

####  3.1.2. Classifying models

######  Lemma 3.7.

The definition $HF^{*}_{K}(P,a)$ is independent of the choice of the
classifying space $EK$ and the finite smooth approximation $\\{EK_{n}\\}_{n}$.

###### Proof.

Let $E^{\prime}K$ be another classifying space with finite smooth
approximation $\\{E^{\prime}K_{n^{\prime}}\\}_{n^{\prime}}$. For a
$K$-manifold $M$, we denote $(M\times EK_{n}\times
E^{\prime}K_{n^{\prime}})/K$ by $M_{borel,n,n^{\prime}}$. We use the
convention that $M_{borel,0,n^{\prime}}=(M\times E^{\prime}K_{n^{\prime}})/K$
and $M_{borel,n,0}=(M\times EK_{n})/K$.

In particular, we have the projection map $h_{n}:B_{borel,n,n^{\prime}}\to
B_{borel,0,n^{\prime}}$. The admissible bundle $P_{borel,n,n^{\prime}}\to
B_{borel,n,n^{\prime}}$ is precisely the pull-back of the admissible bundle
$P_{borel,0,n^{\prime}}\to B_{borel,0,n^{\prime}}$ along $h_{n}$. Therefore,
by Proposition 3.6, we have

$HF^{*}(P_{borel,0,n^{\prime}},a)\simeq\varprojlim_{n}HF^{*}(P_{borel,n,n^{\prime}},a)$

for any $n^{\prime}$. As a result, we have

$\varprojlim_{n^{\prime}}HF^{*}(P_{borel,0,n^{\prime}},a)=\varprojlim_{n^{\prime}}\varprojlim_{n}HF^{*}(P_{borel,n,n^{\prime}},a)=\varprojlim_{n}HF^{*}(P_{borel,n,0},a)$

showing the independence of the choice of the classifying space and finite
smooth approximation. ∎

####  3.1.3. Product structure

Another important property of $SH^{*}_{K}(P)$ is that it admits a product
structure. To see this, note that we have the following commutative diagram

(47) ${HF(P_{n+1},a_{1})\times
HF(P_{n+1},a_{2})}$${HF(P_{n+1};a_{0})}$${HF(P_{n},a_{1})\times
HF(P_{n},a_{2})}$${HF(P_{n};a_{0})}$

when $a_{0}>a_{1}+a_{2}$, where the rows are the product operations from
Proposition 2.21. The commutativity of (47) comes from the compatibility
between pull-back maps and the product structure, which can be proved in the
same way as Lemma 2.30 and we leave the details to readers. Therefore, by
taking the inverse limit in $n$, we get a product operation

$HF_{K}(P,a_{1})\times HF_{K}(P,a_{2})\to HF_{K}(P;a_{0})$

The compatibility with continuation maps (see Proposition 2.21) imply that we
can take direct limit and get a product structure on $SH^{*}_{K}(P)$. The
product structure on $SH^{*}_{K}(P)$ is graded-commutative and associative.

####  3.1.4. Weyl group action

Recall that $G$ is a connected compact Lie group which acts on $M$ (see
Section 2). Let $T\subset G$ be a maximal torus and $N(T)$ be the normalizer
of $T$. Let $W=N(T)/T$ be the Weyl group. Let $P\to B$ be an admissible bundle
with a compatible $G$ action.

######  Lemma 3.8.

For any constant
$a\in\mathbb{R}_{>0}\setminus\operatorname{Spec}(\partial\bar{M},\alpha)$, we
have

(48) $\displaystyle HF^{*}_{G}(P,a)\simeq HF^{*}_{N(T)}(P,a)$

###### Proof.

For each $n$, the map $(P\times EG_{n})/N(T)\to(P\times EG_{n})/G$ is a fibre
bunle with fibre $G/N(T)$. Since
$H^{*}(G/N(T);\mathbb{C})=H^{*}(\text{point};\mathbb{C})$, we can apply Lemma
2.31 to get the isomorphism $HF^{*}((P\times EG_{n})/G,a)\simeq
HF^{*}((P\times EG_{n})/N(T),a)$. The result follows from passing to inverse
limit. ∎

######  Lemma 3.9.

For any constant
$a\in\mathbb{R}_{>0}\setminus\operatorname{Spec}(\partial\bar{M},\alpha)$, we
have

(49) $\displaystyle HF^{*}_{T}(P,a)^{W}\simeq HF^{*}_{N(T)}(P,a)$

###### Proof.

The map

$(P\times EG_{n})/T\to(P\times EG_{n})/N(T)$

is a covering map which covers the corresponding covering map of their bases
$(B\times EG_{n})/T\to(B\times EG_{n})/N(T)$. We can use an admissible base
triple and Floer data on $(P\times EG_{n})/T\to(B\times EG_{n})/T$ that come
from pulling back an admissible base triple and a compatible Floer data on
$(P\times EG_{n})/N(T)\to(B\times EG_{n})/N(T)$. The compatibility of Floer
data on $(P\times EG_{n})/N(T)\to(B\times EG_{n})/N(T)$ implies that the pull-
back data on $(P\times EG_{n})/T\to(B\times EG_{n})/T$ is also compatible.
Moreover, there is a isomorphism of

(50) $\displaystyle CF^{*}((P\times EG_{n})/T,a)^{W}\simeq CF^{*}((P\times
EG_{n})/N(T),a)$

coming from sending $(c,x)\in CF^{*}((P\times EG_{n})/N(T),a)$ to the sum over
its lifts devided by $|W|$. It induces an isomorphism on the cohomology
$H(CF^{*}((P\times EG_{n})/T,a)^{W})\simeq CF^{*}((P\times EG_{n})/N(T),a)$.
Moreover, since $\Lambda$ has characteristic $0$, we have

$H(CF^{*}((P\times EG_{n})/T,a)^{W})=HF^{*}((P\times EG_{n})/T,a)^{W}$

By passing to the inverse limit, we get the result.

∎

##  4\. Equivariant Seidel morphism

Let $T\subset G$ be a maximal torus. In this section, we are going to
construct an equivariant Seidel map:

(51) $\displaystyle\mathcal{S}:\hat{H}_{*}^{T}(\Omega G)\otimes
SH^{*}_{T}(M)\to SH^{*}_{T}(M).$

The papers [Sav08], [ChiHong3], [GMP22] consider similar constructions in the
context of quantum cohomology for closed monotone, symplectic manifolds, and
[CL22] uses an algebraic approach to give a similar construction for quantum
$K$-theory of $G/P$. As explained in the introduction, a crucial technical
difference for $SH^{*}_{T}(M)$ is to achieve the maximum principle for the
cylindrical Hamiltonians obtained by pulling back along a family of loops (cf.
the discussion after (54) below).

###  4.1. Overview

As the construction of (51) involves a number of steps, we devote this
subsection to providing an overview of our work. The actual construction is
obtained by a smooth finite dimensional approximation of the overview here.

We work with an alternative model for $\hat{H}_{*}^{T}(\Omega G)$,
$H_{*}^{geo,T}(\Omega G)$, which represents cycles as tuples $(B,\alpha,f)$
such that $B$ is a smooth oriented, closed manifold with $T$-action,
$\alpha\in H_{T}^{*}(B,\mathbb{Z}),$ and $f:B\to\Omega G$ is a smooth,
$T$-equivariant map (see §4.4). These cycles are considered up to an
appropriate notion of equivalence. To construct (51), it suffices to
construct, for any $T$-equivariant smooth map $f:B\to\Omega G$, a map:

(52) $\displaystyle\mathcal{S}_{f}:H_{T}^{*}(B)\times SH_{T}^{*}(M)\to
SH_{T}^{*}(M)$

which respects the various equivalence relations imposed in geometric
homology. Let

$P_{B}:=(B\times M)_{borel}=(B\times ET\times M)/T.$

For a cylindrical Hamiltonian $H\in C^{\infty}(S^{1}\times P_{B})$, let
$\tilde{H}:S^{1}\times B\times ET\times M\to\mathbb{R}$ be its lift. We define

(53) $\displaystyle\begin{split}&f^{*}\tilde{H}:S^{1}\times B\times ET\times
M\to\mathbb{R}\\\ &f^{*}\tilde{H}_{t}(b,y,m)=\tilde{H}_{t}(y,(f(b)(t))\cdot
m)-K_{f(b),t}((f(b)(t))\cdot m)\end{split}$

where $K_{f(b)}:S^{1}\times M\to\mathbb{R}$ is the mean-normalized Hamiltonian
(see the definition after (6)) generating the based loop $f(b)\in\Omega
G\subset\Omega\operatorname{Ham}(M)$. The Hamiltonian $f^{*}\tilde{H}$ is
$T$-invariant and it descends to a cylindrical Hamiltonian, denoted by
$f^{*}H$, on $P_{B}$.

The key ingredient (beyond those already introduced) needed for (52) is the
tautological isomorphism

(54) $\displaystyle\begin{split}\mathcal{C}:&HF^{*}(P_{B};f^{*}H)\simeq
HF^{*}(P_{B};H).\end{split}$

A priori, even if $H$ is compatible with $\eta:B_{borel}\to\mathbb{R}$, this
doesn’t imply that $f^{*}H$ is compatible with $\eta$. This is because even if
$H$ has a constant slope $\mathbf{s}_{H_{b}}\equiv a\in\mathbb{R}_{\geq
0}\setminus\operatorname{Spec}(\partial\bar{M},\alpha)$ for all $b\in
B_{borel}$, $f^{*}H$ does not have to have constant slope due to the second
term in (53). However, we will show in §4.2.2 and §4.2.3 that there are
sufficiently many $H$ such that both $H$ and $f^{*}H$ are compatible with some
$\eta$, and both $\mathbf{s}_{H}$ and $\mathbf{s}_{f^{*}H}$ are bounded
functions even though $P_{B}$ is infinite dimensional.

Let $H^{\prime}$ be a cylindrical Hamiltonian on $P_{B}$ with constant slope
$a^{\prime}$ such that

$H^{\prime}\leq_{P_{B}}f^{*}H.$

We can define a map

(55) $\displaystyle H_{T}^{*}(B)\times HF_{T}^{*}(M;a^{\prime})\to
HF_{T}^{*}(M;f^{*}H)$

by composing the acceleration map from $H^{\prime}$ to $f^{*}H$ with the Floer
theoretic pull-back

(56) $\displaystyle H_{T}^{*}(B)\times HF_{T}^{*}(M;a^{\prime})\to
HF^{*}(P_{B};H^{\prime})$

More precisely, the map (56) is the composition of the Floer theoretic pull-
back map induced by $(B\times ET\times M\times ET)/T\to B_{borel}\times
M_{borel}$ and the isomorphism induced by the pull-back along the map
$(B\times ET\times M\times ET)/T\to P_{B}$.

On the other hand, let $H^{\prime\prime}$ be a cylindrical equivariant
Hamiltonian on $P_{B}$ with constant slope $a^{\prime\prime}$ such that

$H\leq_{P_{B}}H^{\prime\prime}.$

We can define a map

(57) $\displaystyle HF^{*}(P_{B};H)\to HF_{T}^{*}(M;a^{\prime\prime})$

by composing the acceleration map from $H$ to $H^{\prime\prime}$ and the Floer
theoretic push-forward

(58) $\displaystyle HF^{*}(P_{B};H^{\prime\prime})\to
HF_{T}^{*}(M;a^{\prime\prime})$

The map (58) is induced by pushing forward along the composition map
$P_{B}\to(\Omega G\times M)_{borel}\to M_{borel}$. By composing (55), (54) and
(57), we get

(59) $\displaystyle H_{T}^{*}(B)\times HF_{T}^{*}(M;a^{\prime})\to
HF_{T}^{*}(M;a^{\prime\prime})$

Letting $a^{\prime}$ and $a^{\prime\prime}$ go to infinity, (59) becomes (52).

######  Remark 4.1.

We emphasize that some of the maps above do not preserve gradings. The
necessary grading shifts for these maps are discussed in the subsequent
subsections.

###  4.2. Tautological isomorphism

In this subsection, we are going to prove the isomorphism (54).

####  4.2.1. Identifying Hamiltonian loops

We start with some basic properties. The following lemma is standard. A proof
can be found in e.g. [OhBook1, Section 2.3].

######  Lemma 4.2.

Let $H,H^{\prime}:S^{1}\times M\to\mathbb{R}$ be Hamiltonian functions. Denote
their time $t$ Hamiltonian flow by $\phi^{t}_{H}$ and $\phi^{t}_{H^{\prime}}$
respectively. Then

  1. (1)

$(\phi^{t}_{H}\circ\phi_{H^{\prime}}^{t})_{t\in S^{1}}$ is generated by
$(H\\#H^{\prime})_{t}(x):=H_{t}(x)+H^{\prime}_{t}((\phi_{H}^{t})^{-1}(x))$

  2. (2)

$(\phi^{-t}_{H})_{t\in S^{1}}$ is generated by
$\overline{H}_{t}(x):=-H_{t}(\phi_{H}^{t}(x))$

  3. (3)

for any $\psi\in\operatorname{Symp}(M)$, $\psi^{-1}\circ\phi^{t}_{H}\circ\psi$
is generated by $H_{t}(\psi(x))$

Let $\tilde{H}:S^{1}\times B\times ET\times M\to\mathbb{R}$ be a Hamiltonian
function. We define $f^{*}\tilde{H}:S^{1}\times B\times ET\times
M\to\mathbb{R}$ by (53). Notice that if $\tilde{H}$ is an $T$-equivariant
Hamiltonian function (i.e. $\tilde{H}_{t}(gb,gy,gm)=\tilde{H}_{t}(b,y,m)$ for
all $g\in T$), then so does $f^{*}\tilde{H}$ because

$\displaystyle(f^{*}\tilde{H})_{t}(gb,gy,gm)$
$\displaystyle=\tilde{H}_{t}(gb,gy,(f(gb)(t))m)-K_{f(gb),t}((f(gb)(t))gm)$
$\displaystyle=\tilde{H}_{t}(gb,gy,(gf(b)(t)g^{-1})gm)-K_{gf(b)g^{-1},t}((gf(b)(t)g^{-1})gm)$
$\displaystyle=\tilde{H}_{t}(gb,gy,g(f(b)(t))m)-K_{f(b),t}(g^{-1}(gf(b)(t)g^{-1})gm)$
$\displaystyle=(f^{*}\tilde{H})_{t}(b,y,m)$

######  Lemma 4.3 (cf. [SeidelRep], Lemma 2.3).

Let $\tilde{H}:S^{1}\times B\times ET\times M\to\mathbb{R}$ be a Hamiltonian
function. For any $(b,y)\in B\times ET$, if $(x(t))_{t\in S^{1}}$ is a
Hamiltonian orbit of $\tilde{H}(b,y,\cdot)$, then the loop
$(t\mapsto(f(b)(t))^{-1}x(t))_{t\in S^{1}}$ is a Hamiltonian orbit of
$(f^{*}\tilde{H})(b,y,\cdot)=\tilde{H}(b,y,f(b)(t)\cdot)-K_{f(b),t}(f(b)(t)\cdot)$.
Moreover, it defines a bijective correspondence between the Hamiltonian
orbits.

###### Proof.

Let $\gamma(t)=f(b)(t)$. By Lemma 4.2(2), $\gamma(t)^{-1}$ is generated by

(60) $\displaystyle(t,m)\mapsto-K_{\gamma,t}(\gamma(t)\cdot m)$

By applying Lemma 4.2(1) to the composition of $\gamma(t)^{-1}$ and
$\phi_{\tilde{H}}^{t}$, we know that $\gamma(t)^{-1}\phi_{\tilde{H}}^{t}$ is
generated by $\tilde{H}(b,t,f(b)(t)\cdot)-K_{f(b),t}(f(b)(t)\cdot)$. It
follows that

(61)
$\displaystyle\phi_{\tilde{H}(b,t,f(b)(t)\cdot)-K_{f(b),t}(f(b)(t)\cdot)}^{t}x(0)=\gamma(t)^{-1}\phi_{\tilde{H}}^{t}(x(0))=\gamma(t)^{-1}x(t)$

and hence the loop $\gamma(t)^{-1}x(t)$ is a Hamiltonian orbit of
$\tilde{H}(b,t,f(b)(t)\cdot)-K_{f(b),t}(f(b)(t)\cdot)$.

The other direction of the bijective correspondence can be proved analogously.
∎

Let $m\in M$, $b\in B$ and consider the loop
$c=(t\mapsto(f(b)(t))^{-1}m)_{t\in S^{1}}$. As discussed in the paragraph
after Definition 2.9, we have chosen a homotopy class of trivialization of the
canonical bundle of $M$. Let $\iota(f)$ be the Conley-Zehnder index of the
linearization of $(f(b)(t))^{-1}$ along the loop $c$ with respect to the
homotopy class of trivialization (more precisely, since $D(f(b)(1))^{-1}$ is
the identity, $\iota(f)$ is defined as $2$ times the Maslov index, see
[SalamonLecture, Section 2.4]). By continuity, $\iota(f)$ is independent of
$b\in B$ and the point $m\in M$.

######  Lemma 4.4 (cf. [SeidelRep], Lemma 2.6).

Let $(x(t))_{t\in S^{1}}$ be a Hamiltonian orbit of $\tilde{H}(b,y,\cdot)$.
The grading of $(t\mapsto(f(b)(t))^{-1}x(t))_{t\in S^{1}}$ is the sum of the
grading of $(x(t))_{t\in S^{1}}$ and $\iota(f)$.

###### Proof.

This follows from the loop property of the Conley-Zehnder index (see
[SalamonLecture, Section 2.4]). It can be proved by noticing that
$(t\mapsto(f(b)(t))^{-1}x(t))_{t\in S^{1}}$ is homotopic to the concatenation
of $(t\mapsto x(t))_{t\in S^{1}}$ and $(t\mapsto(f(b)(t))^{-1}x(0))$, so the
result follows from the additivity of the Conley-Zehnder index for a path of
symplectic matrices and a loop of symplectic matrices.

∎

####  4.2.2. A good class of admissible base triples

Let $B$ be a finite dimensional closed smooth $T$-manifold. A continuous map
$f:B\to\Omega G$ is called smooth if the map $B\times S^{1}\to G$ given by
$(b,t)\mapsto(f(b))(t)$ is smooth. Let $f:B\to\Omega G$ be a $T$-equivariant
smooth map. Let $P_{B,n}:=(B\times M)_{borel,n}=(B\times ET_{n}\times M)/T$
and $B_{n}:=B_{borel,n}$. We want to consider a good class of admissible base
triple $(\eta_{n},g_{n},\nabla_{n})$ of $P_{B,n}\to B_{n}$ as follows.

Let $\eta_{BT}:BT\to\mathbb{R}$ be a Morse function (i.e a sequence of Morse
functions $\eta_{BT,n}:BT_{n}\to\mathbb{R}$ such that
$\eta_{BT,n+1}|_{BT_{n}}=\eta_{BT,n}$ and
$\operatorname{critp}(\eta_{BT,n+1})\cap
BT_{n}=\operatorname{critp}(\eta_{BT,n})$ for all $n$). Let $g_{BT}$ be a
Riemannian metric on $BT$ (i.e. a sequence of Riemannian metrics $g_{BT,n}$ of
$BT_{n}$ that is compatible under embeddings $BT_{n}\to BT_{n+1}$). We can
choose $\eta_{BT}$ and $g_{BT}$ such that any gradient trajectory of
$\eta_{BT,n}$ which starts in $BT_{k}\subset BT_{n}$ and tangent to $BT_{k}$,
for some $k<n$, coincides with the gradient trajectory of $\eta_{BT,k}$ in
$BT_{k}$ for all time. Moreover, there is no gradient trajectory which goes
from a critical point of $\eta_{BT,n}$ in $BT_{k}$ to a critical point of
$\eta_{BT,n}$ in $BT_{n}\setminus BT_{k}$ for all $n>k$ (cf. Example 2.28).

Let $\nabla_{T}$ be a connection of the principal $T$-bundle $ET\to BT$ (i.e.
a sequence of connections $\nabla_{T,n}$ of $ET_{n}\to BT_{n}$ that is
compatible under embeddings $ET_{n}\to ET_{n+1}$) such that it is flat near
critical points of $\eta_{BT}$. It defines a decomposition of the tangent
space $T(ET_{n})=T^{vert}ET_{n}\oplus T^{hor}ET_{n}$, where $T^{vert}ET_{n}$
is the kernel of $T(ET_{n})\to T(BT_{n})$ and $T^{hor}ET_{n}$ is the
horizontal subbundle determined by $\nabla_{T,n}$. The connection
$\nabla_{T,n}$ induces a connection for the associated bundle $B_{n}\to
BT_{n}$ with the holonomy group in $T$, and the horizontal subbundle
$T^{hor}B_{n}$ is given by the projection of $T^{hor}ET_{n}$ to $TB_{n}$ under
$T(B\times ET_{n})\to TB_{n}$ (it is well-defined because $g\cdot
T^{hor}_{x}ET_{n}=T^{hor}_{gx}ET_{n}$ for all $g\in T$). We choose a
Riemannian metric $g_{n}$ on $B_{n}$ such that $T^{hor}B_{n}$ is orthogonal to
the vertical subbundle, $(B_{n},g_{n})\to(BT_{n},g_{BT})$ is a Riemannian
submersion, and over the region $U\subset BT$ where the connection is flat,
$g_{n}$ is the product of a $T$-invariant metric $g_{B}$ on $B$ and the metric
$g_{BT}|_{U}$ on $U$.

We use $\nabla_{T}$ again to induce a connection $\nabla_{n}$ of $P_{B,n}\to
B_{n}$ by requiring that the horizontal subbundle is the projection of
$TB\oplus T^{hor}ET_{n}$ to $TP_{B,n}$ under

$T(B\times ET_{n}\times M)\to TP_{B,n}$

######  Lemma 4.5.

Let $c:\mathbb{R}/\mathbb{Z}\to BT_{n}$ be a loop, $c^{\prime}:[0,1]\to B_{n}$
be a horizontal lift of $c$ and $c^{\prime\prime}:[0,1]\to P_{B,n}$ be a
horizontal lift of $c^{\prime}$ with respect to $\nabla_{n}$. Let $(B\times
M)_{c(0)}$ be the fibre of the natural map $P_{B,n}\to BT_{n}$ over
$c(0)=c(1)$. Then $c^{\prime\prime}(0),c^{\prime\prime}(1)\in(B\times
M)_{c(0)}$ lie in the same $T$-orbit, where $T$ acts diagonally.

###### Proof.

Consider the connection $\nabla$ of $P_{B,n}\to BT_{n}$ whose horizontal
subbundle is given by the projection of $T^{hor}ET_{n}$ to $TP_{B_{n}}$ under
$T(B\times ET_{n}\times M)\to TP_{B,n}$. The holonomy group for this
connection is $T$. It is easy to check that $c^{\prime\prime}$ is a horizontal
lift of $c$ with respect to $\nabla$ because both the connections of $B_{n}\to
BT_{n}$ and $P_{B,n}\to B_{n}$ are induced by $\nabla_{T}$. As a result,
$c^{\prime\prime}(0)$ and $c^{\prime\prime}(1)$ lie in the same $T$-orbit.

∎

Let $\eta_{n}^{\prime}$ be the pull-back of $\eta_{BT,n}$ under the Riemannian
submersion $B_{n}\to BT_{n}$. Gradient trajectories of $\eta_{n}^{\prime}$ are
horizontal lifts of gradient trajectories of $\eta_{BT,n}$.

######  Lemma 4.6.

Let $f:B\to\Omega G$ be a $T$-equivariant map and $K_{f,n}:S^{1}\times
P_{B,n}\to\mathbb{R}$ be the generating Hamilotnian function. Let
$\tau^{\prime}:\mathbb{R}\to B_{n}$ be a gradient trajectory of
$\eta_{n}^{\prime}$. Then $K_{f,n}$ is covariantly constant along
$\tau^{\prime}$ with respect to the connection $\nabla_{n}$

###### Proof.

Let $\tau^{\prime\prime}:\mathbb{R}\to P_{B,n}$ be a horizontal lift of
$\tau^{\prime}$ with respect to $\nabla_{n}$. We need to show that
$K_{f,n}(t,\tau^{\prime\prime}(s))$ is independent of $s$.

Recall that the $T$-invariant lift $K_{F,n}:S^{1}\times B\times ET_{n}\times
M$ of $K_{f,n}$ is defined by

$K_{F,n}(t,b,x,m):=K_{f(b),t}((f(b)(t))m)$

and the $T$-invariance means

$\displaystyle K_{F,n}(t,gb,gx,gm)$
$\displaystyle=K_{f(gb),t}((f(gb)(t))gm)=K_{f(gb),t}((f(gb)(t))gm)$
$\displaystyle=K_{gf(b)g^{-1},t}(g(f(b)(t))g^{-1}gm)=K_{f(b),t}(g^{-1}g(f(b)(t))g^{-1}gm)$
$\displaystyle=K_{F,n}(t,b,x,m)$

Note that $K_{F,n}$ also $T$-invariant along the $ET_{n}$ direction (i.e.
$K_{F,n}(t,b,x,m)=K_{F,n}(t,b,gx,m)$), which is an additional feature that is
not due to coming from lifting from $P_{B,n}$. Combining both, we have
$K_{F,n}(t,b,x,m)=K_{F,n}(t,gb,x,gm)$ for all $g\in T$.

By Lemma 4.5, $\tau^{\prime\prime}$ is actually a horizontal lift of a
gradient trajectory $\tau:\mathbb{R}\to BT_{n}$ of $\eta_{BT,n}$. Since
$K_{F,n}$ is independent of $ET_{n}$ and invariant under the diagonal $T$
action on $B\times M$, together with the fact that the diagonal $T$ action is
precisely the holonomy group of $P_{B,n}\to BT_{n}$, we conclude that
$K_{f,n}(t,\tau^{\prime\prime}(s))$ is independent of $s$.

∎

Note that $\eta_{n}^{\prime}$ is only a Morse-Bott function so
$(\eta_{n}^{\prime},g_{n},\nabla_{n})$ is not an admissible base triple. We
want to Morsify $\eta_{n}^{\prime}$ to $\eta_{n}$ so that we get an admissible
base triple, and at the same time still have some control on the derivative of
$K_{f,n}$ along gradient trajectory of $\eta_{n}$ with respect to the
connection $\nabla_{n}$.

Before we explain this, we need to introduce the notion of a good pair.

######  Definition 4.7.

Let $f:B\to\Omega G$ be a $T$-equivariant map and
$\eta_{B}^{\prime}:B\to\mathbb{R}$ be a $T$-invariant Morse-Bott function. We
call $(f,\eta_{B}^{\prime})$ a good pair if $f$ is a constant function near
each connected critical submanifold of $\eta_{B}^{\prime}$.

######  Lemma 4.8.

Given a $T$-equivariant smooth map $f:B\to\Omega G$, we can homotope $f$ to
another $T$-equivariant smooth map $f^{\prime}:B\to\Omega G$ such that there
is a $T$-invariant Morse-Bott function $\eta_{B}^{\prime}$ making
$(f^{\prime},\eta_{B}^{\prime})$ a good pair.

###### Proof.

Let $\eta_{B}^{\prime}:B\to\mathbb{R}$ be a $T$-invariant Morse-Bott function
such that every connected component of its critical Morse-Bott submanifolds is
isomorphic to $T/H$ for some closed subgroup $H$ of $T$ (see [Wasserman, Lemma
4.8] for its existence and genericity).

Let $C\simeq T/H$ be one of the connected components. Note that $f|_{T/H}$
factors through $\Omega C_{G}^{0}(H)\subset\Omega G$, where $C_{G}^{0}(H)$ is
the identity component of the centralizer $C_{G}(H)$ of $H$ in $G$. Note also
that $C_{G}(T)=T\subset C_{G}^{0}(H)$ is a maximal torus in $C_{G}^{0}(H)$.
Therefore, $\pi_{1}(T)$ surjects onto $\pi_{1}(C_{G}^{0}(H))$. Let $p\in T/H$
so $f(p)\in\Omega C_{G}^{0}(H)$. We can find a smooth map
$F_{p}:[0,1]\to\Omega C_{G}^{0}(H)$ such that $F_{p}(0)=f(p)$ and
$F_{p}(1)\in\Omega C_{G}(T)=\Omega T$ because of the surjectivity of
$\pi_{1}(T)\to\pi_{1}(C_{G}^{0}(H))$. Let $F:[0,1]\times T/H\to\Omega
C_{G}^{0}(H)$ be the $T$-equivariant map given by $F(s,g\cdot
p):=gF_{p}(s)g^{-1}\in\Omega C_{G}^{0}(H)$ for all $g\in T$ and $s\in[0,1]$.
In other words, $F$ is a $T$-equivariant homotopy from $F(0,\cdot)=f$ to the
map $F(1,\cdot)$ which lands in $\Omega T$. In particular, $F(1,g\cdot
p)=F(1,p)$ for all $g\in T$. We want to use this homotopy to homotope $f$ to
another $T$-equivariant map such that it is constantly equal to $F(1,p)$ in a
neighborhood of $C$.

To do that, let $N$ be a $T$-invariant neighborhood of $C$. By choosing a
$T$-invariant metric on $B$ and using the exponential map, we can identify $N$
as the total space of a $T$-equivariant normal bundle over $C$. Let
$\pi_{N}:N\to C$ be the projection map. Let $r:N\to\mathbb{R}_{\geq 0}$ be the
distance function from $C$. Let $\rho_{\epsilon}:\mathbb{R}_{\geq
0}\to\mathbb{R}_{\geq 0}$ be a smooth function such that
$\rho_{\epsilon}(s)=0$ near $s=0$ and $\rho_{\epsilon}(s)=s$ when
$s>\epsilon>0$. For $\epsilon>0$ being sufficiently small, we define a
$T$-equivariant smooth map $c_{\epsilon}:N\to N$ by
$b\mapsto\rho_{\epsilon}(r(b))b$. This map collapses a small neighborhood of
$C$ to $C$. We define $f_{c}:=f\circ c$, which is a $T$-equivariant smooth map
that is $T$-equivariant homotopic to $f$. Moreover, by definition, we know
that $f_{c}$ factors through $\pi_{N}$ in a small neighborhood $N_{C}$ of $C$.
Let $\rho^{C}_{\delta}:[0,\delta]\to[0,1]$ be a smooth function such that
$\rho^{C}_{\delta}(s)=1$ near $s=0$ and $\rho^{C}_{\delta}(s)=0$ near
$s=\delta$. For $\delta$ being sufficiently small, we define
$f^{\prime}:B\to\Omega G$ by $f^{\prime}(b):=f_{c}(b)$ if $r(b)\geq\delta$,
and $f^{\prime}(b):=F(\rho^{C}_{\delta}(r(b)),\pi_{N}(b))$ if
$r(b)\leq\delta$. Clearly, $f^{\prime}$ is $T$-equivariant homotopic to $f$,
and $f^{\prime}$ is a constant near $C$.

By applying this procedure to every connected component of the critical
submanifolds of $\eta_{B}^{\prime}$, we get a good pair
$(f^{\prime},\eta_{B}^{\prime})$ as desired.

∎

Let $(g_{n},\nabla_{n})$ be as above. We are now ready to introduce
$\eta_{n}$, which is a special form of Morsification of $\eta_{n}^{\prime}$
that makes use of $(f,\eta_{B}^{\prime})$.

Since $(f,\eta_{B}^{\prime})$ is a good pair, we can find a $T$-invariant
neighborhood $N_{\eta}$ of $\operatorname{critp}(\eta_{B}^{\prime})$ such that
$f|_{N_{\eta}}$ is locally constant. Let $\eta_{B}:B\to\mathbb{R}$ be a
Morsification of $\eta_{B}^{\prime}$ such that
$\eta_{B}(b)=\eta_{B}^{\prime}(b)$ if $b\notin Int(N_{\eta})$.

Let $U_{BT,n}$ be a small neighborhood of the critical points of $\eta_{BT,n}$
where $\nabla_{T}$ is flat. Let $U_{n}\subset B_{n}$ be the preimage of
$U_{BT,n}$ under $B_{n}\to BT_{n}$. Since $\nabla_{n}$ and the connection of
$B_{n}\to BT_{n}$ are both induced by $\nabla_{T}$, we can identify $U_{n}$ as
$B\times U_{BT,n}$ and trivialize the bundle $P_{B,n}$ over $U_{n}$ as
$B\times M\times U_{BT,n}$.

Let $\eta_{n}^{\prime}$ be the Morse-Bott function on $B_{n}$ above. The
critical submanifolds of $\eta_{n}^{\prime}$ are contained in $U_{n}=B\times
U_{BT,n}$, and are of the form $B\times\operatorname{critp}(\eta_{BT,n})$.
Note that, there is a choice in the identification of $B$ coming from the
choice of trivialization, and it is canonical only up to an element in $T$.
Our argument below works for any such choice. We Morsify $\eta_{n}^{\prime}$
inside $U_{n}$ by adding a function of the form $\epsilon\chi(u)\eta_{B}(b)$
for $(b,u)\in B\times U_{BT,n}$, where $0<\epsilon\ll 1$, $\chi$ is a bump
function and $\eta_{B}$ is the Morse function constructed above. We can choose
$\chi:U_{BT,n}\to[0,1]$ such that over each connected component of $U_{BT,n}$,
it ony depends on the distance from the corresponding critical point of
$\eta_{BT,n}$, the only critical values are $\\{0,1\\}$ and
$\chi^{-1}(1)=\operatorname{critp}(\eta_{BT,n})$. Denote the Morse funciton
$\eta_{n}^{\prime}+\epsilon\chi(u)\eta_{B}(b)$ by $\eta_{n}$. In the next
subsection, we will use the admissible basse triple
$(\eta_{n},g_{n},\nabla_{n})$.

We end this subsection with the following observation.

######  Lemma 4.9.

Let $(f,\eta_{B}^{\prime})$ be a good pair and $\eta_{B},g_{B}$ be defined as
above. For any constant $c\in\mathbb{R}$, there is a $T$-invariant function
$\mathbf{s}\in C^{\infty}(S^{1}\times B)$ such that $\mathbf{s}>c$,
$\mathbf{s}(b)\notin\operatorname{Spec}(\partial\bar{M},\alpha)$ for any
$b\in\operatorname{critp}(\eta_{B})$, and for any gradient trajectory
$\tau_{B}$ of $\eta_{B}$ with respect to $g_{B}$, we have

(62)
$\displaystyle\frac{d}{ds}\mathbf{s}(t,\tau_{B}(s))\leq-\max_{m\in\partial\bar{M}}\left|\frac{d}{ds}\mathbf{s}_{K_{f(\tau_{B}(s))},t}(m)\right|$

for all $t\in S^{1}$.

###### Proof.

To see that $\mathbf{s}$ exists even though $\eta_{B}$ is NOT $T$-invariant,
we argue as follows. Recall that $(f,\eta_{B}^{\prime})$ is a good pair and
$f|_{N_{\eta}}$ is locally constant. It implies that $K_{f(b),t}(m)$ is
locally independent of $b\in N_{\eta}$. Therefore, the RHS of (65) is $0$ when
$\tau_{B}(s)\in N_{\eta}$. When $\tau_{B}(s)\notin N_{\eta}$, $\tau_{B}(s)$ is
also a gradient trajectory of $\eta_{B}^{\prime}$, and $\eta_{B}^{\prime}$ is
$T$-invariant. Therefore, it suffices to find a $T$-invariant function
$\mathbf{s}\in C^{\infty}(S^{1}\times B)$ such that it is locally constant in
$N_{\eta}$, $\mathbf{s}>c$,
$\mathbf{s}(b)\notin\operatorname{Spec}(\partial\bar{M},\alpha)$ for any
$b\in\operatorname{critp}(\eta_{B})$, and for any gradient trajectory
$\tau_{B}^{\prime}$ of $\eta_{B}^{\prime}$, we have

(63)
$\displaystyle\frac{d}{ds}\mathbf{s}(t,\tau_{B}^{\prime}(s))\leq-\max_{m\in\partial\bar{M}}\left|\frac{d}{ds}\mathbf{s}_{K_{f(\tau_{B}^{\prime}(s))},t}(m)\right|$

but this is easy because $\eta_{B}^{\prime}$ is $T$-invariant, $N_{\eta}$ is
$T$-invariant and $g_{B}$ is also $T$-invariant. ∎

####  4.2.3. A good class of admissible Hamiltonians

Let $f^{*}0$ be the Hamiltonian function on $P_{B}$ given by (53) with $H=0$.
It is descended from $-K_{F,n}(t,b,y,m)$ and
$K_{F,n}(t,b,y,m):=K_{f(b),t}((f(b)(t))m)$ is independent of $ET_{n}$.
Therefore, for any $b\in B_{borel}$, there is $b^{\prime}\in B$ such that

$\max_{(t,m)\in
S^{1}\times\partial\bar{M}}|\mathbf{s}_{(f^{*}0)_{b}}(m)|=\max_{(t,m)\in
S^{1}\times\partial\bar{M}}|\mathbf{s}_{K_{f(b^{\prime}),t}}(m)|$

Since $S^{1}\times B\times\partial\bar{M}$ is compact, the value

(64) $\displaystyle c_{f,K}:=\sup_{b\in B_{borel}}\max_{(t,m)\in
S^{1}\times\partial\bar{M}}|\mathbf{s}_{(f^{*}0)_{b}}(m)|=\sup_{b^{\prime}\in
B}\max_{(t,m)\in
S^{1}\times\partial\bar{M}}|\mathbf{s}_{K_{f(b^{\prime}),t}}(m)|$

is finite.

Given a $T$-invariant function $\mathbf{s}\in C^{\infty}(S^{1}\times B)$, we
can pull it back to get a $T$-invariant function on $S^{1}\times B\times ET$.
It descends to a function on $S^{1}\times B_{borel}$ which we denote by
$\mathbf{s}_{borel}$.

######  Proposition 4.10.

Let $(f,\eta_{B}^{\prime})$ be a good pair. Then there is a constant $C_{f}>0$
depending only on $(f,\eta_{B}^{\prime})$ with the following property. There
is an admissible base triple $(\eta,g,\nabla)$ of $P_{B}\to B_{borel}$ (i.e. a
sequence of admissible base triples $(\eta_{n},g_{n},\nabla_{n})$ of
$P_{B,n}\to B_{n}$) such that for any $c\in\mathbb{R}_{>0}$, there is a
cylindrical Hamiltonian $A_{f}\in C^{\infty}(S^{1}\times P_{B})$ (i.e a
sequence of cylindrical Hamiltonian $A_{f,n}$ on $P_{B,n}$ such that
$A_{f,n+1}|_{P_{B,n}}=A_{f,n}$) and a $T$-invariant function $\mathbf{s}$
obtained by Lemma 4.9 such that

  * •

$\mathbf{s}>c_{f,K}+c$, and

  * •

$\mathbf{s}_{(A_{f})_{b}}=\mathbf{s}_{borel}(b)$ for all $b\in B_{borel}$, and

  * •

both $f^{*}A_{f,n}$ and $A_{f,n}$ are compatible with
$(\eta_{n},g_{n},\nabla_{n})$ for all $n$, and

  * •

$c\leq\mathbf{s}_{A_{f}},\mathbf{s}_{f^{*}A_{f}}\leq c+C_{f}$.

###### Proof.

Recall that $g_{B}$ is a $T$-invariant metric on $B$ such that
$g_{n}|_{U_{n}}=g_{B}+g_{BT}|_{U_{BT,n}}$. By Lemma 4.9, we can pick a
$T$-invariant function $\mathbf{s}\in C^{\infty}(S^{1}\times B)$ such that
$\mathbf{s}>c_{f,K}+c$, and for any gradient trajectory $\tau_{B}$ of
$\eta_{B}$ with respect to $g_{B}$, we have

(65)
$\displaystyle\frac{d}{ds}\mathbf{s}(t,\tau_{B}(s))\leq-\max_{m\in\partial\bar{M}}\left|\frac{d}{ds}\mathbf{s}_{K_{f(\tau_{B}(s))},t}(m)\right|$

for all $t\in S^{1}$.

Since $\mathbf{s}$ is $T$-invariant, it induces a function
$\mathbf{s}_{borel}\in C^{\infty}(S^{1}\times B_{borel})$. We claim that

(66)
$\displaystyle\frac{d}{ds}\mathbf{s}_{borel}(t,\tau(s))\leq-\max_{m\in\partial\bar{M}}\left|\frac{d}{ds}\mathbf{s}_{(f^{*}0)_{\tau(s)}}(t,m)\right|$

for all gradient trajectory $\tau:\mathbb{R}\to B_{n}$ of $\eta_{n}$, for all
$t\in S^{1}$, and for all $n$. To see why, note that outside $U_{n}$, $\tau$
is also a gradient trajectory of $\eta_{n}^{\prime}$ because
$\epsilon\chi(u)\eta_{B}(b)$ is supported inside $U_{n}$. Moreover, gradient
trajectories of $\eta_{n}^{\prime}$ are horizontal lifts of gradient
trajectory of $\eta_{BT,n}$. By Lemma 4.6, the RHS is $0$. The LHS is also $0$
because $\mathbf{s}_{borel}$ is also covariantly constant. Inside $U_{n}$, we
use the trivialization $U_{n}=B\times U_{BT,n}$ so for $(b,u)\in B\times
U_{BT,n}$, we have

(67) $\displaystyle\mathbf{s}_{borel}(t,(b,u))=\mathbf{s}(t,b)$

and

(68)
$\displaystyle\mathbf{s}_{(f^{*}0)_{(b,u)}}(t,m)=\mathbf{s}_{K_{f(b),t}}(m)$

Both of them are independent of $u\in U_{BT,n}$. On the other hand, we have

$d\eta_{n}=d\eta_{n}^{\prime}(u)+\epsilon\chi(u)d\eta_{B}(b)+\epsilon
d\chi(u)\eta_{B}(b)$

The gradient $grad(\eta_{n})$ of $\eta_{n}$ is therefore lying inside
$\epsilon\chi(u)grad(\eta_{B}(b))+TU_{BT,n}$. By (65), (67) and (68), we
conclude that (66) is true.

Now, we would like to choose $A_{f}$ such that
$\mathbf{s}_{(A_{f})_{b}}(m)=\mathbf{s}_{borel}(b)$ for all
$m\in\partial\bar{M}$. This choice of $A_{f}$ trivially satisfies the second
bullet of the proposition.

For the last bullet, note that

$c_{f,K}+c\leq\min_{B}\mathbf{s}\leq\mathbf{s}_{A_{f}}\leq\max_{B}\mathbf{s}$

Therefore, we have

$c\leq\mathbf{s}_{f^{*}A_{f}}\leq\max_{B}\mathbf{s}+c_{f,K}$

So the last bullet is satisfied with $C_{f}=\max_{B}\mathbf{s}+c_{f,K}-c$.
Recall that, $c_{f,K}$ is a constant which only depends on
$(f,\eta_{B}^{\prime})$. Therefore, in order to find a $C_{f}$ which depends
only on $(f,\eta_{B}^{\prime})$ we need to give a uniform upper bound for
$\max_{B}\mathbf{s}-c$ that is independent of $c$, and only depends on
$(f,\eta_{B}^{\prime})$. This uniform upper bound exists because, if
$\mathbf{s}>c_{f,K}+c$ is obtained from Lemma 4.9, then for any other
$c^{\prime}>0$, $\mathbf{s}-c+c^{\prime}>c_{f,K}+c^{\prime}$ almost satisfy
all the conditions in Lemma 4.9 except possibly that
$(\mathbf{s}-c+c^{\prime})(b)$ might lie in the spectrum
$\operatorname{Spec}(\partial\bar{M},\alpha)$ for some
$b\in\operatorname{critp}(\eta_{B})$. Therefore, one can $T$-equivariantly
perturb $\mathbf{s}-c+c^{\prime}$ to get a $T$-invariant function
$\mathbf{s}^{\prime}>c_{f,K}+c^{\prime}$ which satisfy all the conditions in
Lemma 4.9. The perturbation can be as small as we want so
$\max_{B}\mathbf{s}-c$ and $\max_{B}\mathbf{s}^{\prime}-c^{\prime}$ can be
made as close to each other as we want. This implies that there is a uniform
upper bound for $\max_{B}\mathbf{s}-c$ that is independent of $c$, which in
turn implies that we can find a $C_{f}$ we want which depends only on
$(f,\eta_{B}^{\prime})$.

It remains to show that it also fulfills the third bullet. Recall the
condition of compatiblity with $(\eta_{n},g_{n},\nabla_{n})$ from Definition
2.7. Our choice of $\mathbf{s}$ is intensionally chosen such that the third
bullet of Definition 2.7 is satisfied for both $A_{f}$ and $f^{*}A_{f}$ (see
(66)).

The first bullet of Definition 2.7 can be achieved because
$\mathbf{s}:S^{1}\times B\to\mathbb{R}$ is chosen such that over the critical
points of $\eta_{B}$, $\mathbf{s}$ does not lie in the spectrum of the Reeb
flow, so a generic choice of cylindrical $A_{f}$ with
$\mathbf{s}_{(A_{f})_{b}}=\mathbf{s}_{borel}(b)$ will be non-degenerate over
the critical points of $\eta$. When $\mathbf{s}_{(A_{f})_{b}}$ is non-
degenerate over the critical points of $\eta$, so is true for
$\mathbf{s}_{(f^{*}A_{f})_{b}}$.

The second bullet of Definition 2.7 can be achieved because
$\mathbf{s}_{borel}$ is locally constant in $N_{\eta}\times U_{BT,n}\subset
B\times U_{BT,n}=U_{n}$, so we can pick $A_{f,n}$ which is locally constant
over $N_{\eta}\times U_{BT,n}$, which contains all the critical points of
$\eta_{n}$. Since $f$ is also locally constant over $N_{\eta}$, $f^{*}A_{f,n}$
will also be locally constant over $N_{\eta}\times U_{BT,n}$.

The last bullet of Definition 2.7 can be achieved by generic fibrewise-
compactly supported perturbation of $A_{f}$ outside $U_{n}$. It will imply
that so is true for $f^{*}A_{f}$.

∎

We have the following tautological isomorphism.

######  Proposition 4.11.

Let $B$ be a finite dimensional closed smooth $T$-manifold. Let $H$ be a
cylindrical Hamiltonian of $P_{B}$ such that both $f^{*}H$ and $H$ are
compatible with $(\eta_{n},g_{n},\nabla_{n})$. Then we have an isomorphism

(69) $\displaystyle\mathcal{C}_{f}:HF^{*}(P_{B,n},f^{*}H)\simeq
HF^{*-\iota(f)}(P_{B,n};H)$

By taking inverse limit in $n$, we get the isomorphism (54).

######  Remark 4.12.

In fact, Proposition 4.11 is true even if $H$ is compatible with
$(\eta_{n},g_{n},\nabla_{n})$ but $f^{*}H$ is not. The only difference is that
when $f^{*}H$ is not compatible with $(\eta_{n},g_{n},\nabla_{n})$, we cannot
apply Lemma 2.12 to get the well-definedness of $HF(P_{B,n},f^{*}H)$. But as
we shall see from the proof below, the identification
$CF(P_{B,n},f^{*}H)\simeq CF(P_{B,n};H)$ proves that $HF(P_{B,n},f^{*}H)$ is
well-defined in a posterori.

###### Proof of Proposition 4.11.

The strategy to prove (69) is to establish a cochain level isomorphism with
respect to appropriate auxilary data. Let $\tilde{\Phi}_{t}:B\times ET\times
M\to B\times ET\times M$ be $\tilde{\Phi}_{t}(b,y,m)=(b,y,f(b)(t)m)$. It
satisfies

(70)
$\displaystyle\tilde{\Phi}_{t}(gb,gy,gm)=(gb,gy,f(gb)(t)gm)=(gb,gy,gf(b)(t)m)$

so it descends to a map $\Phi_{t}:P_{B}\to P_{B}$ for all $t\in S^{1}$.

By Lemma 4.3, $\Phi_{t}$ provides a bijection between the generators of
$CF(P_{B,n},f^{*}H)$ and $CF(P_{B,n},H)$.

Let $(J_{b})_{b\in B_{n}}$ be a generic $S^{1}$-dependent fibrewise almost
complex structure on $P_{B,n}$ that is compatible with the fibrewise
symplectic form and is of contact type. The differential in the Floer complex
$CF(P_{B,n};H)$ is defined by counting solutions of coupled equations with
gradient trajectories on $BT_{n}$ with respect to $(\eta_{n},g_{n})$, and
Floer solution liftings of the gradient trajectories with respect to
$(\nabla_{n},H,J)$.

If we use $(\Phi_{t}^{*}J_{b})_{b\in B_{n}}$ to define the Floer differential
of $CF(P_{B,n},f^{*}H)$, then we will have a bijective correspondence between
the solutions contributing to the differential of $CF(P_{B,n},f^{*}H)$ and
solutions contributing to the differential of $CF(P_{B,n};H)$. More precisely,
if $(\tau(s),u(s,t))$ is a solution contributing to the differential of
$CF(P_{B,n},H)$, then $(\tau(s),(\Phi_{t})^{-1}\circ u(s,t))$ will be a
solution contributing to the differential of $CF(P_{B,n},f^{*}H)$.

Moreover, by trivializing $P_{B}$ along $\tau$ and applying the Lemma 4.13
below, we see that the correspondence preserves the topological energy
$E((\Phi_{t})^{-1}\circ u(s,t))=E(u(s,t))$. Therefore, we get the isomorphism
(see Lemma 4.4 for the grading shift).

∎

######  Lemma 4.13.

Let $u:\mathbb{R}\times S^{1}\to M$ be a Floer solution with respect to
$H=(H_{s,t})_{(s,t)\in\mathbb{R}\times S^{1}}\in C^{\infty}(\mathbb{R}\times
S^{1}\times M)$. Let $\gamma\in\Omega G$ and $v(s,t)=\gamma(t)^{-1}u(s,t)$.
Then $E(v)=E(u)$.

###### Proof.

We have

$\displaystyle\partial_{s}v(s,t)$
$\displaystyle=(D\gamma(t))^{-1}\partial_{s}u(s,t)$
$\displaystyle\partial_{t}v(s,t)$
$\displaystyle=(D\gamma(t))^{-1}\partial_{t}u(s,t)+X_{\gamma(t)^{-1}}(v(s,t))$

where $X_{\gamma(t)^{-1}}$ is the Hamiltonian vector field generating
$\gamma^{-1}$ at time $t$ (recall from Lemma 4.2(2) the generating Hamiltonian
of $\gamma^{-1}$). Therefore,

$\displaystyle\int_{\mathbb{R}\times S^{1}}v^{*}\omega$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}\int_{0}^{1}\omega((D\gamma(t))^{-1}\partial_{s}u(s,t),(D\gamma(t))^{-1}\partial_{t}u(s,t)+X_{\gamma(t)^{-1}}(v(s,t))dtds$
$\displaystyle=$ $\displaystyle\int_{\mathbb{R}\times
S^{1}}u^{*}\omega+\int_{-\infty}^{\infty}\int_{0}^{1}\omega(\partial_{s}u(s,t),(D\gamma(t))X_{\gamma(t)^{-1}}(v(s,t))dtds$
$\displaystyle=$ $\displaystyle\int_{\mathbb{R}\times
S^{1}}u^{*}\omega+\int_{-\infty}^{\infty}\int_{0}^{1}d(-K_{\gamma,t}(u(s,t)))(\partial_{s}u(s,t))dtds$
$\displaystyle=$ $\displaystyle\int_{\mathbb{R}\times
S^{1}}u^{*}\omega+\int_{0}^{1}K_{\gamma,t}(u(-\infty,t))dt-\int_{0}^{1}K_{\gamma,t}(u(\infty,t))dt$

where $u(\pm\infty,t):=\lim_{s\to\pm\infty}u(s,t)$. As a result,

$\displaystyle E(v)$ $\displaystyle=$ $\displaystyle\int_{\mathbb{R}\times
S^{1}}v^{*}\omega+\int_{0}^{1}(H_{-\infty,t}-K_{\gamma,t})(\gamma(t)v(-\infty,t))dt-\int_{0}^{1}(H_{\infty,t}-K_{\gamma,t})(\gamma(t)v(\infty,t))dt$
$\displaystyle=$ $\displaystyle\int_{\mathbb{R}\times
S^{1}}u^{*}\omega+\int_{0}^{1}H_{-\infty,t}(u(-\infty,t))dt-\int_{0}^{1}H_{\infty,t}(u(\infty,t))dt$
$\displaystyle=$ $\displaystyle E(u)$

∎

The isomorphism (69) is compatible with the product structure as follow.

######  Proposition 4.14.

Let $B$ be a finite dimensional closed smooth $T$-manifold. Let $H$ be a
cylindrical Hamiltonian of $P_{B}$ such that both $f^{*}H$ and $H$ are
compatible with $(\eta_{n},g_{n},\nabla_{n})$. Let $H^{\prime}$ be another
cylindrical Hamiltonian of $P_{B}$ that is of constant slope
$\mathbf{s}^{\prime}$. Let $H^{\prime\prime}$ be a cylindrical Hamiltonian of
$P_{B}$ such that both $f^{*}H^{\prime\prime}$ and $H^{\prime\prime}$ are
compatible with $(\eta_{n},g_{n},\nabla_{n})$,
$\max\mathbf{s}_{H}+\max\mathbf{s}_{H^{\prime}}\leq\min\mathbf{s}_{H^{\prime\prime}}$
and
$\max\mathbf{s}_{f^{*}H}+\max\mathbf{s}_{H^{\prime}}\leq\min\mathbf{s}_{f^{*}H^{\prime\prime}}$.
Then the following diagram commutes

${HF(P_{B,n},f^{*}H)\times
HF(P_{B,n},H^{\prime})}$${HF(P_{B,n};f^{*}H^{\prime\prime})}$${HF(P_{B,n},H)\times
HF(P_{B,n},H^{\prime})}$${HF(P_{B,n};H^{\prime\prime})}$

###### Sketch of proof.

Similar to the proof of Proposition 4.11, we want to construct
$\Phi=(\Phi_{z,t}:P_{B}\to P_{B})_{z\in\Sigma,t\in S^{1}}$ to give a chain
level identification of the product operation on the first row and on the
second row. More precisely, over the first positive cylindrical end and the
negative cylindrical end of $\Sigma$, we choose $\tilde{\Phi}_{z,t}:B\times
ET\times M\to B\times ET\times M$ to be
$\tilde{\Phi}_{\epsilon_{i}(s,t),t}(b,y,m)=(b,y,f(b)(t)m)$. Over the second
positive cylindrical end, we choose
$\tilde{\Phi}_{\epsilon_{2}(s,t),t}(b,y,m)=(b,y,m)$. There is no homotopical
obstruction to extends the definition of $\tilde{\Phi}_{z,t}$ over the
complement of cylindrical ends and at the same time having
$\tilde{\Phi}_{z,t}$ to be $T$-equivariant for all $z,t$. For example, we can
start with a $s$-invariant $\tilde{\Phi}$ over a cylinder and then isotope it
such that it is the identity in some $s$-invariant neighborhood. Adding a
puncture in this $s$-invariant neighborhood and introducing a cylindrical end
$\epsilon_{2}$ near the puncture will give what we want. The $T$-equivariant
$\tilde{\Phi}$ descends to $\Phi$. We can choose all the auxiliary data in the
definition of the two product operations to be intertwined by $\Phi$.

∎

Let $(f,\eta_{B}^{\prime})$ be a good pair. For each
$a\in\mathbb{R}_{>0}\setminus\operatorname{Spec}(\partial\bar{M},\alpha)$, we
can apply Proposition 4.10 to find an $A_{f}$ with the listed properties.
Together with Proposition 4.11 and the continuation maps 2.16, we get

(71) $\displaystyle\mathcal{C}_{a,f,A_{f}}:$ $\displaystyle HF(P_{B},a)\to
HF(P_{B},f^{*}A_{f})\simeq HF(P_{B};A_{f})\to HF(P_{B};a+C_{f}+\epsilon_{a})$

where $\epsilon_{a}\geq 0$ is a small number such that