#  Hamiltonian Floer theory on surfaces:
Linking, positively transverse foliations and spectral invariants

Dustin Connery-Grigg This work was supported by the Fonds de recherche du
Québec – Nature et technologies, Grant # 260640

###### Abstract

We develop connections between the qualitative dynamics of Hamiltonian
isotopies on a surface $\Sigma$ and their chain-level Floer theory using ideas
drawn from Hofer-Wysocki-Zehnder’s theory of finite energy foliations. We
associate to every collection of capped $1$-periodic orbits which is
‘maximally unlinked relative the Morse range’ a singular foliation on
$S^{1}\times\Sigma$ which is positively transverse to the vector field
$\partial_{t}\oplus X^{H}$ and which is assembled in a straight-forward way
from the relevant Floer moduli spaces. This provides a Floer-theoretic method
for producing foliations of the type which appear in Le Calvez’s theory of
positively transverse foliations for surface homeomorphisms. Additionally, we
provide a purely topological characterization of those Floer chains which both
represent the fundamental class in $CF_{*}(H,J)$, and which lie in the image
of some chain-level PSS map. This leads to the definition of a novel family of
spectral invariants which share many of the same formal properties as the Oh-
Schwarz spectral invariants, and we compute the novel spectral invariant
associated to the fundamental class in entirely dynamical terms. This
significantly extends a project initiated by Humilière-Le Roux-Seyfaddini in
[12].

###### Contents

  1. 1 Introduction
    1. 1.1 Structure of this paper
    2. 1.2 Acknowledgments
  2. 2 Capped braids and homological linking
    1. 2.1 Capped Braids
    2. 2.2 Linking of capped loops with close strands
    3. 2.3 The homological linking number for capped braids
  3. 3 Elements of Floer theory and linking
    1. 3.1 Floer theory
      1. 3.1.1 Continuation maps
      2. 3.1.2 The PSS isomorphism
      3. 3.1.3 The Gromov trick
    2. 3.2 Asymptotic analysis for pseudoholomorphic cylinders
      1. 3.2.1 Winding of eigenvectors of $A_{x,J}$
  4. 4 Constructing chain-level continuation maps with prescribed behaviour
    1. 4.1 Perturbing positively intersecting strands to solve Floer’s equation near the intersection
    2. 4.2 Proof of Theorem 4.4
    3. 4.3 From models of cobordisms to regular models
    4. 4.4 Appendix
      1. 4.4.1 A short fibration argument
      2. 4.4.2 A proof of Lemma 4.9
  5. 5 Application: A new spectral invariant and its computation on surfaces
    1. 5.1 A new spectral invariant: definitions and properties
      1. 5.1.1 Basic Properties of $c_{im}$
      2. 5.1.2 A new spectral norm
    2. 5.2 Computing $c_{im}([\Sigma];H)$ on surfaces
    3. 5.3 Dynamical Consequences
    4. 5.4 On the equivalence $c_{OS}=c_{im}$
  6. 6 Application: Positively transverse foliations from Floer theory
    1. 6.1 Foliated sectors and Floer moduli spaces as leaf spaces
    2. 6.2 The restricted complex associated to a capped braid
      1. 6.2.1 Maximal unlinkedness relative the Morse range
      2. 6.2.2 Dominating Morse functions and dominating homotopies
    3. 6.3 Construction and properties of $\mathcal{F}^{\hat{X}}$
      1. 6.3.1 $\mathcal{F}^{\hat{X}}$ as negative gradient flow-lines of the restricted action functional
      2. 6.3.2 $\mathcal{F}^{\hat{X}}$ as a positively transverse foliation
    4. 6.4 Consequences for the structure of Hamiltonian isotopies
      1. 6.4.1 Comparison to Le Calvez’s theory of transverse foliations
  7. 7 Appendix: Stefan-Sussmann Foliations

##  1 Introduction

The goal of this paper is to develop a certain picture — the possibility of
which was initially suggested by the work of Humilière, Seyfaddini and Le Roux
in [12] — for Hamiltonian dynamics on surfaces via a set of tools which are
well-suited to the study of the relationship between their dynamics and the
structure of their Floer complexes. More precisely, recall that an isotopy

$\displaystyle\phi^{H}:[0,1]\times\Sigma$ $\displaystyle\rightarrow\Sigma$

of a symplectic surface $(\Sigma,\omega)$ is called Hamiltonian if it is
realized as the flow of the periodic family of vector fields
$(X^{H}_{t})_{t\in S^{1}}$, where each vector field satisfies

$\displaystyle\omega(X^{H}_{t},\cdot)$ $\displaystyle=-dH_{t}(\cdot)$

for some smooth family of functions $(H_{t}:\Sigma\rightarrow\mathbb{R})_{t\in
S^{1}}$. Modulo some genericity conditions, to each such isotopy and any
family $(J_{t})_{t\in S^{1}}$ of $\omega$-compatible almost complex
structures, we may associate a complex $CF_{*}(H,J)$ which is generated by the
set $\widetilde{Per}_{0}({H})$ of capped $1$-periodic orbits of $\phi^{H}$.
This complex has a $\mathbb{Z}$-grading $\mu_{CZ}$ which counts the amount of
‘symplectic winding’ that occurs locally about the orbit throughout the
isotopy, and whose differential counts the number of ‘Floer cylinders’
$u:\mathbb{R}\times S^{1}\rightarrow\Sigma$ which connect these orbits.
Together with $\mu_{CZ}$, any collection of capped orbits
$\hat{X}\subset\widetilde{Per}_{0}({H})$ such that the underlying orbits $x,y$
of any two elements $\hat{x},\hat{y}\in\hat{X}$ are distinct forms a geometric
object which we call an indexed capped braid $(\hat{X},\mu_{CZ})$. We can
think of the topology of the indexed capped braids which make up
$\widetilde{Per}_{0}({H})$ as encoding the topological structure of
$1$-periodic orbits of $(\phi^{H}_{t})_{t\in[0,1]}$ along with the coarse
local structure of the isotopy near these orbits. The broad question broached
by this work may be stated as follows

What relations does the topological structure of $\widetilde{Per}_{0}({H})$
impose on the (filtered) algebraic structure of $CF_{*}(H,J)$ and vice versa?

Clearly, this question is only particularly meaningful in low-dimensions where
the orbits may twist about one another in homotopically interesting ways, but
in this situation it proves surprisingly fruitful and sheds considerable light
on the possible dynamics of generic Hamiltonian isotopies on surfaces.

To wit, we will say that a capped braid $\hat{X}$ is unlinked if the capping
disks of the strands may be chosen such that their graphs in
$D^{2}\times\Sigma$ are disjoint. An indexed capped braid
$\hat{X}\subset\widetilde{Per}_{0}({H})$ is maximally unlinked relative the
Morse range if $\hat{X}$ is unlinked, every (capped) strand of $\hat{X}$ has
index lying in the set $\\{-1,0,1\\}$ and $\hat{X}$ is maximal among all
collections of capped orbits in $\widetilde{Per}_{0}({H})$ having these two
properties. We write $murm(H)$ to denote the set of such capped braids. As a
consequence of the theory developed herein, we obtain the following structural
result for generic Hamiltonians $H$ on an arbitrary symplectic surface.

######  Theorem A.

Let $H\in C^{\infty}(S^{1}\times\Sigma)$ be a non-degenerate Hamiltonian, and
let $J\in C^{\infty}(S^{1};\mathcal{J}_{\omega}(\Sigma))$ be such that $(H,J)$
is Floer regular. For any capped braid $\hat{X}\in murm(H)$, we may construct
an oriented singular foliation $\mathcal{F}^{\hat{X}}$ of $S^{1}\times\Sigma$
with the following properties

  1. 1.

The singular leaves of $\mathcal{F}^{\hat{X}}$ are precisely the graphs of the
orbits in $\hat{X}$.

  2. 2.

The regular leaves are precisely the annuli parametrized by maps

$\displaystyle\check{u}:\mathbb{R}\times S^{1}$ $\displaystyle\rightarrow
S^{1}\times\Sigma$ $\displaystyle(s,t)$ $\displaystyle\mapsto(t,u(s,t)).$

for some $u\in\widetilde{\mathcal{M}}(\hat{x},\hat{y};H,J)$, and some
$\hat{x},\hat{y}\in\hat{X}$.

  3. 3.

The vector field $\check{X}^{H}(t,z)=\partial_{t}\oplus X^{H}_{t}(z)$ is
positively transverse to every regular leaf of $\mathcal{F}^{\hat{X}}$.

Similar foliations play a crucial role in Bramham’s recent celebrated
construction of periodic approximations for irrational pseudo-rotations of the
disk in [2]. Our approach gives both the existence of such foliations for
generic Hamiltonians, and moreover ties their behaviour directly to the
structure of the Floer complex and the dynamics of the Hamiltonian isotopy.

The structure of the foliations $\mathcal{F}^{\hat{X}}$ could be a priori
rather complicated, however, with Theorem A in hand, we can define the
$\hat{X}$-restricted action functional $A^{\hat{X}}\in
C^{\infty}(S^{1}\times\Sigma)$ by
$A^{\hat{X}}(t,u(s,t))=\mathcal{A}_{H}(\hat{u}_{s})$, for
$\mathcal{A}_{H}:\widetilde{\mathcal{L}_{0}}(\Sigma)\rightarrow\mathbb{R}$ the
Hamiltonian action functional on the space of capped loops (the capping of
$u_{s}$ is naturally induced by the cappings of the limiting orbits).
$A^{\hat{X}}$ turns out to be a Morse-Bott function, and if we define
$A^{\hat{X}}_{t}\in C^{\infty}(\Sigma)$, $t\in S^{1}$, to be its restriction
to the fiber $\\{t\\}\times\Sigma$, we obtain an $S^{1}$-family of Morse
functions, such that the negative gradient flow of $A^{\hat{X}}_{t}$ provides
a singular foliation which coincides with the foliation
$\mathcal{F}^{\hat{X}}_{t}$ given by intersecting $\mathcal{F}^{\hat{X}}$ with
the fiber over $t\in S^{1}$. Sliding the fiber $\\{0\\}\times\Sigma$ along the
circles $t\mapsto(t,u_{s}(t))$ provides a loop $(\psi^{\hat{X}}_{t})_{t\in
S^{1}}$, and we prove

######  Theorem B.

For every $t\in S^{1}$, $\mathcal{F}^{\hat{X}}_{t}$ is a singular foliation of
Morse type. Moreover, the loop $(\psi^{\hat{X}}_{t})_{t\in S^{1}}$ is a
contractible loop of diffeomorphisms such that the orbits of
$(\psi^{\hat{X}})^{-1}\circ\phi^{H}$ are positively transverse to the
foliation $\mathcal{F}^{\hat{X}}_{0}$.

We thereby reduce the study of the qualitative dynamics of the isotopy
$\phi^{H}$ to the much better understood situation of dynamics which are
positively transverse to a Morse-type foliation. Note that similar foliations
(with a weaker notion of positive transversality) have been constructed by Le
Calvez in [15] for Hamiltonian homeomorphisms, and play a central role in the
forcing theory developed in [17]. The above result can be viewed as giving a
Floer-theoretic construction of certain of Le Calvez’s foliations, along with
additional insight into their structure in the smooth case (see Section 6.4.1
for discussion of this point) . We also obtain as a corollary the following
somewhat surprising structural result about the topology of the braid
generated by the $1$-periodic orbits of $H$ (a capped braid is said to be
linked if it is not unlinked):

######  Theorem C.

Let $H$ be non-degenerate, $\hat{X}=\\{[x_{i}(t),w_{i}(se^{2\pi
it})]\\}_{i=1}^{k}\in murm(H)$ and for any $m\in\mathbb{Z}_{>0}$, denote by
$\hat{X}^{\sharp m}=\\{[x_{i}(mt),w_{i}(se^{2m\pi
it})]\\}_{i=1}^{k}\subset\widetilde{Per}_{0}({H^{\sharp m}})$ its $m$-fold
iterate. For any $\hat{y}\in\widetilde{Per}_{0}({H^{\sharp m}})$ with
$y(t)\neq x_{i}(mt)$ for all $i=1,\ldots,k$ and $t\in S^{1}$, $\hat{X}^{\sharp
m}\cup\hat{y}$ is linked. In particular every $\hat{X}\in murm(H)$ is
maximally unlinked as a subset of $\widetilde{Per}_{0}({H})$.

Remark that this is essentially the same relationship as exists between the
periodic orbits of an autonomous Morse Hamiltonian and its critical points.
Taken together with the observations that every capped orbit
$\hat{x}\in\widetilde{Per}_{0}({H})$ with index in the Morse range lies in
some $\hat{X}\in murm(H)$ and that any two orbits in the Morse range which are
connected by a Floer cylinder are unlinked (see Corollary 6.4 in Section 6.1),
the above results imply that the problem of understanding the structure of
$CF_{*}(H,J)$ in the Morse range may be decomposed into the problem of
understanding the finitely many Morse-type foliations
$\\{\mathcal{F}^{\hat{X}}_{0}\\}_{\hat{X}\in murm(H)}$ on $\Sigma$ which are
associated to the Morse functions $\\{A^{\hat{X}}_{0}\\}_{\hat{X}\in
murm(H)}$. Moreover, for each $\hat{X}\in murm(H)$, the qualitative dynamics
of the isotopy $(\psi^{\hat{X}})^{-1}\circ\phi^{H}$ relative to the foliation
$\mathcal{F}^{\hat{X}}_{0}$ are similar in fundamental ways to the dynamics of
an autonomous Morse Hamiltonian relative to the foliations provided by its
negative gradient trajectories.

The techniques developed in this paper also enable a second, logically
distinct — though thematically related — line of enquiry, examining the
relationship between the qualitative dynamics of a Hamiltonian isotopy and
those aspects of the filtered Floer complex which are ‘probeable’ by chain-
level PSS maps. To be more precise, recall that there is a natural way to
identify the quantum homology of a symplectic manifold with the Floer homology
of a generic Floer pair $(H,J)$ given by the PSS isomorphism originally
introduced in [24] (see also [27]). This isomorphism is induced on homology by
choosing some ancillary data $\mathcal{D}$ — which includes a Morse-Smale pair
$(f,g)$ and so provides a Morse-theoretic model for the quantum chain complex
$QC_{*}(f,g)=C^{Morse}(f,g)\otimes\Lambda_{\omega}$ whose homology computes
the quantum homology of $(M,\omega)$ — and this data is then used to construct
a chain morphism

$\displaystyle\Phi^{PSS}_{\mathcal{D}}:QC_{*}(f,g)$ $\displaystyle\mapsto
CF_{*-n}(H,J).$

The induced map on homology is an isomorphism and is independent of the
ancillary data $\mathcal{D}$. Our next result provides, in the case of
surfaces, a purely topological characterization of those non-trivial Floer
cycles in $CF_{*}(H,J)$ which lie in the image some chain-level PSS map and
which represent the fundamental class. Note that for $\sigma=\sum
a_{\hat{x}}\hat{x}\in CF_{*}(H,J)$, $\text{supp
}\hat{\sigma}:=\\{\hat{x}:\;a_{\hat{x}}\neq 0\\}$ may be thought of as a
capped braid.

######  Theorem D.

Let $\sigma\in CF_{1}(H,J)$. $\sigma$ is a non-trivial cycle such that
$\sigma\in\text{im}\;\Phi^{PSS}_{\mathcal{D}}$ for some regular PSS data
$\mathcal{D}$ if and only if $\text{supp }\sigma$ is a maximal positive capped
braid relative index $1$.

See Section 5.2 for the definition of a ‘maximal positive capped braid
relative index $1$’. The salient point is that this condition depends only on
the qualitative dynamics of the Hamiltonian isotopy generated by $H$. This
result is something of a novelty, since in general, it is very difficult
(unless we are in the case of a $C^{2}$-small autonomous Hamiltonian) to
identify candidates for cycles which represent the fundamental class in Floer
homology. The above theorem provides us with a wealth of different such
candidates having nice dynamical properties. Our methods also permit a similar
type of characterization to be given for cycles which lie in the image of some
chain-level PSS map and which represent the point class, although this
characterization is somewhat more involved and we do not present it here.
Interestingly though, there seem to be fundamental obstructions to using the
same approach to characterize Floer cycles of middle degree lying in the image
of some chain-level PSS map.

Theorem D motivates us to examine, for general symplectic manifolds, the
quantity $c_{im}(\alpha;H)$ obtained by examining the infimal action level
required to represent some non-zero quantum homology class $\alpha$ in the
filtered Floer complex of $H$ via some chain-level PSS map. It turns out that
these quantities define spectral invariants which are intimately related to
the geometry of Hamiltonian fibrations over disks and which we term the PSS-
image spectral invariants. Theorem D thereby provides us with a purely
topological formula for $c_{im}([\Sigma];H)$ on arbitrary surfaces (here
$\hat{X}\in mp_{(1)}(H)$ if $\hat{X}=\\{\hat{x}_{1},\ldots,\hat{x}_{k}\\}$ is
a maximally positive capped braid relative index $1$)

######  Corollary.

Let $H$ be non-degenerate.

$\displaystyle c_{im}([\Sigma];H)$ $\displaystyle=\inf_{\hat{X}\in
mp_{(1)}(H)}\sup_{\hat{x}\in\hat{X}}\mathcal{A}_{H}(\hat{x}).$

The above result is quite similar in spirit to the characterization of the
usual Oh-Schwarz spectral invariant associated to the fundamental class (see
Section 3.1.2 for the definition) established by Humilière-Le Roux-Seyfaddini
in [12] for autonomous Hamiltonians on aspherical surfaces (indeed, in many
ways this work grows out of an attempt to understand what is going on in [12]
in more directly geometric terms) and, in fact, we will see in Section 5.4
that their work implies that the two quantities agree in this setting.
Intriguingly, the class of orbits over which the minimax procedure is
performed in the work of Humilière-Le Roux-Seyfaddini is quite different from
the class of orbits which we consider; it would be interesting to understand
the geometry behind this equivalence better. For instance, does it hold for a
wider class of Hamiltonians than simply the autonomous ones? And if not, what
dynamical phenomena are responsible for its failure?

In virtue of their definition, the PSS-image spectral invariants share many of
the desirable formal properties of the usual Oh-Schwarz spectral invariants
(see Section 3.1.2 for the definition), and so one can perform many of the
same arguments with them, but with the added advantage that we have a better
understanding of the relationship between PSS-image spectral invariants and
the dynamical behaviour of the system under study — at least on surfaces. For
instance, we may use the PSS-image spectral invariants to define a
symplectically bi-invariant norm on $Ham(M,\omega)$ by

$\displaystyle\gamma_{im}(\phi)$
$\displaystyle:=\inf\;c_{im}([M];H)+c_{im}([M];\bar{H}),$

where the infimum is taken over Hamiltonians $H$ such that $\phi^{H}_{1}=\phi$
and which are normalized so that $\int_{M}H_{t}\;\omega^{n}=0$ for all $t\in
S^{1}$. In the case of surfaces, the fact that the PSS-image spectral
invariants share enough of the formal properties satisfied by the Oh-Schwarz
spectral invariants allows Seyfaddini’s argument establishing the
$C^{0}$-continuity of the Oh-Schwarz spectral norm on surfaces (see Theorem 3
in [29]) to carry through for $\gamma_{im}$. As a consequence, we obtain

######  Theorem E.

On surfaces, the symplectically bi-invariant norm $\gamma_{im}$ is both
$C^{0}$-continuous and Hofer-continuous. Moreover, if $H$ is non-degenerate,
then

$\displaystyle\gamma_{im}(\phi)$ $\displaystyle=\inf_{\hat{X}\in
mp_{(1)}(H)}\sup_{\hat{x}\in\hat{X}}\mathcal{A}_{H}(\hat{x})-\sup_{\hat{X}\in
mn_{(-1)}(H)}\inf_{\hat{x}\in\hat{X}}\mathcal{A}_{H}(\hat{x}),$

for $H$ any normalized Hamiltonian such that $\phi^{H}_{1}=\phi$. (Here
$mn_{(-1)}(H)$ denotes the set of all capped braids
$\hat{X}\subset\widetilde{Per}_{0}({H})$ which are ‘maximally negative
relative index $-1$’. See Section 5.2 for the definition).

The main interest in the above theorem, apart from the intrinsic interest of
having a dynamically defined symplectically bi-invariant norm on Hamiltonian
diffeomorphisms, is that the structure of the sets $mp_{(1)}(H)$ and
$mn_{(-1)}(H)$ can change dramatically as one interpolates between
Hamiltonians (in either the $C^{0}$ or the Hofer topology), however, the above
theorem guarantees that the quantity obtained by the specified mini-max
procedure over these collections remains continuous regardless. Finally, as a
last example of the sort of information that can be drawn from Theorem D, we
may exploit the computability of our newfound spectral invariants together
with their close relationship to the Oh-Schwarz spectral invariants, in order
to extract the following dynamical controls over the commutator lengths of
homotopy classes $\tilde{\phi}\in\widetilde{Ham}(S^{2},\omega)$ from Entov’s
work in [4].

######  Theorem F.

Assume that $H\in C^{\infty}(S^{1}\times S^{2})$ is non-degenerate and
normalized to that $\int H_{t}\omega=0$ for all $t\in S^{1}$, then

$\displaystyle\min\big{\\{}\min_{\hat{X}\in
mp_{(1)}(H)}\max_{\hat{x}\in\hat{X}}\mathcal{A}_{H}(\hat{x}),-\max_{\hat{X}\in
mn_{(-1)}(H)}\min_{\hat{x}\in\hat{X}}\mathcal{A}_{H}(\hat{x})\big{\\}}$
$\displaystyle<-kArea(S^{2},\omega)$

only if the commutator length of $\tilde{\phi}^{H}$ in
$\widetilde{Ham}(S^{2})$ is strictly greater than $2k+1$.

###  1.1 Structure of this paper

Sections 2, 3 and 4 develop the main conceptual and technical tools in this
work, which we then apply in Sections 5 and 6 to deduce the results laid out
in the previous section. These latter two sections are independent and may be
read in any order, but both depend heavily on the first three sections.

Section 2 introduces the basic notion of capped braids and related basic
language. The most essential development in this section is the definition of
a relative invariant for capped braids which associates to any pair of capped
$k$-braids $(\hat{X},\hat{Y})$ an integer
$L_{0}(\hat{X},\hat{Y})\in\mathbb{Z}$. We call this quantity the homological
linking number of $\hat{X}$ and $\hat{Y}$. If we think of the (uncapped)
braids $X$ and $Y$ as being graphs — suggestively called ‘strands’ — in
$S^{1}\times\Sigma$ of loops in $\Sigma$, then roughly speaking,
$L_{0}(\hat{X},\hat{Y})$ counts the number of signed intersections that occur
between the strands if we start with the braid $X$ and attempt to continuously
deform it to the braid $Y$, subject to a certain homotopy condition coming
from the cappings of $\hat{X}$ and $\hat{Y}$.

Section 3 introduces the elements of Floer theory of which we will have need,
and explains the relevance of the homological linking number to the study of
the Floer theory of low-dimensional systems. More precisely, one of the
central geometric ideas in this work is that we may interpret collections of
Floer-type cylinders as providing such deformations between the capped braid
$\hat{X}$ formed by the capped orbits at their negative ends and the capped
braid $\hat{Y}$ formed by the capped orbits at their positive ends. When we do
this, the Gromov trick combined with positivity of intersections in dimension
$4$ implies that the homological linking number $L_{0}(\hat{X},\hat{Y})$ must
be non-negative, which gives strong topological restrictions on the putative
existence of collections of Floer cylinders in terms of the relative
topologies of the capped braids formed by their negative and positive ends. In
the contact setting on $S^{3}$, the fact that positivity of intersection for
pseudo-holomorphic cylinders in dimension $4$ is related to a monotonicity
phenomenon for winding numbers seems to be a fairly well-appreciated
phenomenon, going back at least to the work of Hofer-Wysocki-Zehnder in [8].
The idea of relating the global topology of the braids formed by periodic
orbits of a Hamiltonian to Floer theory via positivity of intersection
arguments also has some precedent on the disk in [35], in which the authors
use Floer homology to define braid invariants and obtain a forcing theory for
periodic orbits of Hamiltonian. The approach sketched so far exhibits a slight
deficiency when studying the deformations of braids induced by general
collections of Floer-type cylinders however, in the sense that, in general,
one may have multiple Floer cylinders emerging from, or converging to, the
same orbit and it is not necessarily clear how one should treat this
situation.

In order to deal with cylinders which emerge from or converge to the same
orbit, we make use of the analysis of the relative asymptotic behaviour of
pseudo-holomorphic curves developed by Siefring in [30], which pairs with work
of Hofer-Wysocki-Zehnder in [8] to connect the Conley-Zehnder index of an
orbit to bounds on the winding behaviour of pairs of cylinders which emerge
from or converge to that orbit. In the contact setting, these sorts of bounds
(in the non-relative case), along with the insight that under appropriate
index conditions on the asymptotic orbits families of pseudoholomorphic curves
automatically form local foliations in the symplectization of a contact
manifold, go back to the pioneering work of Hofer-Wysocki-Zehnder in [9], [8],
[10] and [11]. Siefring has also more recently put this circle of ideas to use
in [31] to define an intersection number for arbitrary pseudoholomorphic
curves in $4$-dimensional symplectic cobordisms which is invariant under
homotopy. In any case, we may thereby split the analysis of the behaviour of
general collections of Floer cylinders into two portions: an asymptotic
portion on the ends where the linking behaviour of cylinders asymptotic to the
same orbit is controlled by the Conley-Zehnder index, and a compact portion
which interpolates between two capped braids such that their intersections are
controlled by the homological linking number of the capped braids at each end
of this compact portion.

Section 4 is the last of the technical sections, and introduces our technique
for designing Floer continuation maps such that certain moduli spaces are non-
empty. The basic idea is to show that, given any collection of smooth
cylinders which topologically could be a collection of ($s$-dependent) Floer-
type cylinders (in the sense that their graphs intersect only positively and
they exhibit the appropriate asymptotic winding behaviour), we may — after a
series of suitable small perturbations — always find regular Floer homotopy
data $(\mathcal{H},\mathbb{J})$ such that the perturbed cylinders are all
$(\mathcal{H},\mathbb{J})$-Floer.

In Section 5, we introduce the PSS-image spectral invariants and prove Theorem
D and its various consequences. The main line of the proof of Theorem D is as
follows. The fact that the cycle in question lies in the image of some chain-
level PSS map implies the existence of various PSS disks, which may be thought
of as providing a deformation from a trivial capped braid to the support of
the cycle in question. The theory from Sections 2 and 3 may then be used to
force the support of this cycle to be positive, and when combined with the
cycle’s homological non-triviality this theory forces the support to be
maximally positive (relative index $1$). Conversely, in order to deduce that
every such capped braid lies in the image of some chain level PSS map, it is
sufficient to construct an appropriate continuation map from a small Morse
function which sends the fundamental class of the Morse function to a cycle
supported on the capped braid in question. The technique for designing
continuation maps with prescribed behaviour developed in Section 4 may be seen
to provide precisely such a continuation map.

Section 6 proves Theorems A, B and C. Theorem A is the main theorem from which
the others follow as ready consequences. The rough structure of its proof is
as follows. The theory developed in Section 3 may be used to show that if two
capped orbits $\hat{x},\hat{y}$ have Conley-Zehnder indices lying in the Morse
range with $\widetilde{\mathcal{M}}(\hat{x},\hat{y};H,J)$, then maps of the
form $(s,t)\mapsto(t,u(s,t))$ for
$u\in\widetilde{\mathcal{M}}(\hat{x},\hat{y};H,J)$ foliate some subset
$W(\hat{x},\hat{y})\subset S^{1}\times\Sigma$. This fact was essentially
discovered in the contact setting in [10] (see also [11]). We then study
conditions under which collections $\hat{X}$ of capped orbits having indices
lying in the Morse range may be chosen so that these foliated subsets piece
together to form the desired foliation $\mathcal{F}^{\hat{X}}$. In order to do
this, it turns out to be convenient to associate a ‘restricted complex’
$C_{*}(\hat{X})$ to a capped braid $\hat{X}\subset\widetilde{Per}_{0}({H})$
whose ‘differential’ counts the Floer cylinders running between capped orbits
in $\hat{X}$ of index difference $1$. In general, this is not actually a
differential and $C_{*}(\hat{X})$ is not actually a chain complex. However, we
show that when $\hat{X}\in murm(H)$, $CF_{*}(\hat{X};H,J):=C_{*}(\hat{X})$ is
a chain complex (although it is not, in general, a subcomplex of
$CF_{*}(H,J)$). In order to study $CF_{*}(\hat{X};H,J)$, we introduce a useful
class of continuation maps which we call $\hat{X}$-dominating continuation
maps from (resp. to) $\hat{X}$-dominating Morse functions. Roughly speaking,
these are continuation maps such that for each $\hat{x}\in\hat{X}$, we can
find some continuation moduli space
$\mathcal{M}(\hat{p},\hat{x};\mathcal{H},\mathbb{J})$ (resp.
$\mathcal{M}(\hat{x},\hat{p};\mathcal{H},\mathbb{J})$) which is non-empty. The
techniques of Section 4 are used to establish their existence. The crucial
point is that if $h_{\mathcal{H},\mathbb{J}}$ is a $\hat{X}$-dominating
continuation map from a $\hat{X}$-dominating Morse function $f$, then the map

$\displaystyle\pi^{\hat{X}}\circ
h_{\mathcal{H},\mathbb{J}}:CF_{*}(f,J^{-})\rightarrow$ $\displaystyle
CF_{*}(H,J)\rightarrow CF(\hat{X};H,J),$

where $\pi^{\hat{X}}:CF_{*}(H,J)\rightarrow CF(\hat{X};H,J)$ is the map which
is the identity of $\Lambda_{\omega}\langle\hat{x}\rangle_{\hat{X}}$ and sends
everything else to $0$, is a chain map which induces an injection on homology.
Moreover, such maps interact nicely with the cap action on the Floer chain
complex in the sense that for a generic smooth cycle $\alpha^{\\#}$,

$\displaystyle(\pi^{\hat{X}}\circ
h_{\mathcal{H},\mathbb{J}})(\alpha^{\\#}\cap\hat{z})$
$\displaystyle=\alpha^{\\#}\cap(\pi^{\hat{X}}\circ
h_{\mathcal{H},\mathbb{J}})(\hat{z})$

at the chain level. Consequently, if $\alpha^{\\#}$ is taken to be a generic
point $p\in\Sigma$ this implies that when $\sum\hat{z}\in CF_{*}(f,J^{-})$
represents the fundamental class

$\displaystyle p\cap(\pi^{\hat{X}}\circ
h_{\mathcal{H},\mathbb{J}})(\sum\hat{z})$ $\displaystyle=(\pi^{\hat{X}}\circ
h_{\mathcal{H},\mathbb{J}})(p\cap\sum\hat{z}).$

Since $[p\cap\sum\hat{z}]=[pt]\in QH_{0}(\Sigma)\simeq HF_{-1}(f)$ and
$\pi^{\hat{X}}\circ h_{\mathcal{H},\mathbb{J}}$ is injective on homology, the
left-hand side of the above equation is non-zero. This implies that through
any generic $(t,p)\in S^{1}\times\Sigma$, we can find
$\hat{x},\hat{y}\in\hat{X}$ and a Floer cylinder
$u\in\widetilde{\mathcal{M}}(\hat{x},\hat{y};H,J)$ such that
$(t,u(0,t))=(t,p)$. Consequently, when $\hat{X}\in murm(H)$, the union of the
subspaces $W(\hat{x},\hat{y})$ over all $\hat{x},\hat{y}\in\hat{X}$ forms a
foliated open, dense set of $S^{1}\times\Sigma$. We may then use an argument
pioneered in [11] to establish that, in fact, all of $S^{1}\times\Sigma$ is
foliated. The positive transversality property for $\mathcal{F}^{\hat{X}}$ is
a direct consequence of the $\omega$-compatibility of the almost complex
structure and the fact that the regular leaves are parametrized by Floer
cylinders.

###  1.2 Acknowledgments

The author would like to thank his PhD supervisor François Lalonde for his
enthusiastic support and encouragement throughout this project. He would also
like to thank Egor Shelukhin for his helpful comments and feedback on an
earlier version of this paper, as well as Jordan Payette for many stimulating
conversations on the subject matter. Finally, particular thanks are due to
Vincent Humilière, Sobhan Seyfaddini and Frédéric Le Roux for their remarks on
an earlier version of this paper which identified a significant error in a
purported computation of the Oh-Schwarz spectral invariant. Additional thanks
are due to Vincent Humilière for directing my attention to Yan’s work on
torsion-low isotopies in [36] and suggesting their relationship to the
foliations $\mathcal{F}^{\hat{X}}_{0}$ constructed in Section 6.

##  2 Capped braids and homological linking

In this section we introduce the basic notion of capped braids, their
appropriate notion of equivalence ($0$-homotopy), and an important relative
invariant of a pair of capped braids with the same number of strands which we
call the homological linking number. These notions will later serve to encode
the topology of the capped $1$-periodic orbits of a non-degenerate Hamiltonian
system, and give controls on the behaviour of Floer-type cylinders which may
run between various collections of such capped orbits.

The basic definitions required for our work with capped braids are presented
in §2.1. §2.2 explains a mild generalization of the classical winding number
of loops in $\mathbb{R}^{2}$ to capped loops whose underlying loops are close
in the loop space. This generalization becomes important later on in the work
when we consider the asymptotics of Floer-type cylinders which emerge or
converge to the same orbit. §2.3 introduces the homological linking number and
establishes its basic properties including how it recovers the generalized
winding number in the setting of §2.3.

For the duration of this paper, $\Sigma$ will always denote a smooth
symplectic surface $(\Sigma,\omega)$, $\mathcal{L}(M)$ the (smooth) loop space
of the manifold $M$, $\mathcal{L}_{0}(M)$ its space of contractible loops and
$\widetilde{\mathcal{L}_{0}}(M)$ its Novikov covering space (see [18], Section
12.1).

Throughout this work, there is a certain amount of juggling of different
perspectives on the same objects that will be necessary. In particular, though
our main objects of study are isotopies on some surface $\Sigma$ — and so the
initial arena in which the action takes place is $2$ dimensional — it will
frequently be useful to work on the $3$-dimensional mapping torus

$\displaystyle\check{\Sigma}$ $\displaystyle:=S^{1}\times\Sigma$

as well as the $4$-dimensional space

$\displaystyle\widetilde{\Sigma}$ $\displaystyle:=\mathbb{R}\times
S^{1}\times\Sigma.$

In order to have our notation be suggestive of these perspectival shifts, maps
taking values in $\check{\Sigma}$ will frequently be adorned with a
$\check{\cdot}$, while maps taking values in $\widetilde{\Sigma}$ will be
adorned with a $\tilde{\cdot}$. The most frequent use-cases for this notation
will be the following. For $x\in\mathcal{L}(M)$, we write
$\check{x}(t):=(t,x(t))$ for its graph in $\check{M}=S^{1}\times M$. For
$u:I\times S^{1}\rightarrow M$, where $I\subseteq\mathbb{R}$, we write
$\tilde{u}(s,t):=(s,t,u(s,t))$ for its graph in $\widetilde{M}=I\times
S^{1}\times M$ and $\check{u}(s,t):=(t,u(s,t))$ for the projection of this
graph onto $\check{M}$.

###  2.1 Capped Braids

######  Definition 2.1.

For any $k\in\mathbb{N}$, we define the $k$-configuration space

$\displaystyle C_{k}(\Sigma)$
$\displaystyle:=\\{(z_{1},\ldots,z_{k})\in\Sigma^{k}:(i\neq j)\Rightarrow
z_{i}\neq z_{j}\\}$

######  Definition 2.2.

An (ordered) $k$-braid is an element
$X=(x_{1},\ldots,x_{k})\in\mathcal{L}(C_{k}(\Sigma))$. Denote by
$B^{k}(\Sigma)$ the space of ordered $k$-braids. The loop $x_{i}$ is called
the $i$-th strand of $X$, for $i=1,\ldots,k$.

######  Definition 2.3.

An unordered $k$-braid is an element $[X]\in\mathcal{L}(C_{k}(\Sigma))/S_{k}$,
where $S_{k}$ acts by permutation of coordinates. Such unordered braids may be
identified with certain finite subsets of $\mathcal{L}(\Sigma)$.

######  Remark.

We raise the distinction between ordered and unordered braids here mainly to
flag for the reader that we will make no real effort outside of this section
to separate these two concepts. In particular, we will routinely treat ordered
braids as finite subsets of $\mathcal{L}(\Sigma)$ and perform set-wise
operations on them, when properly speaking we should be speaking of the
unordered braids which they represent. We will moreover speak simply of
‘braids’ relying on the context to make clear whether these braids are ordered
or unordered. For the remainder of this section, we will make a clear
distinction between ordered and unordered braids, mainly to convince the
suspicious reader that nothing essential is lost in making this elision.

######  Definition 2.4.

The graph $\check{X}$ of an (ordered) $k$-braid is the set-valued map
$\check{X}(t)=\sqcup_{i=1}^{k}\check{x}(t)\subseteq S^{1}\times\Sigma$, $t\in
S^{1}$. The graph of an unordered braid $[X]$ is the graph of some (hence
every) representative $X$ of $[X]$.

######  Definition 2.5.

An ordered $k^{\prime}$-braid $Y\in B^{k^{\prime}}(\Sigma)$ is an ordered sub-
braid of $X\in B^{k}(\Sigma)$ if $Y\subseteq X$, as an ordered set. An
unordered braid $[Y]$ is a sub-braid of $[X]$ if $[Y]\subseteq[X]$ as sets.
There is an obvious partial ordering on the collection of sub-braids of $X$
(resp. of $[X]$).

######  Definition 2.6.

$X\in B^{k}(\Sigma)$ is contractible if each strand of $X$ is a contractible
loop. We write $B_{0}^{k}(\Sigma)$ for the space of contractible ordered
$k$-braids. $[X]\in B^{k}(\Sigma)/S_{k}$ is contractible if some (hence every)
representative $X$ of $[X]$ is contractible.

######  Definition 2.7.

A continuous map $h:[0,1]\rightarrow B^{k}(\Sigma)$ with $h(0)=X$, $h(1)=Y$ is
a braid homotopy from $X$ to $Y$. When such a map exists, we shall say that
$X$ and $Y$ are braid homotopic, denoted $X\simeq Y$. The map $(s,t)\mapsto
h_{i}(s,t)$ is called the $i$-th strand of $h$. To any braid homotopy, we
associate its graph
$\tilde{h}(s,t)=\sqcup_{i=1}^{k}\tilde{h}_{i}(s,t)\subseteq[0,1]\times
S^{1}\times\Sigma$, $(s,t)\in[0,1]\times S^{1}$.

######  Definition 2.8.

An ordered braid $X$ will be said to be trivial if all of its strands are
constant maps. We will sometimes write $0\in B^{k}(\Sigma)$ to stand for some
fixed but arbitrary trivial braid, when the particular choices of the constant
maps are unimportant. An unordered braid $[X]$ is trivial if some (hence
every) ordered representative is trivial.

######  Definition 2.9.

$X\in B^{k}(\Sigma)$ is unlinked if $X\simeq 0$. An ordered braid is linked if
it is not unlinked. An unordered braid is unlinked (resp. linked) if some,
hence every, ordered representative is unlinked (resp. linked).

######  Definition 2.10.

A continuous map $h:[0,1]\rightarrow\mathcal{L}(\Sigma)^{k}$ with $h(0)=X\in
B^{k}(\Sigma)$, $h(1)=Y\in B^{k}(\Sigma)$ will be called a braid cobordism if
there exists some $\delta>0$ such that $h(s)\in B^{k}(\Sigma),\;\forall
s\in(0,\delta)\cup(1-\delta,1)$.

######  Remark.

We will frequently find ourselves concerned with maps
$h:I\rightarrow\mathcal{L}(\Sigma)^{k}$, where $I=\mathbb{R}$ or $I=[a,b]$ for
some $a,b\in\mathbb{R}$, and in the case that $I=\mathbb{R}$, it will always
be the case that $h$ extends continuously to a map
$\bar{\mathbb{R}}\rightarrow\mathcal{L}(\Sigma)^{k}$ such that on some
neighbourhood of $\pm\infty$, the graphs of the strands of $h$ do not
intersect. In such a case, we will speak freely of ‘the’ braid cobordism
induced by $h$, which is simply any braid cobordism $h\circ\varphi$, where
$\varphi:\bar{I}\rightarrow[0,1]$ is any orientation-preserving
diffeomorphism.

######  Definition 2.11.

An (ordered) capped $k$-braid $\hat{X}$ is an equivalence class $[X,\vec{w}]$
where $X\in B_{0}^{k}(\Sigma)$ and $\vec{w}=(w_{1},\cdots,w_{k})$ with
$w_{i}:D^{2}\rightarrow\Sigma$ a capping disk for the $i$-the strand of $X$,
subject to the equivalence relation
$[X,\vec{w}]\sim[X^{\prime},\vec{w}^{\prime}]$ if and only if $X=X^{\prime}$
and $[w_{i}]\\#(-[w_{i}^{\prime}])=0\in\pi_{2}(\Sigma)$ for each
$i=1,\cdots,k$. The space of ordered capped $k$-braids is denoted by
$\widetilde{B}_{0}^{k}(\Sigma)$. The capped loop
$\hat{x}_{i}=[x_{i},w_{i}]\in\widetilde{\mathcal{L}_{0}}(\Sigma)$ is called
the $i$-th strand of $\hat{X}$. The notion of capped sub-braids
$\hat{Y}\subseteq\hat{X}$ is defined in the obvious way.

######  Remark.

The distinction between ordered capped braids and unordered capped braids
obtains here as well, and we adopt parallel conventions as those discussed in
the case of braids in Remark Remark.

$\pi_{2}(\Sigma)^{k}$ acts on $\widetilde{B}_{0}^{k}(\Sigma)$ by the obvious
‘gluing of spheres’:

$\displaystyle(A_{1},\ldots,A_{k})\cdot([x_{1},w_{1}],\ldots,[x_{k},w_{k}])$
$\displaystyle=([x_{1},A_{1}\\#w_{1}],\ldots,[x_{k},A_{k}\\#w_{k}]),$

where here we abuse notation slightly by thinking of
$A_{i}\in\pi_{2}(\Sigma,x_{i}(0))$ as being both a homotopy class of maps, as
well as a particular choice of a representative from that class. This action
does not descend to an action on $\widetilde{B}_{0}^{k}(\Sigma)/S_{k}$.
However, if we denote by
$Fix_{S_{k}}(\pi_{2}(\Sigma)^{k})\simeq\pi_{2}(\Sigma)$ the set of fixed
points of the action of the symmetric group on $\pi_{2}(\Sigma)^{k}$ by
permutation of coordinates, we obtain a well-defined induced action by
$Fix_{S_{k}}(\pi_{2}(\Sigma)^{k})$ on unordered braids given by
$(A,\ldots,A)\cdot[\hat{X}]=[(A,\ldots,A)\cdot\hat{X}]$, for
$A\in\pi_{2}(\Sigma)$.

######  Definition 2.12.

A trivial braid $0\in B^{k}(\Sigma)$ has a naturally associated capping
$\hat{0}\in\widetilde{B}_{0}^{k}(\Sigma)$ given by capping each strand of $0$
with the constant capping. We call any such braid a trivial capped braid. When
the particular components of a trivial capped braid are unimportant, we denote
some fixed but arbitrary capped braid by the symbol $\hat{0}$. An unordered
capped braid is said to be trivial if some (hence every) ordered
representative is trivial.

######  Definition 2.13.

For $A=(A_{1},\cdots,A_{k})\in\pi_{2}(\Sigma)^{k}$, an ordered braid cobordism
$h$ from $X$ to $Y$ will be called an $A$-cobordism from $[X,\vec{w}]$ to
$[Y,\vec{v}]$ if $[w_{i}]\\#[h_{i}]\\#(-[v_{i}])=A_{i}$, for all
$i=1,\cdots,k$. Whenever such a map exists, $[X,\vec{w}]$ and $[Y,\vec{v}]$
will be said to be $A$-cobordant. This notion descends to unordered capped
braids provided that $A\in Fix_{S_{k}}(\pi_{2}(\Sigma)^{k})$.

######  Definition 2.14.

If $u:[0,1]\rightarrow\mathcal{L}_{0}(\Sigma)$ is homotopy from $x$ to $y$,
then for any choice of cappings $\hat{x}=[x,w_{x}]$ and $\hat{y}=[y,w_{y}]$,
$u$ is an $A$-cobordism from $\hat{x}$ to $\hat{y}$, for
$A=[w_{x}]\\#[u]\\#(-[w_{y}])\in\pi_{2}(\Sigma)$. Moreover, for any
$s\in[0,1]$, there are two natural choices of cappings for the loop
$u_{s}\in\mathcal{L}_{0}(\Sigma)$. Namely, if we write
$\alpha^{s}(\tau):=u(s\cdot\tau,t)$ and $\beta(\tau):=u(1-(1-s)\cdot\tau,t)$
for $s\in[0,1]$, then we may associate to $u_{s}$ either of the cappings
$[u_{s},w_{x}\\#\alpha^{s}]$ or $[u_{s},w_{y}\\#\beta^{s}]$, and these two
cappings are obviously related by
$A\cdot[u_{s},w_{y}\\#\beta^{s}]=[u_{s},w_{x}\\#\alpha^{s}]$. Consequently, if
$u$ is a $0$-homotopy between $\hat{x}$ and $\hat{y}$, these two cappings
agree and we may associate a unique capping

$\displaystyle\hat{u}_{s}:=[u_{s},w_{x}\\#\alpha^{s}]=[u_{s},w_{y}\\#\beta^{s}]$

to each $u_{s}$ in this case. We will call such a capping the natural capping
of $u_{s}$ whenever $u$ is such a $0$-homotopy.

######  Definition 2.15.

An $A$-cobordism $h$ from $\hat{X}=[X,\vec{w}]$ to $\hat{Y}=[Y,\vec{v}]$ is
called an $A$-homotopy if $h$ is in addition a braid homotopy from $X$ to $Y$.
In such a situation, we will say that $\hat{X}$ and $\hat{Y}$ are
$A$-homotopic, and we will denote the relation by $\hat{X}\simeq_{A}\hat{Y}$.
This notion descends to unordered capped braids, provided that $A\in
Fix_{S_{k}}(\pi_{2}(\Sigma))$.

######  Definition 2.16.

$\hat{X}\in\widetilde{B}_{0}^{k}(\Sigma)$ is unlinked if
$\hat{X}\simeq_{0}\hat{0}$. An unordered capped braid $[\hat{X}]$ is unlinked
if some (hence every) ordered representative is unlinked. The notion of
linkedness for a capped braid or for a pair of capped braids is defined as in
the case of braids.

###  2.2 Linking of capped loops with close strands

For use in the sequel, we explain here a minor adaptation of the classical
linking number of two loops in the plane $x,y:S^{1}\rightarrow\mathbb{R}^{2}$
to two capped loops $\hat{x}$ and $\hat{y}$ such that the underlying loops $x$
and $y$ lie sufficiently close to each other in an arbitrary symplectic
surface $(\Sigma,\omega)$.

To any $x\in\mathcal{L}_{0}(\Sigma)$, we may associate the set

$\displaystyle S_{x}$ $\displaystyle:=\\{y\in\mathcal{L}_{0}(\Sigma):\exists
t\in S^{1}\;\text{such that}\;x(t)=y(t)\\},$

with the property that $\mathcal{L}_{0}(\Sigma)\setminus S_{x}$ consists of
precisely those loops $y$ such that $(x,y)\in\mathcal{L}_{0}(\Sigma)^{2}$ is a
braid.

We fix some family $J=(J_{t})_{t\in S^{1}}$ of $\omega$-compatible almost
complex structures, and let $g_{J}=(g_{J_{t}})_{t\in S^{1}}$ denote the
associated family of compatible metrics. This data provides us with an
exponential neighbourhood $\mathcal{O}\subset\mathcal{L}_{0}(M)$ of $x$, along
with a diffeomorphism

$\displaystyle Exp:U\subset\Gamma^{\infty}(x^{*}T\Sigma)$
$\displaystyle\rightarrow\mathcal{O}$ $\displaystyle\xi$
$\displaystyle\mapsto\\{t\mapsto exp^{J_{t}}_{x(t)}(\xi(t))\\}$

from a neighbourhood of the zero section onto $\mathcal{O}$.

Remark that any choice of a lift
$\hat{x}=[x,\alpha]\in\widetilde{\mathcal{L}_{0}}(\Sigma)$ of $x$ gives rise
to a lift $\tilde{\mathcal{O}}_{\alpha}$ of $\mathcal{O}$, and
$T_{\hat{x}}\widetilde{\mathcal{L}_{0}}(\Sigma)\simeq(x^{*}T\Sigma,J,\omega)$
comes equipped with a homotopically unique unitary trivialization

$\displaystyle T_{\hat{x}}:S^{1}\times(\mathbb{R}^{2},J_{0},\omega_{0})$
$\displaystyle\rightarrow(x^{*}T\Sigma,J,\omega),$

provided by any unitary trivialization which extends over the capping. For any
$y\in\mathcal{O}\setminus S_{x}$ and any capping
$\hat{x}_{\alpha}:=[x,\alpha]$ of $x$, let $\hat{y}_{\alpha}$ denote the
unique lift of $y$ lying in $\widetilde{\mathcal{O}}_{\alpha}$. We define the
linking number of $\hat{x}_{\alpha}$ and $\hat{y}_{\alpha}$ as

$\displaystyle\ell(\hat{x}_{\alpha},\hat{y}_{\alpha})$
$\displaystyle:=wind((T_{\hat{x}}^{-1}\circ\widetilde{Exp}^{-1})(\hat{y}_{\alpha})),$

where $wind(\xi)$ denotes the classical winding number of a non-vanishing
family of vectors $t\mapsto\xi(t)$ for $t\in S^{1}$ in $\mathbb{R}^{2}$. Note
that for $A\in\pi_{2}(\Sigma)$, we clearly have

$\displaystyle\ell(A\cdot\hat{x}_{\alpha},A\cdot\hat{y}_{\alpha})$
$\displaystyle=\ell(\hat{x}_{\alpha},\hat{y}_{\alpha})+c_{1}(A),$

and it is moreover not hard to show that $\ell$ symmetric in its arguments. In
order to extend this definition to arbitrary cappings of the loops $x$ and
$y$, let $A,B\in\pi_{2}(\Sigma)$ and define

$\displaystyle\ell(A\cdot\hat{x}_{\alpha},B\cdot\hat{y}_{\alpha})$
$\displaystyle:=\ell(\hat{x}_{\alpha},\hat{y}_{\alpha})+\frac{1}{2}(c_{1}(A)+c_{1}(B)).$

It is not hard to check that this definition does not depend on the choice of
$\alpha$, nor the choice of compatible almost complex structure
$J=(J_{t})_{t\in S^{1}}$, and agrees with the previous definition in the case
that $\hat{x}$ and $\hat{y}$ are close in
$\widetilde{\mathcal{L}_{0}}(\Sigma)$. A more geometric view of this formula
will be provided in the following two sections.

###  2.3 The homological linking number for capped braids

######  Definition 2.17.

Let $\hat{X},\hat{Y}\in\widetilde{B}_{0}^{k}(\Sigma)$ and
$A\in\pi_{2}(\Sigma)^{k}$. We define the homological ($A$)-linking number of
$\hat{Y}$ relative to $\hat{X}$

$\displaystyle L_{A}(\hat{X};\hat{Y})$ $\displaystyle:=\sum_{1\leq i<j\leq
k}\\#(\tilde{h}_{i}\pitchfork\tilde{h}_{j}),$

where $h=(h_{1},\cdots,h_{k})$ is any $A$-cobordism from $\hat{X}$ to
$\hat{Y}$ such that the graphs of the strands of $h$ in $[0,1]\times
S^{1}\times\Sigma$ are all pairwise transverse, and
$\\#(\tilde{h}_{i}\pitchfork\tilde{h}_{j})$ denotes the signed count of the
intersections of the graphs $\tilde{h}_{i}$ and $\tilde{h}_{j}$ (recall that
in our setting $\Sigma$ carries the orientation induced by $\omega$).

######  Remark.

The above definition may be generalized straightforwardly by replacing the
cylinder $[0,1]\times S^{1}$ with a surface $S_{g,k^{-},k^{+}}$ of genus $g$,
having $k^{-}$ negatively oriented boundary components and $k^{+}$ positively
oriented boundary components. This provides a family of homotopy invariants
for collections of $k^{-}$ ‘input’ and $k^{+}$ ‘output’ capped braids in the
obvious way. Much of the theory developed in this work can be adapted in a
straightforward way to use such invariants to extract information about the
field-theoretic operations in Floer theory described in [24] at the chain
level for Hamiltonian isotopies on surfaces, but we will not pursue this
extension in this paper.

The following proposition summarizes the main properties of the homological
linking number which we will need in our investigations.

######  Proposition 2.18.

For any $\hat{X},\hat{Y},\hat{Z}\in\widetilde{B}_{0}^{k}(\Sigma)$ and
$A,B\in\pi_{2}(\Sigma)^{k}$ we have that:

  1. 1.

$L_{A}(\hat{X},\hat{Y})$ is well-defined.

  2. 2.

For any $\sigma\in S_{k}$, $L_{\sigma\cdot
A}(\sigma\cdot\hat{X};\sigma\cdot\hat{Y})=L_{A}(\hat{X};\hat{Y})$.

  3. 3.

$L_{A}(\hat{X},\hat{Y})+L_{B}(\hat{Y},\hat{Z})=L_{A+B}(\hat{X},\hat{Z})$.

  4. 4.

If $\hat{X}$ and $\hat{Y}$ are $A$-homotopic, then $L_{A}(\hat{X},\hat{Y})=0$.

  5. 5.

$L_{A}(\hat{X},\hat{Y})=-L_{-A}(\hat{Y},\hat{X})$.

  6. 6.

$L_{A}(\hat{X},B\cdot\hat{Y})=L_{A+B}(\hat{X},\hat{Y})$.

  7. 7.

$L_{0}(\hat{X},A\cdot\hat{X})=(k-1)\sum_{i=1}^{k}\frac{c_{1}(A_{i})}{2}$.

###### Proof.

  1. 1.

That $L_{A}(\hat{X},\hat{Y})$ is well-defined follows from the standard
transversality arguments that are typical in differential topology.
Alternately, one may simply note that for any braid cobordism
$h=(h_{1},\cdots,h_{k})$ and any $i=1,\cdots,k$, the graph of $h_{i}$ in
$[0,1]\times S^{1}\times\Sigma$ defines a compact surface with boundary
$S_{i}\subset[0,1]\times S^{1}\times\Sigma$ which induces a well-defined
element of

$\displaystyle[S_{i}]$ $\displaystyle\in H_{2}([0,1]\times
S^{1}\times\Sigma;\check{X}\sqcup\check{Y}),$

where $\check{X}$ and $\check{Y}$ denote the graphs of the braids $X$ and $Y$
respectively, thought of as submanifolds lying in $\\{0\\}\times
S^{1}\times\Sigma$ and $\\{1\\}\times S^{1}\times\Sigma$, respectively. The
intersection product of such classes is well-defined and

$\displaystyle L_{A}(\hat{X},\hat{Y})$ $\displaystyle=\sum_{1\leq i<j\leq
k}[S_{i}]\cup[S_{j}],$

which, since $[0,1]\times S^{1}\times(\Sigma,\omega)$ is canonically oriented
by $(ds\wedge dt)\wedge\omega$, is obviously precisely what is computed by the
sum of pairwise intersection numbers of the graphs when these are transverse.

  2. 2.

This statement follows upon remarking that for any $\sigma\in S_{k}$ and any
$A\in\pi_{2}(\Sigma)^{k}$, $h=(h_{1},\ldots,h_{k})$ is an $A$-cobordism from
$\hat{X}$ to $\hat{Y}$ if and only if $\sigma\cdot
h:=(h_{\sigma^{-1}(1)},\ldots,h_{\sigma^{-1}(k)})$ is a $(\sigma\cdot
A)$-cobordism from $\sigma\cdot\hat{X}$ to $\sigma\cdot\hat{Y}$.

  3. 3.

This is straightforward and follows directly from concatenating an
$A$-cobordism from $\hat{X}$ to $\hat{Y}$ with a $B$-cobordism from $\hat{Y}$
to $\hat{Z}$.

  4. 4.

An $A$-cobordism $h=(h_{1},\cdots,h_{k})$ is an $A$-homotopy precisely when
the graphs of the $h_{i}$, $i=1,\cdots,k$ are all disjoint. Clearly this
implies $L_{A}(\hat{X},\hat{Y})=0$.

  5. 5.

This follows immediately from noting that $h$ is an $A$-cobordism from
$\hat{X}$ to $\hat{Y}$ if and only if $\bar{h}(s,t):=h(1-s,t)$ is a
$(-A)$-cobordism from $\hat{Y}$ to $\hat{X}$.

  6. 6.

This follows immediately from the equivalence of the homotopy conditions

$\displaystyle\alpha_{i}\\#[h_{i}]\\#-\beta_{i}$
$\displaystyle=A_{i}\\#B_{i}=A_{i}+B_{i},\;\text{and}$
$\displaystyle\alpha_{i}\\#[h_{i}]\\#-(A_{i}\cdot\beta_{i})$
$\displaystyle=B_{i}.$

That is, $h$ is an $A+B$-cobordism from $\hat{X}$ to $\hat{Y}$ if and only if
$h$ is also a $B$-cobordism from $\hat{X}$ to $A\cdot\hat{Y}$.

  7. 7.

We note first that item $(2)$ implies that

$\displaystyle
L_{0}(\hat{0},\hat{X})+L_{0}(\hat{X},A\cdot\hat{X})+L_{0}(A\cdot\hat{X},A\cdot\hat{0})$
$\displaystyle=L_{0}(\hat{0},A\cdot\hat{0}).$

Next, items $(4)$ and $(5)$ imply that

$\displaystyle L_{0}(A\cdot\hat{X},A\cdot\hat{0})$
$\displaystyle=L_{A}(A\cdot\hat{X},\hat{0})$
$\displaystyle=-L_{-A}(\hat{0},A\cdot\hat{X})$
$\displaystyle=-L_{0}(\hat{0},\hat{X}),$

whence we need only show that the desired formula holds when
$\hat{X}=\hat{0}$. To reduce to an even simpler case, let us write $A$ as

$\displaystyle(A_{1},\cdots,A_{k})$
$\displaystyle=(A_{1},0,\cdots,0)+(0,A_{2},0,\cdots,0)+\cdots+(0,0,\cdots,A_{k})$
$\displaystyle=A_{1}^{\prime}+\cdots+A_{k}^{\prime}.$

By items $(2)$ and $(5)$, demonstrating the desired equality is therefore
equivalent to showing that

$\displaystyle L_{0}(\hat{0},A_{i}^{\prime}\cdot\hat{0})$
$\displaystyle=(k-1)\cdot\frac{c_{1}(A_{i})}{2}$

for any $i=1,\cdots,k$. In what follows, let
$(\hat{p}_{1},\cdots,\hat{p}_{k})=\hat{0}$ represent the trivial capped braid.
Since the statement is trivial when $\Sigma\neq S^{2}$, as then
$\pi_{2}(\Sigma)=0$ and every capped braid is $0$-homotopic to itself, we now
suppose $\Sigma=S^{2}$. For $m\in\mathbb{Z}$, if
$u_{i}:(S^{2},\ast)\rightarrow(\Sigma,p_{i})$ represents
$A_{i}=m[S^{2}]\in\pi_{2}(\Sigma,p_{i})$, we may pull $u_{i}$ back along the
quotient $[0,1]\times S^{1}\rightarrow S^{2}$ (given by collapsing the
boundary circles to points) to a map which we will denote

$\displaystyle h_{i}:[0,1]\times S^{1}$ $\displaystyle\rightarrow\Sigma.$

If we take $h$ to be the $0$-cobordism from $\hat{0}$ to
$A_{i}^{\prime}\cdot\hat{0}$ given by $h_{i}$ as the $i$-th strand and the
constant strand $h_{j}(s,t)\equiv p_{j}$ for all other strands $j\neq i$, then
the important point is that $PD(c_{1})=2[S^{2}]$ and hence the intersection of
the graph of $h_{i}$ with the constant cylinder $h_{j}(s,t)\equiv p_{j}$ for
$j\neq i$ contributes precisely

$\displaystyle(u_{i*}[S^{2}])\cap[p_{j}]$ $\displaystyle=m[S^{2}]$
$\displaystyle=\frac{c_{1}(A_{i})}{2}$

to the sum defining $L_{0}(\hat{0},A_{i}^{\prime}\cdot\hat{0})$, and such
intersections are the only ones that occur, since all other strands are
constant and disjoint. The desired equality follows.

∎

######  Proposition 2.19.

For $A\in\pi_{2}(\Sigma)^{k}$, and
$[\hat{X}],[\hat{Y}]\in\widetilde{B}_{0}^{k}(\Sigma)/S_{k}$, the function
$L_{A}([\hat{X}];[\hat{Y}]):=L_{A}(\hat{X};\hat{Y})$, is well-defined.

###### Proof.

The previous proposition implies that for any $\sigma,\tau\in S_{k}$, we have

$\displaystyle L_{A}(\sigma\cdot\hat{X};\tau\cdot\hat{Y})$
$\displaystyle=L_{A}(\hat{0};\hat{0})-L_{0}(\hat{0};\sigma\cdot\hat{X})+L_{0}(\hat{0};\tau\cdot\hat{Y}),$

so it suffices to show that the expression $L_{0}(\hat{0};\sigma\cdot\hat{X})$
is independent of $\sigma\in S_{k}$. To see this, note that item $2$ of the
previous proposition, together with the fact that $0\in
Fix_{S_{k}}(\pi_{2}(\Sigma)^{k})$ implies that
$L_{0}(\hat{0};\sigma\cdot\hat{X})=L_{0}(\sigma^{-1}\cdot\hat{0};\hat{X})$.
Moreover, it is easy to see that $\hat{0}=(\hat{p}_{1},\ldots,\hat{p}_{k})$ is
$0$-homotopic to
$\sigma\cdot\hat{0}=(\hat{p}_{\sigma^{-1}(1)},\ldots,p_{\sigma^{-1}(k)})$ for
any $\sigma\in S_{k}$ (simply choose $k$ paths
$s\mapsto\gamma_{i}(s)\in\Sigma$, $s\in[0,1]$ from $p_{i}$ to
$p_{\sigma^{-1}(i)}$ such that $\gamma_{i}(s)=\gamma_{j}(s)$ implies $i=j$ for
all $s\in[0,1]$ and define the $i$-th strand of the $0$-homotopy to be
$h_{i}(s,t)=\gamma_{i}(s)$) and consequently,
$L_{0}(\sigma^{-1}\cdot\hat{0};\hat{X})=L_{0}(\hat{0};\hat{X})$, which is
independent of $\sigma\in S_{k}$. ∎

######  Proposition 2.20.

Let $\hat{X}=(\hat{x}_{1},\hat{x}_{2})\in\widetilde{B}_{0}^{2}(\Sigma)$ with
$x_{2}$ lying in some exponential of neighbourhood of $x_{1}$ in
$\mathcal{L}_{0}(\Sigma)$ (as in Section 2.2), then
$L_{0}(\hat{0};\hat{X})=\ell(\hat{x}_{1},\hat{x}_{2})$.

###### Proof.

As $L_{0}(\hat{0};\hat{X})$ depends on $\hat{X}$ only up to a $0$-homotopy, we
may assume without loss of generality that $x_{1}$ is a constant loop.
Moreover, noting that if $A,B\in\pi_{2}(\Sigma)$, then

$\displaystyle L_{0}(\hat{0};(A,B)\cdot\hat{X})-L_{0}(\hat{0};\hat{X})$
$\displaystyle=\frac{1}{2}(c_{1}(A)+c_{1}(B))$
$\displaystyle=\ell(A\cdot\hat{x}_{1},B\cdot\hat{x}_{2})-\ell(\hat{x}_{1},\hat{x}_{2}),$

and so it suffices to prove the statement in the case in which $\hat{x}_{1}$
is a trivially capped constant loop and $\hat{x}_{2}$ lies inside an
exponential neighbourhood of $\tilde{x}_{1}$ in
$\widetilde{\mathcal{L}_{0}}(\Sigma)$.

As discussed in Section 2.2, $T_{\hat{x}}\widetilde{\mathcal{L}_{0}}(\Sigma)$
is naturally identified (up to a homotopy of trivializations) with
$\Gamma^{\infty}(S^{1}\times\mathbb{R}^{2})$ and so (in the notation of that
section) we may write $\tilde{x}_{2}$ in local coordinates as

$\displaystyle v(t)$
$\displaystyle:=(T_{\hat{x}_{1}}^{-1}\circ\widetilde{Exp}^{-1})(\tilde{x}_{2}),$

and we have that $\ell(\hat{x}_{1},\hat{x}_{2})=wind(v_{2})$ by definition. By
the capped braid homotopy-invariance of the homological linking number, and
the homotopy invariance of the winding number in
$\mathbb{R}^{2}\setminus\\{0\\}$, we may assume that

$\displaystyle v(t)$ $\displaystyle=r_{0}e^{2\pi ilt}\in\mathbb{C},\;\forall
t\in[0,1],$

for some small $r_{0}>0$ and $l=\ell(\hat{x}_{1};\hat{x}_{2})$. Taking the
homotopies $h_{1}(s,t)\equiv 0$,
$h_{2}(s,t)=(1-s)\frac{-r_{0}}{2}+s\frac{r_{0}}{2}e^{2\pi ilt}$, we see that
their graphs $\tilde{h}_{1}$ and $\tilde{h}_{2}$ intersect only if $l\neq 0$,
and in that case intersections occur when $s=\frac{1}{2}$ and
$t=0,\frac{1}{l},\cdots,\frac{l-1}{l}$. Moreover, using polar coordinates
$(r,\theta)$ on $D^{2}$, since

$\displaystyle\partial_{s}h_{2}$ $\displaystyle=\partial_{r}$
$\displaystyle\partial_{t}h_{2}$ $\displaystyle=l\partial_{\theta},$

each intersection transverse and has orientation $sign(l)$. Consequently

$\displaystyle L_{0}(\hat{0};\hat{X})$
$\displaystyle=l=\ell(\hat{x}_{1},\hat{x}_{2}),$

proving the claim. ∎

The following proposition is not used anywhere in this paper, but we include
it because it provides some intuitive justification for the relationship
between the homological linking number of capped braids symplectic
measurements on surfaces: the area functional on the space of capped loops may
be computed at a given capped loop $\hat{x}$ as simply being the average
linking number of $\hat{x}$ with a generic set of points in the surface
(viewed as trivially capped loops).

######  Proposition 2.21.

Suppose that $\hat{\gamma}=[\gamma,w]$ is a capped loop such that $\gamma$ is
smooth, then

$\displaystyle\int_{D^{2}}w^{*}\omega$
$\displaystyle=\int_{\Sigma\setminus\text{im}\;\gamma}\ell(\hat{\gamma},\hat{x})\omega,$

where for $x\in\Sigma$, $\hat{x}=[x,x]$ denotes the trivially capped constant
loop based at $x$.

###### Proof.

Suppose without loss of generality that $w:D^{2}\rightarrow\Sigma$ is a smooth
map with $w(0)=\gamma(0)$. The main point is to notice that for any
$x\in\Sigma\setminus\text{im}\;\gamma$ such that $w$ is transversal to $x$, we
have that $\ell(\hat{\gamma},\hat{x})=deg(w)_{x}$, where $deg(w)_{x}$ denotes
the local degree of $w$ at $x$. Indeed, transversality of $w$ to $x$ implies
the transversality of the maps into $[0,1]\times S^{1}\times\Sigma$ defined by
$\tilde{w}(s,t)=(s,t,w(se^{2\pi it}))$ and $\tilde{x}(s,t)=(s,t,x)$, for
$(s,t)\in[0,1]\times S^{1}$, and obviously the algebraic count of the
intersection number between these two graphs is identical with $deg(w)_{x}$.
One immediate consequence is that $\ell(\hat{\gamma},\hat{x})=0$ for all
$x\not\in\text{im}\;w$, so it suffices to show that

$\displaystyle\int_{D^{2}}w^{*}\omega$
$\displaystyle=\int_{\text{im}\;w\setminus\text{im}\;\gamma}\ell(\hat{\gamma},\hat{x})\omega.$

To this end, denote by $S(w)\subseteq D^{2}$ the set of points such that $Dw$
is not of full rank and define $G(w)\subseteq D^{2}$ to be
$G(w)=w^{-1}(\text{im}\;\gamma)\setminus S(w)$. Note that $G(w)$ is a set of
measure $0$, since we may realize $G(w)$ as the projection onto
$D^{2}\setminus S(w)$ of $(w|_{D^{2}\setminus
S(w)}\times\gamma)^{-1}(\triangle)$ which is a submanifold of $D^{2}\setminus
S(w)$ of codimension $2$. Next note that we must have
$\int_{S(w)}w^{*}\omega=0$, since $w^{*}\omega$ vanishes on $S(w)$.
Consequently, writing $N:=S(w)\cup G(w)$, we note that
$\text{im}\;\gamma\subseteq w(N)$ and so it suffices to establish that

$\displaystyle\int_{D^{2}\setminus N}w^{*}\omega$
$\displaystyle=\int_{w(D^{2}\setminus
N)}\ell(\hat{\gamma},\hat{x})\omega=\int_{w(D^{2}\setminus
N)}deg(w)_{x}\omega.$

The local degree is a locally constant function of $x$, and so we obtain

$\displaystyle\int_{w(D^{2}\setminus N)}deg(w)_{x}\omega$
$\displaystyle=\sum_{C\in\pi_{0}(w(D^{2}\setminus
N))}deg(w)|_{C}\int_{C}\omega=\sum_{C\in\pi_{0}(w(D^{2}\setminus
N))}\int_{w^{-1}(C)}w^{*}\omega=\int_{D^{2}\setminus N}w^{*}\omega$

as claimed. ∎

##  3 Elements of Floer theory and linking

This section serves to collect and review the necessary facts that we will
need from Hamiltonian Floer theory, along with the analysis of the asymptotic
behaviour of Floer-type cylinders that proves crucial to our study. §3.1
collects the Floer-theoretic preliminaries. §3.2 explains how we may combine a
result formulated by Siefring in [30], which relates the asymptotic behaviour
or Floer-type cylinders tending to an orbit $x$ to the eigenvectors of the so-
called asymptotic operator associated to $x$, with Hofer-Wysocki-Zehnder’s
study in [8] of the winding behaviour of such eigenvectors, in order to obtain
information about the asymptotics of Floer-type cylinders. We then explain how
we may combine this information with the homological linking number of §2.3 to
obtain control over the relative topology of collections of capped braids
having Floer-type cylinders running between them. The results of the present
section form the technical core of the rest of the work.

###  3.1 Floer theory

In this section, we give a rapid overview of the elements of Floer theory of
which we will have need, mainly to fix notation and conventions. For a more
detailed treatment, see [1] or [26] for standard accounts of Hamiltonian Floer
theory (see also [7] for its adaptation to the weakly monotone case) and [27],
[24] or [13] for a more detailed treatment of how Floer theory fits into a
field theory over surfaces. Throughout, we assume that $(M,\omega)$ is a semi-
positive compact symplectic manifold of dimension $2n$, while
$\mathcal{J}(M,\omega)$ denotes the space of all smooth $\omega$-compatible
almost complex structures. For convenience, we work only with
$\mathbb{Z}_{2}$-coefficients, but this restriction is inessential.

A smooth (not necessarily autonomous) Hamiltonian function

$\displaystyle H:S^{1}\times M$ $\displaystyle\rightarrow\mathbb{R}$
$\displaystyle(t,x)$ $\displaystyle\mapsto H_{t}(x)$

induces a time-dependent vector field $(X^{t}_{H})_{t\in[0,1]}$ on $M$ defined
by the relation

$\displaystyle\omega(X^{t}_{H},-)$ $\displaystyle=-dH_{t}.$

The Hamiltonian isotopy obtained as the flow by this vector field is denoted
$\phi^{H}:=(\phi^{H}_{t})_{t\in[0,1]}$. A Hamiltonian $H$ is said to be
normalized if $\int_{M}H_{t}\omega^{n}=0$ for all $t\in S^{1}$. There is a
group structure Hamiltonian functions $C^{\infty}(S^{1}\times M)$ given by the
operation

$\displaystyle(H\\#K)(t,x)$
$\displaystyle:=H(t,x)+K(t,(\phi^{H}_{t})^{-1}(x)),$

which is such that $\phi^{H\\#K}_{t}=\phi^{H}_{t}\circ\phi^{K}_{t}$. The
inverse of $H$ with respect to this relation is given by

$\displaystyle\bar{H}(t,x)$ $\displaystyle=-H(t,\phi^{H}_{t}(x)),$

which generates the isotopy $t\mapsto(\phi^{H}_{t})^{-1}$.

The Hamiltonian $H$ defines a corresponding action functional on the space of
capped loops

$\displaystyle\mathcal{A}_{H}:\widetilde{\mathcal{L}_{0}}(M)$
$\displaystyle\rightarrow\mathbb{R}$ $\displaystyle[\gamma,v]$
$\displaystyle\mapsto\int_{0}^{1}H_{t}(\gamma(t))\;dt-\int_{D^{2}}v^{*}\omega.$

We write

$\displaystyle\widetilde{Per}_{0}({H})$
$\displaystyle:=Crit\;\mathcal{A}_{H},\;\text{and}$ $\displaystyle Per_{0}(H)$
$\displaystyle:=\pi(\widetilde{Per}_{0}({H}))\subseteq\mathcal{L}_{0}(M),$

noting that the latter consists precisely of the contractible $1$-periodic
orbits of $\phi^{H}$, while the former consists of capped such periodic
orbits.

$H$ is said to be non-degenerate if for all $x\in Per_{0}(H)$,
$(D\phi^{H}_{1})_{x(0)}$ has no eigenvalues equal to $1$. When $H$ is non-
degenerate, there exists a well-defined Conley-Zehnder index

$\displaystyle\mu=\mu_{CZ}:\widetilde{Per}_{0}({H})$
$\displaystyle\rightarrow\mathbb{Z}.$

See [25] for details on the definition of $\mu$. We shall normalize the
Conley-Zehnder index by insisting that if $H$ is a $C^{2}$-small Morse
function and $x$ a critical point of $H$, then

$\displaystyle\mu(\hat{x})$ $\displaystyle=\mu_{Morse}(x)-n,$

where $\mu_{Morse}$ is the Morse index of $x$, and $\hat{x}$ denotes the
trivial capping of the constant orbit $x$. For $k\in\mathbb{Z}$, and any
$\hat{P}\subseteq\widetilde{Per}_{0}({H})$ we define $\hat{P}_{(k)}$ to be the
collection of capped orbits in $\hat{P}$ with Conley-Zehnder index $k$.

Given $\hat{x}^{\pm}=[x^{\pm},w^{\pm}]\in\widetilde{Per}_{0}({H})$, we write
$C^{\infty}(\mathbb{R}\times S^{1};M)_{\hat{x}^{-},\hat{x}^{+}}$ for the
subspace of $C^{\infty}(\mathbb{R}\times S^{1};M)$ consisting of cylinders
which induce a $0$-homotopy from $\hat{x}^{-}$ to $\hat{x}^{+}$. Letting

$\displaystyle\mathcal{E}$ $\displaystyle\rightarrow
C^{\infty}(\mathbb{R}\times S^{1};M)_{\hat{x}^{-},\hat{x}^{+}}$

be the infinite dimensional vector bundle with fiber
$\mathcal{E}_{u}=\Gamma^{\infty}(u^{*}TM)$ at $u$, any smooth $S^{1}$-family
$J=(J_{t})_{t\in S^{1}}\subseteq\mathcal{J}(M,\omega)$, permits the definition
of the Floer operator, which is the section

$\displaystyle\mathcal{F}_{H,J}:C^{\infty}(\mathbb{R}\times
S^{1};M)_{\hat{x}^{-},\hat{x}^{+}}$ $\displaystyle\rightarrow\mathcal{E}$
$\displaystyle u$ $\displaystyle\mapsto\partial_{s}u+J(\partial_{t}u-X_{H}).$

After passing to appropriate Banach space completions (see Section 8.2 of [1]
for instance), $\mathcal{F}_{H,J}$ defines a Fredholm operator with index
$\mu(\hat{x}^{-})-\mu(\hat{x}^{+})$. The intersection of $\mathcal{F}_{H,J}$
with the $0$-section gives rise to Floer’s equation

$\displaystyle\partial_{s}u+J_{t}(\partial_{t}u-X_{H}^{t})$ $\displaystyle=0$
(1)

for smooth maps $u:\mathbb{R}\times S^{1}\rightarrow M$. If we define the
energy of $u\in C^{\infty}(\mathbb{R}\times S^{1};M)$ by

$\displaystyle E(u)$ $\displaystyle:=\int_{\mathbb{R}\times
S^{1}}\|\partial_{s}u\|^{2}_{J_{t}}\;dt\;ds,$

then the finite energy solutions of Floer’s equation may be thought of as the
projections to $M$ of negative gradient flow lines of $\mathcal{A}_{H}$ with
respect to the $L^{2}$-metric on $\widetilde{\mathcal{L}_{0}}(M)$ induced by
$J$. It follows easily from this that if $u\in C^{\infty}(\mathbb{R}\times
S^{1};M)_{\hat{x}^{-},\hat{x}^{+}}$ is such a finite energy solution, then
$E(u)=\mathcal{A}_{H}(\hat{x}^{-})-\mathcal{A}_{H}(\hat{x}^{+})$.

For any $\hat{x}^{\pm}\in\widetilde{Per}_{0}({H})$, we define define
$\widetilde{\mathcal{M}}(\hat{x}^{-},\hat{x}^{+};H,J)$ to be the zero set of
$\mathcal{F}_{H,J}$ on $C^{\infty}(\mathbb{R}\times
S^{1};M)_{\hat{x}^{-},\hat{x}^{+}}$. It carries an obvious $\mathbb{R}$-action
given by translation in the $s$-coordinate. The reduced moduli space is
defined by

$\displaystyle\mathcal{M}(\hat{x},\hat{y};H,J)$
$\displaystyle:=\widetilde{\mathcal{M}}(\hat{x},\hat{y};H,J)/\mathbb{R}.$

######  Definition 3.1.

A pair $(H,J)$ with $H$ and $J$ as above, $H$ non-degenerate, are said to be
Floer regular if the intersection of the Floer operator with the $0$-section
is transverse (after taking appropriate Banach space completions) for all
$\hat{x},\hat{y}\in\widetilde{Per}_{0}({H})$ with
$\mu(\hat{x})-\mu(\hat{y})\leq 2$.

For $H$ non-degenerate, let $\mathcal{J}^{nd}(H)\subseteq
C^{\infty}(S^{1};\mathcal{J}(M,\omega))$ denote the space of $S^{1}$-families
of complex structures such that $(H,J)$ is Floer regular.
$\mathcal{J}^{nd}(H)$ is residual in
$C^{\infty}(S^{1};\mathcal{J}(M,\omega))$.

If $(H,J)$ is Floer regular, then $\mathcal{M}(\hat{x},\hat{y};H,J)$ is a
compact manifold of dimension $0$ whenever $\mu(\hat{x})-\mu(\hat{y})=1$, and
in this case we may define the Floer chain complex

$\displaystyle CF_{*}(H,J)$
$\displaystyle:=\mathbb{Z}_{2}\langle\hat{x}\rangle_{\hat{x}\in\widetilde{Per}_{0}({H})},$

which is graded by $\mu$ and which has differential defined on generators by

$\displaystyle\partial_{H,J}\hat{x}$
$\displaystyle:=\sum_{\mu(\hat{x})-\mu(\hat{y})=1}n(\hat{x},\hat{y})\hat{y},$

with $n(\hat{x},\hat{y})$ being the mod $2$ count of elements in
$\mathcal{M}(\hat{x},\hat{y};H,J)$. The homology of this complex $HF_{*}(H)$
is the Floer homology of $H$ and is independent of the choice of $J$.

The Floer complex has the structure of a filtered complex, with the filtration
coming from the action functional. Explicitly, for
$\sigma=\sum_{\hat{x}\in\widetilde{Per}_{0}({H})}a_{\hat{x}}\hat{x}\in
CF_{*}(H,J)$, we define

$\displaystyle\text{supp }\sigma$
$\displaystyle:=\\{\hat{x}\in\widetilde{Per}_{0}({H}):a_{\hat{x}}\neq 0\\},$

and we define the level of $\sigma$ to be

$\displaystyle\lambda_{H}(\sigma)$ $\displaystyle:=\sup_{\hat{x}\in\text{supp
}\sigma}\mathcal{A}_{H}(\hat{x}).$

Let $H^{S}_{2}(M)$ denote the image in $H_{2}(M;\mathbb{Z})$ of the Hurewicz
morphism, and let $\Gamma_{\omega}:=H^{S}_{2}(M)/\ker c_{1}\cap\ker[\omega]$.
We define the Novikov ring

$\displaystyle\Lambda_{\omega}$
$\displaystyle:=\\{\sum_{A\in\Gamma_{\omega}}\lambda_{A}e^{A}:\lambda_{A}\in\mathbb{Z}_{2},\\#\\{\lambda_{A}\neq
0,\omega(A)\leq c\\}<\infty,\;\text{for all }c\in\mathbb{R}\\}.$

This is a graded commutative ring with grading given by declaring
$deg(e^{A}):=2c_{1}(A)$. $CF_{*}(H,J)$ is a $\Lambda_{\omega}$-module where
the action of $\Lambda_{\omega}$ is defined on generators $\hat{x}=[x,v]$ of
$CF_{*}(H,J)$ and $e^{A}$ of $\Lambda_{\omega}$ by
$e^{A}\cdot\hat{x}:=[x,A\\#v]$, and extended linearly. Note that we have the
relations

$\displaystyle\mu(e^{A}\cdot\hat{x})$ $\displaystyle=\mu(\hat{x})-2c_{1}(A),$
$\displaystyle\mathcal{A}_{H}(e^{A}\cdot\hat{x})$
$\displaystyle=\mathcal{A}_{H}(\hat{x})-\omega(A).$

Remark that the $\Lambda_{\omega}$ action needn’t preserve the filtration.

It is a standard fact in Floer theory that if $f\in C^{\infty}(M)$ is a
sufficiently $C^{2}$-small Morse function and $J\in\mathcal{J}(M,\omega)$ is
such that $(f,g_{J})$ is Morse-Smale, then Floer chain complex of $(f,J)$ may
be identified (after a grading shift) with the quantum chain complex of
$(f,g_{J})$, which is by definition the Morse complex of $(f,g_{J})$ with
coefficients in the Novikov ring, ie.

$\displaystyle CF_{*}(f,J)$
$\displaystyle=QC_{*+n}(f,g_{J}):=(C^{Morse}(f,g_{J})\otimes\Lambda_{\omega})_{*+n}.$

Taking homology then gives a natural identification with the quantum homology
of $(M,\omega)$:

$\displaystyle HF_{*}(f)=H_{*+n}(M;\Lambda_{\omega})=QH_{*+n}(M,\omega).$

####  3.1.1 Continuation maps

######  Definition 3.2.

For $X$ a smooth manifold, a function $F\in C^{\infty}(\mathbb{R}\times X)$ is
said to be $T$-adapted for $T\in(0,\infty)$ if $(\partial_{s}F)_{s_{0}}\equiv
0$ for all $|s_{0}|\geq T$. $F$ is said to be adapted if it is $T$-adapted for
some $T$. For $X=C^{\infty}(S^{1}\times M)$, and $H^{\pm}\in
C^{\infty}(S^{1}\times M)$, we denote by $\mathscr{H}(H^{-},H^{+})$ the space
of adapted homotopies $\mathcal{H}$ having
$\lim_{s\rightarrow\pm\infty}\mathcal{H}(s)\equiv H^{\pm}$. We make a similar
definition for $\mathscr{J}(J^{-},J^{+})$ in the case where
$X=C^{\infty}(S^{1};\mathcal{J}(M,\omega))$.

######  Definition 3.3.

A pair $(\mathcal{H},\mathbb{J})$ is an adapted homotopy of Floer data from
$(H^{-},J^{-})$ to $(H^{+},J^{+})$ if
$\mathcal{H}=(H^{s}_{t})\in\mathscr{H}(H^{-},H^{+})$ and
$\mathbb{J}=(J^{s}_{t})\in\mathscr{J}(J^{-},J^{+})$. We will write
$\mathscr{H\mspace{-8.0mu}J}(H^{-},J^{-};H^{+},J^{+})$ for the collection of
all such adapted homotopies, often omitting the dependence on
$(H^{\pm},J^{\pm})$ if it is clear from context.

Just as in the $s$-independent case, for any adapted homotopy of Floer data
$(\mathcal{H},\mathbb{J})$, we obtain a corresponding Floer operator
$\mathcal{F}_{\mathcal{H},J}$. For any pair
$\hat{x}^{\pm}\in\widetilde{Per}_{0}({H^{\pm}})$, consideration of the zeros
of $\mathcal{F}_{\mathcal{H},\mathbb{J}}$ along $C^{\infty}(\mathbb{R}\times
S^{1};M)_{\hat{x}^{-},\hat{x}^{+}}$ gives rise to the $s$-dependent Floer
equation

$\displaystyle\partial_{s}u+J^{s}_{t}(\partial_{t}u-X_{H_{t}^{s}})$
$\displaystyle=0,$ (2)

and everything proceeds as before, with the proviso that now, if $u\in
C^{\infty}(\mathbb{R}\times S^{1};M)_{\hat{x}^{-},\hat{x}^{+}}$ solves
Equation 2, then its energy is given by

$E(u)=\mathcal{A}_{H^{-}}(\hat{x}^{-})-\mathcal{A}_{H}^{+}(\hat{x}^{+})+\int_{-\infty}^{\infty}\int_{0}^{1}(\partial_{s}\mathcal{H})(s,t,u(s,t))\;dtds.$
(3)

The moduli space $\mathcal{M}(\hat{x}^{-},\hat{x}^{+};\mathcal{H},J)$ is
defined to be the zero set of $\mathcal{F}_{\mathcal{H},J}$ on
$C^{\infty}(\mathbb{R}\times S^{1};M)_{\hat{x}^{-},\hat{x}^{+}}$.

######  Remark.

When $(H^{-},J^{-})=(H^{+},J^{+})$, then the $s$-independent homotopy
$(\mathcal{H},\mathbb{J})=(H^{-},J^{-})=(H^{+},J^{+})$ is a special case of an
adapted homotopy. In this case,
$\mathcal{M}(\hat{x},\hat{y};\mathcal{H},J)=\widetilde{\mathcal{M}}(\hat{x},\hat{y};H^{\pm},J^{\pm})$.
In the sequel, when we speak of adapted homotopies of Floer data, this case is
included.

######  Definition 3.4.

Given $(H^{\pm},J^{\pm})$ Floer regular,
$\hat{x}^{\pm}\in\widetilde{Per}_{0}({H^{\pm}})$, and
$(\mathcal{H},\mathbb{J})\in\mathscr{H\mspace{-8.0mu}J}$, we will say that
$(\mathcal{H},\mathbb{J})$ is $(\hat{x}^{-},\hat{x}^{+})$-regular if
$\mathcal{F}_{\mathcal{H},\mathbb{J}}$ is transverse to the zero section along
$C^{\infty}(\mathbb{R}\times S^{1};M)_{\hat{x}^{-},\hat{x}^{+}}$. We denote
the collection of all such adapted homotopies by
$\mathscr{H\mspace{-8.0mu}J}^{reg}_{\hat{x}^{-},\hat{x}^{+}}$.
$(\mathcal{H},\mathbb{J})$ will be said to be Floer-regular if it is
$(\hat{x}^{-},\hat{x}^{+})$-regular whenever
$\mu(\hat{x}^{-})-\mu(\hat{x}^{+})\leq 1$. We denote the space of Floer-
regular adapted homotopies from $(H^{-},J^{-})$ to $(H^{+},J^{+})$ by
$\mathscr{H\mspace{-8.0mu}J}^{reg}(H^{-},J^{-};H^{+},J^{+})$, suppressing the
dependence on $(H^{\pm},J^{\pm})$ when no confusion will arise.

For any fixed $\mathbb{J}\in\mathscr{J}(J^{-},J^{+})$, the set
$\mathscr{H}^{reg}(\mathbb{J};H^{-},H^{+})\subseteq\mathscr{H}(H^{-},H^{+})$
of adapted homotopies $\mathcal{H}$ such that $(\mathcal{H},\mathbb{J})$ is
Floer regular is residual.

For $(\mathcal{H},\mathbb{J})\in\mathscr{H\mspace{-8.0mu}J}^{reg}$, the spaces
$\mathcal{M}(\hat{x}^{-},\hat{x}^{+};\mathcal{H},J)$ are all compact manifolds
of dimension $0$ whenever $\mu^{H^{-}}(\hat{x}^{-})=\mu^{H^{+}}(\hat{x}^{+})$,
and so we may define the continuation morphism

$\displaystyle h_{\mathcal{H},J}:CF_{*}(H^{-},J^{-})$
$\displaystyle\rightarrow CF_{*}(H^{+},J^{+})$

on generators by setting

$\displaystyle h_{\mathcal{H},\mathbb{J}}(x^{-})$
$\displaystyle:=\sum_{\mu(\hat{x}^{-})-\mu(\hat{x}^{+})=0}n(\hat{x}^{-},\hat{x}^{+})\hat{x}^{+},$

where $n(\hat{x}^{-},\hat{x}^{+})$ is the mod $2$ count of elements in the
moduli space $\mathcal{M}(\hat{x}^{-},\hat{x}^{+};\mathcal{H},\mathbb{J})$.
The continuation morphism is a morphism of complexes, which descends to an
isomorphism at the level of homology. Moreover any two continuation maps
between $(H^{-},J^{-})$ and $(H^{+},J^{+})$ define the same map at the level
of homology, and further these isomorphism satisfy the obvious composition law

$\displaystyle h_{21}\circ h_{10}$ $\displaystyle=h_{20},$

where $h_{ji}:HF(H_{i})\rightarrow HF(H_{j})$.

####  3.1.2 The PSS isomorphism

There is another type of morphism of chain complexes, in some way related to
the continuation morphisms, which will concern us in this work. Introduced in
[24] (see also [27]), PSS maps may be viewed as a variant on Floer
continuation maps, with the exception that they consider adapted homotopies of
Floer data from $(0,J^{-})$ — for $J^{-}\in\mathcal{J}(M,\omega)$ an
autonomous almost complex structure —to the Floer pair $(H,J)$ whose Floer
complex we are studying. The fact that the $0$ function is a heavily
degenerate Hamiltonian forces some modifications to the definition of the
chain morphism. Explicitly, we write

$\displaystyle\mathscr{H\mspace{-8.0mu}J}^{PSS}(H,J)$
$\displaystyle:=\\{(J^{-};\mathcal{H},J):J^{-}\in\mathcal{J}(M,\omega),\;(\mathcal{H},\mathbb{J})\in\mathscr{H\mspace{-8.0mu}J}(0,J^{-};H,J)\\}$

and the set of all PSS data for the pair $(H,J)$ is

$\displaystyle PSS(H,J)$
$\displaystyle:=\\{(f,g;J^{-};\mathcal{H},\mathbb{J})\in C^{\infty}(M)\times
Met(M)\times\mathscr{H\mspace{-8.0mu}J}^{PSS}(H,J):\;(f,g)\text{ is Morse-
Smale}\\},$

where $Met(M)$ denotes the space of smooth Riemannian metrics on $M$. There is
a residual set $PSS_{reg}(H,J)\subset PSS_{reg}(H,J)$ of regular PSS data such
that for any $\mathcal{D}=(f,g;J^{-};\mathcal{H},\mathbb{J})\in
PSS_{reg}(H,J)$, we may define a morphism of chain complexes

$\displaystyle\Phi^{PSS}_{\mathcal{D}}:(C^{Morse}(f,g)\otimes\Lambda_{\omega})_{*+n}$
$\displaystyle\rightarrow CF_{*}(H,J)$ $\displaystyle p\otimes e^{A}$
$\displaystyle\mapsto\sum_{A\in\Gamma_{\omega}}\sum_{\hat{x}}n(p,(-A)\cdot\hat{x})\hat{x},$

where for $q\in Crit(f)_{k-n}$,
$\hat{y}=[y,v]\in\widetilde{Per}_{0}({H})_{k}$, $n(q,\hat{y})$ denotes the
$\mathbb{Z}/2\mathbb{Z}$-count of elements in the $0$-dimensional moduli space

$\displaystyle\mathcal{M}(q,\hat{y};\mathcal{D})$

of finite energy maps $u\in C^{\infty}(\mathbb{R}\times S^{1};M)$ which are
$(\mathcal{H},\mathbb{J})$-Floer and which satisfy

$\displaystyle\lim_{s\rightarrow-\infty}u(s,t)$ $\displaystyle\in
W^{u}(q;f,g),$ $\displaystyle\lim_{s\rightarrow\infty}u(s,t)$
$\displaystyle=y(t),\;\text{and}$
$\displaystyle[\bar{u}]\\#[v]=0\in\Gamma_{\omega},$

where $W^{u}(q;f,g)$ is the unstable manifold of $q$ with respect to $(f,g)$
and $\bar{u}:D^{2}\rightarrow M$ is the disc map obtained by completing $u$ to
a continuous map $\bar{u}:\bar{\mathbb{R}}\times S^{1}\rightarrow M$ (note
that since $u$ has finite energy and is by hypothesis $J^{-}$-holomorphic in
some neighbourhood of $\\{-\infty\\}\times S^{1}$ for some fixed
$J^{-}\in\mathcal{J}(M,\omega)$, removal of singularities for pseudo-
holomorphic maps implies that there is some $m\in W^{u}(q,f,g)$ such that
$\lim_{s\rightarrow-\infty}u(s,t)=m$ for all $t\in S^{1}$). The map
$\Phi^{PSS}_{\mathcal{D}}$ descends to a map in homology

$\displaystyle\Phi^{PSS}_{*}:QH_{*+n}(M,\omega)$ $\displaystyle\rightarrow
HF_{*}(H)$

which is independent of the regular PSS data. It turns out (see [24] for more
discussion on this point as well as the definitions of the relevant
operations) that $\Phi^{PSS}_{*}$ is an isomorphism which intertwines the
quantum product with the pair-of-pants product in Floer homology. This
isomorphism permits the definition of the Oh-Schwarz spectral invariants

$\displaystyle c_{OS}:QH_{*}(M,\omega)\setminus\\{0\\}\times
C^{\infty}(S^{1}\times M)$ $\displaystyle\rightarrow\mathbb{R}$
$\displaystyle(\alpha,H)$ $\displaystyle\mapsto c_{OS}(\alpha;H).$

Introduced by Schwarz in the symplectically aspherical case [28] and extended
by Oh to more general symplectic manifolds in [20], these invariants are
defined by

$\displaystyle c_{OS}(\alpha;H)$
$\displaystyle:=\inf\\{\lambda_{H}(\sigma):\sigma\in
CF_{*}(H,J),\;[\sigma]=\Phi^{PSS}_{*}(\alpha)\\}.$

A cycle $\sigma\in CF_{*}(H,J)$ such that $[\sigma]=\Phi^{PSS}_{*}(\alpha)$
and $\lambda_{H}(\sigma)=c_{OS}(H;\alpha)$ is called tight (for
$c_{OS}(\alpha;H)$). It is a non-trivial fact that such cycles always exist
(see [34] or [21]).

The Oh-Scwharz spectral invariants are $C^{0}$-continuous in their Hamiltonian
argument, take values in the spectrum of their Hamiltonian argument, and
satisfy a bevy of formal properties (see Theorem III in [20] for a
representative list, for example) which make them useful in studying the
behaviour of Hamiltonian diffeomorphisms.

####  3.1.3 The Gromov trick

We conclude this section by recalling the so-called ‘Gromov trick’, which
forms the basis of much of this paper by establishing that we may use pseudo-
holomorphic techniques to analyze the graphs of Floer-type cylinders.

######  Theorem 3.5 (1.4.C’. in [6]).

Let $(\mathcal{H},\mathbb{J})$ be an adapted homotopy of Floer data, then
there exists a unique almost complex structure $\tilde{\mathbb{J}}$ on
$\mathbb{R}\times S^{1}\times M$ with the property that a graph

$\displaystyle\tilde{u}:\mathbb{R}\times S^{1}$
$\displaystyle\rightarrow\mathbb{R}\times S^{1}\times M$ $\displaystyle(s,t)$
$\displaystyle\mapsto(s,t,u(s,t))$

is $(j_{0},\tilde{\mathbb{J}})$-holomorphic if and only if $u$ satisfies
Equation 2, where $j_{0}$ denotes the standard complex structure on the
cylinder.

###  3.2 Asymptotic analysis for pseudoholomorphic cylinders

The main analytic fact that gives us control over the asymptotic winding
behaviour of Floer cylinders, as well as that of vector fields lying in the
the kernel of the Floer differential, is the following theorem which describes
the asymptotic behaviour of solutions to an appropriately perturbed Cauchy-
Riemann equation. This result is originally due to [19], although the version
we reproduce here for the convenience of the reader is from the appendix of
[30]

######  Theorem 3.6.

Let $w:[0,\infty)\times S^{1}\rightarrow\mathbb{R}^{2n}$ satisfy the equation

$\displaystyle\partial_{s}w+J_{0}\partial_{t}w+(S(t)-\Delta(s,t))w$
$\displaystyle=0,$ (4)

where $S:S^{1}\rightarrow End(\mathbb{R}^{2n})$ is a smooth family of
symmetric matrices and

$\displaystyle\Delta:[0,\infty)\times S^{1}$ $\displaystyle\rightarrow
End(\mathbb{R}^{2n})$

is smooth. Suppose that for $\beta\in\mathbb{N}^{2},$ there exist constants
$M_{\beta},d>0$ such that

$\displaystyle|(\partial^{\beta}\Delta)(s,t)|$ $\displaystyle\leq
M_{\beta}e^{-ds},\;\text{and}$ $\displaystyle|(\partial^{\beta}w)(s,t)|$
$\displaystyle\leq M_{\beta}e^{-ds}.$

Then either $w\equiv 0$ or $w(s,t)=e^{\lambda s}(\xi(t)+r(s,t))$, where
$\lambda$ is a negative eigenvalue of the self-adjoint operator

$\displaystyle\mathbf{A}:H^{1}(S^{1};\mathbb{R}^{2n})\subseteq
L^{2}(S^{1};\mathbb{R}^{2n})$ $\displaystyle\rightarrow
L^{2}(S^{1};\mathbb{R}^{2n})$ $\displaystyle h$ $\displaystyle\mapsto-
J_{0}(\partial_{t}-J_{0}S)h,$

$\xi:S^{1}\rightarrow\mathbb{R}^{2n}$ is an eigenvector of $\mathbf{A}$ with
eigenvalue $\lambda$, and $r$ satisfies the decay estimates

$\displaystyle|(\partial^{\beta}r)(s,t)|$ $\displaystyle\leq
e^{-d^{\prime}s}M_{\beta}^{\prime}$

for $d^{\prime},M_{\beta}^{\prime}>0$, for all $\beta\in\mathbb{N}^{2}$.

This theorem is useful in the following setting. Let $(H,J)$ be Floer regular.
To any $x\in Per_{0}(H)$, we may assign the asymptotic operator

$\displaystyle A_{x,J}:\Gamma(x^{*}TM)\rightarrow\Gamma(x^{*}TM)$

as follows. Viewing $\xi\in\Gamma(x^{*}TM)$ as a section of the vertical
tangent bundle $\mathcal{V}|_{\check{x}}\leq T(S^{1}\times M)|_{\check{x}}$
along the graph $\check{x}$ of $x$, we let $\check{X}_{H}:=\partial_{t}\oplus
X_{H}\in\mathcal{X}(S^{1}\times M)$, and we view $J=(J_{t})_{t\in S^{1}}$ as
an endomorphism of the vertical tangent bundle by setting
$\check{J}_{t,x}:=J_{t}(x)$. $A_{x,J}$ is then defined by setting
$A_{x,J}(\xi):=-\check{J}\mathcal{L}_{\check{X}_{H}}\xi$, where
$\mathcal{L}_{X}Y$ denotes the Lie derivative of $Y$ along $X$. $A_{x,J}$
extends to an unbounded self-adjoint operator with discrete spectrum (still
denoted $A_{x,J}$) from $W^{1,2}(x^{*}TM)$ to $L^{2}(x^{*}TM)$.

By taking an exponential chart as in Section 2.2 on a neighbourhood
$\tilde{\mathcal{O}}$ of $\hat{x}\in\widetilde{Per}_{0}({H})$, Floer’s
equation may be written in the local coordinates provided by this chart in the
form of Equation 4, with $A_{x,J}$ being sent via these coordinates to
$\mathbf{A}$. Following [30], we define

######  Definition 3.7.

Let $x\in\mathcal{L}_{0}(M)$ and suppose that
$\lim_{s\rightarrow\infty}u_{s}\equiv x$ for a map $u:\mathbb{R}\times
S^{1}\rightarrow M$. For any $R>0$, a map

$\displaystyle U^{+}:[R,\infty)$ $\displaystyle\rightarrow\Gamma(x^{*}TM)$

will be said to be a positive asymptotic representative of $u$ if
$u(s,t)=Exp(U^{+}(s))(t)$ for all $(s,t)\in[R,\infty)\times S^{1}$, where
$Exp$ is as in Section 2.2. The notion of a negative asymptotic representative
of $u$,

$\displaystyle U^{-}:(-\infty,-R]$ $\displaystyle\rightarrow\Gamma(x^{*}TM)$

is defined in the obvious analogous manner.

Every Floer-type cylinder considered in this paper admits, due to exponential
convergence at the ends, essentially unique positive and negative asymptotic
representatives, determined up to a restriction of the domains of $U^{\pm}$ to
larger values of $|R|$.

The main result that we will need from [30] is the following (this is Theorem
$2.2$ of the quoted work, paraphrased for our setting)

######  Theorem 3.8.

Let $(H,J)$ be Floer regular, $x\in Per_{0}(H)$ and let $u,v$ solve Equation 2
for $s>>0$ (resp. for $s<<0$), where the adapted homotopy used in defining
Equation 2 satisfies $(H^{+},J^{+})=(H,J)$ (resp. $(H^{-},J^{-})=(H,J)$).
Suppose moreover that $u_{s}$ and $v_{s}$ both converge to $x$ as
$s\rightarrow\infty$ (resp. $s\rightarrow-\infty$). Let $U$ and $V$ be
positive (resp. negative) asymptotic representatives of $u$ and $v$
respectively. Then either $U\equiv V$ or there exists a strictly negative
(resp. strictly positive) eigenvalue $\lambda\in\sigma(A_{x,J})$ and an
eigenvector $\xi$ with eigenvalue $\lambda$ such that

$\displaystyle(U-V)(s,t)$ $\displaystyle=e^{\lambda s}(\xi(t)+r(s,t)),$

where the remainder term satisfies the decay estimates
$|\nabla_{s}^{i}\nabla_{t}^{i}r(s,t)|\leq M_{ij}e^{-ds}$ for all
$(i,j)\in\mathbb{N}^{2}$ and some $M_{i,j},d>0$ (resp. $d<0$).

Whenever $u,v$ and $x$ are as above, we will write $\xi^{+}_{u,v}$ (resp.
$\xi^{-}_{u,v}$) for the eigenvectors of $A_{x,J}$ whose existence is
guaranteed by the above theorem. We will call $\xi^{\pm}_{u,v}$ the positive
(resp. negative) asymptotic eigenvector of $v$ relative $u$. Note that the
above result only requires that $u$ and $v$ solve Equation 2 on some
neighbourhood of $s=\infty$ (resp. $s=-\infty$), and that, for
$(\mathcal{H},\mathbb{J})\in\mathscr{H\mspace{-8.0mu}J}$, the trivial cylinder
$v(s,t)=x(t)$ is always a solution to Equation 2 outside some compact set. We
will write $\xi^{\pm}_{u}:=\xi^{\pm}_{u,x}$ and call these the (positive and
negative) asymptotic eigenvectors of $u$.

This asymptotic information becomes especially useful when combined with the
following fact (see [8] p. 285 or [30] p.1637).

######  Proposition 3.9.

If $\xi\in\Gamma(x^{*}TM)$ is an eigenvector of $A_{x,J}$, then $\xi(t)\neq 0$
for all $t\in S^{1}$.

######  Corollary 3.10.

Let $u,v:\mathbb{R}\times S^{1}\rightarrow M$ be distinct finite energy
solutions of Equation 2, then there is a compact subset
$K\subseteq\mathbb{R}\times S^{1}$ such that $u(s,t)=v(s,t)$ only if $(s,t)\in
K$.

These results become even stronger in the case when $\dim M=2$, as in this
case Proposition 3.9 implies that eigenvectors of the asymptotic operator have
a well-defined winding number, once we fix a trivialization of $x^{*}TM$ via a
choice of capping disk. More precisely, when $M=\Sigma$, if
$\hat{x}\in\widetilde{\mathcal{L}_{0}}(\Sigma)$, and
$T_{\hat{x}}:S^{1}\times(\mathbb{R}^{2},\omega_{0})\rightarrow(x^{*}T\Sigma,\omega)$
is a symplectic trivialization as in Section 2, then for any $\xi\in A_{x,J}$,
the map $t\mapsto T_{\hat{x}}(t)^{-1}\xi(t)$ has a well-defined winding number
$wind(\xi;\hat{x})$, by Proposition 3.9. Proposition 2.20 then implies

######  Corollary 3.11.

Let $u,v$ be distinct finite energy solutions of Equation 2 with

$\displaystyle\lim_{s\rightarrow-\infty}u_{s}$
$\displaystyle=\lim_{s\rightarrow-\infty}v_{s}=x.$

Then there exists $R>0$ such that for all $s<-R$ and any capping
$\hat{x}=[x,\alpha]$, we have

$\displaystyle\ell(\hat{v}_{s}^{\alpha},\hat{u}^{\alpha}_{s})$
$\displaystyle=wind(\xi_{u,v}^{-};\hat{x}),$

where $\hat{u}^{\alpha}_{s}$ (resp. $\hat{v}^{\alpha}_{s}$) denotes the
capping of $u_{s}$ (resp. $v_{s}$) such that $[x,\alpha]$ and $\hat{u}_{s}$
(resp. $\hat{v}_{s}$) are $0$-homotopic. The analogous statement when
$\lim_{s\rightarrow\infty}u_{s}=\lim_{s\rightarrow\infty}v_{s}=x$ also holds.

If we combine the positivity of intersection of holomorphic curves in
dimension $4$ with the foregoing discussion, we arrive at the principal point
of this section

######  Lemma 3.12.

Let $(H^{\pm},J^{\pm})$ be Floer regular,
$(\mathcal{H},\mathbb{J})\in\mathscr{H\mspace{-8.0mu}J}$ and let $u,v\in
C^{\infty}(\mathbb{R}\times S^{1};\Sigma)$ be distinct finite energy solutions
to Equation 2 for $(\mathcal{H},\mathbb{J})$. Then for any lifts
$\hat{u},\hat{v}$ of $u,v:\mathbb{R}\rightarrow\mathcal{L}_{0}(\Sigma)$, the
function $\ell_{\hat{u},\hat{v}}(s):=\ell(\hat{u}_{s},\hat{v}_{s})$ is non-
decreasing, locally constant, and well-defined for all but finitely many
values $s\in\mathbb{R}$. Moreover, for $s,s^{\prime}\in
dom(\ell_{\hat{u},\hat{v}})$, with $s<s^{\prime}$,
$\ell_{\hat{u},\hat{v}}(s)\neq\ell_{\hat{u},\hat{v}}(s^{\prime})$ if and only
if there exists $s_{0}\in(s,s^{\prime})$ and some $t_{0}\in S^{1}$ such that
$u(s_{0},t_{0})=v(s_{0},t_{0})$.

###### Proof.

That $\ell_{\hat{u},\hat{v}}$ has only finitely many points at which it is
ill-defined follows the fact that, by definition,
$\ell(\hat{u}_{s_{0}},\hat{v}_{s_{0}})$ is undefined only when there exists
$t_{0}\in S^{1}$ such that $u(s_{0},t_{0})=v(s_{0},t_{0})$. By Corollary 3.10,
the set of all such $(s_{0},t_{0})\in\mathbb{R}\times S^{1}$ must lie inside
some compact set, and we may then apply Theorem 3.5 to choose an almost
complex structure on $\mathbb{R}\times S^{1}\times\Sigma$ such that the graphs
$\tilde{u}$ and $\tilde{v}$ are pseudoholomorphic, whence all such
intersections must be isolated, and so finite in number. That
$\ell_{\hat{u},\hat{v}}$ is non-decreasing follows by applying the positivity
of intersections for holomorphic curves in dimension $4$ at these
intersections, which implies moreover that any such intersection contributes
strictly positively to the change in $\ell_{\hat{u},\hat{v}}(s)$ as $s$
increases and passes from one connected component of
$dom(\ell_{\hat{u},\hat{v}})$ to another. ∎

######  Definition 3.13.

For $(H^{\pm},J^{\pm})$ Floer regular,
$(\mathcal{H},\mathbb{J})\in\mathscr{H\mspace{-8.0mu}J}$ and
$u,v\in\mathcal{M}(\hat{x},\hat{y};\mathcal{H},\mathbb{J})$, $u\neq v$, we
define

$\displaystyle\ell_{\pm\infty}(u,v)$
$\displaystyle:=\lim_{s\rightarrow\pm\infty}\ell(\hat{u}_{s},\hat{v}_{s}),$

where $\hat{u}_{s}$ and $\hat{v}_{s}$ are the natural cappings of $u_{s}$ and
$v_{s}$ (cf. Definition 2.14).

Note that the previous lemma implies that these quantities exist and are
finite. Indeed, if $\hat{u}_{s}$ and $\hat{v}_{s}$ tend to $\hat{x}$ as
$s\rightarrow\pm\infty$, then
$\ell_{\pm\infty}(u,v)=wind(\xi^{\pm}_{u,v};\hat{x})$ by Corollary 3.11, while
if $\lim_{s\rightarrow\pm\infty}u_{s}=\hat{x}$ and
$\lim_{s\rightarrow\pm\infty}v_{s}=\hat{y}$ with $x\neq y$, then
$\ell_{\pm\infty}(u,v)=\ell(\hat{x},\hat{y})$.

####  3.2.1 Winding of eigenvectors of $A_{x,J}$

We summarize here some necessary facts from [8] on the winding numbers of
eigenvectors of $A_{x,J}$ which appeared in the previous subsection (while [8]
works in the aspherical case, our previous discussion makes clear how this
winding number depends on the choice of cappings of $x$ and $y$ and this is
all that is needed to extend the results to capped orbits). For a loop
$x\in\mathcal{L}_{0}(\Sigma)$, let $\pi_{2}(\Sigma;x)$ denote the set of
homotopy classes of capping disks for $x$.

######  Proposition 3.14.

Let $x\in Per_{0}(H)$ with $H$ non-degenerate and
$J:S^{1}\rightarrow\mathcal{J}(\Sigma,\omega)$ arbitrary. There is a well-
defined function

$\displaystyle W=W_{x,J}:\pi_{2}(\Sigma;x)\times\sigma(A_{x,J})$
$\displaystyle\rightarrow\mathbb{Z}$ $\displaystyle(\alpha,\lambda)$
$\displaystyle\mapsto wind(T_{[x,\alpha]}^{-1}\circ\xi),$

where $\xi\in\Gamma(x^{*}T\Sigma)$ is any eigenvector with eigenvalue
$\lambda$. Moreover, $W$ satisfies the following properties

  1. 1.

For any $\alpha\in\pi_{2}(\Sigma;x)$,
$\lambda<\lambda^{\prime}\;\Rightarrow\;W(\alpha,\lambda)\leq
W(\alpha,\lambda^{\prime})$.

  2. 2.

For any $\alpha\in\pi_{2}(\Sigma;x)$, and any $k\in\mathbb{Z}$,
$\sum_{\lambda\in W_{\alpha}^{-1}(k)}\dim E_{\lambda}=2$,
where $W_{\alpha}(\lambda)=W(\alpha,\lambda)$, and $E_{\lambda}$ is the
eigenspace associated to the eigenvalue $\lambda$.

  3. 3.

For any $A\in\pi_{2}(\Sigma)$,
$W(A\cdot\alpha,\lambda)=W(\alpha,\lambda)+c_{1}(A)$.

In view of the control over the sign of the eigenvalue provided by Theorem
3.8, combined with the monotonicity of the winding number provided by item
$(1)$ of the above proposition, we make the following

######  Definition 3.15.

For $(H,J)$ and $\hat{x}=[x,\alpha]\in Per_{0}(H)$ as above, define

$\displaystyle a(\hat{x})=a(\hat{x};H)$
$\displaystyle:=\sup_{\lambda\in\sigma(A_{x,J})\cap(-\infty,0)}W(\alpha,\lambda)$
$\displaystyle b(\hat{x})=b(\hat{x};H)$
$\displaystyle:=\inf_{\lambda\in\sigma(A_{x,J})\cap(0,\infty)}W(\alpha,\lambda).$

Remark that we have, by the monotonicity of $W$ and by Theorem 3.8, that

######  Corollary 3.16.

Let $(H^{\pm},J^{\pm})$ be Floer regular,
$(\mathcal{H},\mathbb{J})\in\mathscr{H\mspace{-8.0mu}J}$,
$x^{\pm}_{i}\in\widetilde{Per}_{0}({H^{\pm}})$ for $i=0,1$.

  1. 1.

If
$u_{i}\in\mathcal{M}(\hat{x}_{0}^{-},\hat{x}_{i}^{+};\mathcal{H},\mathbb{J})$,
$i=0,1$, then $b(\hat{x}_{0}^{-})\leq\ell_{-\infty}(u_{0},u_{1})$.

  2. 2.

If
$u_{i}\in\mathcal{M}(\hat{x}_{i}^{-},\hat{x}_{0}^{+};\mathcal{H},\mathbb{J})$,
$i=0,1$, then $\ell_{\infty}(u_{0},u_{1})\leq a(\hat{x}^{+}_{0})$.

The result which relates this discussion to the behaviour of the Floer complex
is the following

######  Theorem 3.17 ([8] Theorem 3.10).

$\displaystyle-\mu(\hat{x})$ $\displaystyle=2a(\hat{x})+p(\hat{x}),$ (5)

where $p([x,\alpha])=0$ if there exists
$\lambda\in\sigma(A_{x,J})\cap(0,\infty)$ such that
$W(\alpha,\lambda)=a(\hat{x})$ and $p([x,\alpha])=1$ otherwise.

######  Remark.

Note that our sign convention for the Conley-Zehnder index is the negative of
the convention adopted in [8].

######  Corollary 3.18.

$\displaystyle-\mu(\hat{x})$ $\displaystyle=a(\hat{x})+b(\hat{x})$

###### Proof.

The point is that $b(\hat{x})=a(\hat{x})+p(\hat{x})$. To see this write
$\hat{x}=[x,\alpha]$, and let $\lambda=\max\\{\nu\in\sigma(A_{x,J}):\nu<0\\}$
be such that $W(\alpha,\lambda)=a(\hat{x})$.

Suppose first that $p(\hat{x})=0$, then by the definition of $p(\hat{x})$,
there exists $\lambda^{\prime}\in\sigma(A_{x,J})\cap(0,\infty)$ such that
$W(\alpha,\lambda^{\prime})=a(\hat{x})$. By item (2) of Proposition 3.14, it
follows that $E_{\lambda}$ and $E_{\lambda^{\prime}}$ are $1$-dimensional and
that $W(\alpha,\nu)=a(\hat{x})$ for $\nu\in\sigma(A_{x,J})$ if and only if
$\nu\in\\{\lambda,\lambda^{\prime}\\}$. The monotonicity of the winding number
expressed in item (1) of Proposition 3.14 then implies that if
$\nu\in\sigma(A_{x,J})\cap(0,\infty)$ is distinct from $\lambda^{\prime}$,
then $W(\alpha,\lambda^{\prime})<W(\alpha,\nu)$, and consequently we see that

$\displaystyle W(\alpha,\lambda^{\prime})$ $\displaystyle=b(\hat{x}).$

Suppose next that $p(\hat{x})=1$ so that, by the monotonicity of item (1) in
Proposition 3.14, we have that $W(\alpha,\lambda^{\prime})>a(\hat{x})$ for all
$\lambda^{\prime}\in\sigma(A_{x,J})\cap(0,\infty)$. Item (2) of Proposition
3.14 implies that there exists $\lambda^{\prime}\in\sigma(A_{x,J})$ such that
$W(\alpha,\lambda^{\prime})=a(\hat{x})+1$ and the monotonicity of the winding
number implies that such $\lambda^{\prime}$ must satisfy
$\lambda^{\prime}>\lambda$. Since $\lambda$ is by definition the largest
negative eigenvalue of $A_{x,J}$, it follows from the non-degeneracy of
$A_{x,J}$ (which is equivalent to the non-degeneracy of $x\in Per_{0}(H)$ as a
$1$-periodic orbit of $H$) that $\lambda^{\prime}$ is positive, and so clearly
we must have $b(\hat{x})=W(\alpha,\lambda^{\prime})=a(\hat{x})+1$. ∎

######  Lemma 3.19.

Let $k\in\mathbb{Z}$.

  1. 1.

If $\mu(\hat{x};H)\in\\{2k-1,2k\\}$, then $a(\hat{x};H)=-k$.

  2. 2.

If $\mu(\hat{x};H)\in\\{2k,2k+1\\}$, then $b(\hat{x};H)=-k$.

###### Proof.

We prove (1) with the argument for (2) being entirely analogous (but
additionally making use of the fact that $b(\hat{x})=a(\hat{x})+p(\hat{x})$
established in the proof of Corollary 3.18). Theorem 3.17 states that
$-\mu(\hat{x})=2a(\hat{x})+p(\hat{x})$, where $p(\hat{x})\in\\{0,1\\}$ is the
parity of $\mu(\hat{x})$. Consequently, if $\mu(\hat{x})=2k-1$, we see that

$\displaystyle 1-2k$ $\displaystyle=2a(\hat{x})+1$

and therefore $a(\hat{x})=-k$. Similarly, if $\mu(\hat{x})=2k$, then

$\displaystyle-2k$ $\displaystyle=2a(\hat{x})$

and so $a(\hat{x})=-k$, as claimed. ∎

######  Corollary 3.20.

Let $(H^{\pm},J^{\pm})$ be Floer regular,
$(\mathcal{H},\mathbb{J})\in\mathscr{H\mspace{-8.0mu}J}(H^{-},J^{-};H^{+},J^{+})$
and suppose that $\hat{x}^{\pm}\in\widetilde{Per}_{0}({H^{\pm}})$ satisfy
$\mu(\hat{x}^{\pm})=2k+1$ for some $k\in\mathbb{Z}$. Then
$|\mathcal{M}(\hat{x}^{-},\hat{x}^{+};\mathcal{H},\mathbb{J})|\in\\{0,1\\}$.

###### Proof.

Suppose for the sake of contradiction that there exist
$u,v\in\mathcal{M}(\hat{x}^{-},\hat{x}^{+};\mathcal{H},\mathbb{J})$, $u\neq
v$. Then we have by Lemma 3.12 and Lemma 3.19

$\displaystyle-k=b(\hat{x}^{-})\leq\ell_{-\infty}(u,v)$
$\displaystyle\leq\ell_{\infty}(u,v)\leq a(\hat{x}^{+})=-k-1,$

which is a contradiction. The lemma follows. ∎

Recall from Section 3.1 that to any
$(\mathcal{H},\mathbb{J})\in\mathscr{H\mspace{-8.0mu}J}(H^{-},J^{-};H^{+},J^{+})$,
we associate the operator
$\mathcal{F}_{\mathcal{H},\mathbb{J}}:C^{\infty}(\mathbb{R}\times
S^{1};\Sigma)\rightarrow\mathcal{E}$. Whenever
$\mathcal{F}_{\mathcal{H},\mathbb{J}}(u)=0$, for $u\in
C^{\infty}(\mathbb{R}\times S^{1};\Sigma)_{\hat{x}^{-},\hat{x}^{+}}$,
$T_{u}\mathcal{E}$ splits canonically as $T_{u}C^{\infty}(\mathbb{R}\times
S^{1};\Sigma)_{\hat{x}^{-},\hat{x}^{+}}\oplus\mathcal{E}_{u}$. In such a case,
we denote by $D\mathcal{F}_{\mathcal{H},\mathbb{J}}$ the projection of the
differential of $\mathcal{F}_{\mathcal{H},\mathbb{J}}$ onto $\mathcal{E}_{u}$,
and we call $D\mathcal{F}_{\mathcal{H},\mathbb{J}}$ the linearized Floer
operator. The transversality of $\mathcal{F}_{\mathcal{H},\mathbb{J}}$ to the
$0$-section of $\mathcal{E}$ at $u$ is equivalent to the surjectivity of
$(D\mathcal{F}_{\mathcal{H},\mathbb{J}})_{u}$, which is in turn related to the
behaviour of its kernel by the Fredholm property. The following result is
essentially proved in [8] as Proposition $5.6$ and serves to give significant
control over elements in the kernel of the linearized Floer operator. We give
a simple proof here in the Floer-theoretic setting for the convenience of the
reader.

######  Proposition 3.21.

Let $(H^{\pm},J^{\pm})$ be Floer regular, let
$(\mathcal{H},\mathbb{J})\in\mathscr{H\mspace{-8.0mu}J}$,
$u\in\mathcal{M}(\hat{x}^{-},\hat{x}^{+};\mathcal{H},\mathbb{J})$, and let
$\xi\in\ker(D\mathcal{F}_{\mathcal{H},\mathbb{J}})_{u}$. Suppose that
$\xi\not\equiv 0$ and denote by $Z(\xi)$ the algebraic count of the number of
zeros of $\xi$, then $Z(\xi)$ is finite and satisfies the inequality $0\leq
Z(\xi)\leq a(\hat{x}^{+})-b(\hat{x}^{-})$.

###### Proof.

It is a standard result in Floer theory (see for instance [26], Section $2.2$)
that for any
$u\in\mathcal{M}(\hat{x}^{-},\hat{x}^{+};\mathcal{H},\mathbb{J})$, any element
$\xi\in\ker(D\mathcal{F}_{\mathcal{H},\mathbb{J}})_{u}$ may be expressed (with
respect to the unitary trivialization $\Phi:\mathbb{R}\times
S^{1}\times(\mathbb{R}^{2},J_{0})\rightarrow u^{*}(T\Sigma,J)$ along $u$
induced by the cappings of $\hat{x}^{-}$ and $\hat{x}^{+}$) as solving an
equation of the form

$\displaystyle\partial_{s}\xi+J_{0}\partial_{t}\xi+S\xi$ $\displaystyle=0,$

where we may write $S$ on the positive and negative ends as
$S^{\pm}(s,t)=\Phi^{-1}A_{x^{\pm},J}-\Delta^{\pm}(s,t)$, with $\Delta^{\pm}$
satisfying the decay estimates of Theorem 3.6. Consequently, any
$\xi\in\ker(D\mathcal{F})_{u}$ must be non-vanishing outside of some compact
neighbourhood of $\mathbb{R}\times S^{1}$, and the Carlemann similarity
principle, combined with positivity of intersections of holomorphic curves in
dimension $4$, implies that $Z(\xi)$ is finite and non-negative.

To see that $Z(\xi)\leq a(\hat{x}^{+})-b(\hat{x}^{-})$, we take $R>0$
sufficiently large so that $\xi$ is non-vanishing outside of $(-R,R)\times
S^{1}$ and consider the homotopy of $2$-braids in $\mathbb{R}^{2}$ induced by
$h(s)=(0,\Phi^{-1}\xi_{s})\in\mathcal{L}_{0}(\mathbb{R}^{2})^{2}$,
$s\in[-R,R]$. Theorem 3.6 implies that for $R>0$ sufficiently large,

$\displaystyle\ell(0,\xi_{-R})=wind(\Phi^{-1}\xi_{-R})$ $\displaystyle\geq
b(\hat{x}^{-}),\;and$ $\displaystyle\ell(0,\xi_{R})=wind(\Phi^{-1}\xi_{R})$
$\displaystyle\leq a(\hat{x}^{+}),$

(since $\mathbb{R}^{2}$ is aspherical, we omit any mention of cappings), and
the algebraic count zeros of $\xi$ correspond to the algebraic count of the
intersections of the graphs of the strands of $h$ from which the proposition
follows. ∎

The following corollary is essentially the linear analogue of Corollary 3.20.

######  Corollary 3.22.

Let $(H^{\pm},J^{\pm})$ be Floer regular, let
$(\mathcal{H},\mathbb{J})\in\mathscr{H\mspace{-8.0mu}J}$, and let
$\hat{x}^{\pm}\in\widetilde{Per}_{0}({H^{\pm}})$ have
$\mu(\hat{x}^{\pm})=2k+1$ for some $k\in\mathbb{Z}$. Then
$(\mathcal{H},\mathbb{J})$ is $(\hat{x}^{-},\hat{x}^{+})$-regular.

###### Proof.

If $(H^{-},J^{-})=(H^{+},J^{+})$ and $(\mathcal{H},\mathbb{J})$ is the
constant homotopy of Floer data, then this is automatic by Floer regularity of
$(H^{\pm},J^{\pm})$. The statement is also vacuously true if
$\mathcal{M}(\hat{x}^{-},\hat{x}^{+};\mathcal{H},\mathbb{J})$ is empty. So we
may suppose that $(\mathcal{H},\mathbb{J})$ is not $\mathbb{R}$-invariant and
that there exists some
$u\in\mathcal{M}(\hat{x}^{-},\hat{x}^{+};\mathcal{H},\mathbb{J})$. To see that
$u$ is regular, note that $D\mathcal{F}_{\mathcal{H},\mathbb{J}}$ has Fredholm
index $0$, and so to prove that $(D\mathcal{F}_{\mathcal{H},\mathbb{J}})_{u}$
is surjective, it suffices to show that its kernel vanishes. Suppose to the
contrary that $\xi\in\ker(D\mathcal{F}_{\mathcal{H},\mathbb{J}})_{u}$ is a
non-trivial vector field along $u$. By the previous proposition, we must have
that the algebraic count of its zeros $Z(\xi)$ satisfies

$\displaystyle 0\leq$ $\displaystyle Z(\xi)\leq
a(\hat{x}^{+})-b(\hat{x}^{-})=-k-(k-1)=-1,$

where the second to last inequality follows from Lemma 3.19. Clearly this is a
contradiction, so we conclude that
$\ker(D\mathcal{F}_{\mathcal{H},\mathbb{J}})_{u}=0$ and so $u$ is regular. ∎

In the case of even index orbits, we have a somewhat weaker conclusion that
will still be of use to us

######  Corollary 3.23.

Let $(H^{\pm},J^{\pm})$ be Floer regular, let
$(\mathcal{H},\mathbb{J})\in\mathscr{H\mspace{-8.0mu}J}$, and let
$\hat{x}^{\pm}\in\widetilde{Per}_{0}({H^{\pm}})$ have $\mu(\hat{x}^{\pm})=2k$
for some $k\in\mathbb{Z}$. Then for every
$u\in\mathcal{M}(\hat{x}^{-},\hat{x}^{+};\mathcal{H},\mathbb{J})$,

$\displaystyle\dim\ker(D\mathcal{F}_{\mathcal{H},\mathbb{J}})_{u}$
$\displaystyle\leq 1$

###### Proof.

As in the proof of the previous corollary, it suffices to consider the case
where $(\mathcal{H},\mathbb{J})$ is not $\mathbb{R}$-invariant and
$\mathcal{M}(\hat{x}^{-},\hat{x}^{+};\mathcal{H},\mathbb{J})$ is non-empty.
Letting $u\in\mathcal{M}(\hat{x}^{-},\hat{x}^{+};\mathcal{H},\mathbb{J})$,
consider the behaviour of some
$\eta_{1},\eta_{2}\in\ker(D\mathcal{F}_{\mathcal{H},\mathbb{J}})_{u}$. Let

$\displaystyle\Phi:\mathbb{R}\times S^{1}\times\mathbb{C}$
$\displaystyle\rightarrow u^{*}T\Sigma$

be a unitary trivialization of the tangent bundle along $u$ which extends over
the cappings of $\hat{x}^{\pm}$. Since $\eta_{1}$ and $\eta_{2}$ satisfy an
appropriate perturbed Cauchy-Riemann equation, Theorem 3.6 implies that
$v_{i}(s,t):=(\Phi^{-1}\circ\eta_{i})(s,t)$, $i=1,2$ satisfies

$\displaystyle\lim_{s\rightarrow-\infty}v_{i}(s,t)=\Phi^{-1}(-\infty,t)\circ\xi_{i}(t)$

for $\xi_{i}\in\Gamma^{\infty}((x^{-})^{*}T\Sigma)$ an eigenvector of the
asymptotic operator $A_{x^{-},J^{-}}$ associated to $x^{-}$, whose associated
eigenvalue is positive. Let $\lambda_{0}$ be the smallest positive eigenvalue
of $A_{x^{-},J^{-}}$ and note that every eigenvector $\xi^{\prime}\in
E_{\lambda_{0}}$ is such that $\Phi^{-1}(-\infty,t)\circ\xi^{\prime}(t)$ has
winding number $-k$ by Corollary 3.18, and by the same reasoning as in the
proof of that corollary, $\dim E_{\lambda_{0}}=1$. We claim first that
$\xi_{i}\in E_{\lambda_{0}}$ for $i=1,2$. Indeed, suppose not, then without
loss of generality we may suppose that $\xi_{1}\in E_{\lambda}$ for
$\lambda>\lambda_{0}$. This implies that
$wind(\Phi^{-1}(-\infty,t)\circ\xi_{1}(t))\geq-k+1$, by points $(1)$ and $(2)$
of Proposition 3.14 combined with the fact that
$a(\hat{x}^{-})=b(\hat{x}^{-})$ (so the only other eigenspace whose
eigenvector have winding $-k$ is the eigenspace associated to the largest
negative eigenvalue of $A_{x^{-},J^{-}}$). Whence,

$\displaystyle wind(\Phi^{-1}(-R,t)\circ\eta_{1}(-R,t))$
$\displaystyle\geq-k+1$

for $R>0$ sufficiently large. As in the proof of Proposition 3.21, we have
that

$\displaystyle Z(\eta_{1})$ $\displaystyle\leq
a(\hat{x}^{+})-wind(\Phi^{-1}(-R,t)\circ\eta_{1}(-R,t))$

for $R>0$ sufficiently large, but this implies

$\displaystyle Z(\eta_{1})$ $\displaystyle\leq-k-(-k+1)=-1$

which violates the positivity of $Z(\eta_{1})$. So $\xi_{i}\in
E_{\lambda_{0}}$ for $i=1,2$.

Now consider $\delta:=\eta_{1}-\eta_{2}$. $\delta$ once again satisfies a
perturbed Cauchy-Riemann equation satisfying the required asymptotic decay
conditions, because $\eta_{1}$ and $\eta_{2}$ do. Let
$\xi_{\delta}(t)=\lim_{s\rightarrow-\infty}\delta(s,t)$ be the associated
negative asymptotic eigenvector. We claim that $\xi_{\delta}\in
E_{\lambda_{0}}$. Indeed, if not then we may repeat the above argument with
$\delta$ in the place of $\eta_{1}$ and $\xi_{\delta}$ in the place of
$\xi_{1}$ to derive a contradiction. It follows that the map which sends
$\eta\in\ker(D\mathcal{F}_{\mathcal{H},\mathbb{J}})_{u}$ to its negative
asymptotic eigenvector in $E_{\lambda_{0}}$ is linear, and it is injective by
Theorem 3.6 (since the negative asymptotic eigenvector of $\eta$ is zero if
and only if $\eta$ vanishes on the negative end, and thus by the Carlemann
similarity principle, $\eta$ vanishes everywhere). Since $\dim
E_{\lambda_{0}}=1$, this proves the claim. ∎

##  4 Constructing chain-level continuation maps with prescribed behaviour

In this section, we introduce a technique for designing chain-level
continuation maps such that certain $0$-dimensional moduli spaces may be
guaranteed to be non-empty. For the remainder of the section, we fix Floer
regular $(H^{\pm},J^{\pm})\in C^{\infty}(S^{1}\times M)\times
C^{\infty}(S^{1};\mathcal{J}(M,\omega))$

######  Definition 4.1.

We will say that a finite collection of smooth maps $\\{u_{i}:\mathbb{R}\times
S^{1}\rightarrow\Sigma\\}$, $i=1,\ldots,k$ is a model for a continuation
cobordism (from $(H^{-},J^{-})$ to $(H^{+},J^{+})$) if there exists
$(\mathcal{H},\mathbb{J})\in\mathscr{H\mspace{-8.0mu}J}(H^{-},J^{-};H^{+},J^{+})$
such that $u_{i}$ is an $(\mathcal{H},\mathbb{J})$-Floer cylinder with finite
energy for each $i=1,\ldots,k$. Such a model for a continuation cobordism will
be called an $(\mathcal{H},\mathbb{J})$-model.

If we are doing Floer theory on a surface, then the results of Section 3.2
tell us that the graphs $\tilde{u}_{i}$, when restricted to a sufficiently
large compact set $[-K,K]\times S^{1}$, define a braid cobordism having only
positive intersections (outside this compact set, the maps $u_{i}$ solve the
$s$-independent Floer equations on the ends, and their behaviour is controlled
by the winding behaviour of the eigenvectors of the asymptotic operator as
discussed in Section 3.2). It will be convenient to make the following

######  Definition 4.2.

Let $h:[0,1]\rightarrow\mathcal{L}(C_{k}(\Sigma))$ be a braid cobordism. We
will say that $h$ is a positive (resp. negative) cobordism if for $1\leq
i<j\leq k$, the graphs $\tilde{h}_{i},\tilde{h}_{j}:[0,1]\times
S^{1}\rightarrow[0,1]\times S^{1}\times\Sigma$ are transverse, and every
intersection is positive (resp. negative).

The main result of this section is to show that this essentially topological
condition, combined with the obvious necessary condition on the asymptotic
behaviour of the strands, is sufficient to guarantee the existence of some
regular homotopy of Floer data $(\mathcal{H},J)$ such that each cylinder is
$(\mathcal{H},J)$-Floer.

######  Definition 4.3.

We will say that a finite collection of smooth maps $u_{i}:\mathbb{R}\times
S^{1}\rightarrow\Sigma$, $i=1,\ldots,k$, is a pre-model for a continuation
cobordism (from $(H^{-},J^{-})$ to $(H^{+},J^{+})$) if they satisfy the
following

  1. 1.

There exists $K>0$ such that the maps $u_{i}|_{[-K,K]}$ are the strands of a
positive braid cobordism such that the graphs of these strands intersect
transversally.

  2. 2.

For the same $K$ as above, $u_{i}|_{(-\infty,-K]}$ has finite
$(H^{-},J^{-})$-energy and satisfies the $(H^{-},J^{-})$-Floer equation, while
$u_{i}|_{[K,\infty)}$ has finite $(H^{+},J^{+})$ energy and satisfies the
$(H^{+},J^{+})$-Floer equation, for each $i=1,\ldots,k$.

And we aim to show

######  Theorem 4.4.

If $u_{i}:\mathbb{R}\times S^{1}\rightarrow\Sigma$, $i=1,\ldots,k$ defines a
pre-model for a continuation cobordism from $(H^{-},J^{-})$ to
$(H^{+},J^{+})$, then there exists $u_{i}^{\prime}:\mathbb{R}\times
S^{1}\rightarrow\Sigma$, $i=1,\ldots,k$ which is a model for a continuation
cobordism from $(H^{-},J^{-})$ to $(H^{+},J^{+})$ and such that each
$u_{i}^{\prime}$ differs from $u_{i}$ only by a small homotopy inside a
compact set of $(-K,K)\times S^{1}$ (where here $K>0$ is as in the definition
of a pre-model for a continuation cobordism).

The principal virtue of this result is that the existence or non-existence of
pre-models may largely be reduced to the question of the relative topologies
of the collections of capped orbits (qua capped braids) being connected by
these cylinders, along with asymptotic information provided by the asymptotic
operator.

The argument itself is straightforward and rather hands-on. Our goal is to
show that given a pre-model for a cobordism from $(H^{-},J^{-})$ to
$(H^{+},J^{+})$, we may build a homotopy of Floer data
$(\mathcal{H},J)\in\mathscr{H\mspace{-8.0mu}J}(H^{-},J^{-};H^{+},J^{+})$ such
that each $u_{i}$ is $(\mathcal{H},J)$-Floer. When
$(u_{1}(s_{0},t_{0}),\ldots,u_{k}(s_{0},t_{0}))\in C^{k}(\Sigma)$, it is not
difficult to simply choose $(\mathcal{H}^{s_{0}}_{t_{0}},J^{s_{0}}_{t_{0}})$
such that each $u_{i}$ satisfies the relevant Floer-equation at
$(s_{0},t_{0})$, and this assignment may be made smoothly in $(s,t)$ as long
as $u_{i}(s,t)\neq u_{j}(s,t)$ for $i\neq j$. The difficulty, therefore,
arises when the graphs of the cylinders intersect. To resolve this, in Section
4.1 we explain how to perturb the cylinders in a neighbourhood $V$ of an
intersection point to another pre-model which satisfies the Floer equations
for a judiciously chosen pair $(\mathcal{H}^{V},J^{V})$ on $V$. In Section
4.2, we prove Theorem 4.4 by performing this perturbation at every
intersection point, and then arguing that we may extend the locally
judiciously chosen s-dependent Hamiltonian-almost complex structure pairs to
an adapted homotopy of Floer data $(\mathcal{H},J)$ such that the resulting
perturbed cylinders are all $(\mathcal{H},J)$-Floer. Finally, in Section 4.3,
we address the question of obtaining model cobordisms for regular homotopies
of Floer data from the existence of the model cobordisms constructed in
Theorem 4.4. This gives method for designing Floer continuation maps which
have prescribed chain-level behaviour.

###  4.1 Perturbing positively intersecting strands to solve Floer’s equation
near the intersection

Let $R(\kappa):=\\{z=s+it\in\mathbb{C}:|s|\leq\kappa,|t|\leq\kappa\\}$, and
denote by $R=R(1)$ the closed square of side-length $2$ centered at the
origin.

######  Lemma 4.5.

Let $u,v:D\rightarrow M^{4}$ be smooth embeddings which are transversal and
which intersect positively at $u(0)=v(0)=p\in M$. Suppose that $v$ is
$J$-holomorphic on some neighbourhood of $p$ for some almost complex structure
$J$ on $M$. Then for any $C^{0}$-neighbourhood $\mathcal{U}$ of $u$ in
$C^{\infty}(D;M)$, there exists $\kappa>0$, $\delta>0$ and
$u^{\prime}\in\mathcal{U}$ such that

  1. 1.

$u^{\prime}(0)=p$,

  2. 2.

for $z\in D(\kappa+\delta)$, $u^{\prime}(z)=v(z)$ only if $z=0$,

  3. 3.

$u^{\prime}$ is $J$-holomorphic on $D(\kappa)$,

  4. 4.

$u^{\prime}=u$ on $D(\kappa+\delta)\setminus D(\kappa)$.

Moreover, if $\mathcal{F}$ is a smooth, oriented codimension $2$ foliation of
$M$ such that $T\mathcal{F}$ is $J$-invariant, and both $u$ and $v$ intersect
the leaves of $\mathcal{F}$ positively and transversally, then $u^{\prime}$
may be taken to be transverse to leaves of $\mathcal{F}$.

###### Proof.

By the local existence theorem for holomorphic curves (see theorem $3.1.1$ in
[32], for instance), there exists a $\rho>0$ and a $J$-holomorphic map
$f:R(\rho)\rightarrow M$ such that $f(0)=p$ and
$(\partial_{s}f)_{p}=(\partial_{s}u)_{p}$. Up to taking $\rho$ to be smaller
and restricting the domain of $f$, we may assume that $f$ is an embedding.
Note that, $(\partial_{s}u)_{p}=(\partial_{s}f)_{p}\neq 0$ and hence
$(\partial_{t}f)_{p}\neq 0$, so $f$ and $v$ intersect transversally and
positively, since both are $J$-holomorphic. Note that similar reasoning shows
that $f$ is positively transverse to the leaf of $\mathcal{F}$ passing through
$p$, and so $f$ is positively transverse to $\mathcal{F}$ for all $z\in
R(\rho)$ sufficiently close to $0$. We will construct $u^{\prime}$ by
interpolating between $u$ and $f$. In order to construct this interpolation,
it will be useful to introduce a convenient local coordinate system.

To this end, let us note that there is a neighbourhood $U\subset M$ of $p$, an
$\epsilon>0$, and an orientation-preserving chart

$\displaystyle\phi:U$ $\displaystyle\rightarrow\mathbb{C}\times\mathbb{C},$

such that for all $z\in D(\epsilon)$, $(\phi\circ v)(z)=(z,0)$. Moreover, if
the foliation $\mathcal{F}$ is as in the statement of the lemma, then the fact
that $v$ intersects positively and transversally with the leaves of
$\mathcal{F}$ implies that $\phi$ may be chosen such that for any leaf
$F\in\mathcal{F}$, $\phi(F\cap U)\subset\\{z_{0}\\}\times\mathbb{C}$ for some
$z_{0}\in\mathbb{C}$. That is, $\phi$ locally diffeomorphically sends leaves
of $\mathcal{F}$ into the fibers of the projection map onto the first
coordinate. Since $u$ and $f$ are both positively transverse to $v$, up to
shrinking $\epsilon>0$, for $z\in D(\epsilon)$ we may write

$\displaystyle f_{loc}(z):=(\phi\circ f)(z)=(z,h_{0}(z)),$ $\displaystyle
u_{loc}(z):=(\phi\circ u)(z)=(z,h_{1}(z))$

for $h_{0},h_{1}:D(\epsilon)\rightarrow\mathbb{C}$ smooth functions such that
$h_{0}(0)=h_{1}(0)=0$, $(\partial_{s}h_{0})_{0}=(\partial_{s}h_{1})_{0}\neq
0$, and $\\{(\partial_{s}h_{i})_{0},(\partial_{t}h_{i})_{0}\\}$ is a
positively-oriented basis for $\mathbb{C}$ for $i=0,1$. Next, for
$\tau\in[0,1]$, let us define $h_{\tau}:D(\epsilon)\rightarrow\mathbb{C}$ by

$\displaystyle h_{\tau}(z)$ $\displaystyle:=(1-\tau)h_{0}(z)+\tau h_{1}(z).$

Note that $(\partial_{s}h_{\tau})_{0}$ is constant in $\tau$, and so it is
easy to see that

$\displaystyle\\{(\partial_{s}h_{\tau})_{0},(\partial_{t}h_{\tau})_{0}\\}$
$\displaystyle=\\{(\partial_{s}h_{0})_{0},(1-\tau)(\partial_{t}h_{0})_{0}+\tau(\partial_{t}h_{1})_{0}\\}$

is positively oriented, as the set of all vectors $w\in\mathbb{C}$ such that
$\\{(\partial_{s}h_{\tau})_{0},w\\}$ is positively oriented is clearly a
convex set and contains both $(\partial_{t}h_{0})_{0}$ and
$\tau(\partial_{t}h_{1})_{0}$ by hypothesis. Consequently, for each
$\tau\in[0,1]$ there exists $\epsilon_{\tau}\in(0,\epsilon)$, varying
continuously with $\tau$, such that $h_{\tau}(z)=0$ if and only if $z=0$ for
all $z\in D(\epsilon_{\tau})$. Posing

$\displaystyle\epsilon^{\prime}$
$\displaystyle:=\inf_{\tau\in[0,1]}\epsilon_{\tau},$

we see that $\epsilon^{\prime}>0$ and the maps $u_{\tau}(z):=(z,h_{\tau}(z))$
intersect $D(\epsilon^{\prime})\times\\{0\\}$ only in $(0,0)$ for all
$\tau\in[0,1]$. Fix $\kappa\in(0,\epsilon^{\prime})$ and some
$\delta\in(0,\epsilon^{\prime}-\kappa)$, and let
$\beta:D(\epsilon^{\prime})\rightarrow[0,1]$ be a smooth radially non-
decreasing function which is identically $0$ on $D(\kappa)$ and identically
$1$ outside of $D(\kappa+\delta)$. For $z\in D(\epsilon^{\prime})$, we define

$\displaystyle
u^{\prime}_{loc}(z):=u_{\beta(z)}(z)=(1-\beta(z))f_{loc}+\beta(z)u_{loc}.$

Note that $\phi^{-1}\circ u^{\prime}_{loc}$ then obviously satisfies the first
three items listed in the lemma, by construction, and is moreover obviously
transverse to leaves of $\mathcal{F}$ by construction, since $\phi^{-1}$ sends
graphs of maps $\mathbb{C}\rightarrow\mathbb{C}$ to submanifolds transverse to
leaves of $\mathcal{F}$ by our choice of $\phi$. We then obtain the claimed
$u^{\prime}:D\rightarrow M$ by setting

$u^{\prime}(z)=\begin{cases}(\phi^{-1}\circ u^{\prime}_{loc})(z)&z\in
D(\epsilon^{\prime})\\\ u(z)&z\in D\setminus D(\epsilon^{\prime}).\end{cases}$
(6)

It is then clear from construction that $u^{\prime}$ satisfies the four
properties listed in the lemma, and is transverse to the leaves of
$\mathcal{F}$. Moreover, because $f(0)=u(0)=p$ and $f$ and $u$ are both
locally Lipchitz (being smooth functions), $u^{\prime}$ may obviously be taken
to be as $C^{0}$-close to $u$ as we wish, simply by taking $\kappa$ and
$\delta$ smaller in the preceding argument if necessary. ∎

######  Corollary 4.6.

Let $u,v:\mathbb{R}\times S^{1}\rightarrow\Sigma$ be smooth maps such that
their graphs $\tilde{u},\tilde{v}:\mathbb{R}\times
S^{1}\rightarrow\mathbb{R}\times S^{1}\times\Sigma$ intersect positively and
transversally at $(s_{0},t_{0},p)\in\mathbb{R}\times S^{1}\times\Sigma$. Then
for any neighbourhood $V\subset\mathbb{R}\times S^{1}$ of $(s_{0},t_{0})$,
there exists an open set $U\subset\bar{U}\subset V$ containing
$(s_{0},t_{0})$, smooth maps

$\displaystyle\mathcal{H}_{\bar{U}}:\bar{U}\rightarrow C^{\infty}(\Sigma),$
$\displaystyle J_{\bar{U}}:\bar{U}\rightarrow\mathcal{J}(\Sigma,\omega),$

and a smooth map $u^{\prime}:\mathbb{R}\times S^{1}\rightarrow\Sigma$ agreeing
with $u$ outside of $V$, such that $\tilde{u}^{\prime}$ intersects $\tilde{v}$
positively and transversally at $(s_{0},t_{0},p)$, and such that both
$u^{\prime}$ and $v$ solve the $(\mathcal{H}_{\bar{U}},J_{\bar{U}})$-Floer
equation for $(s,t)\in\bar{U}$.

###### Proof.

All that needs to be shown here is that for $V$ a small enough neighbourhood
of $(s_{0},t_{0})$, we may always choose $\mathcal{H}_{V}:V\rightarrow
C^{\infty}(\Sigma)$ and $J_{V}:V\rightarrow\mathcal{J}(\Sigma,\omega)$ such
that $v$ is $(\mathcal{H}_{V},J_{V})$-Floer on $V$. Once this is established,
we simply apply the above lemma to $\tilde{u}$ and $\tilde{v}$ using Gromov’s
trick to construct an almost complex structure $\tilde{J}^{\mathcal{H}}$ on
$V\times\Sigma$ such that maps solve the $(\mathcal{H}_{V},J_{V})$-Floer
equations if and only if their graphs are $\tilde{J}^{\mathcal{H}}$
holomorphic over $V$. The above lemma gives us an open set
$U\subset\bar{U}\subset V$ containing $(s_{0},t_{0})$ and a map
$\tilde{u}^{\prime}:\mathbb{R}\times S^{1}\rightarrow\mathbb{R}\times
S^{1}\times\Sigma$ which may be taken to be transverse to the fibers of the
projection map $\mathbb{R}\times S^{1}\times\Sigma\rightarrow\mathbb{R}\times
S^{1}$ — and thus may be taken to be the graph of some function
$u^{\prime}:\mathbb{R}\times S^{1}\rightarrow\Sigma$ — which is
$\tilde{J}^{\mathcal{H}}$ holomorphic on $\bar{U}$, and which agrees with
$\tilde{u}$ outside of $V$. We then simply take
$\mathcal{H}_{\bar{U}}:=\mathcal{H}_{V}|_{\bar{U}}$,
$J_{\bar{U}}:=J_{V}|_{\bar{U}}$.

To see that $\mathcal{H}_{V}:V\rightarrow C^{\infty}(\Sigma)$ and
$J_{V}:V\rightarrow\mathcal{J}(\Sigma,\omega)$ may be chosen as claimed, let
$J_{V}$ be arbitrary, and define

$\displaystyle X:V\rightarrow v^{*}T\Sigma|_{V}$ $\displaystyle(s,t)$
$\displaystyle\mapsto(\partial_{t}v-J\partial_{s}v)_{(s,t)}.$

It’s straightforward to see that we may choose $\mathcal{H}_{V}$ to satisfy
$X^{\mathcal{H}_{V}}(v(s,t))=X(s,t)$ for all $(s,t)\in V$. Indeed, define

$\displaystyle\mathcal{H}(X)$ $\displaystyle:=\\{(s,t,H)\in V\times
C^{\infty}(\Sigma):X^{H}(v(s,t))=X(s,t)\\}.$

It is not difficult to verify that $\pi:\mathcal{H}(X)\rightarrow V$ is a
locally trivial fibration (see Corollary 4.13 in the appendix to this
section), with $\pi$ being the obvious projection map. The fiber over $z\in V$
is diffeomorphic to the subspace of functions on $\Sigma$ having a critical
point at $v(z)$. This shows that the fibers are contractible, and so there
exists a section $\mathcal{H}^{V}$. By construction, $v$ is
$(\mathcal{H}_{V},J_{V})$-Floer on $V$. ∎

###  4.2 Proof of Theorem 4.4

We note first that for any pre-model for a continuation cobordism, there are
only ever finitely many intersection points where the strands must be
perturbed to make them locally satisfy Floer’s equation for some choice of
$(\mathcal{H},J)\in\mathscr{H\mspace{-8.0mu}J}(H^{-},J^{-};H^{+},J^{+})$ in a
neighbourhood of the intersection point.

######  Proposition 4.7.

If $u_{i}:\mathbb{R}\times S^{1}\rightarrow\Sigma$, $i=1,\ldots,k$ is a pre-
model for a continuation cobordism, then

$\displaystyle I$ $\displaystyle:=\\{(s,t)\in[-K,K]\times
S^{1}:u_{i}(s,t)=u_{j}(s,t),1\leq i<j\leq k\\},$

where $K$ is as in the definition of a pre-model for a continuation cobordism,
contains only finitely many points and lies in the interior of $[-K,K]\times
S^{1}$.

###### Proof.

That $I$ is disjoint from the boundary of $[-K,K]\times S^{1}$ follows
directly from the fact that, by hypothesis
$h(s):=(u^{s}_{1},\ldots,u^{s}_{k})$, $s\in[-K,K]$ defines a braid cobordism
(here $u^{s}_{i}$ denotes the loop $t\mapsto u_{i}(s,t)$). As a consequence,
$(u^{\pm K}_{1},\ldots,u^{\pm K}_{k})$ are braids, and therefore their graphs
do not intersect. Consequently, $I$ has no accumulation points on
$\\{-K,K\\}\times S^{1}$, and therefore, there exists some $\delta>0$ such
that $I\subset[-K+\delta,K-\delta]\times S^{1}$. Since the intersections of
the graphs of the $u_{i}$ are assumed to be transverse, all points in $I$ must
be isolated, and consequently, $I$ is a finite collection of points. ∎

Let us now prove Theorem 4.4.

###### Proof.

By Lemma 4.7, $I\subset[-K,K]\times S^{1}$ consists of only finitely many
points, and the graphs of the $u_{i}$ do not intersect over $\\{-K,K\\}\times
S^{1}$. By Corollary 4.6 for each $z\in I$, there exists an arbitrarily small
open set $U_{z}$ containing $z$, local sections

$\displaystyle\mathcal{H}^{z}_{loc}:\bar{U}_{z}$ $\displaystyle\rightarrow
C^{\infty}(\Sigma)$ $\displaystyle J^{z}_{loc}:\bar{U}_{z}$
$\displaystyle\rightarrow\mathcal{J}(\Sigma,\omega),$

and small perturbations $u_{i}^{\prime}$ of each $u_{i}$, $i=1,\ldots,k$ such
that each $u_{i}^{\prime}$ satisfies the
$(\mathcal{H}^{z}_{loc},J^{z}_{loc})$-Floer equation on $\bar{U}_{z}$. Let us
write

$\displaystyle V$ $\displaystyle:=([-K,K]\times S^{1})\setminus\bigcup_{z\in
I_{K}}U_{z}.$

Our goal is to find smooth maps

$\displaystyle\mathcal{H}^{V}:V$ $\displaystyle\rightarrow C^{\infty}(\Sigma)$
$\displaystyle J^{V}:V$ $\displaystyle\rightarrow\mathcal{J}(\Sigma,\omega),$

which agree with $\mathcal{H}^{z}_{loc}$ and $J^{z}_{loc}$ respectively on
$\partial\bar{U}_{z}$, which in addition agree with $H^{\pm}$ and with
$J^{\pm}$ respectively on $\\{\pm K\\}\times S^{1}$, and such that, moreover,
each $u_{i}^{\prime}$ is $(\mathcal{H}^{V},J^{V})$-Floer on $V$. Clearly, if
this can be done, these data will patch together to give a homotopy
$(\mathcal{H},J)$ of Floer data between $(H^{-},J^{-})$ and $(H^{+},J^{+})$
such that each $u_{i}^{\prime}$ is $(\mathcal{H},J)$-Floer on all of
$\mathbb{R}\times S^{1}$, and we will be done.

To see that $(\mathcal{H}^{V},J^{V})$ may be chosen as desired we argue
essentially identically to the proof of Corollary 4.6. First take $J^{V}$ to
be any smooth map $J^{V}:V\rightarrow\mathcal{J}(\Sigma,\omega)$ which
restricts to the desired almost complex structures on $\partial V$. Clearly,
this can be done, since the space of $\omega$-compatible almost complex
structures is contractible. Next, for each $i=1,\ldots,k$, we define a section
of $(u_{i}^{\prime}|_{V})^{*}T\Sigma$ by

$\displaystyle X_{i}:V$ $\displaystyle\mapsto(u_{i}^{\prime}|_{V})^{*}T\Sigma$
$\displaystyle(s,t)$
$\displaystyle\mapsto(\partial_{t}u_{i}^{\prime}-J^{V}\partial_{s}u_{i}^{\prime})_{(s,t)},$

and we define $\mathcal{H}(J^{V})\rightarrow V$ to be the fiber bundle with
fiber

$\displaystyle\mathcal{H}(\vec{X})_{s,t}$ $\displaystyle:=\\{H\in
C^{\infty}(\Sigma):X^{H}(u_{i}^{\prime}(s,t))=X_{i}(s,t),\;\forall(s,t)\in
V,\;\forall i=1,\ldots,k\\}.$

It is not difficult to see that $\mathcal{H}(\vec{X})$ is a locally trivial
fibration (apply Corollary 4.13 from the appendix in the setting where the
underlying symplectic manifold is the configuration space $C_{k}(\Sigma)$) and
that each $\mathcal{H}^{z}_{loc}|_{\partial\bar{U}_{z}}$ defines section of
$\mathcal{H}(\vec{X})$ over $\partial\bar{U}_{z}$, in addition to $(\pm
K,t)\mapsto H^{\pm}_{t}$ defining a section over $\\{\pm K\\}\times S^{1}$.
Moreover, each fiber of $\mathcal{H}(\vec{X})$ is diffeomorphic to the space
of functions on $\Sigma$ having critical points at $k$ distinct points. In
particular, $\mathcal{H}(\vec{X})$ has contractible fibers and therefore
admits a smooth section extending the given aforementioned data on $\partial
V$, completing the proof. ∎

###  4.3 From models of cobordisms to regular models

Our ultimate goal is to use the topological construction of pre-models for
cobordisms to guarantee the more rigid existence of Floer continuation maps
with prescribed behaviour. As such, we will need to consider issues of
regularity.

######  Definition 4.8.

Let $(H^{\pm},J^{\pm})$ be Floer regular. A model for a continuation cobordism
from $(H^{-},J^{-})$ to $(H^{+},J^{+})$ will be called regular if it is an
$(\mathcal{H},\mathbb{J})$-model for some
$(\mathcal{H},\mathbb{J})\in\mathscr{H\mspace{-8.0mu}J}_{reg}(H^{-},J^{-};H^{+},J^{+})$.

The following lemma follows essentially by combining Corollaries 3.22 and 3.23
with results which are likely folkloric: that local transversality of the
$s$-dependent Floer operator at some cylinder $u$ only requires perturbations
with support contained in an appropriately dense open neighbourhood of the
graph of $u$, and the fact that near a minimally degenerate Floer cylinder of
index $0$ (ie. when $\dim\ker(D\mathcal{F}_{\mathcal{H},\mathbb{J}})_{u})=1$),
the universal moduli space locally has the structure of a ‘fold’ singularity
over the space of perturbation data. I am unaware of a place in the published
literature where such results are stated, so for the convenience of the
reader, a detailed proof sketch is provided in the appendix to this section
(see Section 4.4.2).

######  Lemma 4.9.

Let $(H^{\pm},J^{\pm})$ be Floer-regular,
$(\mathcal{H},J)\in\mathscr{H\mspace{-8.0mu}J}(H^{-},J^{-};H^{+},J^{+})$,
$\hat{x}^{\pm}\in\widetilde{Per}_{0}({H^{\pm}})$, and $u\in