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## Abstract

A contact structure on a manifold is a maximally non-integrable hyperplane
field in the tangent bundle. Contact manifolds are generalizations of constant
energy hypersurfaces in Hamiltonian phase space, and have many applications to
classical mechanics and dynamical systems. Cylindrical contact homology is a
homology theory associated to a contact manifold, whose chain complex is
generated by periodic trajectories of a Reeb vector field, and whose
differential counts pseudoholomorphic cylinders between generating orbits. We
compute the cylindrical contact homology of the links of the simple
singularities. These 3-manifolds are contactomorphic to $S^{3}/G$ for finite
subgroups $G\subset\text{SU}(2)$. We perturb the degenerate contact form on
$S^{3}/G$ with a Morse function, invariant under the corresponding
$H\subset\text{SO}(3)$ symmetries of $S^{2}$, to achieve nondegeneracy up to
an action threshold. The cylindrical contact homology is recovered by taking a
direct limit of the action filtered homology groups. The ranks of this
homology are given in terms of the number of irreducible representations of
$G$, namely $|\text{Conj}(G)|$, demonstrating a dynamical form of the McKay
correspondence. Furthermore, the homology encodes information regrading the
orbifold base $S^{2}/H$ over which $S^{3}/G$ fibers.

## Acknowledgments

An elder graduate student once suggested that a Cantor function acts as an
appropriate model of progress in a PhD program; advancement feels stalled
almost all of the time, save a handful of bursts of mathematical insight.
Peers, mentors, and loved ones make the uneventful periods bearable, and
without their support, the dissertation cannot arrive. Thus, some
acknowledgments are in order:

First, to the person I owe everything – thank you Mom. You have given me not
only the ultimate gift of life, but one that is both full and stable. You
trained me to be a responsible and hardworking man. Second, to my sister
Natalia: thank you for the perspective that comes from sharing a childhood.
That my best friend is also my sibling is not something to take for granted.

To Tatia Totorica, my first calculus teacher, thank you. The impact you made
on me as a role model during an impressionable time of life eclipses the value
of the powerful mathematical tools you gave me. You are my guiding example of
an effective instructor when I teach calculus classes of my own.

I owe the majority of my development as a mature mathematician to my first
mathematical parents, Iva Stavrov and Paul Allen. I am part of a flock of
students that can call themselves empathetic and curious thinkers after
working under your wings. Both of you will find influences of your teaching
strewn throughout this work, including my stubborn unwillingness to denote the
gradient of $f$ by $\nabla f$.

To Frank Morgan, a mathematical parent and chosen family member, thank you.
You are a father to many. I will always consider your guidance when making the
big kinds of decisions. I also give gratitude for my mathematical and chosen
sister, Lea Kenigsberg. If it were not for you, I would not have a strong case
for the notion of fate. I do know deeply that our paths were meant to cross,
and I think brightly on the future of our friendship and where it will take
us.

To all of the graduate students in the basement of Herman Brown Hall, where
would I be without your support? Firstly, to my algebraic geometer office
mates Austen James and Zac Spaulding – thank you for your patience in my
perpetual demands that we take the ground field to be ${\mathbb{C}}$ in our
discussions. Thank you Giorgio Young for looking over my Sobolev spaces, and
for the laughs. Thank you Alex Nolte for your willingness to chat about math
and more. To those unnamed, thank you.

The most important thanks goes to my advisor, Jo Nelson. Thank you for
teaching me the most beautiful branch of mathematics that I have encountered.
I hold a deep appreciation for Floer theory and how it fits into the field of
mathematics as a whole. I am proud to have contributed to this branch of math
and to have made it more accessible for students to come - certainly, I could
not have made this contribution without your patience and direction.

Finally, I must thank Tina Fey and include my favorite quote from the 2004
film _Mean Girls_ , which has guided me in a surprising number and variety of
situations -

“All you can do in life is try to solve the problem in front of you.”

\- Cady Heron

###### Contents

  1. Abstract
  2. Acknowledgments
  3. 1 Introduction
    1. 1.1 McKay correspondence and links of simple singularities
    2. 1.2 Definitions and overview of cylindrical contact homology
    3. 1.3 Main result and connections to other work
    4. 1.4 Structure of proof of main theorem
  4. 2 Background: orbifold Morse homology and Conley-Zehnder indices
    1. 2.1 Morse and orbifold Morse theory
    2. 2.2 Asymptotic operators and spectral flows
    3. 2.3 The Conley-Zehnder index
    4. 2.4 Cylindrical contact homology as an analogue of orbifold Morse homology
  5. 3 Geometric setup and dynamics
    1. 3.1 Spherical geometry and associated Reeb dynamics
    2. 3.2 Geometry of $S^{3}/G$ and associated Reeb dynamics
    3. 3.3 Cylinders over orbifold Morse trajectories
  6. 4 Computation of filtered cylindrical contact homology
    1. 4.1 Cyclic subgroups
    2. 4.2 Binary dihedral groups ${\mathbb{D}}^{*}_{2n}$
    3. 4.3 Binary polyhedral groups ${\mathbb{T}}^{*}$, ${\mathbb{O}}^{*}$, and ${\mathbb{I}}^{*}$
    4. 4.4 Construction of $H$-invariant Morse-Smale functions
    5. 4.5 Orbifold and contact interplays: an example
  7. 5 Direct limits of filtered cylindrical contact homology
    1. 5.1 Holomorphic buildings in cobordisms
    2. 5.2 Homotopy classes of Reeb orbits and proof of Proposition 5.4
      1. 5.2.1 Cyclic subgroups
      2. 5.2.2 Binary dihedral groups ${\mathbb{D}}_{2n}^{*}$
      3. 5.2.3 Binary polyhedral groups ${\mathbb{T}}^{*}$, ${\mathbb{O}}^{*}$, and ${\mathbb{I}}^{*}$
  8. A Appendix
    1. A.1 Signature of crossing forms
    2. A.2 When contractible Reeb orbits have index 3 and fail to be embedded

##  1 Introduction

The scope of this project combines a wide palate of distinct mathematical
fields. In this dissertation, our studies range from surface singularities and
varieties in algebraic geometry, to Fredholm theory and Sobolev spaces in
analysis; from trees and diagrams in combinatorics, to flows and critical
point theory in differential geometry; from characteristic classes and
covering spaces in algebraic topology, to representations and characters in
finite group theory; from classical mechanics and Hamiltonian systems in
dynamics, to the groups and their associated algebras in Lie theory.

Each of these topics and objects plays a role in either contextualizing or
proving Theorem 1.5; given any _link of a simple singularity_ , $L$, our
result produces the _cylindrical contact homology of $L$_, a graded
${\mathbb{Q}}$-vector space associated to $L$, denoted $CH_{*}(L)$. This
infinite dimensional space is the homology of a chain complex whose generators
are periodic trajectories of a Hamiltonian system, and whose differential
counts cylindrical solutions to a geometric partial differential equation.

This cylindrical contact homology falls under the wide umbrella of _Floer
theory_. Loosely speaking, Floer theories are (co)-homology theories whose
chain complexes are generated by solutions to differential equations (critical
points of functions, fixed points of diffeomorphisms, or periodic trajectories
of a geometric flow, for example), and the differentials count notions of flow
lines between these generating objects. Examples of Floer theories include
Hamiltonian-Floer homology ([F1], [F2], [F3], [S]), embedded contact homology
([H]), symplectic homology ([BO], [FH], [FHW], [Si], [V]), and symplectic
field theory ([EGH], [BEHWZ], [W3]). These theories have been used to prove
existence results in dynamics - for example, by establishing lower bounds on
the number of fixed points of a Hamiltonian symplectomorphism, or the number
of closed embedded Reeb orbits in contact manifolds.

In Section 1, we first review some algebraic geometry necessary to explore the
geometric objects of study in this paper, and to see how the result fits into
a classical story via the McKay correspondence. The rest of the section is
devoted to outlining the general construction of cylindrical contact homology,
and the proof of Theorem 1.5. Section 2 establishes background for the
homology computed by Theorem 1.5, and is meant to provide the reader with a
more complete picture of cylindrical contact homology and its motivations.
Then, Sections 3, 4, and 5 fill in the technical details of the proof of the
main theorem.

###  1.1 McKay correspondence and links of simple singularities

One major way in which algebraic geometry diverges from its differential
counterpart is that the study permits its geometric objects to exhibit
singular points. In the study of 2-dimensional complex varieties, there is a
notion of a _simple singularity_. A simple singularity is locally modeled by
the isolated singular point $[(0,0)]$ of the variety ${\mathbb{C}}^{2}/G$, for
a finite nontrivial subgroup $G\subset\text{SU}(2)$. The natural action of $G$
on ${\mathbb{C}}[u,v]$ has an invariant subring, generated by three monomials,
$m_{i}(u,v)$ for $i=1,2,3$. These $m_{i}$ satisfy a minimal polynomial
relation,

$f_{G}(m_{1}(u,v),m_{2}(u,v),m_{3}(u,v))=0,$

for some nonzero $f_{G}\in{\mathbb{C}}[z_{1},z_{2},z_{3}]$, whose complex
differential has an isolated 0 at the origin. The polynomial $f_{G}$ provides
an alternative perspective of the simple singularity as a hypersurface in
${\mathbb{C}}^{3}$. Specifically, the map

${\mathbb{C}}^{2}/G\to
V:=f^{-1}_{G}(0),\,\,\,\,\,[(u,v)]\mapsto(m_{1}(u,v),m_{2}(u,v),m_{3}(u,v))$

defines an isomorphism of complex varieties, ${\mathbb{C}}^{2}/G\simeq V$. In
this manner, we produce a hypersurface singularity given any finite nontrivial
$G\subset\text{SU}(2)$.

Conversely, we can recover the isomorphism class of $G$ from $V$ by studying
the _Dynkin diagram_ associated to the minimal resolution of a simple
singularity, using the _McKay correspondence_. We detail this process now.
Suppose a variety $V$ exhibits a simple singularity at the point $p\in V$.
There exists a _minimal resolution_ , denoted $(\widetilde{V},\pi,Z)$, of
$(V,p)$. Explicitly, this means that $\widetilde{V}$ is a complex
2-dimensional variety, $\pi:\widetilde{V}\to V$ is birational,
$\pi^{-1}(p)=Z\subset\widetilde{V}$, $\widetilde{V}$ is smooth near $Z$, and
$\pi$ away from $Z$ provides an isomorphism of varieties;

$\pi:\widetilde{V}\setminus Z\xrightarrow{\sim}V\setminus\\{p\\}.$

The resolution $(\widetilde{V},\pi,Z)$ is unique (or _minimal_) in the sense
that any other smoothing of $(V,p)$ factors through $\pi$.

We call $Z$ the _exceptional divisor_ of the resolution; $Z$ is the transverse
intersection of a finite number of copies of
${\mathbb{C}}P^{1}\subset\widetilde{V}$, we enumerate these as
$Z_{1},Z_{2},\dots,Z_{n}$. The _Dynkin diagram_ associated to $(V,p)$ is the
finite graph whose vertex $v_{i}$ is labeled by the sphere $Z_{i}$, and
$v_{i}$ is adjacent to $v_{j}$ if and only if $Z_{i}$ transversely intersects
with $Z_{j}$. In this way, we we associate to any simple singularity $(V,p)$ a
graph, $\Gamma(V,p)$. It is a classical fact that $\Gamma(V,p)$ is isomorphic
to one of the $A_{n}$, $D_{n}$, or the $E_{6}$, $E_{7}$, or $E_{8}$ graphs
(see [Sl, §6]), depicted in Figure 1. This provides the $A_{n}$, $D_{n}$,
$E_{6}$, $E_{7}$, and $E_{8}$ singularities, alternatively known as the
_Klein_ , _Du Val_ , or ADE singularities.

Figure 1: The Dykin diagrams. The diagrams $A_{n}$ and $D_{n}$ both feature
$n$ nodes

Types of finite subgroups of $\text{SU}(2)$: These graphs simultaneously
classify the types of conjugacy classes of finite subgroups $G$ of
$\text{SU}(2)$. To illuminate our discussion of these conjugacy classes, it is
helpful to recall the double cover of Lie groups
$P:\text{SU}(2)\to\text{SO}(3)$ (described in coordinates in (13)).

  1. 1.

A finite subgroup $G$ could be _cyclic_ of order $n$, written
$G\cong{\mathbb{Z}}_{n}$. If $G$ is cyclic, then its image $H:=P(G)$ under $P$
is also cyclic. If $|G|$ is even, then $2|H|=|G|$, otherwise $|H|=|G|$. The
natural action of $H$ on $S^{2}\subset{\mathbb{R}}^{3}$ is by rotations about
some axis. The Dynkin diagram that will be associated to
$G\cong{\mathbb{Z}}_{n}$ is $A_{n-1}$.

  2. 2.

A finite subgroup $G$ could be conjugate to the _binary dihedral group_ ,
denoted ${\mathbb{D}}^{*}_{2n}\subset\text{SU}(2)$, for some integer $n>1$.
The binary dihedral group satisfies $|{\mathbb{D}}_{2n}^{*}|=4n$ and has $n+3$
conjugacy classes. We provide explicit matrix generators of
${\mathbb{D}}_{2n}^{*}$ in Section 4.2, and descriptions of the $n+3$
conjugacy classes in Table 8. The group
${\mathbb{D}}_{2n}:=P({\mathbb{D}}^{*}_{2n})\subset\text{SO}(3)$ is the
_dihedral group_ , and has $2n$ elements. Explicit matrix generators of
${\mathbb{D}}_{2n}$ are also provided in Section 4.2. The natural
${\mathbb{D}}_{2n}$-action on $S^{2}\subset{\mathbb{R}}^{3}$ is given by
symmetries of a regular $2n$-gon whose vertices are contained in a great
circle of $S^{2}$. The Dynkin diagram that will be associated to
$G\cong{\mathbb{D}}_{2n}^{*}$ is $D_{n+2}$.

  3. 3.

Finally a finite subgroup $G$ is _binary polyhedral_ if the image group
$P(G)\subset\text{SO}(3)$ can be identified with the symmetry group of a
regular polyhedron, inscribed in $S^{2}$. This polyhedron is either a
tetrahedron, an octahedron, or an icosahedron (the cube and dodecahedron can
be omitted by duality), in which case the symmetry group
$P(G)\subset\text{SO}(3)$ is denoted ${\mathbb{T}}$, ${\mathbb{O}}$, or
${\mathbb{I}}$, and the group order of $P(G)$ is 12, 24, or 60111These facts
may easily be checked by considering an geometric orbit-stabilizer argument..
In each respective case, we write $G={\mathbb{T}}^{*}$, ${\mathbb{O}}^{*}$, or
${\mathbb{I}}^{*}$; $|G|$ is 24, 48, or 120, and $G$ has 7, 8 , or 9 conjugacy
classes. The Dynkin diagram that will be associated to
$G\cong{\mathbb{T}}^{*}$, ${\mathbb{O}}^{*}$, or ${\mathbb{I}}^{*}$ is
$E_{6}$, $E_{7}$, or $E_{8}$.

It is a classical fact that any finite subgroup $G\subset\text{SU}(2)$ must be
either cyclic, conjugate to ${\mathbb{D}}_{2n}^{*}$, or is a binary polyhedral
group ([Z, §1.6]). Associated to each type of finite subgroup
$G\subset\text{SU}(2)$ is a finite graph, $\Gamma(G)$. The vertices of
$\Gamma(G)$ are in correspondence with the nontrivial irreducible
representations of $G$ (of which there are $|\text{Conj}(G)|-1$, where
$\text{Conj}(G)$ denotes the set of conjugacy classes of a group $G$), and
adjacency is determined by coefficients of tensor products of representations.
For more details regarding this construction of $\Gamma(G)$, consult [St].
Again, $\Gamma(G)$ is isomorphic to one of the $A_{n}$, $D_{n}$, or the
$E_{6}$, $E_{7}$, or $E_{8}$ graphs. Table 1 lists $\Gamma(G)$.

$G\subset\text{SU}(2)$ |  $|\text{Conj}(G)|-1$ |  $\Gamma(G)$
---|---|---
${\mathbb{Z}}_{n}$ |  $n-1$ |  $A_{n-1}$
${\mathbb{D}}_{2n}^{*}$ |  $n+2$ |  $D_{n+2}$
${\mathbb{T}}^{*}$ |  $6$ |  $E_{6}$
${\mathbb{O}}^{*}$ |  $7$ |  $E_{7}$
${\mathbb{I}}^{*}$ |  $8$ |  $E_{8}$

Table 1: Dykin diagrams associated to finite subgroups $G\subset\text{SU}(2)$.
See Section 5.2

for more discussion regarding $\text{Conj}(G)$, the set of conjugacy classes
of $G$.

The McKay correspondence tells us that, given any variety $V$ exhibiting a
simple singularity $p\in V$, and given any finite subgroup $G$ of
$\text{SU}(2)$, then $\Gamma(V,p)=\Gamma(G)$ if and only if a neighborhood of
$p$ in $V$ is isomorphic to a neighborhood of $[(0,0)]$ in
${\mathbb{C}}^{2}/G$.

The _link_ of the singularity $V=f^{-1}_{G}(0)\subset{\mathbb{C}}^{3}$ is the
3-dimensional contact manifold $L:=S^{5}_{\epsilon}(0)\cap V$, with contact
structure $\xi_{L}:=TL\cap J_{{\mathbb{C}}^{3}}(TL)$, where
$J_{{\mathbb{C}}^{3}}$ is the standard integrable almost complex structure on
${\mathbb{C}}^{3}$, and $\epsilon>0$ is small. Analogously to the isomorphism
of varieties ${\mathbb{C}}^{2}/G\cong V$, we have a contactomorphism
$(S^{3}/G,\xi_{G})\simeq(L,\xi_{L})$, where $\xi_{G}$ on $S^{3}/G$ is the
descent of the standard contact structure $\xi$ on $S^{3}$ to the quotient by
the $G$-action.

We adapt a method of computing the cylindrical contact homology of
$(S^{3}/G,\xi_{G})$ as a direct limit of filtered homology groups, described
by Nelson in [N2]. This process uses a Morse function, invariant under a
symmetry group, to perturb a degenerate contact form. We prove a McKay
correspondence result, that the ranks of the cylindrical contact homology of
these links are given in terms of the number of conjugacy classes of the group
$G$. Our computations agree with McLean and Ritter’s work, which provides the
positive $S^{1}$-equivariant symplectic cohomology of the crepant resolution
of ${\mathbb{C}}^{n}/G$ similarly in terms of the number of conjugacy classes
of the finite $G\subset\text{SU}(n)$, [MR, Corollary 2.11].

###  1.2 Definitions and overview of cylindrical contact homology

Let $M$ be a $2n+1$ dimensional smooth manifold, and let $\xi\subset TM$ be a
rank $2n$ sub-bundle of $TM$. We say that $\xi$ is a _contact structure_ on
$M$ if, for any 1-form $\lambda$ that locally describes $\xi$222We say that
$\lambda$ describes $\xi$ if, for all $p$ where $\lambda$ is defined,
$\text{ker}(\lambda_{p})=\xi_{p}$. Note that such a $\lambda$ may only be
locally defined. There exists a global 1-form describing $\xi$ precisely when
the quotient line bundle $TM/\xi\to M$ is trivial.,

$\lambda\wedge(d\lambda)^{n}\in\Omega^{2n+1}(M)$

is nowhere vanishing. In this case, we say $(M,\xi)$ is a _contact manifold_.
If the contact structure $\xi$ is defined by a global 1-form $\lambda$, then
we say that $\lambda$ is a _contact form_ on $M$ for $\xi$.

A contact distribution $\xi$ is _maximally non-integrable_ , meaning there
does not anywhere exist a hypersurface $S$ in $M$ whose tangent space agrees
with $\xi$ at every point of $S$. When $M$ is 3-dimensional, the condition
that $\xi$ is a contact structure is equivalent to

$[X,Y]_{p}\notin\xi_{p},$

for all $p\in M$ and all pointwise-linear independent sections $X$ and $Y$ of
$\xi$ defined locally near $p$. If $\lambda$ is a contact form on $M$ defining
contact distribution $\xi$, and if $f:M\to{\mathbb{R}}$ is a smooth, nowhere
vanishing function on $M$, then $f\lambda$ is also a contact form on $M$
defining the same contact structure $\xi$.

A _contactomorphism_ $f$ from contact manifold $(M_{1},\xi_{1})$ to contact
manifold $(M_{2},\xi_{2})$ is a diffeomorphism $f:M_{1}\to M_{2}$ such that
$f_{*}(\xi_{1})=\xi_{2}$. That is, a contactomorphism takes vectors in the
contact distribution of the first manifold to vectors in the contact
distribution of the second manifold. If $\lambda_{i}$ is a contact form on
$M_{i}$ defining the contact structure $\xi_{i}$, then we say that $f:M_{1}\to
M_{2}$ is a _strict_ contactomorphism if $f^{*}\lambda_{2}=\lambda_{1}$.
Strict contactomorphisms are always contactomorphisms.

######  Example 1.1.

Take $M={\mathbb{R}}^{2n+1}$ with coordinates
$\\{(z,x_{1},y_{1},\cdots,x_{n},y_{n})\\}$. Set

$\lambda_{0}:=dz+\sum_{k=1}^{n}x_{k}dy_{k}.$

Now we have that

$d\lambda_{0}=\sum_{k=1}^{n}dx_{k}\wedge
dy_{k},\,\implies\lambda_{0}\wedge(d\lambda_{0})^{n}=dz\wedge dx_{1}\wedge
dy_{1}\wedge\cdots\wedge dx_{n}\wedge dy_{n}\neq 0,$

and we see that $\lambda_{0}$ is a contact form on $M$. For any
$z_{0}\in{\mathbb{R}}$, the translation

$f_{z_{0}}:M\to
M,\,\,\,f_{z_{0}}(z,x_{1},\dots,y_{n})=(z+z_{0},x_{1},\dots,y_{n})$

defines a strict contactomorphism of $M$.

Contact manifolds are the odd-dimensional siblings of _symplectic manifolds_.
A symplectic form $\omega$ on a manifold $W$ is a closed, nondegenerate
2-form. If $W$ admits a symplectic form then we say that $(W,\omega)$ is a
_symplectic manifold_. A symplectic manifold $W$ is necessarily even
dimensional, and we write $\text{dim}(W)=2n$. Furthermore, $\omega^{n}$ is a
volume form on $W$. A diffeomorphism $f$ from symplectic manifold
$(W_{1},\omega_{1})$ to symplectic manifold $(W_{2},\omega_{2})$ is a
_symplectomorphism_ if $f^{*}\omega_{2}=\omega_{1}$.

######  Example 1.2.

Take $W={\mathbb{R}}^{2n}$ with coordinates
$(x_{1},y_{1},\cdots,x_{n},y_{n})$. Set

$\omega_{0}:=\sum_{k=1}^{n}dx_{k}\wedge dy_{k}.$

Now we have that

$\omega_{0}^{n}=dx_{1}\wedge dy_{1}\wedge\cdots\wedge dx_{n}\wedge dy_{n}\neq
0.$

Thus, $\omega_{0}$ is a symplectic form on $W$.

There are substantial interplays between symplectic and contact geometry;

  1. 1.

Given a $2n+1$-dimensional contact manifold $(M,\xi=\text{ker}(\lambda))$, the
_symplectization_ of $M$ is the $2n+2$-dimensional manifold
${\mathbb{R}}\times M$ with symplectic form
$\omega:=d(e^{r}\pi_{M}^{*}\lambda)$, where $\pi_{M}:W\to M$ is the canonical
projection to $M$, and $r$ is the ${\mathbb{R}}$-coordinate.

  2. 2.

_Contact-type hypersurfaces_ in symplectic manifolds have induced contact
structures, inherited from the ambient symplectic form. These hypersurfaces
generalize the notion of constant energy level sets in phase space. For
example, $S^{3}$ is a contact-type hypersurface in
$({\mathbb{R}}^{4},\omega_{0})$, which we define and work with in Section 3.1.

  3. 3.

A _prequantization bundle_ is a contact manifold $(Y,\lambda)$ presenting as a
principal $S^{1}$-bundle over a symplectic manifold,

$\pi:(Y,\lambda)\to(V,\omega),$

satisfying $\pi^{*}\omega=-d\lambda$. For more details, see ([MS, Example
3.5.11]). The Hopf fibration, $\mathfrak{P}:S^{3}\to S^{2}$, detailed in
Section 3.1, is an example of a prequantization bundle.

  4. 4.

On the linear level, if $\lambda$ is a contact form on $M$, then for any $p\in
M$, $d\lambda|_{\xi_{p}}$ defines a _linear symplectic form_ on the even
dimensional $\xi_{p}\subset T_{p}M$. Consequently, we say that
$(\xi,d\lambda|_{\xi})$ is an example of a _symplectic vector bundle_ over
$M$.

In the interest of presenting an outline of cylindrical contact homology, let
$(Y,\xi)$ be a closed contact 3-manifold with defining contact form $\lambda$.
Most of the following objects and constructions described also exist in
arbitrary odd dimensions, but the specialization to $2n+1=3$ simplifies some
key parts.

The contact form $\lambda$ on $Y$ determines a smooth vector field,
$R_{\lambda}$, called the _Reeb vector field_ , which uniquely satisfies
$\lambda(R_{\lambda})=1$ and $d\lambda(R_{\lambda},\cdot)=0$. For
$t\in{\mathbb{R}}$, let $\phi^{t}:Y\to Y$ denote the time $t$ Reeb flow.
Explicitly, this is a 1-parameter family of diffeomorphisms of $Y$ satisfying

$\tfrac{d}{dt}\phi^{t}|_{p}=R_{\lambda}(\phi^{t}(p))\in
T_{\phi^{t}(p)}Y,\,\,\,\text{for all}\,\,p\in\ Y.$

A Reeb orbit is a map $\gamma:{\mathbb{R}}/T{\mathbb{Z}}\to Y$ with
$\gamma^{\prime}(t)=R_{\lambda}(\gamma(t))$, for some $T>0$. This positive
quantity $T$ is called the _action_ (or sometimes _period_) of $\gamma$,
denoted $\mathcal{A}(\gamma)$, and can always be computed via an integral;

$T=\mathcal{A}(\gamma)=\int_{\gamma}\lambda.$

We consider Reeb orbits up to reparametrization, meaning that, given some
fixed $s\in{\mathbb{R}}$, we do not distinguish between the Reeb orbit
$t\mapsto\gamma(t)$ and the Reeb orbit $t\mapsto\gamma(t+s)$, for
$t\in{\mathbb{R}}/T{\mathbb{Z}}$. Let $\mathcal{P}(\lambda)$ denote the set of
Reeb orbits of $\lambda$ up to reparametrization. If
$\gamma\in\mathcal{P}(\lambda)$ and $m\in{\mathbb{N}}$, then the $m$-fold
iterate of $\gamma$, a map $\gamma^{m}:{\mathbb{R}}/mT{\mathbb{Z}}\to Y$, is
the precomposition of $\gamma$ with the $m$-fold cover,
${\mathbb{R}}/mT{\mathbb{Z}}\to{\mathbb{R}}/T{\mathbb{Z}}$, and is also an
element of $\mathcal{P}(\lambda)$. The orbit $\gamma$ is _embedded_ when
$\gamma:{\mathbb{R}}/T{\mathbb{Z}}\to Y$ is injective. If $\gamma$ is the
$m$-fold iterate of an embedded Reeb orbit, then $m(\gamma):=m$ is the
_multiplicity_ of $\gamma$.

The linearized return map of $\gamma\in\mathcal{P}(\lambda)$, denoted
$P_{\gamma}$, is the restriction of the time $T$ linearized Reeb flow to a
linear symplectomorphism of $(\xi_{\gamma(0)},d\lambda|_{\xi_{\gamma(0)}})$,
defined only after selection of a basepoint $\gamma(0)\in\text{im}(\gamma)$.
That is,

$P_{\gamma}:=d\phi^{T}|_{\xi_{\gamma(0)}}:\xi_{\gamma(0)}\to\xi_{\gamma(0)}.$

Two linearized return maps corresponding to different choices of basepoint
differ by conjugation, so the spectrum of $P_{\gamma}$ is independent of this
choice. We say $\gamma$ is _nondegenerate_ if
$1\notin\text{Spec}(P_{\gamma})$, and that $\lambda$ is _nondegenerate_ if all
$\gamma\in\mathcal{P}(\lambda)$ are nondegenerate. The product of eigenvalues
of $P_{\gamma}$ equals 1: if both eigenvalues are unit complex numbers, then
$\gamma$ is _elliptic_ , and if both eigenvalues are positive (negative) real
numbers, then $\gamma$ is _positive (negative) hyperbolic_. An orbit
$\gamma\in\mathcal{P}(\lambda)$ is _bad_ if it is an even iterate of a
negative hyperbolic orbit, otherwise, $\gamma$ is _good_. In Section 2.4 we
provide a bit of discussion as to which characteristics of bad Reeb orbits
give them their name. Let
$\mathcal{P}_{\text{good}}(\lambda)\subset\mathcal{P}(\lambda)$ denote the set
of good Reeb orbits.

For a nondegenerate contact form $\lambda$, define the ${\mathbb{Q}}$-vector
space generated by $\mathcal{P}_{\text{good}}(\lambda)$:

$CC_{*}(Y,\lambda):={\mathbb{Q}}\langle\mathcal{P}_{\text{good}}(\lambda)\rangle.$

We equip this vector space with a grading using the _Conley-Zehnder index_ ,
which we describe now. Given $\gamma\in\mathcal{P}(\lambda)$, a _symplectic
trivialization_ $\tau$ of $(\gamma^{*}\xi,d\lambda|_{\xi})$ is a
diffeomorphism

$\tau:\gamma^{*}\xi\to{\mathbb{R}}/T{\mathbb{Z}}\times{\mathbb{R}}^{2},$

that restricts to linear symplectomorphisms
$(\xi_{\gamma(t)},d\lambda|_{\xi_{\gamma(0)}})\to(\\{t\\}\times{\mathbb{R}}^{2},\omega_{0}),$
where $\omega_{0}$ is the standard linear symplectic form on
${\mathbb{R}}^{2}$. Symplectic trivializations of
$(\gamma^{*}\xi,d\lambda|_{\xi})$ always exist. Now, given
$\gamma\in\mathcal{P}(\lambda)$ and symplectic trivialization $\tau$, the
_Conley-Zehnder index_ , $\mu_{\text{CZ}}^{\tau}(\gamma)\in{\mathbb{Z}}$, is
an integer that describes the rotation of the Reeb flow along $\gamma$
according to the trivialization, and depends only on the homotopy class of
$\tau$. We develop multiple constructions and perspectives of the Conley-
Zehnder index in Section 2.3.

A bundle automorphism $J$ of a rank $2m$ symplectic vector bundle $(E,\omega)$
over a manifold $X$ is called a _complex structure_ on $E$ if
$J^{2}=-\text{Id}$. The complex structure $J$ is $\omega$-_compatible_ if
$\omega(\cdot,J\cdot)$ defines a $J$-invariant inner product on the fibers of
$E$. The collection of $\omega$-compatible complex structures on $E$ is
denoted $\mathcal{J}(\omega)$, and is non-empty and contractible ([MS,
Proposition 2.6.4]). Any choice of $J\in\mathcal{J}(\omega)$ endows $E$ with
the structure of a rank $m$ complex vector bundle, inducing Chern classes
$c_{i}(E)\in H^{2i}(X;{\mathbb{Z}})$, for $i\in{\mathbb{N}}$. Contractibility
of $\mathcal{J}(\omega)$ ensures that these cohomology classes are independent
in choice of $J$. Because $(\xi,d\lambda|_{\xi})$ defines a rank 2 symplectic
vector bundle over $Y$, this discussion tells us that the first Chern class
$c_{1}(\xi)\in H^{2}(Y;{\mathbb{Z}})$ is well-defined. We say a complex
structure $J$ on $\xi$ is $\lambda$-compatible if it is
$d\lambda|_{\xi}$-compatible, and we set
$\mathcal{J}(\lambda):=\mathcal{J}(d\lambda|_{\xi})$. We call $\xi$ a complex
line bundle even without a $J$ specified.

If $\langle c_{1}(\xi),\pi_{2}(Y)\rangle=0$ and if
$\mu_{\text{CZ}}^{\tau}(\gamma)\geq 3$ for all contractible
$\gamma\in\mathcal{P}(\lambda)$, with any $\tau$ extendable over a disc, we
say the nondegenerate contact form $\lambda$ is _dynamically convex_. The
complex line bundle $\xi$ admits a global trivialization if $c_{1}(\xi)=0$,
which is unique up to homotopy333This is due to the fact that two global
trivializations of $\xi$ differ by a smooth map $Y\to{\mathbb{C}}^{*}$, which
must nullhomotopic if the abelianization of $\pi_{1}(Y)$ is completely
torsion. Furthermore, for a topological space $X$ with finitely generated
integral homology groups, the integer $\beta_{i}(X)$ denotes the rank of the
free part of $H_{i}(X;{\mathbb{Z}}).$ if $\beta_{1}(Y)=0$. In this case, the
integral grading $|\gamma|$ of the generator
$\gamma\in\mathcal{P}(\lambda)_{\text{good}}$ is defined to be
$\mu_{\text{CZ}}^{\tau}(\gamma)-1$ for any $\tau$ induced by a global
trivialization of $\xi$. The links $(S^{3}/G,\xi_{G})$ that we study in this
paper all feature $c_{1}(\xi_{G})=0$ and $\beta_{1}(S^{3}/G)=0$, and thus
$\xi_{G}$ admits a global trivialization (explicitly described in Section 3),
unique up to homotopy, inducing a canonical ${\mathbb{Z}}$-grading on the
chain complex.

Any $J\in\mathcal{J}(\lambda)$ can be uniquely extended to an
${\mathbb{R}}$-invariant, $\omega=d(e^{r}\pi_{Y}^{*}\lambda)$-compatible
almost complex structure on the symplectization ${\mathbb{R}}\times Y$,
mapping $\partial_{r}\mapsto R_{\lambda}$, which we continue to call $J$. A
smooth map

$u:{\mathbb{R}}\times S^{1}\to{\mathbb{R}}\times Y,\,\,(s,t)\mapsto u(s,t)$

is $J$-_holomorphic_ if $du\circ j=J\circ du$, where $j$ is the standard
complex structure on the cylinder ${\mathbb{R}}\times S^{1}$, satisfying
$j\partial_{s}=\partial_{t}$, and $S^{1}={\mathbb{R}}/{\mathbb{Z}}$. The map
$u$ is said to be positively (negatively) asymptotic to the Reeb orbit
$\gamma$ if $\pi_{Y}\circ u(s,\cdot)$ converges uniformly to a parametrization
of $\gamma$ as $s\to\infty$ ($s\to-\infty$) and if $\pi_{{\mathbb{R}}}\circ
u(s,t)\to\pm\infty$ as $s\to\pm\infty$.

Given $\gamma_{+}$ and $\gamma_{-}$ in $\mathcal{P}(\lambda)$,
$\mathcal{M}^{J}(\gamma_{+},\gamma_{-})$ denotes the set of $J$-holomorphic
cylinders positively (negatively) asymptotic to $\gamma_{+}$ ($\gamma_{-}$),
modulo holomorphic reparametrization of $({\mathbb{R}}\times S^{1},j)$. Note
that ${\mathbb{R}}$ acts on $\mathcal{M}^{J}(\gamma_{+},\gamma_{-})$ by
translation on the target. If $m\in{\mathbb{N}}$ and
$u\in\mathcal{M}^{J}(\gamma_{+},\gamma_{-})$, we can precompose $u$ with the
unbranched degree $m$-holomorphic covering map of cylinders, and we denote the
composition by $u^{m};$

$u^{m}:{\mathbb{R}}\times S^{1}\xrightarrow{m\,:\,1}{\mathbb{R}}\times
S^{1}\xrightarrow{u}{\mathbb{R}}\times Y.$

This $m$-_fold cover_ $u^{m}$ is $J$-holomorphic and is an element of
$\mathcal{M}^{J}(\gamma_{+}^{m},\gamma_{-}^{m})$.

A $J$-holomorphic cylinder $u$ is _somewhere injective_ if there exists a
point $z$ in the domain for which $u^{-1}(u(z))=\\{z\\}$ and $du_{z}\neq 0$.
Every $J$-holomorphic cylinder is the $m$-fold cover of a somewhere injective
$J$-holomorphic cylinder $v$, written $u=v^{m}$ ([N1, Theorem 3.7]). Define
the _multiplicity_ of $u\in\mathcal{M}^{J}(\gamma_{+},\gamma_{-})$, $m(u)$, to
be this integer $m$. It is true that $m(u)$ divides both $m(\gamma_{\pm})$ (a
fact that can be seen by considering the number of preimages of a single point
in the image of the Reeb orbits), but it need not be the case that
$\text{GCD}(m(\gamma_{+}),m(\gamma_{-}))=m(u)$, as discussed in Remark 4.8.

Take a cylinder $u\in\mathcal{M}^{J}(\gamma_{+},\gamma_{-})$, and
symplectically trivialize $\gamma_{\pm}^{*}\xi$ via $\tau_{\pm}$. Let $\tau$
denote the trivialization over the disjoint union of Reeb orbits. The
_Fredholm index of_ $u$ is an integer associated to the cylinder which is
related to the dimension of the moduli space in which $u$ appears.
Computationally, this index may be expressed as

$\text{ind}(u)=\mu_{\text{CZ}}^{\tau_{+}}(\gamma_{+})-\mu_{\text{CZ}}^{\tau_{-}}(\gamma_{-})+2c_{1}(u^{*}\xi,\tau),$

where $c_{1}(u^{*}\xi,\tau)$ is a relative first Chern number which vanishes
when $\tau$ extends to a trivialization of $u^{*}\xi$ (see [H, §3.2] for a
full definition). This term should be thought of as a correction term,
determining how compatible the trivializations $\tau_{\pm}$ of
$\gamma_{\pm}^{*}\xi$ are. That is, one could make the difference
$\mu_{\text{CZ}}^{\tau_{+}}(\gamma_{+})-\mu_{\text{CZ}}^{\tau_{-}}(\gamma_{-})$
as arbitrarily large or small as desired, by choosing $\tau_{\pm}$
appropriately.444This is possible by the fact that Conley-Zehnder indices
arising from two different choices of trivialization of $\xi$ along a single
Reeb orbit differ by twice the degree of the associated overlap map (see [W3,
Exercise 3.37]). The value of $c_{1}(u^{*}\xi,\tau)$ accommodates for this
freedom so that $\text{ind}(u)$ is independent in the choice of
trivializations. Of course, when $c_{1}(\xi)=0$, a global trivialization
$\tau$ exists. For this $\tau$, and for any cylinder $u$, the integer
$c_{1}(u^{*}\xi,\tau)$ is always 0, and the Fredholm index of $u$ is simply

$\text{ind}(u)=\mu_{\text{CZ}}(\gamma_{+})-\mu_{\text{CZ}}(\gamma_{-}),$

where $\mu_{\text{CZ}}(\gamma)$ is implicitly computed with respect to the
global trivialization.

For $k\in{\mathbb{Z}}$,
$\mathcal{M}_{k}^{J}(\gamma_{+},\gamma_{-})\subset\mathcal{M}^{J}(\gamma_{+},\gamma_{-})$
denotes those cylinders with $\text{ind}(u)=k$. If
$u\in\mathcal{M}_{k}^{J}(\gamma_{+},\gamma_{-})$ is somewhere injective and
$J$ is generic, then $\mathcal{M}^{J}_{k}(\gamma_{+},\gamma_{-})$ is a
manifold of dimension $k$ near $u$ (see [W2, §8]).

######  Remark 1.3.

If $\text{GCD}(m(\gamma_{+}),m(\gamma_{-}))=1$, then every cylinder
$u\in\mathcal{M}^{J}(\gamma_{+},\gamma_{-})$ must have $m(u)=1$, and
particular, is somewhere injective, implying
$\mathcal{M}^{J}_{k}(\gamma_{+},\gamma_{-})$ is a $k$-dimensional manifold.

######  Definition 1.4.

(Trivial cylinders) Given a Reeb orbit $\gamma:{\mathbb{R}}/T{\mathbb{Z}}\to
Y$ with period $T>0$, we define the _trivial cylinder_ over $\gamma$ by

$u_{\gamma}:{\mathbb{R}}\times{\mathbb{R}}/{\mathbb{Z}}\to{\mathbb{R}}\times
Y,\,\,(s,t)\mapsto(T\cdot s,\gamma(T\cdot t)).$

Note that

$\frac{\partial u_{\gamma}}{\partial s}(s,t)=T\partial_{r}\in
T_{p}({\mathbb{R}}\times Y),\,\,\,\text{and}\,\,\,\frac{\partial
u_{\gamma}}{\partial t}(s,t)=TR_{\lambda}\in T_{p}({\mathbb{R}}\times Y),$

where $p:=u_{\gamma}(s,t)$, and $r$ is the ${\mathbb{R}}$-coordinate on the
symplectization. Importantly, this provides that (for any choice of complex
structure $J\in\mathcal{J}(\lambda)$)

$J\frac{\partial u_{\gamma}}{\partial
s}=JT\partial_{r}=TJ\partial_{r}=TR_{\lambda}=\frac{\partial
u_{\gamma}}{\partial t},$

implying that $J\circ du_{\gamma}=du_{\gamma}\circ j$. The cylinder
$u_{\gamma}$ is called the _trivial cylinder_ associated to $\gamma$, and is
asymptotic to $\gamma$ at $\pm\infty$. By appealing to Stokes’ theorem and
action of Reeb orbits, one concludes that any other parametrized
$J$-holomorphic cylinder asymptotic to $\gamma$ at both ends is a holomorphic
reparametrization of $u_{\gamma}$ so that $\mathcal{M}^{J}(\gamma,\gamma)$ is
a singleton set. Note that the ${\mathbb{R}}$-action on
$\mathcal{M}^{J}(\gamma,\gamma)$ is trivial and that the Fredholm index of
$u_{\gamma}$ is 0.

Suppose that for every pair of good Reeb orbits $\gamma_{\pm}$ that
$\mathcal{M}_{1}^{J}(\gamma_{+},\gamma_{-})/{\mathbb{R}}$ is a compact, zero
dimensional manifold. In this desired case, the differential

$\partial:CC_{*}(Y,\lambda)\to CC_{*-1}(Y,\lambda)$

between two generators $\gamma_{\pm}\in\mathcal{P}_{\text{good}}(\lambda)$ may
be defined as the signed and weighted count of
$\mathcal{M}_{1}^{J}(\gamma_{+},\gamma_{-})/{\mathbb{R}}$:

$\langle\partial\gamma_{+},\gamma_{-}\rangle=\sum_{u\in\mathcal{M}_{1}^{J}(\gamma_{+},\gamma_{-})/{\mathbb{R}}}\epsilon(u)\frac{m(\gamma_{+})}{m(u)}\in{\mathbb{Z}}.$
(1)

Here, the values $\epsilon(u)\in\\{\pm 1\\}$ are induced by a choice of
coherent orientations (see [BM]). Recall that $m(u)$ divides
$m(\gamma_{\pm})$, and so this count is always an integer. In favorable
circumstances, $\partial^{2}=0$ (for example, if $\lambda$ is dynamically
convex, [HN, Theorem 1.3]), and the resulting homology is denoted
$CH_{*}(Y,\lambda,J)$. Under additional hypotheses, this homology is
independent of contact form $\lambda$ defining $\xi$ and generic $J$ (for
example, if $\lambda$ admits no contractible Reeb orbits, [HN2, Corollary
1.10]), and is denoted $CH_{*}(Y,\xi)$. This is the _cylindrical contact
homology_ of $(Y,\xi)$. Upcoming work of Hutchings and Nelson will show that
$CH_{*}(Y,\xi)=CH_{*}(Y,\lambda,J)$ is independent of dynamically convex
$\lambda$ and generic $J$.

###  1.3 Main result and connections to other work

The link of the $A_{n}$ singularity is shown to be contactomorphic to the lens
space $L(n+1,n)$ in [AHNS, Theorem 1.8]. More generally, the links of simple
singularities $(L,\xi_{L})$ are shown to be contactomorphic to quotients
$(S^{3}/G,\xi_{G})$ in [N3, Theorem 5.3], where $G$ is related to $L$ via the
McKay correspondence. Theorem 1.5 computes the cylindrical contact homology of
$(S^{3}/G,\xi_{G})$ as a direct limit of action filtered homology groups.

######  Theorem 1.5.

Let $G\subset\mbox{\em SU}(2)$ be a finite nontrivial group, and let
$m=|\mbox{\em Conj}(G)|\in{\mathbb{N}}$, the number of conjugacy classes of
$G$. The cylindrical contact homology of $(S^{3}/G,\xi_{G})$ is

$CH_{*}(S^{3}/G,\xi_{G}):=\varinjlim_{N}CH_{*}^{L_{N}}(S^{3}/G,\lambda_{N},J_{N})\cong\bigoplus_{i\geq
0}{\mathbb{Q}}^{m-2}[2i]\oplus\bigoplus_{i\geq
0}H_{*}(S^{2};{\mathbb{Q}})[2i].$

The directed system of action filtered cylindrical contact homology groups

$CH_{*}^{L_{N}}(S^{3}/G,\lambda_{N},J_{N}),\,\,\,\text{for}\,\,N\in{\mathbb{N}},$

is described in Section 1.4. Upcoming work of Hutchings and Nelson will show
that the direct limit of this system is an invariant of $(S^{3}/G,\xi_{G})$,
in the sense that it is isomorphic to $CH_{*}(S^{3}/G,\lambda,J)$ where
$\lambda$ is any dynamically convex contact form on $S^{3}/G$ with kernel
$\xi_{G}$, and $J\in\mathcal{J}(\lambda)$ is generic.

The brackets in Theorem 1.5 describe the degree of the grading. For example,
${\mathbb{Q}}^{8}[5]\oplus H_{*}(S^{2};{\mathbb{Q}})[3]$ is a ten dimensional
space with nine dimensions in degree 5, and one dimension in degree 3. By the
classification of finite subgroups $G$ of $\text{SU}(2)$ (see Table 1), the
following enumerates the possible values of $m=|\text{Cong}(G)|$:

  1. 1.

If $G$ is cyclic of order $n$, then $m=n$.

  2. 2.

If $G$ is binary dihedral, written $G\cong{\mathbb{D}}^{*}_{2n}$ for some $n$,
then $m=n+3$.

  3. 3.

If $G$ is binary tetrahedral, octahedral, or icosahedral, written
$G\cong{\mathbb{T}}^{*},{\mathbb{O}}^{*}$, or ${\mathbb{I}}^{*}$, then
$m=7,8$, or $9$, respectively.

######  Remark 1.6.

Theorem 1.5 can alternatively be expressed as

$CH_{*}(S^{3}/G,\xi_{G})\cong\begin{cases}{\mathbb{Q}}^{m-1}&*=0,\\\
{\mathbb{Q}}^{m}&*\geq 2\,\,\mbox{and even}\\\ 0&\mbox{else.}\end{cases}$

In this form, we can compare the cylindrical contact homology to the positive
$S^{1}$-equivariant symplectic cohomology of the crepant resolutions $Y$ of
the singularities ${\mathbb{C}}^{n}/G$ found by McLean and Ritter. Their work
shows that these groups with ${\mathbb{Q}}$-coefficients are free
${\mathbb{Q}}[u]$-modules of rank equal to $m=|\text{Conj}(G)|$, where
$G\subset\text{SU}(n)$ and $u$ has degree 2 [MR, Corollary 2.11]. By [BO], it
is expected that these Floer theories are isomorphic.

The cylindrical contact homology in Theorem 1.5 contains information of the
manifold $S^{3}/G$ as a Seifert fiber space, whose $S^{1}$-action agrees with
the Reeb flow of a contact form defining $\xi_{G}$. Viewing the manifold
$S^{3}/G$ as a Seifert fiber space, which is an $S^{1}$-bundle over an
orbifold surface homeomorphic to $S^{2}$ (see Diagram 23 and Figure 13), the
copies of $H_{*}(S^{2};{\mathbb{Q}})$ appearing in Theorem 1.5 may be
understood as the _orbifold Morse homology_ of this orbifold base $S^{2}/H$
(see Section 2.1 for the construction of this chain complex). Each orbifold
point $p$ with isotropy order $n_{p}\in{\mathbb{N}}$ corresponds to an
exceptional fiber, $\gamma_{p}$, in $S^{3}/G$, which may be realized as an
embedded Reeb orbit. The generators of the ${\mathbb{Q}}^{m-2}[0]$ term are
the iterates

$\gamma_{p}^{k},\,\,\,\text{for $p$ an orbifold point, and
}\,\,k=1,2,\dots,n_{p}-1,$

so that the dimension $m-2=\sum_{p}(n_{p}-1)$ of this summand can be regarded
as a kind of total isotropy of the base. Note that all but finitely many terms
$n_{p}-1$ are nonzero, for $p$ ranging in $S^{2}/H$. The subsequent terms
${\mathbb{Q}}^{m-2}[2i]$ are generated by a similar set of $m-2$ iterates of
orbits $\gamma_{p}$.

######  Remark 1.7.

Recent work of Haney and Mark computes the cylindrical contact homology in
[HM] of a family of Brieskorn manifolds $\Sigma(p,q,r)$, for $p$, $q$, $r$
relatively prime positive integers satisfying
$\frac{1}{p}+\frac{1}{q}+\frac{1}{r}<1$, using methods from [N2]. Their work
uses a family of _hypertight_ contact forms, whose Reeb orbits are non-
contractible. These manifolds are also Seifert fiber spaces, whose cylindrical
contact homology features summands arising from copies of the homology of the
orbit space, as well as summands from the total isotropy of the orbifold.

###  1.4 Structure of proof of main theorem

We now outline the proof of Theorem 1.5. Section 3 explains the process of
perturbing a degenerate contact form $\lambda_{G}$ on $S^{3}/G$ using an
orbifold Morse function. Given a finite, nontrivial subgroup
$G\subset\text{SU}(2)$, $H$ denotes the image of $G$ under the double cover of
Lie groups $P:\text{SU}(2)\to\text{SO}(3)$. By Lemma 3.1, the quotient by the
$S^{1}$-action on the Seifert fiber space $S^{3}/G$ may be identified with a
map $\mathfrak{p}:S^{3}/G\to S^{2}/H$. This $\mathfrak{p}$ fits into a
commuting square of topological spaces (Diagram (23)) involving the Hopf
fibration $\mathfrak{P}:S^{3}\to S^{2}$.

An $H$-invariant Morse-Smale function on
$(S^{2},\omega_{\text{FS}}(\cdot,j\cdot))$ (constructed in Section 4.4)
descends to an _orbifold Morse function_ , $f_{H}$, on $S^{2}/H$, in the
language of [CH]. Here, $\omega_{\text{FS}}$ is the Fubini-Study form on
$S^{2}\cong{\mathbb{C}}P^{1}$, and $j$ is the standard integrable complex
structure. By Lemma 3.4, the Reeb vector field of the $\varepsilon$-perturbed
contact form
$\lambda_{G,\varepsilon}:=(1+\varepsilon\mathfrak{p}^{*}f_{H})\lambda_{G}$ on
$S^{3}/G$ is the descent of the vector field

$\frac{R_{\lambda}}{1+\varepsilon\mathfrak{P}^{*}f}-\varepsilon\frac{\widetilde{X_{f}}}{(1+\varepsilon\mathfrak{P}^{*}f)^{2}}$

to $S^{3}/G$. Here, $\widetilde{X_{f}}\in\xi\subset TS^{3}$ is a
$\mathfrak{P}$-horizontal lift of the Hamiltonian vector field $X_{f}$ of $f$
on $S^{2}$, computed with respect to $\omega_{\text{FS}}$, and with the
convention that $\iota_{X_{f}}\omega_{\text{FS}}=-df$. Thus, the
$\widetilde{X_{f}}$ term vanishes along exceptional fibers $\gamma_{p}$ of
$S^{3}/G$ projecting to orbifold critical points $p\in S^{2}/H$ of $f_{H}$,
implying that these parametrized circles and their iterates $\gamma_{p}^{k}$
are Reeb orbits of $\lambda_{G,\varepsilon}$. Lemma 3.15 computes
$\mu_{\text{CZ}}(\gamma_{p}^{k})$ in terms of $k$ and the Morse index of
$f_{H}$ at $p$, with respect to a global trivialization $\tau_{G}$ of
$\xi_{G}$ (see (11)).

Given a contact 3-manifold $(Y,\lambda)$ and $L>0$, we let
$\mathcal{P}^{L}(\lambda)\subset\mathcal{P}(\lambda)$ denote the set of orbits
$\gamma$ with $\mathcal{A}(\gamma)=\int_{\gamma}\lambda<L$. A contact form
$\lambda$ is $L$-_nondegenerate_ when all $\gamma\in\mathcal{P}^{L}(\lambda)$
are nondegenerate. If $\langle c_{1}(\xi),\pi_{2}(Y)\rangle=0$ and
$\mu_{\text{CZ}}^{\tau}(\gamma)\geq 3$, for all contractible
$\gamma\in\mathcal{P}^{L}(\lambda)$, with any symplectic trivialization $\tau$
extendable over a disc, we say that the $L$-nondegenerate contact form
$\lambda$ is $L$-_dynamically convex_. By Lemma 3.15, given $L>0$, all
$\gamma\in\mathcal{P}^{L}(\lambda_{G,\varepsilon})$ are nondegenerate and
project to critical points of $f_{H}$ under $\mathfrak{p}$, for
$\varepsilon>0$ small. This lemma allows for the computation in Section 4 of
the _action filtered_ cylindrical contact homology, which we describe now.
Recall from Section 1.2, for
$\gamma_{\pm}\in\mathcal{P}_{\text{good}}(\lambda)$ with
$\mu_{\text{CZ}}(\gamma_{+})-\mu_{\text{CZ}}(\gamma_{-})=1$, the differential
$\partial$ of cylindrical contact homology is given by

$\langle\partial\gamma_{+},\gamma_{-}\rangle=\sum_{u\in\mathcal{M}_{1}^{J}(\gamma_{+},\gamma_{-})/{\mathbb{R}}}\epsilon(u)\frac{m(\gamma_{+})}{m(u)}.$
(2)

If $\mathcal{A}(\gamma_{+})<\mathcal{A}(\gamma_{-})$,
$\mathcal{M}^{J}(\gamma_{+},\gamma_{-})$ is empty (because action decreases
along holomorphic cylinders in a symplectization, by Stokes’ Theorem),
ensuring that this sum is zero. Thus, after fixing $L>0$, $\partial$ restricts
to a differential, $\partial^{L}$, on the subcomplex generated by
$\gamma\in\mathcal{P}_{\text{good}}^{L}(\lambda)$, denoted
$CC_{*}^{L}(Y,\lambda)$, whose homology is denoted $CH_{*}^{L}(Y,\lambda,J)$.

In Section 4, we use Lemma 3.15 to produce a sequence
$(L_{N},\lambda_{N},J_{N})_{N=1}^{\infty}$, where $L_{N}\to\infty$ in
${\mathbb{R}}$, $\lambda_{N}$ is an $L_{N}$-dynamically convex contact form
for $\xi_{G}$, and $J_{N}\in\mathcal{J}(\lambda_{N})$ is generic. By Lemmas
4.2 and 4.4, every orbit
$\gamma\in\mathcal{P}^{L_{N}}_{\text{good}}(\lambda_{N})$ is of even degree,
and so $\partial^{L_{N}}=0$, providing

$CH_{*}^{L_{N}}(S^{3}/G,\lambda_{N},J_{N})\cong{\mathbb{Q}}\langle\,\mathcal{P}_{\text{good}}^{L_{N}}(\lambda_{N})\,\rangle\cong\bigoplus_{i=0}^{2N-1}{\mathbb{Q}}^{m-2}[2i]\oplus\bigoplus_{i=0}^{2N-2}H_{*}(S^{2};{\mathbb{Q}})[2i].$
(3)

Finally, we prove Theorem 5.1 in Section 5, which states that a completed
symplectic cobordism $(X,\lambda,J)$ from $(\lambda_{N},J_{N})$ to
$(\lambda_{M},J_{M})$, for $N\leq M$, induces a homomorphism,

$\Psi:CH_{*}^{L_{N}}(S^{3}/G,\lambda_{N},J_{N})\to
CH_{*}^{L_{M}}(S^{3}/G,\lambda_{M},J_{M})$

which takes the form of the standard inclusion when making the identification
(3). The proof of Theorem 5.1 comes in two steps. First, the moduli spaces of
index 0 cylinders in cobordisms $\mathcal{M}_{0}^{J}(\gamma_{+},\gamma_{-})$
are finite by Proposition 5.5 and Corollary 5.6, implying that the map $\Psi$
is well defined. Second, the identification of $\Psi$ with a standard
inclusion is made precise in the following manner. Given
$\gamma_{+}\in\mathcal{P}^{L_{N}}_{\text{good}}(\lambda_{N})$, there is a
unique $\gamma_{-}\in\mathcal{P}^{L_{M}}_{\text{good}}(\lambda_{M})$ that (i)
projects to the same critical point of $f_{H}$ as $\gamma_{+}$ under
$\mathfrak{p}$, and (ii) satisfies $m(\gamma_{+})=m(\gamma_{-})$. When (i) and
(ii) hold, we write $\gamma_{+}\sim\gamma_{-}$. We argue in Section 5 that
$\Psi$ takes the form $\Psi([\gamma_{+}])=[\gamma_{-}]$, when
$\gamma_{+}\sim\gamma_{-}$.

Theorem 5.1 now implies that the system of filtered contact homology groups is
identified with a sequence of inclusions of vector spaces, providing
isomorphic direct limits:

$\varinjlim_{N}CH_{*}^{L_{N}}(S^{3}/G,\lambda_{N},J_{N})\cong\bigoplus_{i\geq
0}{\mathbb{Q}}^{m-2}[2i]\oplus\bigoplus_{i\geq 0}H_{*}(S^{2};{\mathbb{Q}}).$

##  2 Background: orbifold Morse homology and Conley-Zehnder indices

In Section 1, we alluded to orbifold Morse theory and the Conley-Zehnder index
of Reeb orbits. In this section, we provide background for these notions and
explain how they are related.

###  2.1 Morse and orbifold Morse theory

We first review the ingredients and construction of the Morse chain complex
for smooth manifolds, and then compare with its orbifold cousin. Let $(M,g)$
be a closed Riemannian manifold of dimension $n$, and take a smooth function
$f:M\to{\mathbb{R}}$. The triple $(M,g,f)$ determines a vector field, the
_gradient_ of $f$,

$\text{grad}(f)\in\Gamma(TM),\,\,\,g(\text{grad}(f),v)=df(v),\,\,\text{for}\,\,v\in
TM.$

Let $\text{Crit}(f):=\\{p\in M\,|\,df_{p}=0\\}$ denote the _critical point
set_ of $f$. To each $p\in\text{Crit}(f)$, we associate a linear endomorphism,
$H_{p}(f):T_{p}M\to T_{p}M$, the _Hessian_ of $f$ at $p$, whose eigenvalues
provide information about the behavior of $f$ near $p$. We define the Hessian
now.

The Riemannian metric provides the data of a _connection_ on the tangent
bundle $TM$, written $\nabla$. Given any point $p\in M$ and a smooth vector
field $V$, defined locally near $p$, the connection defines a linear map
$T_{p}M\to T_{p}M$, written $v\mapsto\nabla_{v}V$, which satisfies a Leibnitz
rule. The Hessian at the critical point $p\in\text{Crit}(f)$ is then simply
defined in terms of this connection:

$H_{p}(f):T_{p}M\to T_{p}M\,:=\,v\mapsto\nabla_{v}\,\text{grad}(f).$

The values of $H_{p}(f)$ do not depend on the Riemannian metric, because $p$
is a critical point of $f$. Many quantities in Morse theory do not depend on
the (generic) choice of Riemannian metric $g$; the values of $H_{p}(f)$ are a
first example of this independence.

We say that the critical point $p$ is _nondegenerate_ if $0$ is not an
eigenvalue of $H_{p}(f)$. We say that $f$ is a _Morse function_ on $M$ if all
of its critical points are nondegenerate. Note that the Morse condition is
independent of $g$, that generic smooth functions on $M$ are Morse, and that a
Morse function has a finite critical point set. All of these facts can be seen
by the helpful interpretation of $\text{Crit}(f)$ as the intersection of the
graph of $df$ with the zero section in $T^{*}M$; this intersection is
transverse if and only if $f$ is Morse.

As an endomorphism of an inner product space, $H_{p}(f)$ is self-adjoint, and
thus, its spectrum consists of real eigenvalues. Let

$\text{ind}_{p}(f)\in\\{0,1,\dots,n=\text{dim}(M)\\}$

denote the number of negative eigenvalues of $H_{p}(f)$ (counted with
multiplicity). This is the _Morse index_ of $p$. If $f$ is Morse and
$\text{ind}_{f}(p)=n$, then $p$ is a local maximum of $f$; if $f$ is Morse and
$\text{ind}_{f}(p)=0$ then $p$ is a local minimmum of $f$.

For the rest of this section, unless otherwise stated, we assume that $f$ is
Morse. We now associate to each $p\in\text{Crit}(f)$ the stable and unstable
manifolds of $p$, whose dimension depends on this integer $\text{ind}_{f}(p)$.
For every $t\in{\mathbb{R}}$, we have the diffeomorphism of $M$,
$\phi^{t}:M\to M$, given by the time $t$ flow of the smooth vector field
$-\text{grad}(f)$. Note that this $\phi^{t}$ is defined for all
$t\in{\mathbb{R}}$, because $M$ is closed. Given $p\in\text{Crit}(f)$, define
the subsets

$W^{\pm}(p):=\\{x\in M\,|\,\text{lim}_{t\to\pm\infty}\phi^{t}(x)=p\\}.$

We refer to $W^{-}(p)$ as the _unstable submanifold_ of $p$, and to $W^{+}(p)$
as the _stable submanifold_ of $p$. Indeed, these sets are manifolds; it is a
classical fact of Morse theory that $W^{-}(p)$ is a smoothly embedded
$\text{ind}_{f}(p)$ disc in $M$, and $W^{+}(p)$ is a smoothly embedded
$n-\text{ind}_{f}(p)$ disc.

Fix distinct critical points $p$, $q\in\text{Crit}(f)$. If $W^{-}(p)$ and
$W^{+}(q)$ transversely intersect in $M$, then their intersection is an
$\text{ind}_{f}(p)-\text{ind}_{f}(q)$ dimensional submanifold, admitting a
free ${\mathbb{R}}$-action by flowing along $-\text{grad}(f)$. The set

$\mathcal{M}(p,q):=\big{(}W^{-}(p)\cap W^{+}(q)\big{)}/{\mathbb{R}}$

inherits the topology of an $\text{ind}_{f}(p)-\text{ind}_{f}(q)-1$
dimensional manifold, whose points are identified with unparametrized negative
gradient flow lines of $f$. This space $\mathcal{M}(p,q)$ is referred to as
the _moduli space of flow lines_ from $p$ to $q$.

We say that the pair $(f,g)$ is _Smale_ if $W^{-}(p)$ and $W^{+}(q)$ are
transverse in $M$, for all choices $p$, $q\in\text{Crit}(f)$. Given a Morse
function $f$ on $M$, a generic choice of Riemannian metric $g$ provides a
Smale pair $(f,g)$. Assume that $(f,g)$ is a Morse-Smale pair.

An orientation of each disc $W^{-}(p)\subset M$, for $p$ ranging in
$\text{Crit}(f)$, induces an orientation on all $\mathcal{M}(p,q)$ as smooth
manifolds (even if $M$ is not orientable). For more details, see [S, Exercise
1.11]. In particular, in the 0-dimensional case (that is, for
$\text{ind}_{f}(p)-\text{ind}_{f}(q)=1$), we can associate to each
$x\in\mathcal{M}(p,q)$ a sign, $\epsilon(x)\in\\{\pm 1\\}$.

Additionally, the manifold $\mathcal{M}(p,r)$ admits a compactification by
broken flow lines, providing $\overline{\mathcal{M}(p,r)}$, a manifold with
corners. In the 1-dimensional case (that is, for
$\text{ind}_{f}(p)-\text{ind}_{f}(r)=2$), $\overline{\mathcal{M}(p,r)}$ is an
oriented 1-dimensional manifold with boundary, whose boundary consists of
broken flow lines of the form $(x_{+},x_{-})$, for $x_{+}\in\mathcal{M}(p,q)$
and $x_{-}\in\mathcal{M}(q,r)$, where $q\in\text{Crit}(f)$. We can write this
as

$\partial\mathcal{M}(p,r):=\overline{\mathcal{M}(p,r)}\setminus\mathcal{M}(p,r)=\bigcup_{q\in\text{Crit}(f)}\mathcal{M}(p,q)\times\mathcal{M}(q,r).$
(4)

This is an equality of _oriented_ 0-dimensional manifolds, in the sense that
$(x_{+},x_{-})\in\partial\mathcal{M}(p,r)$ is a _positive_ end of
$\overline{\mathcal{M}(p,r)}$ if and only if
$\epsilon(x_{+})\epsilon(x_{-})=+1$.

We associate to the triple $(M,f,g)$ a chain complex, under the assumptions
that $f$ is Morse and that $(f,g)$ is Smale. Select an orientation on each
$W^{-}(p)$, for $p\in\text{Crit}(f)$. For $k\in{\mathbb{Z}}$, define $CM_{k}$
to be the free abelian group generated by the set
$\text{Crit}_{k}(f):=\\{p\in\text{Crit}(f)\,|\,\text{ind}_{f}(p)=k\\}$. For
each $k\in{\mathbb{Z}}$, we define $\partial:CM_{k}\to CM_{k-1}$ to be the
_signed_ count of negative gradient trajectories between critical points of
index difference 1. Specifically, for $p\in\text{Crit}_{k}(f)$ and
$q\in\text{Crit}_{k-1}(f)$,

$\langle\partial p,q\rangle:=\sum_{x\in\mathcal{M}(p,q)}\epsilon(x).$

We explain why $\partial\circ\partial=0$. Select generators
$p\in\text{Crit}_{k}(f)$ and $r\in\text{Crit}_{k-2}(f)$, and consider the
value:

$\langle\partial^{2}p,r\rangle=\sum_{q\in\text{Crit}(f)}\sum_{x_{+}\in\mathcal{M}(p,q)}\sum_{x_{-}\in\mathcal{M}(q,r)}\epsilon(x_{+})\epsilon(x_{-}).$

We argue that the above integer is always 0, for any choice of $p$ and $r$.
Note that by equation (4), this sum is precisely equal to the signed count of
$\partial\mathcal{M}(p,r)$. It is known that the signed count of the oriented
boundary of any compact, oriented 1-manifold with boundary is zero, implying
that the above sum is zero. This demonstrates that
$\big{(}CM_{*},\partial\big{)}$ defines a chain complex over ${\mathbb{Z}}$.
Denote the corresponding homology groups by $H^{\text{Morse}}_{*}(f,g)$.

The Morse homology is isomorphic to the classical singular homology of $M$
with integer coefficients;

$H^{\text{Morse}}_{*}(f,g)\cong H_{*}(M;{\mathbb{Z}}).$

This is because the pair $(f,g)$ determines a cellular decomposition of $M$;
the set $\text{Crit}_{k}(f)$ is in correspondence with the $k$-cells. To each
$p\in\text{Crit}_{k}(f)$, the associated $k$-cell is the embedded disc
$W^{-}(p)$. In fact, at the level of chain complexes,
$\big{(}CM_{*},\partial\big{)}$ is isomorphic to this cellular chain complex
associated to the pair $(f,g)$. For this reason, $H^{\text{Morse}}_{*}(f,g)$
does not depend on the pair $(f,g)$, and is a topological invariant of $M$.
This invariance is used to prove the _Morse inequalities_ , that the number of
critical points of a Morse function $f$ on a closed manifold $M$ is bounded
below by the sum of the Betti numbers of $M$:

$|\text{Crit}(f)|\geq\sum_{i=0}^{\text{dim}(M)}\beta_{i}(M).$

Orbifold Morse homology. We now extend the discussion of Morse data and the
associated chain complex to _orbifolds_. The reader should consult [CH] for
complete details regarding the orbifold-Morse-Smale-Witten complex. For
simplicity, we take our orbifolds to be of the form $M/H$, where $(M,g)$ is a
closed Riemannian manifold and $H$ is a finite group acting on $M$ by
isometries. Let $f:M\to{\mathbb{R}}$ be a Morse function on $M$ which is
invariant under the $H$-action. The quotient space $M/H$ inherits the
structure of an _orbifold_ , and $f$ descends to an _orbifold Morse function_
, $f_{H}:M/H\to{\mathbb{R}}$. For motivating examples, the reader should
consider $M=S^{2}$ with the round metric, and $H\subset\text{SO}(3)$ is a
finite group of dihedral or polyhedral symmetries of $S^{2}$. See Figures 8,
8, 10, and 10 for depictions of possible $H$-invariant Morse functions on
$S^{2}$ for these cases.

As before, for each $p\in\text{Crit}(f)$, select an orientation of the
embedded disc $W^{-}(p)$. Consider that the action of the stabilizer (or,
_isotropy_) subgroup of $p$,

$H_{p}:=\\{h\in H\,|\,h\cdot p=p\\}\subset H,$

on $M$ restricts to an action on $W^{-}(p)$ by diffeomorphisms. We say that
the critical point $p$ is _orientable_ if this action is by _orientation
preserving_ diffeomorphisms. Let $\text{Crit}^{+}(f)\subset\text{Crit}(f)$
denote the set of orientable critical points.

Note that the $H$-action on $M$ permutes $\text{Crit}(f)$, and the action
restricts to a permutation of $\text{Crit}^{+}(f)$. Furthermore, the index of
a critical point is preserved by the action. Let $\text{Crit}(f_{H})$,
$\text{Crit}^{+}(f_{H})$, and $\text{Crit}^{+}_{k}(f_{H})$ denote the
quotients $\text{Crit}(f)/H$, $\text{Crit}^{+}(f)/H$, and
$\text{Crit}^{+}_{k}(f)/H$, respectively. As in the smooth case, we define the
$k^{\text{th}}$-orbifold Morse chain group, denoted $CM_{k}^{\text{orb}}$, to
be the free abelian group generated by $\text{Crit}_{k}^{+}(f_{H})$. The
differential will be defined by a signed _and weighted_ count of negative
gradient trajectories in $M/H$. The homology of this chain complex is, as in
the smooth case, isomorphic to the singular homology of $M/H$ ([CH, Theorem
2.9]).

######  Remark 2.1.

(Discarding non-orientable critical points to recover singular homology)
Consider that every index 1 critical point of $f:S^{2}\to{\mathbb{R}}$
depicted in Figures 8, 8, 10, and 10 is non-orientable. This is because the
unstable submanifolds associated to each of these critical points is an open
interval, and the action of the stabilizer of each such critical point is a
180-degree rotation of $S^{2}$ about an axis through the critical point. Thus,
this action reverses the orientation of the embedded open intervals. If we
were to include these index 1 critical points in the chain complex, then
$CM^{\text{orb}}_{*}$ would have rank three, with

$CM^{\text{orb}}_{0}\cong CM^{\text{orb}}_{1}\cong
CM^{\text{orb}}_{2}\cong{\mathbb{Z}}$

(in Section 4.5 we fill in these details for $H={\mathbb{T}}$, where
$\text{Crit}(f_{{\mathbb{T}}})=\\{\mathfrak{v},\mathfrak{e},\mathfrak{f}\\}$).
There is no differential on this (purported) chain complex with homology
isomorphic to $H_{*}(S^{2}/H;{\mathbb{Z}})\cong H_{*}(S^{2};{\mathbb{Z}})$.
This last isomorphism holds because $S^{2}/H$ is a topological $S^{2}$ in
these examples.555See Section 4.5 and Figure 13 for an explanation of why
$S^{2}/H$ is a topological $S^{2}$ with three orbifold points. Indeed, the
correct chain complex, obtained by discarding the non-orientable index 1
critical points, has rank two:

$CM_{0}^{\text{orb}}\cong
CM_{2}^{\text{orb}}\cong{\mathbb{Z}},\,\,\,CM_{1}^{\text{orb}}=0$

and has completely vanishing differential, producing homology groups
isomorphic to $H_{*}(S^{2}/H;{\mathbb{Z}})\cong H_{*}(S^{2};{\mathbb{Z}})$.

######  Remark 2.2.

(Discarding non-orientable critical points to orient the gradient
trajectories) Let $p$ and $q$ be orientable critical points in $M$ with Morse
index difference equal to 1. Let $x:{\mathbb{R}}\to M/H$ be a negative
gradient trajectory of $f_{H}$ from $[p]$ to $[q]$. Because $p$ and $q$ are
orientable, the value of $\epsilon(\widetilde{x})\in\\{\pm 1\\}$ is
independent of any choice of lift of $x$ to a negative gradient trajectory
$\widetilde{x}:{\mathbb{R}}\to M$ of $f$ in $M$. We define $\epsilon(x)$ to be
this value. Conversely, if one of the points $p$ or $q$ is non-orientable,
then there exist two lifts of $x$ with opposite signs, making the choice
$\epsilon(x)$ dependent on choice of lift.

We now define $\partial^{\text{orb}}:CM_{k}^{\text{orb}}\to
CM_{k-1}^{\text{orb}}$. Select $p\in\text{Crit}_{k}^{+}(f_{H})$ and
$q\in\text{Crit}_{k-1}^{+}(f_{H})$ and let $\mathcal{M}(p,q)$ denote the set
of negative gradient trajectories (modulo the ${\mathbb{R}}$ action by
translations) in $M/H$ of $f_{H}$ from $p$ to $q$. In favorable circumstances,
$\mathcal{M}(p,q)$ is a finite set of signed points. For a flow line
$x\in\mathcal{M}(p,q)$, let $|H_{x}|$ denote the order of the isotropy
subgroup $H_{a}\subset H$ for any choice of $a\in\text{im}(x)\subset M/H$.
This quantity is independent in choice of $a$ and divides both group orders
$|H_{p}|$ and $|H_{q}|$. We define the
$\langle\partial^{\text{orb}}p,q\rangle$ coefficient to be the following
_signed and weighted_ count of $\mathcal{M}(p,q)$:

$\langle\partial^{\text{orb}}p,q\rangle:=\sum_{x\in\mathcal{M}(p,q)}\epsilon(x)\frac{|H_{p}|}{|H_{x}|}\in{\mathbb{Z}}.$

This differential should be directly compared to the differential of
cylindrical contact homology, equation (1). The similarities between these
differentials will be discussed more in Sections 2.4 and 4.5.

###  2.2 Asymptotic operators and spectral flows

In Morse theory and its orbifold version, the number of negative eigenvalues
of the Hessian determines the integral grading of the chain complex. The
analogue of the Hessian in cylindrical contact homology is the _asymptotic
operator_ associated to a Reeb orbit. This asymptotic operator is a bounded
isomorphism between Hilbert spaces that is similar to the Hessian in that it
is self adjoint, and its spectrum is a discrete subset of ${\mathbb{R}}$.

Let $(M,\lambda)$ be a closed contact manifold of dimension $2n+1$, and let
$\gamma$ be a Reeb orbit. In this section for simplicity, we write
$S^{1}={\mathbb{R}}/{\mathbb{Z}}$ and understand $\gamma$ as a map
$S^{1}\to{\mathbb{R}}$, so that $\tfrac{d}{dt}\gamma(t)=T\cdot R(\gamma(t))$,
where $T$ is the action of $\gamma$, and we use $t$ to denote values in
$S^{1}$. The Reeb orbit provides the following symplectic pullback bundle over
$S^{1}$:

$(\gamma^{*}\xi,d\lambda|_{\xi})\to S^{1},$

where $\xi=\text{ker}(\lambda)$. Consider that the linearization of the Reeb
flow along $\gamma$ induces an ${\mathbb{R}}$-action on $\gamma^{*}\xi$ that
provides a parallel transport of the fibers by linear symplectomorphisms. This
family of parallel transports defines a symplectic connection $\nabla$ on
$(\gamma^{*}\xi,d\lambda|_{\xi})$.

Suppose we have a complex structure $J$ on $\xi$ that is compatible with the
symplectic form $d\lambda|_{\xi}$. Letting $\Gamma(\gamma^{*}\xi)$ denote the
space of smooth sections of the bundle $\gamma^{*}\xi\to S^{1}$, we can define
the _asymptotic operator_ associated to $\gamma$,

$A_{\gamma}:\Gamma(\gamma^{*}\xi)\to\Gamma(\gamma^{*}\xi),\,\,\,\eta\mapsto-J\nabla_{t}\eta.$

The motivation for the definition of the asymptotic operator is that it serves
as the linearization of the positive gradient of a contact action functional
on the loop space (i.e. the operator is a Hessian), a framework that we
outline now.

Consider the loop space $\mathcal{L}(M)=C^{\infty}(S^{1},M)$. We have a
_contact action functional_ :

$\mathcal{A}_{\lambda}:\mathcal{L}(M)\to{\mathbb{R}},\,\,\,\mathcal{A}_{\lambda}(\gamma)=\int_{\gamma}\lambda.$

We want to take a sort of gradient of this functional. For
$\gamma\in\mathcal{L}(M)$, $T_{\gamma}\mathcal{L}(M)=\Gamma(\gamma^{*}TM)$.
Let $E\subset T\mathcal{L}(M)$ denote the sub-bundle whose fiber $E_{\gamma}$
over $\gamma\in\mathcal{L}(M)$ is
$\Gamma(\gamma^{*}\xi)\subset\Gamma(\gamma^{*}TM)$. Define a section $\sigma$
of $E\to\mathcal{L}(M)$ by

$\sigma:\mathcal{L}(M)\to E,\,\,\,\sigma(\gamma)=-J\pi_{\xi}\dot{\gamma},$

where $\pi_{\xi}:TM\to\xi$ is the projection along the Reeb direction.
Importantly, if $\gamma$ is a Reeb orbit of $\lambda$, then for all $t\in
S^{1}$, $\pi_{\xi}(\dot{\gamma}(t))=0$, and we conclude that
$\sigma(\gamma)=0$, for all $\gamma\in\mathcal{P}(\lambda)$. This section
$\sigma$ is a positive gradient of $\mathcal{A}_{\lambda}$ with respect to an
$L^{2}$-inner product (defined below). Finally, $A_{\gamma}$ is obtained by
linearizing $\sigma$ at the Reeb orbit $\gamma$. For a more thorough treatment
of this narrative, see [W3, §3.3]

Our next goal is to extend the domain and range of the asymptotic operator
$A_{\gamma}$ to Banach spaces and apply Fredholm theory. Recall that an inner
product $\langle\cdot\,,\cdot\rangle_{p}$ on $\xi_{p}$ is induced by
$d\lambda|_{\xi_{p}}$ and $J_{p}$, for $p\in M$. We introduce an $L^{2}$-inner
product on $\Gamma(\gamma^{*}\xi)$:

$\eta_{1},\eta_{2}\in\Gamma(\gamma^{*}\xi),\,\,\,\,\,\,\langle\eta_{1},\eta_{2}\rangle:=\int_{t\in
S^{1}}\langle\eta_{1}(t),\eta_{2}(t)\rangle_{\gamma(t)}\,dt.$

The operator $A_{\gamma}$ is symmetric with respect to this inner product.
That is, $\langle
A_{\gamma}\eta_{1},\eta_{2}\rangle=\langle\eta_{1},A_{\gamma}\eta_{2}\rangle$.
This can be shown by integrating by parts and observing that $J_{p}$ is a
$\langle\cdot,\cdot\rangle_{p}$-isometry and that $\nabla$ is a symplectic
connection.

Let $L^{2}(\gamma^{*}\xi)$ denote the completion of $\Gamma(\gamma^{*}\xi)$
with respect to this $L^{2}$-inner product, whose corresponding norm is
denoted $|\cdot|_{L^{2}}$. Let $W^{1,2}(\gamma^{*}\xi)$ denote the completion
of $\Gamma(\gamma^{*}\xi)$ with respect to the norm

$|\eta|_{W^{1,2}}:=|\nabla_{t}\eta|_{L^{2}}+|\eta|_{L^{2}}.$

A _unitary trivialization_ of the Hermitian bundle
$(\gamma^{*}\xi,d\lambda|_{\xi},J)$ (defined below) provides an identification
$\Gamma(\gamma^{*}\xi)\cong C^{\infty}(S^{1},{\mathbb{R}}^{2n})$, and also
isometrically identifies $L^{2}(\gamma^{*}\xi)$ and $W^{1,2}(\gamma^{*}\xi)$
with the classical Banach spaces $L^{2}(S^{1},{\mathbb{R}}^{2n})$ and
$W^{1,2}(S^{1},{\mathbb{R}}^{2n})$, respectively. In this way we regard
$W^{1,2}(\gamma^{*}\xi)$ as a dense subspace of $L^{2}(\gamma^{*}\xi)$.

The asymptotic operator extends to $A_{\gamma}:W^{1,2}(\gamma^{*}\xi)\to
L^{2}(\gamma^{*}\xi)$. We understand $A_{\gamma}$ to be an _unbounded
operator_ on $L^{2}(\gamma^{*}\xi)$, whose dense domain of definition is
$W^{1,2}(\gamma^{*}\xi)\subset L^{2}(\gamma^{*}\xi)$. The operator
$A_{\gamma}$ is self-adjoint. Thus, its spectrum is discrete and consists
entirely of real eigenvalues, with each eigenvalue appearing with finite
multiplicity.

Alternatively, we can understand the asymptotic operator as a _bounded
operator_ between Banach spaces,

$A_{\gamma}:\big{(}W^{1,2}(\gamma^{*}\xi),|\cdot|_{W^{1,2}}\big{)}\to\big{(}L^{2}(\gamma^{*}\xi),|\cdot|_{L^{2}}\big{)}.$

We will see that the bounded operator $A_{\gamma}$ is an example of a
_Fredholm operator_ , meaning its kernel and cokernel are finite dimensional.
The _index_ of a Fredholm operator $T$ is defined to be

$\text{ind}(T):=\text{dim}(\text{ker}\,T)-\text{dim}(\text{coker}\,T).$

######  Proposition 2.3.

The bounded operator $A_{\gamma}$ is Fredholm of index 0.

###### Proof.

Take any unitary trivialization $\tau$ of $\gamma^{*}\xi$:

$\tau:\gamma^{*}\xi\to S^{1}\times{\mathbb{R}}^{2n}.$

Explicitly, this is a diffeomorphism, preserving the fibration over $S^{1}$
that restricts to linear isomorphisms
$\xi_{\gamma(t)}\xrightarrow{\sim}\\{t\\}\times{\mathbb{R}}^{2n}$ which take
$d\lambda|_{\xi}$ and $J$ to the standard symplectic form and complex
structure on $\\{t\\}\times{\mathbb{R}}^{2n}$, for $t\in S^{1}$. For some
smooth, symmetric matrix-valued function on $S^{1}$, depending on the choice
of $\tau$, denoted $t\in S^{1}\mapsto S(t)$, the asymptotic operator takes the
form

$A_{\tau}:W^{1,2}(S^{1},{\mathbb{R}}^{2n})\to
L^{2}(S^{1},{\mathbb{R}}^{2n}),\,\,\,\eta\mapsto-J_{0}\dot{\eta}-S\eta.$

Here, $J_{0}$ is the $2n\times 2n$ matrix defining the standard complex
structure on ${\mathbb{R}}^{2n}$.

We first argue that the map

$W^{1,2}(S^{1},{\mathbb{R}}^{2n})\to
L^{2}(S^{1},{\mathbb{R}}^{2n}),\,\,\,\eta\mapsto\dot{\eta}$

is Fredholm of index 0 by showing that both the kernel and cokernel are
$2n$-dimensional. To show this, it is enough to demonstrate that the operator

$T:W^{1,2}(S^{1},{\mathbb{R}})\to
L^{2}(S^{1},{\mathbb{R}}),\,\,\,T(f)=f^{\prime}$

has 1-dimensional kernel and cokernel. The kernel contains the constant
functions, and conversely, if the weak derivative of $f$ is zero almost
everywhere, then $f$ has a representative that is constant on $S^{1}$ in
$L^{2}(S^{1},{\mathbb{R}})$. This shows $\text{dim}(\text{ker}(T))=1$. To
study the cokernel, note that $f\in L^{2}(S^{1},{\mathbb{R}})$ is in the image
of $T$ if and only if

$\int_{S^{1}}f(t)\,dt=0.$

This means that $\text{im}(T)$ precisely equals those elements $f\in
L^{2}(S^{1},{\mathbb{R}})$ whose first Fourier coefficient vanishes, i.e.
$a_{0}=\tfrac{1}{2}\int_{S^{1}}f(t),\,dt=0$. The orthogonal complement of this
space is precisely the constant functions, and so the cokernel of $T$ is also
1-dimensional.

Next, we use this to argue that the asymptotic operator is Fredholm of index 0
as well. The map

$F:W^{1,2}(S^{1},{\mathbb{R}}^{2n})\to
L^{2}(S^{1},{\mathbb{R}}^{2n}),\,\,\,\eta\mapsto-J_{0}\dot{\eta}$

is also Fredholm of index 0, because fiber-wise post-composition by the
constant $-J_{0}$ is a Banach space isometry of
$L^{2}(S^{1},{\mathbb{R}}^{2n})$. Finally, note that the inclusion of
$W^{1,2}(S^{1},{\mathbb{R}}^{2n})$ into $L^{2}(S^{1},{\mathbb{R}}^{2n})$ is a
_compact operator_ , thus so is the map

$K:W^{1,2}(S^{1},{\mathbb{R}}^{2n})\to
L^{2}(S^{1},{\mathbb{R}}^{2n}),\,\,\,\eta\mapsto-S\eta.$

In total, the asymptotic operator takes the form $A_{\tau}=F+K$, where $F$ is
Fredholm of index 0, and $K$ is a compact operator. Because the Fredholm
property (and its index) are stable under compact perturbations, the
asymptotic operator is a Fredholm operator of index 0. ∎

######  Remark 2.4.

The asymptotic operator $A_{\gamma}$ has trivial kernel (and thus, is an
isomorphism) if and only if $\gamma$ is a nondegenerate Reeb orbit. To see
why, recall that a Reeb orbit is degenerate whenever the linearized return map
has 1 as an eigenvalue. If there is a nonzero 1-eigenvector $v$ in
$\xi_{\gamma(0)}$ of the linearized return map $d\phi^{T}$, then one can
produce a smooth section $\eta$ of the pullback bundle $\gamma^{*}\xi$ by
pushing forward $v$ using the linearized Reeb flow. Importantly, this $\eta$
is indeed periodic because we have that $\eta(1)=d\phi^{T}(v)=v=\eta(0)$. The
resulting section $\eta$ is parallel with respect to $\nabla$ (by definition),
and so $A_{\gamma}(\eta)=-J\nabla_{t}\eta=0$. Conversely, given a nonzero
section $\eta$ in the kernel of $A_{\gamma}$, $\eta(0)$ is a 1-eigenvector of
$d\phi^{T}:\xi_{\gamma(0)}\to\xi_{\gamma(0)}$.

Following [W3, §3.2], we define the _spectral flow_ between two asymptotic
operators, which is the first of two ways that we will describe the Conley-
Zehnder indices of Reeb orbits in Section 2.3. Let $\mathcal{F}$ denote the
collection of Fredholm operators

$A:W^{1,2}(S^{1},{\mathbb{R}}^{2n})\to L^{2}(S^{1},{\mathbb{R}}^{2n}).$

This is an open subset of the space of bounded operators from
$W^{1,2}(S^{1},{\mathbb{R}}^{2n})$ to $L^{2}(S^{1},{\mathbb{R}}^{2n})$, and so
is itself a Banach manifold. Furthermore, the function

$\text{ind}:\mathcal{F}\to{\mathbb{Z}},\,\,A\mapsto\text{ind}(A)$

is locally constant, and so the collection of Fredholm index 0 operators,
$\mathcal{F}^{0}\subset\mathcal{F}$, is an open subset and is itself a Banach
manifold. Let $\mathcal{F}^{0,1}\subset\mathcal{F}^{0}$ denote those Fredholm
index 0 operators whose kernel is 1-dimensional. By [W3, Proposition 3.5],
$\mathcal{F}^{0,1}$ is a smooth Banach submanifold of $\mathcal{F}^{0}$ of
codimension 1. Select two invertible elements, $A_{+}$ and $A_{-}$, of
$\mathcal{F}^{0}$ in the same connected component. Then their _relative
spectral flow_ ,

$\mu^{\text{spec}}_{2}(A_{+},A_{-})\in{\mathbb{Z}}/2{\mathbb{Z}}$

is defined to be the mod-2 count of the number of intersections between a
generic path in $\mathcal{F}^{0}$ from $A_{+}$ to $A_{-}$ and
$\mathcal{F}^{0,1}$ (a quantity independent in generic choice of path).

We can upgrade the relative spectral flow to an integer by restricting our
studies to symmetric operators. An operator
$A:W^{1,2}(S^{1},{\mathbb{R}}^{2n})\to L^{2}(S^{1},{\mathbb{R}}^{2n})$ is
_symmetric_ if $\langle Ax,y\rangle=\langle x,Ay\rangle$ for all $x$ and $y$
in the domain. Let $\mathcal{F}_{\text{sym}}$, $\mathcal{F}_{\text{sym}}^{0}$,
and $\mathcal{F}_{\text{sym}}^{0,1}$ denote the intersections of
$\mathcal{F}$, $\mathcal{F}^{0}$, and $\mathcal{F}^{0,1}$ with the closed
subspace of symmetric operators. Now $\mathcal{F}_{\text{sym}}^{0}$ is a
Banach manifold and $\mathcal{F}_{\text{sym}}^{0,1}$ is a _co-oriented_ 666The
_co-oriented_ condition means that the rank 1 real line bundle
$T\mathcal{F}_{\text{sym}}^{0}/T\mathcal{F}_{\text{sym}}^{0,1}$ over
$\mathcal{F}_{\text{sym}}^{0,1}$ is an _oriented_ line bundle. Given a
differentiable path $(-\epsilon,\epsilon)\to\mathcal{F}_{\text{sym}}^{0}$ that
transversely intersects $\mathcal{F}_{\text{sym}}^{0,1}$ at 0, we can use the
given orientation to determine if the intersection is a _positive_ or
_negative_ crossing. In Remark 2.5, we provide a method to determine the sign
of a transverse intersection. codimension-1 submanifold.

Now, given two symmetric, Fredholm index 0 operators, $A_{+}$ and $A_{-}$,
with trivial kernel in the same connected component of
$\mathcal{F}^{0}_{\text{sym}}$, define

$\mu^{\text{spec}}(A_{+},A_{-})\in{\mathbb{Z}}$

to be the signed count of a generic path in $\mathcal{F}^{0}_{\text{sym}}$
from $A_{+}$ to $A_{-}$ with the co-oriented $\mathcal{F}^{0,1}_{\text{sym}}$.
See Remark 2.5 at the end of this section for algebraic descriptions of the
co-orientation and formulaic means of computing
$\mu^{\text{spec}}(A_{+},A_{-})$.

Recall that unitarily trivialized asymptotic operators associated to Reeb
orbits are examples of symmetric Fredholm index 0 operators. A path of these
operators joining $A_{+}$ to $A_{-}$ is described by a 1-parameter family of
loops of $2n\times 2n$ real symmetric matrices,

$\\{S_{s}\\}_{s\in[-1,1]},\,\,\,S_{s}:S^{1}\to\text{Sym}(2n),\,\,\,t\mapsto
S_{s}(t),$

where the corresponding family of asymptotic operators is

$\\{A_{s}\\}_{s\in[-1,1]}\subset\mathcal{F}^{0}_{\text{sym}},\,\,\,A_{s}(\eta)=-J_{0}\dot{\eta}-S_{s}\eta,$

and $A_{\pm 1}=A_{\mp}$. The spectrum of each $A_{s}$ is real, discrete, and
consists entirely of eigenvalues, with each eigenvalue appearing with finite
multiplicity. There exists a ${\mathbb{Z}}$-family of continuous functions
enumerating the spectra of $\\{A_{s}\\}$:

$\\{e_{j}:[-1,1]\to{\mathbb{R}}\\}_{j\in{\mathbb{Z}}},\,\,\,\text{spec}(A_{s})=\\{e_{j}(s)\\}_{j\in{\mathbb{Z}}},\,\,\,\text{for}\,\,s\in[-1,1]$

with multiplicity, in the sense that for any $e\in{\mathbb{R}}$

$\text{dim}(\text{ker}(A_{s}-e\cdot\iota))=\big{|}\\{j\in{\mathbb{Z}}:e_{j}(s)=e\\}\big{|},$

where $\iota:W^{1,2}(S^{1},{\mathbb{R}}^{2n})\to
L^{2}(S^{1},{\mathbb{R}}^{2n})$ is the standard inclusion. Assuming that
$A_{\pm}$ have trivial kernels, then the spectral flow from $A_{+}$ to $A_{-}$
is then the _net_ number of eigenvalues (with multiplicity) that have changed
from negative to positive:

$\mu^{\text{spec}}(A_{+},A_{-})=\big{|}\\{j\in{\mathbb{Z}}:e_{j}(-1)<0<e_{j}(1)\\}\big{|}-\big{|}\\{j\in{\mathbb{Z}}:e_{j}(-1)>0>e_{j}(1)\\}\big{|}.$

See [W3, Theorem 3.3].

######  Remark 2.5.

(Co-orientation of the hypersurface $\mathcal{F}_{\text{sym}}^{0,1}$) On one
hand, the spectral flow is given as a path’s signed count of intersections
with a co-oriented hypersurface. On the other, the spectral flow equals the
net change in eigenvalues that have changed from negative to positive. This
equality implies that a differentiable path
$A:(-\epsilon,\epsilon)\to\mathcal{F}_{\text{sym}}^{0}$ _positively_ crosses
$\mathcal{F}_{\text{sym}}^{0,1}$ at $0$ if a corresponding 1-parameter family
of eigenvalues changes sign from negative to positive at 0. That is, when
there exists differentiable families $e:(-\epsilon,\epsilon)\to{\mathbb{R}}$,
and $\eta:(-\epsilon,\epsilon)\to W^{1,2}(S^{1},{\mathbb{R}}^{2n})$ nonzero,
with $A_{s}\eta_{s}=e(s)\eta_{s}$, the transverse intersection of $A$ with
$\mathcal{F}_{\text{sym}}^{1,0}$ is positive if and only if $\dot{e}(0)>0$.

We use Remark 2.5 to algebraically refashion our geometric description of
$\mu^{\text{spec}}(A_{+},A_{-})$. Fix invertible asymptotic operators
$A_{\pm}$. Let $A:=\\{A_{s}\\}_{s\in[-1,1]}$ be a generic differentiable path
of asymptotic operators from $A_{+}$ to $A_{-}$ (meaning $A_{\pm 1}=A_{\mp}$)
in $\mathcal{F}_{\text{sym}}^{0}$, so that $A_{s}=-J_{0}\partial_{t}-S_{s}$,
where $S_{s}:S^{1}\to\text{Sym}(2n)$ is a loop of symmetric matrices, for
$s\in[-1,1]$. The path transversely intersects the co-oriented hypersurface
$\mathcal{F}^{0,1}_{\text{sym}}$. A value of $s\in(-1,1)$ for which
$A_{s}\in\mathcal{F}_{\text{sym}}^{0,1}$ is called a _crossing_ for A. Let
$s_{0}\in(-1,1)$ be a crossing for $A$, and define a quadratic form

$\Gamma(A,s_{0}):\text{ker}(A_{s_{0}})\to{\mathbb{R}},\,\,\,\eta\mapsto\langle\dot{A}_{s_{0}}\eta,\eta\rangle_{L^{2}},$

where $\langle\cdot\,,\cdot\rangle_{L^{2}}$ is the standard $L^{2}$-inner
product. Because $A$ is generic, the eigenvalue of $\Gamma(A,s_{0})$ is
nonzero. The sign of the single eigenvalue of $\Gamma(A,s_{0})$, denoted
$\text{Sign}(\Gamma(A,s_{0}))\in\\{\pm 1\\}$, is the signature of the
quadratic form.

######  Claim 2.6.

The path $A$ intersects $\mathcal{F}_{\text{sym}}^{0,1}$ positively at $s_{0}$
if and only if $\text{Sign}(\Gamma(A,s_{0}))=+1$.

###### Proof.

By arguments used in [W3, Theorem 3.3], we have a differentiable path of
nonzero eigenvectors of $A_{s}$,

$\eta:(s_{0}-\epsilon,s_{0}+\epsilon)\to
W^{1,2}(S^{1},{\mathbb{R}}^{2n}),\,\,\,s\mapsto\eta_{s},$

and a path of corresponding eigenvalues

$e:(s_{0}-\epsilon,s_{0}+\epsilon)\to{\mathbb{R}},\,\,\,s\mapsto e(s),$

such that for all $s$,

$A_{s}\eta_{s}=e(s)\eta_{s}.$

We have that $e(s_{0})=0$. For simplicity, we can assume that the eigenvectors
are of unit length, i.e. $|\eta_{s}|_{L^{2}}=1$ for all $s$. Now, take a
derivative in the $s$ variable of the above equation, evaluate at $s_{0}$, and
take an inner product with $\eta_{s_{0}}$ to obtain

$\Gamma(A,s_{0})(\eta_{s_{0}})=\langle\dot{A}_{s_{0}}\eta_{s_{0}},\eta_{s_{0}}\rangle_{L^{2}}=\dot{e}(s_{0}),$

where we have used that $\langle
A_{s_{0}}\dot{\eta}_{s_{0}},\eta_{s_{0}}\rangle=\langle\dot{\eta}_{s_{0}},A_{s_{0}}\eta_{s_{0}}\rangle=0$
by symmetry and that $\eta_{s_{0}}\in\text{ker}(A_{s_{0}})$. Ultimately, this
verifies that the sign of the quadratic form $\Gamma(A,s_{0})$ equals that of
$\dot{e}(s_{0})$, which is the sign of the intersection of the path $A$ with
$\mathcal{F}_{\text{sym}}^{0,1}$ at $s_{0}$. ∎

######  Corollary 2.7.

The spectral flow satisfies

$\mu^{spec}(A_{+},A_{-})=\sum_{s\in(-1,1)}\mbox{\em Sign}(\Gamma(A,s)),$

where $A_{\pm}$ are invertible elements of $\mathcal{F}_{\text{sym}}^{0}$,
$A=\\{A_{s}\\}_{s\in[-1,1]}$ is a generic777See [W3, Appendix C] for a
thorough description of the topology of the space of asymptotic operators and
of genericity in this context., differentiable family of asymptotic operators
from $A_{+}$ to $A_{-}$ (meaning $A_{\pm 1}=A_{\mp}$) of the form
$A_{s}=-J_{0}\partial_{t}-S_{s}$.

###  2.3 The Conley-Zehnder index

In this section, we outline various ways to define and work with Conley-
Zehnder indices. The Conley-Zehnder index is an integer that we assign to a
nondegenerate Reeb orbit $\gamma$ with respect to a choice of trivialization
$\tau$ of $\gamma^{*}\xi$, denoted
$\mu_{\text{CZ}}^{\tau}(\gamma)\in{\mathbb{Z}}$. This integer mirrors the role
of the Morse index associated to a critical point. However, as a standalone
number, the Conley-Zehnder index has less geometric meaning than the Morse
index; rather, the _difference_ of Conley-Zehnder indices is generally the
more valuable quantity to consider.

Let $\gamma$ be a nondegenerate Reeb orbit in contact manifold $(M,\lambda)$
of dimension $2n+1$. Let $J$ be a complex structure on
$\xi=\text{ker}(\lambda)$ that is compatible with $d\lambda|_{\xi}$. Let

$\tau:\gamma^{*}\xi\to S^{1}\times{\mathbb{R}}^{2n}$

be a unitary trivialization of $(\gamma^{*}\xi,d\lambda|_{\xi},J)$. Recall
from Section 2.2 that $\gamma$ and $\tau$ define a nondegenerate asymptotic
operator, denoted $A_{\tau}$

$A_{\tau}:W^{1,2}(S^{1},{\mathbb{R}}^{2n})\to
L^{2}(S^{1},{\mathbb{R}}^{2n}),\,\,\,A_{\tau}(\eta)=-J_{0}\dot{\eta}-S\eta,$

where $S$ is some smooth, $S^{1}$-family of symmetric matrices. Define the
_Conley-Zehnder index_ associated to $\gamma$ with respect to $\tau$:

$\mu_{\text{CZ}}^{\tau}(\gamma):=\mu^{\text{spec}}(A_{\tau},A_{0}).$ (5)

Here, $A_{0}:W^{1,2}(S^{1},{\mathbb{R}}^{2n})\to
L^{2}(S^{1},{\mathbb{R}}^{2n})$ is the reference operator

$A_{0}(\eta)=-J_{0}\dot{\eta}-S_{0}\eta,$

where $S_{0}$ is the constant, symmetric, diagonal matrix
$\text{Id}_{n}\oplus-\text{Id}_{n}$, and $J_{0}$ is the $2n\times 2n$ block
matrix

$J_{0}=\begin{pmatrix}0&-\text{Id}_{n}\\\ \text{Id}_{n}&0\end{pmatrix}.$

Because the spectral flow is defined in terms of an intersection of paths with
a co-oriented hypersurface, we immediately have the following difference
formula by concatenation

$\mu_{\text{CZ}}^{\tau_{+}}(\gamma_{+})-\mu_{\text{CZ}}^{\tau_{-}}(\gamma_{-})=\mu^{\text{spec}}(A_{+},A_{-}),$

where $\gamma_{\pm}$ are nondegenerate Reeb orbits, $\tau_{\pm}$ are unitary
trivializations of $\gamma_{\pm}^{*}\xi$, and
$A_{\pm}:W^{1,2}(S^{1},{\mathbb{R}}^{2n})\to L^{2}(S^{1},{\mathbb{R}}^{2n})$
are the asymptotic operators induced by $\tau_{\pm}$.

There is a second definition of the Conley-Zehnder index, as an integer
assigned to a path of symplectic matrices. The following remark is necessary
in bridging the two perspectives.

######  Remark 2.8.

Let $\Phi:[0,1]\to\text{Sp}(2n)$ be a differentiable, 1-parameter family of
symplectic matrices. Then $S:=-J_{0}\dot{\Phi}\Phi^{-1}$ is a 1-parameter
family of _symmetric_ matrices. Conversely, given a differentiable 1-parameter
family of symmetric matrices $S:[0,1]\to\text{Sym}(2n)$, then the solution
$\Phi$ of the initial value problem

$\dot{\Phi}=J_{0}S\Phi,\,\,\,\Phi(0)=\text{Id}$

is a 1-parameter family of _symplectic_ matrices.

We now define the _Conley-Zehnder index_ associated to an arc of symplectic
matrices $\Phi$, denoted $\mu_{\text{CZ}}(\Phi)\in{\mathbb{Z}}$. We’ll then
use $\mu_{\text{CZ}}(\Phi)$ to provide an alternative definition of
$\mu_{\text{CZ}}^{\tau}(\gamma)$, equivalent to (5). Let
$\Phi:[0,1]\to\text{Sp}(2n)$ be a differentiable path of symplectic matrices
such that $\Phi(0)=\text{Id}$ and $\text{ker}(\Phi(1)-\text{Id})=0$. Let $S$
denote the corresponding path of symmetric matrices, from Remark 2.8. We say
that $t\in(0,1)$ is a _crossing_ for $\Phi$ if $\text{ker}(\Phi(t)-\text{Id})$
is nontrivial. For each crossing $t$, define a quadratic form
$\Gamma(\Phi,t):\text{ker}(\Phi(t)-\text{Id})\to{\mathbb{R}}$ by

$\Gamma(\Phi,t)(v)=\omega_{0}(v,\dot{\Phi}(t)v)=\langle v,S(t)v\rangle,$

where $\omega_{0}$ and $\langle\cdot,\cdot\rangle$ are the standard symplectic
form and inner product on ${\mathbb{R}}^{2n}$. The crossing $t$ is _regular_
if this form is nondegenerate, i.e., 0 is not an eigenvalue. Note that regular
crossings are isolated in $(0,1)$ and that a generic path $\Phi$ has only
regular crossings.

Now, we define the Conley-Zehnder index of a path $\Phi$ with only regular
crossings to be

$\mu_{\text{CZ}}(\Phi):=\frac{1}{2}\text{Sign}(S(0))+\sum_{t\in(0,1)}\text{Sign}\big{(}\Gamma(\Phi,t)\big{)},$
(6)

where $\text{Sign}(\cdot)$ is the number of positive eigenvalues minus the
number of negative eigenvalues, with multiplicity, of either a quadratic form
or a symmetric matrix. The Conley-Zehnder index is invariant under
perturbations relative to the endpoints. Thus, after a small perturbation of
any $\Phi$, we can assume that all crossings are regular.

Geometrically, the Conley-Zehnder index of a path of matrices is a signed
count of its intersection with the set
$X=\\{A\in\text{Sp}(2n):\text{det}(A-\text{Id})=0\\}$. There is a
stratification of $X$ by $\text{dim}(\text{ker}(A-\text{Id}))$. Generically, a
path in $\text{Sp}(2n)$ will transversely intersect $X$ at points in the
largest stratum, the hypersurface $X^{1}=\\{A\in
X:\text{dim}(\text{ker}(A-\text{Id}))=1\\}$.

Let $\gamma:S^{1}\to M$ be a nondegenerate Reeb orbit with action $T$, and let
$\tau$ be a unitary trivialization of $(\gamma^{*}\xi,d\lambda|_{\xi},J)$. For
$t\in[0,1]$, let $\Phi(t)\in\text{Sp}(2n)$ represent the time $T\cdot
t$-linearized Reeb flow from $\xi_{\gamma(0)}$ to $\xi_{\gamma(t)}$ with
respect to the trivialization $\tau$ (we must re-normalize by $T$ because of
our convention in this section of renormalization of the period of $\gamma$ by
$T$). Note that $\text{ker}(\Phi(1)-\text{Id})=0$, because $\gamma$ is
nondegenerate. Now, define the Conley-Zehnder index of $\gamma$ with respect
to $\tau$:

$\mu_{\text{CZ}}^{\tau}(\gamma):=\mu_{\text{CZ}}(\Phi).$ (7)

######  Proposition 2.9.

The definitions (5) and (7) of the Conley-Zehnder index of a Reeb orbit with a
unitary trivialization are equivalent.

###### Proof.

We expand on the arguments used in [S, §2.5] regarding spectral flows in
Hamiltonian Floer theory, which concerns negative gradient trajectories of the
action functional. Note the differences in sign conventions come from the fact
that symplectic field theory concerns _positive_ gradient trajectories. See
[W3, §3.3] for more details on regarding these sign conventions.

Take nondegenerate Reeb orbits $\gamma_{\pm}$ and unitary trivializations
$\tau_{\pm}$ of $\gamma_{\pm}^{*}\xi$. Let
$A_{\pm}=-J_{0}\partial_{t}-S_{\pm}$ be the corresponding asymptotic operators
as in definition (5), let $\Phi_{\pm}$ be the corresponding paths of
symplectic matrices as appearing in definition (7). The paths $\Phi_{\pm}$ are
related to the loops $S_{\pm}$ by Remark 2.8. We first argue that

$\mu_{\text{CZ}}(\Phi_{+})-\mu_{\text{CZ}}(\Phi_{-})=\mu^{\text{spec}}(A_{+},A_{-}).$
(8)

This will demonstrate that the definitions differ by a constant integer. Then
we will show that the definitions agree on a choice of a specific value,
completing the proof.

Let $S:[-1,1]\times S^{1}\to\text{Sym}(2n)$ be a smooth, generic path of loops
of symmetric matrices, $(s,t)\mapsto S_{s}(t)$, where
$S_{s}:S^{1}\to\text{Sym}(2n)$, from $S_{+}$ to $S_{-}$. This means $S_{\pm
1}=S_{\mp}$. The path of loops $\\{S_{s}\\}_{s\in[-1,1]}$ defines a path of
asymptotic operators $A=\\{A_{s}\\}_{s\in[-1,1]}$ from $A_{+}$ to $A_{-}$,
given by $A_{s}=-J_{0}\partial_{t}-S_{s}$, so that $A_{\pm 1}=A_{\mp}$. The
path of loops $\\{S_{s}\\}_{s\in[-1,1]}$ additionally defines a 1-parameter
family of paths of symplectic matrices, $\\{\Phi_{s}\\}_{s\in[-1,1]}$, where
$\Phi_{s}:[0,1]\to\text{Sp}(2n)$, for $s\in[-1,1]$, with
$\Phi_{s}(0)=\text{Id}$, as described in Remark 2.8. We similarly have that
$\Phi_{\pm 1}=\Phi_{\mp}$. Define $\Phi:[-1,1]\times[0,1]\to\text{Sp}(2n)$,
$\Phi(s,t)=\Phi_{s}(t)$. Finally, let $\Psi:[-1,1]\to\text{Sp}(2n)$ be given
by $\Psi(s):=\Phi(s,1)=\Phi_{s}(1)$. Figure 2 illustrates this data.

Figure 2: A rectangle of symplectic matrices on the left, and a corresponding
path of asymptotic operators on the right. Both figures depict two theoretical
$s$ values in $(-1,1)$ that are crossings for both $\Psi$ and $A$ in blue.

See that the concatenation $\Phi_{+}\cdot\Psi$ is a path of symplectic
matrices from Id to $\Phi_{-}(1)$ that is homotopic to $\Phi_{-}$, rel
endpoints. Thus, we have that
$\mu_{\text{CZ}}(\Phi_{-})=\mu_{\text{CZ}}(\Phi_{+}\cdot\Psi)$. By definition
(6),

$\displaystyle\mu_{\text{CZ}}(\Phi_{+}\cdot\Psi)$
$\displaystyle=\frac{1}{2}\text{Sign}(S_{+}(0))+\sum_{t\in(0,1)}\text{Sign}(\Gamma(\Phi_{+},t))+\sum_{s\in(-1,1)}\text{Sign}(\Gamma(\Psi,s))$
$\displaystyle=\mu_{\text{CZ}}(\Phi_{+})+\sum_{s\in(-1,1)}\text{Sign}(\Gamma(\Psi,s)).$
(9)

Let $Z$ denote the sum of signatures appearing on the right in equation (9),
so that

$\mu_{\text{CZ}}(\Phi_{+})-\mu_{\text{CZ}}(\Phi_{-})=\mu_{\text{CZ}}(\Phi_{+})-\mu_{\text{CZ}}(\Phi_{+}\cdot\Psi)=-Z.$

We show that $Z=-\mu^{\text{spec}}(A_{+},A_{-})$, implying equation (8).

For each $s\in(-1,1)$, we construct an isomorphism
$F_{s}:\text{ker}(A_{s})\to\text{ker}(\Psi(s)-\text{Id})$. It’s worth noting
that these vector spaces are generically trivial. First note that an
eigenvector of any asymptotic operator has a smooth representative in
$W^{1,2}(S^{1},{\mathbb{R}}^{2n})$, thus $\text{ker}(A_{s})$ consists entirely
of smooth functions. An element $\eta\in\text{ker}(A_{s})$ is necessarily of
the form $t\mapsto\Phi_{s}(t)\eta(0)$. This is because both $\eta(t)$ and
$\Phi_{s}(t)\eta(0)$ are solutions to the initial value problem

$\dot{Y}(t)=J_{0}S_{s}(t)Y(t),\,\,\,Y(0)=\eta(0),$

thus $\eta(t)=\Phi_{s}(t)\eta(0)$.

Importantly, because $\eta$ is 1-periodic, this implies that

$\eta(0)=\eta(1)=\Phi_{s}(1)\eta(0)=\Psi(s)\eta(0),$

demonstrating that $\eta(0)\in\text{ker}(\Psi(s)-\text{Id})$. Define

$F_{s}:\text{ker}(A_{s})\to\text{ker}(\Psi(s)-\text{Id}),\,\,\,\eta\mapsto\eta(0).$

This linear map $F_{s}$ is invertible, with inverse given by
$v\mapsto\eta_{v}$, where $\eta_{v}(t)=\Phi_{s}(t)v$. In total, the spaces
$\text{ker}(A_{s})$ and $\text{ker}(\Psi(s)-\text{Id})$ are isomorphic for all
$s$. Because $\\{S_{s}\\}_{s\in[-1,1]}$ was chosen generically,
$\text{ker}(A_{s})$ is nonzero only for finitely many $s$, whose dimension is
1 at such values $s$. Furthermore, we have that $F_{s}$ negates the quadratic
forms on $\text{ker}(A_{s})$ and $\text{ker}(\Psi(s)-\text{Id})$, in the sense
that for any $e\in{\mathbb{R}}$,

$\Gamma(\Psi,s)(F_{s}(\eta))=-\Gamma(A,s)(\eta),$

For $\eta\in\text{ker}(A_{s})$. Lemma A.1 proves this fact (for
$\text{dim}(\text{ker}(A_{s}))$ arbitrary) in Appendix A.1. This equality
allows us to conclude

$Z=\sum_{s\in(-1,1)}\text{Sign}(\Gamma(\Psi,s))=-\sum_{s\in(-1,1)}\text{Sign}(\Gamma(A,s))=-\mu^{\text{spec}}(A_{+},A_{-}),$

where we have used the formula derived in Corollary 2.7 in the last equality.
This proves equation 8.

It remains to show that Definitions (5) and (7) agree at a single value.
Consider the asymptotic operator $A_{0}=-J_{0}\partial_{t}-S_{0}$, where
$S_{0}$ is the constant loop of symmetric matrices
$\text{Id}_{n}\oplus-\text{Id}_{n}$. The Conley-Zehnder index with respect to
definition 5 is immediately given by $\mu^{\text{spec}}(A_{0},A_{0})=0$. Let
$\Phi_{0}:[0,1]\to\text{Sp}(2n)$ be the corresponding path of symplectic
matrices, according to Remark 2.8. We argue that
$\mu_{\text{CZ}}(\Phi_{0})=0$, proving that the the definitions agree at a
specified value, and thus, at all values.

An explicit expression for the solution $\Phi_{0}$ of the initial value
problem $\dot{\Phi}_{0}=J_{0}S_{0}\Phi_{0}$, $\Phi_{0}(0)=\text{Id}_{2n}$ is
given in block form by

$\Phi_{0}(t)=\begin{pmatrix}\cosh{(t)}\text{Id}_{n}&\sinh{(t)}\text{Id}_{n}&\\\
\sinh{(t)}\text{Id}_{n}&\cosh{(t)}\text{Id}_{n}\end{pmatrix},$

where we use the block form of $J_{0}$;

$J_{0}=\begin{pmatrix}0&-\text{Id}_{n}\\\ \text{Id}_{n}&0\end{pmatrix}.$

Using formulas for the determinant of a matrix in block form, we see that

$\text{det}(\Phi_{0}(t)-\text{Id}_{2n})=2^{n}(1-\cosh{(t)})^{n}.$

This is 0 only at $t=0$, and so the path $\Phi_{0}$ has no crossings in
$(0,1]$. Finally, by definition (6), we have

$\mu_{\text{CZ}}(\Phi_{0})=\frac{1}{2}\text{Sign}(S_{0}(0))+0=\frac{1}{2}(n-n)=0,$

completing the proof. ∎

There are advantages to understanding both perspectives of
$\mu_{\text{CZ}}^{\tau}(\gamma)$. The spectral flow definition is useful in
seeing that the difference of Conley-Zehnder indices is a net change in
eigenvalues of the corresponding asymptotic operators. This highlights the
idea that the asymptotic operator and Conley-Zehnder index are analogous to
the Hessian and Morse index in the finite dimensional setting. The crossing
form definition is computationally useful, and we make heavy use of it in
Section 4.

######  Remark 2.10.

(Properties of the Conley-Zehnder index of a path of symplectic matrices)
Following [S, §2.4], let $\text{SP}(n)$ denote the collection of paths
$\Phi:[0,1]\to\text{Sp}(2n)$ such that $\Phi(0)=\text{Id}$, and
$\text{det}(\Phi(1)-\text{Id})\neq 0$. Then
$\mu_{\text{CZ}}:\text{SP}(n)\to{\mathbb{Z}}$ satisfies the following
properties.

  1. 1.

Naturality: For any path $N:[0,1]\to\text{Sp}(2n)$, $\mu_{\text{CZ}}(N\Phi
N^{-1})=\mu_{\text{CZ}}(\Phi)$.

  2. 2.

Homotopy: The values of $\mu_{\text{CZ}}$ are constant on the components of
$\text{SP}(n)$.

  3. 3.

Zero: If $\Phi(t)$ has no eigenvalue on the unit circle, for $t>0$, then
$\mu_{\text{CZ}}(\Phi)=0$.

  4. 4.

Product: For $n_{1},n_{2}\in{\mathbb{N}}$ with $n=n_{1}+n_{2}$, using the
standard identification
$\text{Sp}(n)\cong\text{Sp}(n_{1})\oplus\text{Sp}(n_{2})$, we have
$\mu_{\text{CZ}}(\Phi_{1}\oplus\Phi_{2})=\mu_{\text{CZ}}(\Phi_{1})+\mu_{\text{CZ}}(\Phi_{2})$.

  5. 5.

Loop: If $L:[0,1]\to\text{Sp}(2n)$ is a loop with $L(0)=L(1)$ then
$\mu_{\text{CZ}}(L\Phi)=2\mu(L)+\mu_{\text{CZ}}(\Phi)$, where $\mu(L)$ is the
Maslov index of $L$ (see below).

  6. 6.

Signature: For $S\in\text{Sym}(2n)$ with $\|S\|<2\pi$888Here, $\|S\|$ denotes
the norm of $S$ as an operator on ${\mathbb{R}}^{2n}$, or equivalently,
$\text{max}\\{|\lambda|:\lambda\in\text{Spec}(S)\\}$.,
$\mu_{\text{CZ}}(\text{exp}(J_{0}St))=\tfrac{1}{2}\text{Sign}(S)$.

  7. 7.

Determinant:
$(-1)^{n-\mu_{\text{CZ}}(\Phi)}=\text{Sign}(\text{det}(\Phi(1)-\text{Id}))$.

  8. 8.

Inverse:
$\mu_{\text{CZ}}(\Phi^{T})=\mu_{\text{CZ}}(\Phi^{-1})=-\mu_{\text{CZ}}(\Phi)$.

For more details, the reader should consult [SZ, §3]. The Conley-Zehnder index
as a family of functions
$\\{\mu_{\text{CZ}}:\text{SP}(n)\to{\mathbb{Z}}\\}_{n\in{\mathbb{N}}}$ is
uniquely characterized by these properties. The loop property uses the
definition of the _Maslov index_ of a loop of symplectic matrices, $\mu(L)$,
for $L:S^{1}\to\text{Sp}(2n)$, defined as follows. There is a deformation
retraction $r:\text{Sp}(2n)\to\text{U}(n):=\text{Sp}(2n)\cap\text{O}(2n)$
using polar decompositions (see [MS, §2.2]). There exists an identification of
the group $\text{U}(n)$ as we have defined it with the collection of unitary
matrices in $\text{GL}(n,{\mathbb{C}})$ and in this way, we have the
determinant map $\text{det}:\text{U}(n)\to S^{1}\subset{\mathbb{C}}^{*}$. The
Maslov index of $L$, denoted $\mu(L)\in{\mathbb{Z}}$, is the degree of the map
$\text{det}\circ r\circ L:S^{1}\to S^{1}$. For details and applications to
Chern classes, see [MS, §2].

######  Remark 2.11.

(Conley-Zehnder indices and rotation numbers) In the 3-dimensional case, we
may compute Conley-Zehnder indices of trivialized Reeb orbits by using
_rotation numbers_ , which we define now. Let $\gamma$ be a nondegenerate Reeb
orbit of action $T>0$, and let $\tau$ be a symplectic trivialization of
$\gamma^{*}\xi$. Let $\\{\Phi_{t}\\}_{t\in[0,T]}\subset\text{Sp}(2)$ denote
the associated 1-parameter family of symplectic matrices defining the Reeb
flow along $\gamma$ with respect to $\tau$. We define a _rotation number_ ,
$\theta\in{\mathbb{R}}$, depending on $\gamma$ and $\tau$, satisfying
$\mu_{\text{CZ}}^{\tau}(\gamma)=\lfloor\theta\rfloor+\lceil\theta\rceil\in{\mathbb{Z}}$.

  * •

When $\gamma$ is elliptic, then (after perhaps modifying $\tau$ by a homotopy)
$\Phi_{t}$ takes the form
$\Phi_{t}=\text{exp}(2\pi\theta_{t}J_{0})\in\text{Sp}(2)$, where
$t\mapsto\theta_{t}$ is a continuous function $[0,T]\to{\mathbb{R}}$ with
$\theta_{0}=0$, $\theta_{T}\notin{\mathbb{Z}}$. Define $\theta:=\theta_{T}$.

  * •

When $\gamma$ is hyperbolic, there exists some nonzero eigenvector
$v\in{\mathbb{R}}^{2}$ of $\Phi_{T}$. Now, the path
$\\{\Phi_{t}(v)\\}_{t\in[0,T]}\subset{\mathbb{R}}^{2}\setminus\\{(0,0)\\}$
rotates through an angle of $2\pi k$ from $t=0$ to $t=T$. Define $\theta:=k$.
When $\gamma$ is negative hyperbolic, $v$ is an eigenvector whose eigenvalue
is negative, so $\Phi_{T}(v)$ and $v$ point in opposite directions, thus this
$2\pi k$ must be an odd multiple of $\pi$ (i.e. $k$ is a half-integer). When
$\gamma$ is positive hyperbolic, the eigenvalue is positive, so $\Phi_{T}(v)$
and $v$ point in the same direction, so $2\pi k$ is an even multiple of $\pi$
(i.e. $k$ is an integer).

In Sections 4.1, 4.2, and 4.3, we compute the rotation numbers for all of the
elliptic orbits (with respect to a global trivialization of $\xi_{G}$) in our
action filtered chain complexes, and use the fact that
$\mu_{\text{CZ}}^{\tau}(\gamma)=\lfloor\theta\rfloor+\lceil\theta\rceil$ to
compute their Conley-Zehnder indices.

######  Remark 2.12.

(Parity of Conley-Zehnder indices) In the 3-dimensional case, the rotation
numbers provide information regarding the parity of
$\mu_{\text{CZ}}^{\tau}(\gamma)$:

  * •

If $\gamma$ is elliptic, then (for any trivialization $\tau$),
$\theta\in{\mathbb{R}}\setminus{\mathbb{Z}}$, and so the quantity

$\mu_{\text{CZ}}^{\tau}(\gamma)=\lfloor\theta\rfloor+\lceil\theta\rceil=2\lfloor\theta\rfloor+1=2\lceil\theta\rceil-1$

is always odd.

  * •

If $\gamma$ is negative hyperbolic, then (for any trivialization $\tau$),
$\theta$ is a half-integer of the form $\theta=k=k^{\prime}+\tfrac{1}{2}$, for
$k^{\prime}\in{\mathbb{Z}}$, and so the quantity

$\mu_{\text{CZ}}^{\tau}(\gamma)=\lfloor\theta\rfloor+\lceil\theta\rceil=2\theta=2k^{\prime}+1$

is always odd.

  * •

If $\gamma$ is positive hyperbolic, then (for any trivialization $\tau$),
$\theta$ is an integer $k\in{\mathbb{Z}}$, and so the quantity

$\mu_{\text{CZ}}^{\tau}(\gamma)=\lfloor\theta\rfloor+\lceil\theta\rceil=2\theta=2k$

is always even.

######  Remark 2.13.

(Conley-Zehnder indices of iterates) In the 3-dimensional case, we can use
rotation numbers to compare the Conley-Zehnder indices of iterates of Reeb
orbits. For a nondegenerate Reeb orbit $\gamma$, a symplectic trivialization
$\tau$ of $\gamma^{*}\xi$, and a positive integer $k$ such that $\gamma^{k}$
is nondegenerate, there exists a unique pullback trivialization of
$(\gamma^{k})^{*}\xi$, denoted $k\tau$. Let $\theta$ and $\theta^{k}$ denote
the rotation numbers of $\gamma$ and $\gamma^{k}$ with respect to $\tau$ and
$k\tau$, respectively. Then it is true that
$\theta^{k}=k\theta\in{\mathbb{R}}$. Notice that, if $\gamma$ is hyperbolic,
then we have

$\mu_{\text{CZ}}^{k\tau}(\gamma^{k})=2\theta^{k}=2k\theta=k\mu_{\text{CZ}}^{\tau}(\gamma).$

That is, the Conley-Zehnder index of hyperbolic orbits grows linearly with
respect to iteration (when using appropriate trivializations). In particular,
when there exists a global trivialization $\tau$ of $\xi$,

$\mu_{\text{CZ}}(\gamma^{k})=k\cdot\mu_{\text{CZ}}(\gamma),$

for all hyperbolic Reeb orbits, and with $\mu_{\text{CZ}}$ computed implicitly
with respect to the global trivialization $\tau$. Due to the non-linearity of
the floor and ceiling functions, the Conley-Zehnder indices of elliptic orbits
do not necessarily grow linearly with the iterate.

In Sections 4.2 and 4.3, we use this observation to compute the Conley-Zehnder
indices of the hyperbolic Reeb orbits appearing in the action filtered chain
complexes.

###  2.4 Cylindrical contact homology as an analogue of orbifold Morse
homology

Let $X$ be a manifold (or perhaps an orbifold) used to study Morse (or
orbifold Morse) theory, and let $(M,\lambda)$ denote a contact manifold. In a
loose sense, cylindrical contact homology is a kind of Morse homology
performed on $\mathcal{L}(M)$ with respect to the contact action functional,
$\mathcal{A}_{\lambda}(\gamma)=\int_{\gamma}\lambda$. This analogy is
highlighted in Table 2.

Morse Theory |  Cylindrical Contact Homology
---|---
$p\mapsto-\text{grad}(f)_{p}\in T_{p}X$, the negative gradient of $f$ at $p$, is a section of $TX\to X$ which vanishes precisely at Crit(f) |  $\gamma\mapsto-J\pi_{\xi}\dot{\gamma}\in\Gamma(\gamma^{*}\xi)$ is a section of $E\to\mathcal{L}(M)$. This section _resembles_ a positive gradient for $\mathcal{A}_{\lambda}$, and vanishes at Reeb orbits
At each $p\in\text{Crit}(f)$, we have the self adjoint Hessian $H_{p}:T_{p}X\to T_{p}X$, obtained by linearizing $-\text{grad}(f)$ at $p$ |  At each Reeb orbit $\gamma$ we have the self adjoint asymptotic operator $A_{\gamma}:\Gamma(\gamma^{*}\xi)\to\Gamma(\gamma^{*}\xi)$, obtained by linearizing $-J\pi_{\xi}\dot{\gamma}$ at $\gamma$
$x\in\mathcal{M}(p_{+},p_{-})$ is a Morse flow line |  $u\in\mathcal{M}(\gamma_{+},\gamma_{-})$ is a $J$-holomorphic cylinder
$p$ nondegenerate $\iff$ $H_{p}$ nondegenerate; $\text{ind}(x)$ is related to the spectra of $H_{p_{\pm}}$ |  $\gamma$ nondegenerate $\iff$ $A_{\gamma}$ nondegenerate; $\text{ind}(u)$ is related to the spectra of $A_{\gamma_{\pm}}$
Table 2: Analogies between the two theories

Regarding the chain complexes and differentials, there are two immediate
similarities demonstrating that _orbifold_ Morse homology is the correct
analogue of cylindrical contact homology:

(1) The differentials are structurally identical. Indeed, let us compare, for
pairs $(p,q)$ and $(\gamma_{+},\gamma_{-})$ of orientable orbifold critical
points and good Reeb orbits of grading difference equal to 1,

$\langle\partial^{\text{orb}}p,q\rangle=\sum_{x\in\mathcal{M}(p,q)}\epsilon(x)\dfrac{|H_{p}|}{|H_{x}|},\,\,\,\,\,\,\,\,\langle\partial\gamma_{+},\gamma_{-}\rangle=\sum_{u\in\mathcal{M}_{1}^{J}(\gamma_{+},\gamma_{-})/{\mathbb{R}}}\epsilon(u)\dfrac{m(\gamma_{+})}{m(u)}.$

Recall that $|H_{q}|$ is the order of isotropy associated to the orbifold
point $q$, $|H_{x}|$ is the order of isotropy associated to any point in the
image of the flow line $x$, and $m$ denotes the multiplicity of either a Reeb
orbit or a cylinder. Both differentials are counts of integers because
$|H_{x}|$ divides $|H_{p}|$ and $m(u)$ divides $m(\gamma_{+})$. The
differentials use the same _weighted_ counts of 0-dimensional moduli spaces
because of the similarities in the compactifications of the 1-dimensional
moduli spaces. A single broken gradient path or holomorphic building may serve
as the limit of _multiple_ ends of a 1-dimensional moduli space in either
setting. For a thorough treatment of why these signed counts generally produce
a differential that squares to zero, see [CH, Theroem 5.1] (in the orbifold
case) and [HN, §4.3] (in the contact case).

For example, a 1-dimensional moduli space of flow lines/cylinders,
diffeomorphic to an open interval $(0,1)$, could compactify in two
geometrically distinct ways. Such an open interval’s compactification by
broken objects may be either a closed interval $[0,1]$, or be a topological
$S^{1}$. In the latter case, the single added point in the compactification
serves as an endpoint, or limit, of _two_ ends of the open moduli space. The
differential as a weighted count is designed to accommodate this multiplicity.
Cylinders and orbifold Morse flow lines appearing in our contact manifolds
$S^{3}/G$ and orbifolds $S^{2}/H$ will exhibit this phenomenon of an open
interval compactifying to a topological $S^{1}$, this is spelled out
explicitly in Section 4.5 and illustrated in Figure 14.

######  Remark 2.14.

Compare these differentials to those in classical Morse theory and Hamiltonian
Floer homology, which take the simpler forms $\sum_{x}\epsilon(x)$ and
$\sum_{u}\epsilon(u)$, where $x$ ranges over Morse flow lines, and $u$ ranges
over Hamiltonian trajectories in appropriate 0-dimensional moduli spaces. In
both of these theories, the ends of the 1-dimensional moduli spaces are in
_bijective correspondence_ with the broken trajectories, and so the
differentials are only _signed_ , not weighted, counts.

(2) Bad Reeb orbits are analogous to non-orientable critical points. Recall
from Remark 2.2, that in orbifold Morse theory the values $\epsilon(x)$, for
$x\in\mathcal{M}(p,q)$, are only well defined when the flow line interpolates
between _orientable_ critical points. The same phenomenon occurs in
cylindrical contact homology. Loosely speaking, in order to assign a coherent
orientation to a holomorphic cylinder $u$, one needs to orient the asymptotic
operators associated to the Reeb orbits $\gamma_{\pm}$ at both ends by means
of a choice of base-point in the images of $\gamma_{\pm}$. The cyclic deck
group of automorphisms of each Reeb orbit (isomorphic to
${\mathbb{Z}}_{m(\gamma_{\pm})}$) acts on the 2-set of orientations of the
asymptotic operator. If the action is non-trivial (i.e., if the deck group
_reverses_ the orientations), then the Reeb orbit is _bad_ , and there is not
a well defined $\epsilon(u)$, independent of intermediate choices made.

In Section 4.5, we explain that the Seifert projections
$\mathfrak{p}:S^{3}/G\to S^{2}/H$ geometrically realize this analogy between
non-orientable objects. Specifically, the bad Reeb orbits in $S^{3}/G$ are
precisely the ones that project to non-orientable orbifold Morse critical
points in $S^{2}/H$.

Finally, not only do non-orientable objects complicate the assignment of
orientation to the 0-dimensional moduli spaces, their inclusion in the chain
complex of both theories jeopardizes $\partial^{2}=0$, providing significant
reasons to exclude them as generators. In Remark 4.7 of Section 4.5, we will
investigate the $H={\mathbb{T}}$ case, and will see (as we did in Remark 2.2)
what happens if we were to include the non-orientable objects as generators of
the respective chain complexes. In particular, we will produce examples of
(orientable) generators $x$ and $y$ with $|x|-|y|=2$ with $\langle\partial
x,y\rangle\neq 0$.

##  3 Geometric setup and dynamics

We now begin our more detailed studies of the contact manifolds $S^{3}/G$. In
this section we review a process of perturbing degenerate contact forms on
$S^{3}$ and $S^{3}/G$ using a Morse function to achieve nondegeneracy up to an
action threshold, following [N2, §1.5]. This process will aid our main goal of
the section, to precisely identify the Reeb orbits of $S^{3}/G$, and to
compute their Conley-Zehnder indices (Lemma 3.15).

###  3.1 Spherical geometry and associated Reeb dynamics

The round contact form on $S^{3}$, denoted $\lambda$, is defined as the
restriction of the 1-form $\iota_{U}\omega_{0}\in\Omega^{1}({\mathbb{C}}^{2})$
to $S^{3}$, where $\omega_{0}$ is the standard symplectic form on
${\mathbb{C}}^{2}$ and $U$ is the radial vector field in ${\mathbb{C}}^{2}$.
In complex coordinates, we have:

$\omega_{0}=\frac{i}{2}\sum_{k=1}^{2}dz_{k}\wedge
d\overline{z_{k}},\,\,\,U=\frac{1}{2}\sum_{k=1}^{2}z_{k}\partial_{z_{k}}+\overline{z_{k}}\partial_{\overline{z_{k}}},\,\,\,\lambda=\frac{i}{4}\sum_{k=1}^{2}z_{k}\wedge
d\overline{z_{k}}-\overline{z_{k}}dz_{k}.$

After interpreting ${\mathbb{C}}^{2}\cong{\mathbb{R}}^{4}$, we can rewrite
$\omega_{0}$, $U$, and $\lambda$ using real notation:

$\omega_{0}=\sum_{k=1}^{2}dx_{k}\wedge
dy_{k},\,\,\,\,\,\,\,\,U=\frac{1}{2}\sum_{k=1}^{2}x_{k}\partial_{x_{k}}+y_{k}\partial_{y_{k}},\,\,\,\,\,\,\,\,\lambda=\frac{1}{2}\sum_{k=1}^{2}x_{k}dy_{k}-y_{k}dx_{k}.$

The contact manifold $(S^{3},\lambda)$ is a classical example of a contact-
type hypersurface in the symplectic manifold $({\mathbb{R}}^{4},\omega_{0})$;
this means that $S^{3}$ is transverse to the _Liouville vector field_ , $U$,
and inherits contact structure from the contraction of $\omega_{0}$ with $U$.

The diffeomorphism $S^{3}\subset{\mathbb{C}}^{2}\to\text{SU}(2)$ provides
$S^{3}$ with Lie group structure,

$(\alpha,\beta)\in S^{3}\mapsto\begin{pmatrix}\alpha&-\overline{\beta}\,\,\\\
\beta&\overline{\alpha}\end{pmatrix}\in\text{SU}(2),$ (10)

and we see that $e=(1,0)\in S^{3}$ is the identity element. The
$\text{SU}(2)$-action on ${\mathbb{C}}^{2}$ preserves $\omega_{0}$ and $U$,
and so the $\text{SU}(2)$-action restricted to $S^{3}$ preserves $\lambda$.
That is, $\text{SU}(2)$ acts on $S^{3}$ by _strict_ contactomorphisms.

There is a natural Lie algebra isomorphism between the tangent space of the
identity element of a Lie group and its collection of left-invariant vector
fields. The contact plane $\xi_{e}=\text{ker}(\lambda_{e})$ at the identity
element $e=(1,0)\in S^{3}$ is spanned by the tangent vectors
$\partial_{x_{2}}|_{e}=\langle 0,0,1,0\rangle$ and
$\partial_{y_{2}}|_{e}=\langle 0,0,0,1\rangle$, where we are viewing

$\xi_{e}\subset T_{e}S^{3}\subset T_{e}{\mathbb{C}}^{2}\cong
T_{e}{\mathbb{R}}^{4}=\text{Span}_{{\mathbb{R}}}\big{(}\partial_{x_{1}}|_{e},\,\partial_{y_{1}}|_{e},\,\partial_{x_{2}}|_{e},\,\partial_{y_{2}}|_{e}\big{)}.$

Let $V_{1}$ and $V_{2}$ be the left-invariant vector fields corresponding to
$\partial_{x_{2}}|_{e}$ and $\partial_{y_{2}}|_{e}$, respectively. Because
$S^{3}$ acts on itself by _contactomorphisms_ , $V_{1}$ and $V_{2}$ are
sections of $\xi$ and provide a global unitary trivialization of
$(\xi,d\lambda|_{\xi},J_{{\mathbb{C}}^{2}})$, denoted $\tau$:

$\tau:S^{3}\times{\mathbb{R}}^{2}\to\xi,\,\,\,(p,\eta_{1},\eta_{2})\mapsto\eta_{1}V_{1}(p)+\eta_{2}V_{2}(p)\in\xi_{p}.$
(11)

Here, $J_{{\mathbb{C}}^{2}}$ is the standard integrable complex structure on
${\mathbb{C}}^{2}$. Note that $J_{{\mathbb{C}}^{2}}(V_{1})=V_{2}$ everywhere.
Given a Reeb orbit $\gamma$ of any contact form on $S^{3}$, let
$\mu_{\text{CZ}}(\gamma)\in{\mathbb{Z}}$ denote the Conley-Zehnder index of
$\gamma$ with respect to this $\tau$, i.e.
$\mu_{\text{CZ}}(\gamma):=\mu_{\text{CZ}}^{\tau}(\gamma)$. If
$(\alpha,\beta)\in S^{3}$, write $\alpha=a+ib$ and $\beta=c+id$. Then, with
respect to the ordered basis
$(\partial_{x_{1}},\partial_{y_{1}},\partial_{x_{2}},\partial_{y_{2}})$ of
$T_{(\alpha,\beta)}{\mathbb{R}}^{4}$, we have the following expressions

$V_{1}(\alpha,\beta)=\langle-c,d,a,-b\rangle,\,\,\,\,V_{2}(\alpha,\beta)=\langle-d,-c,b,a\rangle.$
(12)

Consider the double cover of Lie groups, $P:\text{SU}(2)\to\text{SO}(3)$:

$\begin{pmatrix}\alpha&-\overline{\beta}\,\,\\\
\beta&\overline{\alpha}\end{pmatrix}\in\text{SU}(2)\xmapsto{P}\begin{pmatrix}|\alpha|^{2}-|\beta|^{2}&2\text{Im}(\alpha\beta)&2\text{Re}(\alpha\beta)\\\
-2\text{Im}(\overline{\alpha}\beta)&\text{Re}(\alpha^{2}+\beta^{2})&-\text{Im}(\alpha^{2}+\beta^{2})\\\
-2\text{Re}(\overline{\alpha}\beta)&\text{Im}(\alpha^{2}-\beta^{2})&\text{Re}(\alpha^{2}-\beta^{2})\end{pmatrix}\in\text{SO}(3).$
(13)

The kernel of $P$ has order 2 and is generated by $-\text{Id}\in\text{SU}(2)$,
the only element of $\text{SU}(2)$ of order 2. A diffeomorphism
${\mathbb{C}}P^{1}\to S^{2}\subset{\mathbb{R}}^{3}$ is given in homogeneous
coordinates ($|\alpha|^{2}+|\beta|^{2}=1$) by

$(\alpha:\beta)\in{\mathbb{C}}P^{1}\mapsto(|\alpha|^{2}-|\beta|^{2},-2\text{Im}(\bar{\alpha}\beta),-2\text{Re}(\bar{\alpha}\beta))\in
S^{2}.$ (14)

We have an $\text{SO}(3)$-action on ${\mathbb{C}}P^{1}$, pulled back from the
$\text{SO}(3)$-action on $S^{2}$ by (14). Lemma 3.1 illustrates how the action
of $\text{SU}(2)$ on $S^{3}$ is related to the action of $\text{SO}(3)$ on
${\mathbb{C}}P^{1}\cong S^{2}$ via $P:\text{SU}(2)\to\text{SO}(3).$

######  Lemma 3.1.

For a point $z$ in $S^{3}$, let $[z]\in{\mathbb{C}}P^{1}$ denote the
corresponding point under the quotient of the $S^{1}$-action on $S^{3}$. Then
for all $z\in S^{3}$, and all matrices $A\in\mbox{\em SU}(2)$, we have

$[A\cdot z]=P(A)\cdot[z]\in{\mathbb{C}}P^{1}\cong S^{2}.$

###### Proof.

First, note that the result holds for the case $z=e=(1,0)\in S^{3}$. This is
because $[e]\in{\mathbb{C}}P^{1}$ corresponds to $(1,0,0)\in S^{2}$ under
(14), and so for any $A$, $P(A)\cdot[e]$ is the first column of the $3\times
3$ matrix $P(A)$ appearing in (13). That is,

$P(A)\cdot[e]=(|\alpha|^{2}-|\beta|^{2},-2\text{Im}(\bar{\alpha}\beta),-2\text{Re}(\bar{\alpha}\beta)),$
(15)

(where $(\alpha,\beta)\in S^{3}$ is the unique element corresponding to
$A\in\text{SU}(2)$, using (10)). By (14), the point (15) equals
$[(\alpha,\beta)]=[(\alpha,\beta)\cdot(1,0)]=[A\cdot e]$, and so the result
holds when $z=e$.

For the general case, note that any $z\in S^{3}$ equals $B\cdot e$ for some
$B\in\text{SU}(2)$, and use the fact that $P$ is a group homomorphism. ∎

The Reeb flow of $\lambda$ is given by the $S^{1}\subset{\mathbb{C}}^{*}$
(Hopf) action, $z\mapsto e^{it}\cdot z$. Thus, all
$\gamma\in\mathcal{P}(\lambda)$ have period $2k\pi$, with linearized return
maps equal to $\text{Id}:\xi_{\gamma(0)}\to\xi_{\gamma(0)}$, and are
degenerate.

######  Notation 3.2.

Following the general recipe of perturbing the degenerate contact form on a
prequantization bundle, outlined in [N2, §1.5], we establish the following
notation:

  * •

$\mathfrak{P}:S^{3}\to S^{2}$ denotes the Hopf fibration,

  * •

$f$ is a Morse-Smale function on $(S^{2},\omega_{\text{FS}}(\cdot,j\cdot))$,
$\text{Crit}(f)$ is its set of critical points,

  * •

for $\varepsilon>0$; $f_{\varepsilon}:=1+\varepsilon f:S^{2}\to{\mathbb{R}}$,
$F_{\varepsilon}:=f_{\varepsilon}\circ\mathfrak{P}:S^{3}\to{\mathbb{R}}$, and
$\lambda_{\varepsilon}:=F_{\varepsilon}\lambda\in\Omega^{1}(S^{3})$,

  * •

$\widetilde{X_{f}}\in\xi$ is the horizontal lift of $X_{f}\in TS^{2}$ using
the fiberwise linear symplectomorphism
$d\mathfrak{P}|_{\xi}:(\xi,d\lambda|_{\xi})\to(TS^{2},\omega_{\text{FS}})$,
where $X_{f}$ denotes the Hamiltonian vector field for $f$ on $S^{2}$ with
respect to $\omega_{\text{FS}}$.

######  Remark 3.3.

Our convention is that for a smooth, real valued function $f$ on symplectic
manifold $(M,\omega)$, the Hamiltonian vector field $X_{f}$ uniquely satisfies
$\iota_{X_{f}}\omega=-df$.

Because $F_{\varepsilon}$ is positive for small $\varepsilon$ and smooth on
$S^{3}$, $\lambda_{\varepsilon}$ is another contact form on $S^{3}$ defining
the same contact distribution $\xi$ as $\lambda$. We refer to
$\lambda_{\varepsilon}$ as the _perturbed contact form_ on $S^{3}$. Although
$\lambda$ and $\lambda_{\varepsilon}$ share the same kernel, their Reeb
dynamics differ.

######  Lemma 3.4.

The following relationship between vector fields on $S^{3}$ holds:

$R_{\lambda_{\varepsilon}}=\frac{R_{\lambda}}{F_{\varepsilon}}-\varepsilon\frac{\widetilde{X_{f}}}{F_{\varepsilon}^{2}}.$

###### Proof.

This is [N2, Prop. 4.10]. Note that the sign discrepancy is a result of our
convention regarding Hamiltonian vector fields, see Remark 3.3. ∎

We now explore how the relationship between vector fields from Lemma 3.4
provides a relationship between Reeb and Hamiltonian flows.

######  Notation 3.5.

(Reeb and Hamiltonian flows). For any $t\in{\mathbb{R}}$,

  * •

$\phi_{0}^{t}:S^{3}\to S^{3}$ denotes the time $t$ flow of the unperturbed
Reeb vector field $R_{\lambda}$,

  * •

$\phi^{t}:S^{3}\to S^{3}$ denotes the time $t$ flow of the perturbed Reeb
vector field $R_{\lambda_{\varepsilon}}$,

  * •

$\varphi^{t}:S^{2}\to S^{2}$ denotes the time $t$ flow of the vector field
$V:=-\frac{\varepsilon X_{f}}{f_{\varepsilon}^{2}}.$

Note that $\phi^{t}_{0}:S^{3}\to S^{3}$ is nothing more than the map $z\mapsto
e^{it}\cdot z$.

######  Lemma 3.6.

For all $t$ values, we have
$\mathfrak{P}\circ\phi^{t}=\varphi^{t}\circ\mathfrak{P}$ as smooth maps
$S^{3}\to S^{2}$.

###### Proof.

Pick $z\in S^{3}$ and let $\widetilde{\gamma}:{\mathbb{R}}\to S^{3}$ denote
the unique integral curve for $R_{\lambda_{\varepsilon}}$ which passes through
$z$ at time $t=0$, i.e., $\widetilde{\gamma}(t)=\phi^{t}(z)$. By Lemma 3.4,
$d\mathfrak{P}$ carries the derivative of $\widetilde{\gamma}$ precisely to
the vector $V\in TS^{2}$. Thus,
$\mathfrak{P}\circ\widetilde{\gamma}:{\mathbb{R}}\to S^{2}$ is the unique
integral curve, $\gamma$, of $V$ passing through $p:=\mathfrak{P}(z)$ at time
$t=0$, i.e., $\gamma(t)=\varphi^{t}(p)$. Combining these facts provides

$\mathfrak{P}(\widetilde{\gamma}(t))=\gamma(t)\,\,\implies\,\,\mathfrak{P}(\phi^{t}(z))=\varphi^{t}(p)\,\,\implies\mathfrak{P}(\phi^{t}(z))=\varphi^{t}(\mathfrak{P}(z)).$

∎

Lemma 3.7 describes the orbits $\gamma\in\mathcal{P}(\lambda_{\varepsilon})$
projecting to critical points of $f$ under $\mathfrak{P}$.

######  Lemma 3.7.

Let $p\in\mbox{\em Crit}(f)$ and take $z\in\mathfrak{P}^{-1}(p)$. Then the map

$\gamma_{p}:[0,2\pi f_{\varepsilon}(p)]\to S^{3},\,\,\,\,t\mapsto
e^{\frac{it}{f_{\varepsilon}(p)}}\cdot z$

descends to a closed, embedded Reeb orbit $\gamma_{p}:{\mathbb{R}}/2\pi
f_{\varepsilon}(p){\mathbb{Z}}\to S^{3}$ of $\lambda_{\varepsilon}$, passing
through point $z$ in $S^{3}$, whose image under $\mathfrak{P}$ is
$\\{p\\}\subset S^{2}$, where $\cdot$ denotes the
$S^{1}\subset{\mathbb{C}}^{*}$ action on $S^{3}$.

###### Proof.

The map ${\mathbb{R}}/2\pi{\mathbb{Z}}\to S^{3}$, $t\mapsto e^{it}\cdot z$ is
a closed, embedded integral curve for the degenerate Reeb field $R_{\lambda}$,
and so by the chain rule we have that
$\dot{\gamma_{p}}(t)=R_{\lambda}(\gamma_{p}(t))/f_{\varepsilon}(p)$. Note that
$\mathfrak{P}(\gamma(t))=\mathfrak{P}(e^{\frac{it}{f_{\varepsilon}(p)}}\cdot
z)=\mathfrak{P}(z)=p$ and, because $\widetilde{X_{f}}(\gamma(t))$ is a lift of
$X_{f}(p)=0$, we have $\widetilde{X_{f}}(\gamma(t))=0$. By the description of
$R_{\lambda_{\varepsilon}}$ in Lemma 3.4, we have
$\dot{\gamma_{p}}(t)=R_{\lambda_{\varepsilon}}(\gamma_{p}(t))$. ∎

Next we set notation to be used in describing the local models for the
linearized Reeb flow along the orbits $\gamma_{p}$ from Lemma 3.7. For
$s\in{\mathbb{R}}$, $\mathcal{R}(s)$ denotes the $2\times 2$ rotation matrix:

$\mathcal{R}(s):=\begin{pmatrix}\cos{(s)}&-\sin{(s)}\\\
\sin{(s)}&\cos{(s)}\end{pmatrix}\in\text{SO}(2).$

Note that $J_{0}=\mathcal{R}(\pi/2)\in\text{SO}(2)$. For
$p\in\text{Crit}(f)\subset S^{2}$, pick coordinates $\psi:{\mathbb{R}}^{2}\to
S^{2}$, so that $\psi(0,0)=p$. Then we let $H(f,\psi)$ denote the Hessian of
$f$ in these coordinates at $p$ as a $2\times 2$ matrix of second partial
derivatives.

######  Notation 3.8.

The term _stereographic coordinates at $p\in S^{2}$_ describes a smooth
$\psi:{\mathbb{R}}^{2}\to S^{2}$ with $\psi(0,0)=p$, that has a factorization
$\psi=\psi_{1}\circ\psi_{0}$, where $\psi_{0}:{\mathbb{R}}^{2}\to S^{2}$ is
the map

$(x,y)\mapsto\frac{1}{1+x^{2}+y^{2}}(2x,2y,1-x^{2}-y^{2}),$

taking $(0,0)$ to $(0,0,1)$, and $\psi_{1}:S^{2}\to S^{2}$ is given by the
action of some element of $\text{SO}(3)$ taking $(0,0,1)$ to $p$. If $\psi$
and $\psi^{\prime}$ are both stereographic coordinates at $p$, then they
differ by a precomposition with some $\mathcal{R}(s)$ in $\text{SO}(2)$. Note
that $\psi^{*}\omega_{\text{FS}}=\frac{dx\wedge dy}{(1+x^{2}+y^{2})^{2}}$
([MS, Ex. 4.3.4]).

Lemma 3.9 describes the linearized Reeb flow of the unperturbed $\lambda$ with
respect to $\tau$.

######  Lemma 3.9.

For any $z\in S^{3}$, the linearization
$d\phi_{0}^{t}|_{\xi_{z}}:\xi_{z}\to\xi_{e^{it}\cdot z}$ is represented by
$\mathcal{R}(2t)$, with respect to ordered bases $(V_{1}(z),V_{2}(z))$ of
$\xi_{z}$ and $(V_{1}(e^{it}\cdot z),V_{2}(e^{it}\cdot z))$ of
$\xi_{e^{it}\cdot z}$.

###### Proof.

Because (i) the $V_{i}$ are $\text{SU}(2)$-invariant, and (ii) the
$\text{SU}(2)$-action commutes with $\phi_{0}^{t}$, we may reduce to the case
$z=e=(1,0)\in S^{3}$. That is, we must show for all $t$ values that

$\displaystyle(d\phi_{0}^{t})_{(1,0)}V_{1}(1,0)$
$\displaystyle=\cos{(2t)}V_{1}(e^{it},0)+\sin{(2t)}V_{2}(e^{it},0)$ (16)
$\displaystyle(d\phi_{0}^{t})_{(1,0)}V_{2}(1,0)$
$\displaystyle=-\sin{(2t)}V_{1}(e^{it},0)+\cos{(2t)}V_{2}(e^{it},0).$ (17)

Note that (17) follows from (16) by applying the endomorphism
$J_{{\mathbb{C}}^{2}}$ to both sides of (16), and noting both that
$d\phi_{0}^{t}$ commutes with $J_{{\mathbb{C}}^{2}}$, and that
$J_{{\mathbb{C}}^{2}}(V_{1})=V_{2}$. We now prove (16).

The coordinate descriptions (12) tell us that $V_{1}(e^{it},0)$ and
$V_{2}(e^{it},0)$ can be respectively written as $\langle
0,0,\cos{t},-\sin{t}\rangle$ and $\langle 0,0,\sin{t},\cos{t}\rangle$. Angle
sum formulas now imply

$\cos{(2t)}V_{1}(e^{it},0)+\sin{(2t)}V_{2}(e^{it},0)=\langle
0,0,\cos{t},\sin{t}\rangle.$

The vector on the right is precisely $(d\phi_{0}^{t})_{(1,0)}V_{1}(1,0)$, and
so we have proven (16). ∎

Proposition 3.10 and Corollary 3.11 conclude our discussion of dynamics on
$S^{3}$.

######  Proposition 3.10.

Fix a critical point $p$ of $f$ in $S^{2}$ and stereographic coordinates
$\psi:{\mathbb{R}}^{2}\to S^{2}$ at $p$, and suppose
$\gamma\in\mathcal{P}(\lambda_{\varepsilon})$ projects to $p$ under
$\mathfrak{P}$. Let $\Phi_{t}\in\mbox{\em Sp}(2)$ denote
$d\phi^{t}|_{\xi_{\gamma(0)}}:\xi_{\gamma(0)}\to\xi_{\gamma(t)}$ with respect
to the trivialization $\tau$. Then $\Phi_{t}$ is a a conjugate of the matrix

$\mathcal{R}\bigg{(}\frac{2t}{f_{\varepsilon}(p)}\bigg{)}\cdot\exp\bigg{(}\frac{-t\varepsilon}{f_{\varepsilon}(p)^{2}}J_{0}\cdot
H(f,\psi)\bigg{)}$

by some element of $\mbox{\em SO}(2)$, which is independent of $t$.

###### Proof.

Let $z:=\gamma(0)$. We linearize the identity
$\mathfrak{P}\circ\phi^{t}=\varphi^{t}\circ\mathfrak{P}$ (Lemma 3.6), restrict
to $\xi_{z}$, and rearrange to recover the equality
$d\phi^{t}|_{\xi_{z}}=a\circ b\circ c:\xi_{z}\to\xi_{\phi^{t}(z)}$, where

$a=(d\mathfrak{P}|_{\xi_{\phi^{t}(z)}})^{-1}:T_{p}S^{2}\to\xi_{\phi^{t}(z)},\,\,b=d\varphi^{t}_{p}:T_{p}S^{2}\to
T_{p}S^{2},\,\,\text{and}\,\,c=d\mathfrak{P}|_{\xi_{z}}:\xi_{z}\to
T_{p}S^{2}.$

Let $v_{i}:=d\mathfrak{P}(V_{i}(z))\in T_{p}S^{2}$, and see that the vector
pairs $(v_{1},v_{2})$ and $(V_{1},V_{2})$ provide an oriented basis of each of
the three vector spaces appearing in the above composition of linear maps. Let
$A$, $B$, and $C$ denote the matrix representations of $a$, $b$, and $c$ with
respect to these ordered bases. We have $\Phi_{t}=A\cdot B\cdot C$. Note that
$C=\text{Id}$. We compute $A$ and $B$:

To compute $A$, recall that $\phi^{t}_{0}:S^{3}\to S^{3}$ denotes the time $t$
flow of the unperturbed Reeb field (alternatively, the Hopf action). Linearize
the equality $\mathfrak{P}\circ\phi^{t}_{0}=\mathfrak{P}$, then use
$\phi_{0}^{t}(z)=\phi^{t/f_{\varepsilon}(p)}(z)$ (Lemma 3.7) and Lemma 3.9 to
find that

$A=\mathcal{R}(2t/f_{\varepsilon}(p)).$ (18)

To compute $B$, note that $\varphi^{t}$ is the Hamiltonian flow of the
function $1/f_{\varepsilon}$ with respect to $\omega_{\text{FS}}$. Is is
advantageous to study this flow using our stereographic coordinates: recall
that $\psi$ defines a symplectomorphism $({\mathbb{R}}^{2},\frac{dx\wedge
dy}{(1+x^{2}+y^{2})^{2}})\to(S^{2}\setminus\\{p^{\prime}\\},\omega_{\text{FS}})$,
where $p^{\prime}\in S^{2}$ is antipodal to $p$ in $S^{2}$ (see Notation 3.8).
Because symplectomorphisms preserve Hamiltonian data, we see that
$\psi^{-1}\circ\varphi^{t}\circ\psi:{\mathbb{R}}^{2}\to{\mathbb{R}}^{2}$ is
given near the origin as the time $t$ flow of the Hamiltonian vector field for
$\psi^{*}(1/f_{\varepsilon})$ with respect to $\frac{dx\wedge
dy}{(1+x^{2}+y^{2})^{2}}$. That is,

$\psi^{-1}\circ\varphi^{t}\circ\psi\,\,\,\,\text{is the time $t$ flow
of}\,\,\,\,\frac{\varepsilon(1+x^{2}+y^{2})^{2}f_{y}}{f_{\varepsilon}^{2}}\partial_{x}-\frac{\varepsilon(1+x^{2}+y^{2})^{2}f_{x}}{f_{\varepsilon}^{2}}\partial_{y}.$

Recall that if $P\partial_{x}+Q\partial_{y}$ is a smooth vector field on
${\mathbb{R}}^{2}$, vanishing at $(0,0)$, then the linearization of its time
$t$ flow evaluated at the origin is represented by the $2\times 2$ matrix
$\exp(tX)$, with respect to the standard ordered basis
$(\partial_{x},\partial_{y})$ of $T_{(0,0)}{\mathbb{R}}^{2}$, where

$X=\begin{pmatrix}P_{x}&P_{y}\\\ Q_{x}&Q_{y}\end{pmatrix}.$

Here, the partial derivatives of $P$ and $Q$ are implicitly assumed to be
evaluated at the origin. In this spirit, set
$P=\frac{\varepsilon(1+x^{2}+y^{2})^{2}f_{y}}{f_{\varepsilon}^{2}}$ and
$Q=\frac{-\varepsilon(1+x^{2}+y^{2})^{2}f_{x}}{f_{\varepsilon}^{2}}$, and we
compute

$X=\begin{pmatrix}P_{x}&P_{y}\\\
Q_{x}&Q_{y}\end{pmatrix}=\frac{\varepsilon}{f_{\epsilon}(p)^{2}}\begin{pmatrix}f_{yx}(0,0)&f_{yy}(0,0)\\\
-f_{xx}(0,0)&-f_{xy}(0,0)\end{pmatrix}=\frac{-\varepsilon}{f_{\varepsilon}(p)^{2}}J_{0}H(f,\psi).$

This implies that
$B=D^{-1}\exp\big{(}\frac{-t\varepsilon}{f_{\varepsilon}(p)^{2}}J_{0}H(f,\psi)\big{)}D$,
where $D$ is a change of basis matrix relating $(v_{1},v_{2})$ and the
pushforward of $(\partial_{x},\partial_{y})$ by $\psi$ in $T_{p}S^{2}$.
Because $\psi$ is holomorphic, $D$ must equal $r\cdot\mathcal{R}(s)$ for some
$r>0$ and some $s\in{\mathbb{R}}$. This provides that

$B=\mathcal{R}(-s)\exp\bigg{(}\frac{-t\varepsilon}{f_{\varepsilon}(p)^{2}}J_{0}H(f,\psi)\bigg{)}\mathcal{R}(s).$
(19)

Finally, we combine (18) and (19) to conclude

$\displaystyle\Phi_{t}=A\cdot B$
$\displaystyle=\mathcal{R}\bigg{(}\frac{2t}{f_{\varepsilon}(p)}\bigg{)}\cdot\mathcal{R}(-s)\exp\bigg{(}\frac{-t\varepsilon}{f_{\varepsilon}(p)^{2}}J_{0}H(f,\psi)\bigg{)}\mathcal{R}(s)$
$\displaystyle=\mathcal{R}(-s)\mathcal{R}\bigg{(}\frac{2t}{f_{\varepsilon}(p)}\bigg{)}\exp\bigg{(}\frac{-t\varepsilon}{f_{\varepsilon}(p)^{2}}J_{0}H(f,\psi)\bigg{)}\mathcal{R}(s).$

∎

######  Corollary 3.11.

Fix $L>0$. Then there exists some $\varepsilon_{0}>0$ such that for
$\varepsilon\in(0,\varepsilon_{0}]$, all Reeb orbits
$\gamma\in\mathcal{P}^{L}(\lambda_{\varepsilon})$ are nondegenerate, take the
form $\gamma_{p}^{k}$, a $k$-fold cover of an embedded Reeb orbit $\gamma_{p}$
as in in Lemma 3.7, and $\mu_{\text{CZ}}(\gamma)=4k+\mbox{\em ind}_{f}(p)-1$
for $p\in\mbox{\em Crit}(f)$.

###### Proof.

That there exists an $\varepsilon_{0}>0$ ensuring $\gamma$ is nondegenerate
and projects to a critical point $p$ of $f$ whenever
$\gamma\in\mathcal{P}^{L}(\lambda_{\varepsilon})$, for
$\varepsilon\in(0,\varepsilon_{0}]$, is proven in [N2, Lemma 4.11]. To compute
$\mu_{\text{CZ}}(\gamma)$, we apply the naturality, loop, and signature
properties of the Conley-Zehnder index (see Remark 2.10) to our path
$\Phi=\\{\Phi_{t}\\}\subset\text{Sp}(2)$ from Proposition 3.10. This family of
matrices has a factorization, up to $\text{SO}(2)$-conjugation,
$\Phi_{t}=L_{t}\Psi_{t}$, where $L$ is the loop of symplectic matrices
${\mathbb{R}}/2\pi kf_{\varepsilon}(p){\mathbb{Z}}\to\text{Sp}(2)$,
$t\mapsto\mathcal{R}(2t/f_{\varepsilon}(p))$, and $\Psi$ is the path of matrix
exponentials
$t\mapsto\exp(\frac{-t\varepsilon}{f_{\varepsilon}(p)^{2}}J_{0}H(f,\psi))$,
where $\psi$ denotes a choice of stereographic coordinates at $p$. In total,

$\displaystyle\mu_{\text{CZ}}(\gamma_{p}^{k})$
$\displaystyle=\mu_{\text{CZ}}(\Phi)$ $\displaystyle=\mu_{\text{CZ}}(L\Psi)$
$\displaystyle=2\mu(L)+\mu_{\text{CZ}}(\Psi)$ (20) $\displaystyle=2\cdot
2k+\frac{1}{2}\text{Sign}(-H(f,\psi))$ (21)
$\displaystyle=4k+\text{ind}_{f}(p)-1.$ (22)

In line (20) we use the Loop property from Remark 2.10, in line (21) we use
the Signature property, and in line (22) we use that
$\text{Sign}(H(f,\psi))=2-2\text{ind}_{f}(p)$. Additionally, $\mu$ in line
(20) denotes the Maslov index of a loop of symplectic matrices, so that
$\mu(L)=2k$ (see Remark 2.10 and [MS, §2]). ∎

###  3.2 Geometry of $S^{3}/G$ and associated Reeb dynamics

The process of perturbing a degenerate contact form on prequantization
bundles, outlined in Section 3.1, is often used to compute Floer theories, for