# Selective symplectic homology with applications to contact non-squeezing

Igor Uljarević

###### Abstract

We prove a contact non-squeezing phenomenon on homotopy spheres that are
fillable by Liouville domains with infinite dimensional symplectic homology:
there exists a smoothly embedded ball in such a sphere that cannot be made
arbitrarily small by a contact isotopy. These homotopy spheres include
examples that are diffeomorphic to standard spheres and whose contact
structures are homotopic to standard contact structures. As the main tool, we
construct a new version of symplectic homology, called _selective symplectic
homology_ , that is associated to a Liouville domain and an open subset of its
boundary. The selective symplectic homology is obtained as the direct limit of
Floer homology groups for Hamiltonians whose slopes tend to $+\infty$ on the
open subset but remain close to 0 and positive on the rest of the boundary.

##  1 Introduction

One of the driving questions in contact geometry is how much it differs from
smooth topology. How far does it go beyond topology? Does it, for instance,
remember not only the shape but also the size of an object? In the absence of
a natural measure, the size in contact geometry can conveniently be addressed
via contact (non-)squeezing. We say that a subset $\Omega_{a}$ of a contact
manifold $\Sigma$ can be contactly squeezed into a subset
$\Omega_{b}\subset\Sigma$ if, and only if, there exists a contact isotopy
$\varphi_{t}:\Sigma\to\Sigma,\>t\in[0,1]$ such that
$\varphi_{0}=\operatorname{id}$ and such that
$\overline{\varphi_{1}(\Omega_{a})}\subset\Omega_{b}$. The most basic examples
of contact manifolds are pessimistic as far as contact geometry and size are
concerned. Namely, every bounded subset of the standard
$\operatorname{\mathbb{R}}^{2n+1}$ (considered with the contact form
$dz+\sum_{j=1}^{n}\left(x_{j}dy_{j}-y_{j}dx_{j}\right)$) can be contactly
squeezed into an arbitrary small ball. This is because the map

$\operatorname{\mathbb{R}}^{2n+1}\to\operatorname{\mathbb{R}}^{2n+1}\quad:\quad(x,y,z)\mapsto\left(k\cdot
x,k\cdot y,k^{2}\cdot z\right)$

is a contactomorphism for all $k\in\operatorname{\mathbb{R}}^{+}$.
Consequently, every subset of a contact manifold whose closure is contained in
a contact Darboux chart can be contactly squeezed into any non-empty open
subset. In other words, contact geometry does not remember the size on a small
scale. Somewhat surprisingly, this is not true on a large scale in general. In
the next theorem, $B(R)$ denotes the ball of radius $R$.

######  Theorem 1.1 (Eliashberg-Kim-Polterovich, Chiu).

The subset $\widehat{B}(R):=B(R)\times\mathbb{S}^{1}$ of
$\mathbb{C}^{n}\times\mathbb{S}^{1}$ can be contactly squeezed into itself via
a compactly supported contact isotopy if, and only if, $R<1$.

This remarkable phenomenon, that may be seen as a manifestation of the
Heisenberg uncertainty principle, was first observed by Eliashberg, Kim, and
Polterovich [7]. They proved the case where either $R<1$ or $R\in\mathbb{N}.$
Chiu [4] extended their result to radii that are not necessarily integer.
Fraser [12] presented an alternative proof of the case of non-integer radii
that is more in line with the techniques used in [7]. (Fraser actually proved
the following formally stronger statement: there does not exist a compactly
supported contactomorphism of $\mathbb{C}^{n}\times\mathbb{S}^{1}$ that maps
the closure of $\widehat{B}(R)$ into $\widehat{B}(R)$ if $R\geqslant 1.$ It
seems not to be known whether the group of compactly supported
contactomorphisms of $\mathbb{C}^{n}\times\mathbb{S}^{1}$ is connected.)
Another proof by Sandon using generating functions can be found in [20].

The present paper provides examples of contact manifolds that are
diffeomorphic to standard spheres and that exhibit non-trivial contact non-
squeezing phenomena.

######  Theorem 1.2.

Let $\Sigma$ be an Ustilovsky sphere. Then, there exist two embedded closed
balls $B_{1},B_{2}\subset\Sigma$ of dimension equal to $\dim\Sigma$ such that
$B_{1}$ cannot be contactly squeezed into $B_{2}$.

An Ustilovky sphere is the $(4m+1)$-dimensional Brieskorn manifold

$\left\\{z=(z_{0},\ldots,z_{2m+1})\in\mathbb{C}^{2m+2}\>|\>z_{0}^{p}+z_{1}^{2}+\cdots+z_{2m+1}^{2}=0\>\&\>\left\lvert
z\right\rvert=1\right\\}$

associated to natural numbers $m,p\in\mathbb{N}$ with $p\equiv\pm 1\pmod{8}$.
The Ustilovsky sphere is endowed with the contact structure given by the
contact from

$\alpha_{p}:=\frac{ip}{8}\cdot\left(z_{0}d\overline{z}_{0}-\overline{z}_{0}dz_{0}\right)+\frac{i}{4}\cdot\sum_{j=1}^{2m+1}\left(z_{j}d\overline{z}_{j}-\overline{z}_{j}dz_{j}\right).$

These Brieskorn manifolds were used by Ustilovsky [26] to prove the existence
of infinitely many exotic contact structures on the standard sphere that have
the same homotopy type as the standard contact structure. The strength of
Theorem 1.2 lies in the topological simplicity of the objects used. A closed
ball embedded in a smooth manifold can always be smoothly squeezed into an
arbitrarily small (non-empty) open subset. Moreover, the obstruction to
contact squeezing in Theorem 1.2 does not lie in the homotopy properties of
the contact distribution. Namely, the contact distribution of an Ustilovsky
sphere for $p\equiv 1\pmod{2(2m)!}$ is homotopic to the standard contact
distribution on the sphere and the contact non-sqeezing on the standard
contact sphere is trivial. A consequence of Theorem 1.2 is a contact non-
squeezing on $\operatorname{\mathbb{R}}^{4m+1}$ endowed with a non-standard
contact structure.

######  Corollary 1.3.

Let $m\in\mathbb{N}$. Then, there exist a contact structure $\xi$ on
$\operatorname{\mathbb{R}}^{4m+1}$ and an embedded $(4m+1)$-dimensional closed
ball $B\subset\operatorname{\mathbb{R}}^{4m+1}$ such that $B$ cannot be
squeezed into an arbitrary open non-empty subset by a compactly supported
contact isotopy of $\left(\operatorname{\mathbb{R}}^{4m+1},\xi\right)$.

The exotic $\operatorname{\mathbb{R}}^{4m+1}$ in Corollary 1.3 is obtained by
removing a point from an Ustilovsky sphere. In fact, the contact non-squeezing
implies that $(\operatorname{\mathbb{R}}^{4m+1},\xi)$ constructed in this way
(although tight) is not contactomorphic to the standard
$\operatorname{\mathbb{R}}^{4m+1}$. A more general result was proven by
Fauteux-Chapleau and Helfer [8] using a variant of contact homology: there
exist infinitely many pairwise non-contactomorphic tight contact structures on
$\operatorname{\mathbb{R}}^{2n+1}$ if $n>1$.

Theorem 1.2 is a consequence of the following theorem about homotopy spheres
that bound Liouville domains with large symplectic homology.

######  Theorem 1.4.

Let $n\geqslant 2$ be a natural number and let $W$ be a $2n$-dimensional
Liouville domain such that $SH_{\ast}(W)$ is infinite dimensional and such
that $\partial W$ is a homotopy sphere. Then, there exist two embedded closed
balls $B_{1},B_{2}\subset\partial W$ of dimension $2n-1$ such that $B_{1}$
cannot be contactly squeezed into $B_{2}$.

The smooth non-squeezing problem for a homotopy sphere is trivial: every non-
dense subset of a homotopy sphere can be smoothly squeezed into an arbitrary
non-empty open subset. This is due to the existence of Morse functions with
precisely two critical points on the homotopy spheres. A smooth squeezing can
be realized by the gradient flow of such a Morse function.

Our non-squeezing results are obtained using a novel version of symplectic
homology, called _selective symplectic homology_ , that is introduced in the
present paper. It resembles the relative symplectic cohomology by Varolgunes
[27], although the relative symplectic (co)homology and the selective
symplectic homology are not quite the same. The selective symplectic homology,
$SH_{\ast}^{\Omega}(W)$, is associated to a Liouville domain $W$ and an open
subset $\Omega\subset\partial W$ of its boundary. Informally,
$SH_{\ast}^{\Omega}(W)$ is defined as the Floer homology for a Hamiltonian on
$W$ that is equal to $+\infty$ on $\Omega$ and to 0 on $\partial
W\setminus\Omega$ (whereas, in this simplified view, the symplectic homology
corresponds to a Hamiltonian that is equal to $+\infty$ everywhere on
$\partial W$). The precise definition of the selective symplectic homology is
given in Section 3 below. What follows is a brief description of the main
properties of the selective symplectic homology.

###  1.1 Canonical identification

Although not reflected in the notation, the group $SH_{\ast}^{\Omega}(W)$
depends only on the completion $\widehat{W}$ and an open subset of the _ideal
contact boundary_ of $\widehat{W}$ (defined in [7, page 1643]). More
precisely, $SH_{\ast}^{\Omega}(W)=SH^{\Omega_{f}}_{\ast}(W^{f}),$ whenever the
pairs $(W,\Omega)$ and $(W^{f},\Omega_{f})$ are $\lambda$-related in the sense
of the following definition.

######  Definition 1.5.

Let $(M,\lambda)$ be a Liouville manifold. Let $\Sigma_{1},\Sigma_{2}\subset
M$ be two hypersurfaces in $M$ that are transverse to the Liouville vector
field. The subsets $\Omega_{1}\subset\Sigma_{1}$ and
$\Omega_{2}\subset\Sigma_{2}$ are said to be $\lambda$-related if each
trajectory of the Liouville vector field either intersects both $\Omega_{1}$
and $\Omega_{2}$ or neither of them.

###  1.2 Continuation maps

To a pair $\Omega_{a}\subset\Omega_{b}$ of open subsets of $\partial W$, one
can associate a morphism

$\Phi=\Phi_{\Omega_{a}}^{\Omega_{b}}:SH_{\ast}^{\Omega_{a}}(W)\to
SH_{\ast}^{\Omega_{b}}(W),$

called _continuation map_. The groups $SH_{\ast}^{\Omega}(W)$ together with
the continuation maps form a directed system of groups indexed by open subsets
of $\partial W$. In other words, $\Phi_{\Omega}^{\Omega}$ is equal to the
identity and
$\Phi_{\Omega_{b}}^{\Omega_{c}}\circ\Phi_{\Omega_{a}}^{\Omega_{b}}=\Phi_{\Omega_{a}}^{\Omega_{c}}$.

###  1.3 Behaviour under direct limits

Let $\Omega_{k}\subset\partial W$, $k\in\mathbb{N}$ be an increasing sequence
of open subsets, i.e. $\Omega_{k}\subset\Omega_{k+1}$ for all
$k\in\mathbb{N}$. Denote $\Omega:=\bigcup_{k=1}^{\infty}\Omega_{k}$. Then, the
map

$\underset{k}{\lim_{\longrightarrow}}\>SH_{\ast}^{\Omega_{k}}(W)\to
SH_{\ast}^{\Omega}(W),$

furnished by continuation maps is an isomorphism. The direct limit is taken
with respect to continuation maps.

###  1.4 Conjugation isomorphisms

The conjugation isomorphism

$\mathcal{C}(\psi):SH_{\ast}^{\Omega_{a}}(W)\to SH_{\ast}^{\Omega_{b}}(W)$

is associated to a symplectomorphism $\psi:\widehat{W}\to\widehat{W}$, defined
on the completion of $W$, that preserves the Liouville form outside of a
compact set. To any such symplectomorphism $\psi$, one can associate a unique
contactomorphism $\varphi:\partial W\to\partial W$, called _ideal restriction_
, such that

$\psi(x,r)=\left(\varphi(x),f(x)\cdot r\right)$

for $r\in\operatorname{\mathbb{R}}^{+}$ large enough and for a certain
positive function $f:\partial W\to\operatorname{\mathbb{R}}^{+}$. The set
$\Omega_{b}$ is the image of $\Omega_{a}$ under the contactomorphism
$\varphi^{-1}:\partial W\to\partial W$. I.e. the conjugation isomorphism has
the following form

$\mathcal{C}(\psi):SH_{\ast}^{\Omega}(W)\to
SH_{\ast}^{\varphi^{-1}(\Omega)}(W),$

where $\varphi$ is the ideal restriction of $\psi$. As a consequence, the
groups $SH^{\Omega}_{\ast}(W)$ and $SH^{\varphi(\Omega)}_{\ast}(W)$ are
isomorphic whenever the contactomorphism $\varphi$ is the ideal restriction of
some symplectomorphism $\psi:\widehat{W}\to\widehat{W}$ (that preserves the
Liouville form outside of a compact set). If a contactomorphism of $\partial
W$ is contact isotopic to the identity, then it is equal to the ideal
restriction of some symplectomorphism of $\widehat{W}$. Hence, if
$\Omega_{a},\Omega_{b}\subset\partial W$ are two contact isotopic open subsets
(i.e. there exists a contact isotopy $\varphi_{t}:\partial W\to\partial W$
such that $\varphi_{0}=\operatorname{id}$ and such that
$\varphi_{1}(\Omega_{a})=\Omega_{b}$), then the groups
$SH_{\ast}^{\Omega_{a}}(W)$ and $SH_{\ast}^{\Omega_{b}}(W)$ are isomorphic.
The conjugation isomorphisms behave well with respect to the continuation
maps, as asserted by the next theorem.

######  Theorem 1.6.

Let $W$ be a Liouville domain, let $\psi:\widehat{W}\to\widehat{W}$ be a
symplectomorphism that preserves the Liouville form outside of a compact set,
and let $\varphi:\partial W\to\partial W$ be the ideal restriction of $\psi$.
Let $\Omega_{a}\subset\Omega_{b}\subset\partial W$ be open subsets. Then, the
following diagram, consisting of conjugation isomorphisms and continuation
maps, commutes

${SH_{\ast}^{\Omega_{a}}(W)}$${SH_{\ast}^{\varphi^{-1}(\Omega_{a})}(W)}$${SH_{\ast}^{\Omega_{b}}(W)}$${SH_{\ast}^{\varphi^{-1}(\Omega_{b})}(W).}$$\scriptstyle{\mathcal{C}(\psi)}$$\scriptstyle{\Phi}$$\scriptstyle{\Phi}$$\scriptstyle{\mathcal{C}(\psi)}$

### Applications

The selective symplectic homology is envisioned as a tool for studying contact
geometry and dynamics of Liouville fillable contact manifolds. The present
paper shows how it can be used to prove contact non-squeezing type of results.
In fact, all the theorems of this paper concerning contact non-squeezing
follow from the following observation.

######  Theorem 1.7.

Let $W$ be a Liouville domain and let $\Omega_{a},\Omega_{b}\subset\partial W$
be open subsets. If the rank of the continuation map
$SH_{\ast}^{\Omega_{b}}(W)\to SH_{\ast}(W)$ is (strictly) greater than the
rank of the continuation map $SH_{\ast}^{\Omega_{a}}(W)\to SH_{\ast}(W),$ then
$\Omega_{b}$ cannot be contactly squeezed into $\Omega_{a}$.

The theory of selective symplectic homology has rich algebraic structure that
is beyond the scope of the present paper. For instance,

  1. 1.

one can construct a persistent module associated to an open subset of a
contact manifold,

  2. 2.

topological quantum field theory operations are well defined on
$SH_{\ast}^{\Omega}(W),$

  3. 3.

it is possible to define transfer morphisms for selective symplectic homology
in analogy to Viterbo’s transfer morphisms for symplectic homology,

  4. 4.

there exist positive selective symplectic homology,
$\mathbb{S}^{1}$-equivariant selective symplectic homology, positive
$\mathbb{S}^{1}$-equivariant selective symplectic homology…

### The structure of the paper

The paper is organized as follows. Section 2 recalls the definition of
Liouville domains and construction of the Hamiltonian-loop Floer homology.
Sections 3 \- 5 define the selective symplectic homology and derive its
properties. Sections 6 \- 8 contain proofs of the applications to the contact
non-squeezing and necessary computations. Section 9 proves technical (and,
perhaps, expected) details required by the theory.

### Acknowledgements

I would like to thank Paul Biran and Leonid Polterovich for their interest in
this work and for valuable suggestions. This research was supported by the
Science Fund of the Republic of Serbia, grant no. 7749891, Graphical Languages
- GWORDS.

##  2 Preliminaries

###  2.1 Liouville manifolds

This section recalls the notions of a Liouville domain and a Liouville
manifold of finite type. Liouville manifolds (of finite type) play the role of
an ambient space in this paper. The selective symplectic homology is built
from objects on a Liouville manifold of finite type.

######  Definition 2.1.

A Liouville manifold of finite type is an open manifold $M$ together with a
1-form $\lambda$ on it such that the following conditions hold.

  1. 1.

The 2-form $d\lambda$ is a symplectic form on $M.$

  2. 2.

There exist a contact manifold $\Sigma$ with a contact form $\alpha$ and a
codimension-0 embedding $\iota:\Sigma\times\operatorname{\mathbb{R}}^{+}\to M$
such that $M\setminus\iota(\Sigma\times\operatorname{\mathbb{R}}^{+})$ is a
compact set, and such that $\iota^{\ast}\lambda=r\cdot\alpha,$ where $r$
stands for the $\operatorname{\mathbb{R}}^{+}$ coordinate.

We will refer to the map $\iota$ as a _conical end_ of the Liouville manifold
$M.$ With slight abuse of terminology, the set
$\iota(\Sigma\times\operatorname{\mathbb{R}}^{+})$ will also be called
_conical end_. A conical end is not unique.

The Liouville vector field, $X_{\lambda},$ of the Liouville manifold
$(M,\lambda)$ of finite type is the complete vector field defined by
$d\lambda(X_{\lambda},\cdot)=\lambda.$ If $\Sigma\subset M$ is a closed
hypersurface that is transverse to the Liouville vector field $X_{\lambda},$
then $\left.\lambda\right|_{\Sigma}$ is a contact form on $\Sigma$ and there
exists a unique codimension-0 embedding
$\iota_{\Sigma}:\Sigma\times\operatorname{\mathbb{R}}^{+}\to M$ such that
$\iota_{\Sigma}(x,1)=x$ and such that
$\iota_{\Sigma}^{\ast}\lambda=r\cdot\left.\lambda\right|_{\Sigma}$.

The notion of a Liouville manifold of finite type is closely related to that
of a Liouville domain.

######  Definition 2.2.

A Liouville domain is a compact manifold $W$ (with boundary) together with a
1-form $\lambda$ such that

  1. 1.

$d\lambda$ is a symplectic form on $W,$

  2. 2.

the Liouville vector field $X_{\lambda}$ points transversely outwards at the
boundary.

The Liouville vector field on a Liouville domain $(W,\lambda)$ is not
complete. The completion of the Liouville domain is the Liouville manifold
$(\widehat{W},\widehat{\lambda})$ of finite type obtained by extending the
integral curves of the vector field $X_{\lambda}$ towads $+\infty.$
Explicitly, as a topological space,

$\widehat{W}\quad:=\quad W\quad\cup_{\partial}\quad(\partial
W)\times[1,+\infty).$

The manifolds $(\partial W)\times[1,+\infty)$ and $W$ are glued along the
boundary via the map

$\partial W\times\\{1\\}\to\partial W\quad:\quad(x,1)\mapsto x.$

The completion $\widehat{W}$ is endowed with the unique smooth structure such
that the natural inclusions $W\hookrightarrow\widehat{W}$ and $\partial
W\times[1,+\infty)\hookrightarrow\widehat{W}$ are smooth embeddings, and such
that the vector field $X_{\lambda}$ extends smoothly to $\partial
W\times[1,+\infty)$ by the vector field $r\partial_{r}.$ (Here, we tacitly
identified $\partial W\times[1,+\infty)$ and $W$ with their images under the
natural inclusions.) The 1-form $\widehat{\lambda}$ is obtained by extending
the 1-form $\lambda$ to $\partial W\times[1,+\infty)$ by
$r\cdot\left.\lambda\right|_{\partial W.}$ The completion of a Liouville
domain is a Liouville manifold of finite type. And, other way around, every
Liouville manifold of finite type is the completion of some Liouville domain.

Let $M$ be a Liouville manifold of finite type, let $W\subset M$ be a
codimension-0 Liouville subdomain, and let $f:\partial
W\to\operatorname{\mathbb{R}}^{+}$ be a smooth function. The completion
$\widehat{W}$ can be seen as a subset of $M$. Throughout the paper, $W^{f}$
denotes the subset of $M$ defined by

$W^{f}:=\widehat{W}\setminus\iota_{\partial W}\big{(}\\{f(x)\cdot
r>1\\}\big{)}.$

Here, $\\{f(x)\cdot r>1\\}$ stands for $\left\\{(x,r)\in\partial
W\times\operatorname{\mathbb{R}}^{+}\>|\>f(x)\cdot r>1\right\\}$. The set
$W^{f}$ is a codimension-0 Liouville subdomain in its own right, and the
completions of $W$ and $W^{f}$ can be identified.

###  2.2 Floer theory

In this section, we recall the definition of the Floer homology for a contact
Hamiltonain , $HF_{\ast}(W,h).$ A contact Hamiltonian is called admissible if
it does not have any $1$-periodic orbits and if it is 1-periodic in the time
variable. The group $HF_{\ast}(W,h)$ is associated to a Liouville domain
$(W,\lambda)$ and to an admissible contact Hamiltonian $h_{t}:\partial
W\to\operatorname{\mathbb{R}}$ that is defined on the boundary of $W.$

The Floer homology for contact Hamiltonians was introduced in [16] by Merry
and the author. It relies heavily on the Hamiltonian loop Floer homology [9]
and symplectic homology [10, 11, 5, 6, 29, 28], especially the version of
symplectic homology by Viterbo [29].

####  2.2.1 Auxiliary data

Let $(W,\lambda)$ be a Liouville domain, and let $h_{t}:\partial
W\to\operatorname{\mathbb{R}}$ be an admissible contact Hamiltonian. The group
$HF_{\ast}(W,h)$ is defined as the Hamiltonian loop Floer homology,
$HF_{\ast}(H,J),$ associated to a Hamiltonian $H$ and an almost complex
structure $J.$ Both $H$ and $J$ are objects on the completion $\widehat{W}=:M$
of the Liouville domain $W.$ Before stating the precise conditions that $H$
and $J$ are assumed to satisfy, we define the set $\mathcal{J}(\Sigma,\alpha)$
of almost complex structures of _SFT type_. Let $\Sigma$ be a contact manifold
with a contact form $\alpha$. The set $\mathcal{J}(\Sigma,\alpha)$ (or simply
$\mathcal{J}(\Sigma)$ when it is clear from the context what the contact form
is equal to) is the set of almost complex structures $J$ on the
symplectization $\Sigma\times\operatorname{\mathbb{R}}^{+}$ such that

  * •

$J$ is invariant under the $\operatorname{\mathbb{R}}^{+}$ action on
$\Sigma\times\operatorname{\mathbb{R}}^{+}$,

  * •

$J(r\partial_{r})=R_{\alpha}$, where $R_{\alpha}$ is the Reeb vector field on
$\Sigma$ with respect to the contact form $\alpha$,

  * •

the contact distribution $\xi:=\ker\alpha$ is invariant under $J$ and
$\left.J\right|_{\xi}$ is a compatible complex structure on the symplectic
vector bundle $(\xi,d\alpha)\to\Sigma$.

The list of the conditions for $(H,J)$ follows.

  1. 1.

(Conditions on the conical end). There exists a positive number
$a\in\operatorname{\mathbb{R}}^{+}$ and a constant
$c\in\operatorname{\mathbb{R}}$ such that

$H_{t}\circ\iota_{\partial W}(x,r)=r\cdot h(x)+c,$

for all $t\in\operatorname{\mathbb{R}}$ and $(x,r)\in\partial
W\times[a,+\infty),$ and such that $\iota_{\partial W}^{\ast}J_{t}$ coincides
with an element of $\mathcal{J}(\partial W)$ on $\partial W\times[a,+\infty)$
for all $t\in\operatorname{\mathbb{R}}$. Here, $\iota_{\partial W}:\partial
W\times\operatorname{\mathbb{R}}^{+}\to M$ is the conical end of $M$
associated to $\partial W.$

  2. 2.

(One-periodicity). For all $t\in\operatorname{\mathbb{R}},$ $H_{t+1}=H_{t}$
and $J_{t+1}=J_{t}.$

  3. 3.

($d\widehat{\lambda}$-compatibility). $d\widehat{\lambda}(\cdot,J_{t}\cdot)$
is a Riemannian metric on $M$ for all $t\in\operatorname{\mathbb{R}}.$

The pair $(H,J)$ that satisfies the conditions above is called _Floer data_
(for the contact Hamiltonian $h$ and the Liouville domain $(W,\lambda)$).
Floer data $(H,J)$ is called _regular_ if additionally the following two
conditions hold.

  1. 4.

(Non-degeneracy). The linear map

$d\phi^{H}_{1}(x)-\operatorname{id}\quad:\quad T_{x}M\to T_{x}M$

is invertible for all fixed points $x$ of $\phi_{1}^{H}.$

  2. 5.

(Regularity). The lineaized operator of the Floer equation

$u:\operatorname{\mathbb{R}}\times(\operatorname{\mathbb{R}}/\mathbb{Z})\to
M,\quad\partial_{s}u+J_{t}(u)(\partial_{t}u-X_{H_{t}}(u))=0$

is surjective.

####  2.2.2 Floer complex

Let $(H,J)$ be regular Floer data. The Floer complex, $CF_{\ast}(H,J),$ is
built up on the contractible 1-periodic orbits of the Hamiltonian $H$. For
every 1-periodic orbit $\gamma$ of the Hamiltonian $H,$ there exists a fixed
point $x$ of $\phi^{H}_{1}$ such that $\gamma(t)=\phi^{H}_{t}(x).$ The degree,
$\deg\gamma=\deg_{H}\gamma,$ of a contractible 1-periodic orbit
$\gamma=\phi^{H}_{\cdot}(x)$ of the Hamiltonian $H$ is defined to be the
negative Conley-Zehnder index of the path of symplectic matrices that is
obtained from $d\phi^{H}_{t}(x)$ by trivializing $TM$ along a disc that is
bounded by $\gamma$ (see [19] for details concerning the Conley-Zehnder
index). Different choices of the capping disc can lead to different values of
the degree, however they all differ by an even multiple of the minimal Chern
number

$N:=\min\left\\{c_{1}(u)>0\>|\>u:\mathbb{S}^{2}\to M\right\\}.$

Therefore, $\deg\gamma$ is well defined as an element of $\mathbb{Z}_{2N}$
(but not as an element of $\mathbb{Z},$ in general). The Floer chain complex
as a group is defined by

$CF_{k}(H,J):=\bigoplus_{\deg\gamma=k}\mathbb{Z}_{2}\left\langle\gamma\right\rangle.$

Since the Floer data $(H,J)$ is regular, the set
$\mathcal{M}(H,J,\gamma^{-},\gamma^{+})$ of the solutions
$u:\operatorname{\mathbb{R}}\times(\operatorname{\mathbb{R}}/\mathbb{Z})\to M$
of the Floer equation

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

that join two 1-periodic orbits $\gamma^{-}$ and $\gamma^{+}$ of $H$ (i.e.
$\displaystyle\lim_{s\mapsto\pm\infty}u(s,t)=\gamma^{\pm}(t)$) is a finite
dimensional manifold (components of which might have different dimensions).
There is a natural $\operatorname{\mathbb{R}}$-action on
$\mathcal{M}(H,J,\gamma^{-},\gamma^{+})$ given by

$\operatorname{\mathbb{R}}\>\times\>\mathcal{M}(H,J,\gamma^{-},\gamma^{+})\quad\mapsto\quad\mathcal{M}(H,J,\gamma^{-},\gamma^{+})\quad:\quad(a,u)\mapsto
u(\cdot+a,\cdot).$

The quotient

$\tilde{\mathcal{M}}(H,J,\gamma^{-},\gamma^{+}):=\mathcal{M}(H,J,\gamma^{-},\gamma^{+})/\mathbb{R}$

of $\mathcal{M}(H,J,\gamma^{-},\gamma^{+})$ by this action is also a finite
dimensional manifold. Denote by
$n(\gamma^{-},\gamma^{+})=n(H,J,\gamma^{-},\gamma^{+})\in\mathbb{Z}_{2}$ the
parity of the number of 0-dimensional components of
$\tilde{\mathcal{M}}(H,J,\gamma^{-},\gamma^{+}).$ The boundary map

$\partial:CF_{k+1}(H,J)\to CF_{k}(H,J)$

is defined on the generators by

$\partial\left\langle\gamma\right\rangle:=\sum_{\tilde{\gamma}}n(\gamma,\tilde{\gamma})\left\langle\tilde{\gamma}\right\rangle.$

If $\deg\gamma\not=\deg\tilde{\gamma}+1$, there are no 0-dimensional
components of $\tilde{\mathcal{M}}(H,J,\gamma^{-},\gamma^{+})$, and therefore,
$n(\gamma,\tilde{\gamma})=0.$ Hence, $\partial$ is well defined. The homology
of the chain complex $CF_{\ast}(H,J)$ is denoted by $HF_{\ast}(H,J).$

####  2.2.3 Continuation maps

Continuation maps compare Floer homologies for different choices of Floer
data. They are associated to generic monotone homotopies of Floer data that
join two given instances of Floer data. We refer to these homotopies as
continuation data.

Let $(H^{-},J^{-})$ and $(H^{+},J^{+})$ be regular Floer data. The
continuation data from $(H^{-},J^{-})$ to $(H^{+},J^{+})$ is a pair
$(\\{H_{s,t}\\},\\{J_{s,t}\\})$ that consists of an $s$-dependent Hamiltonian
$H_{s,t}:M\to\operatorname{\mathbb{R}}$ and a family $J_{s,t}$ of almost
complex structures on $M$ such that the following conditions hold:

  1. 1.

(Homotopy of Floer data). For all $s\in\operatorname{\mathbb{R}},$ the pair
$(H_{s,\cdot},J_{s,\cdot})$ is Floer data (not necessarily regular) for some
contact Hamiltonian.

  2. 2.

(Monotonicity). There exists $a\in\operatorname{\mathbb{R}}^{+}$ such that
$\partial_{s}H_{s,t}(x)\geqslant 0,$ for all $s,t\in\operatorname{\mathbb{R}}$
and $x\in\iota_{\partial W}(\partial W\times[a,+\infty)).$

  3. 3.

($s$-independence at the ends). There exists
$b\in\operatorname{\mathbb{R}}^{+}$ such that $H_{s,t}(x)=H^{\pm}_{t}(x),$ for
all $t\in\operatorname{\mathbb{R}}$ and $x\in M$, if $\pm s\in[b,+\infty)$.

Continuation data $(\\{H_{s,t}\\},\\{J_{s,t}\\})$ is called _regular_ if the
linearized operator of the $s$-dependent Floer equation

$u:\operatorname{\mathbb{R}}\times(\operatorname{\mathbb{R}}/\mathbb{Z})\to
M,\quad\partial_{s}u+J_{s,t}(u)(\partial_{t}u-X_{H_{s,t}}(u))=0$

is surjective.

Given regular continuation data $(\\{H_{s,t}\\},\\{J_{s,t}\\})$ from
$(H^{-},J^{-})$ to $(H^{+},J^{+})$ and 1-periodic orbits $\gamma^{-}$ and
$\gamma^{+}$ of $H^{-}$ and $H^{+},$ respectively, the set of the solutions
$u:\operatorname{\mathbb{R}}\times(\operatorname{\mathbb{R}}/\mathbb{Z})\to M$
of the problem

$\displaystyle\partial_{s}u+J_{s,t}(u)(\partial_{t}u-X_{H_{s,t}}(u))=0,$
$\displaystyle\lim_{s\to\pm\infty}u(s,t)=\gamma^{\pm}(t)$

is a finite dimensional manifold. Its 0-dimensional part is compact, and
therefore, a finite set. Denote by $m(\gamma^{-},\gamma^{+})$ the number of
the 0-dimensional components of this manifold. The continuation map

$\Phi=\Phi(\\{H_{s,t}\\},\\{J_{s,t}\\})\quad:\quad CF_{\ast}(H^{-},J^{-})\to
CF_{\ast}(H^{+},J^{+})$

is the chain map defined on the generators by

$\Phi(\gamma^{-}):=\sum_{\gamma^{+}}m(\gamma^{-},\gamma^{+})\left\langle\gamma^{+}\right\rangle.$

The map $HF_{\ast}(H^{-},J^{-})\to HF_{\ast}(H^{+},J^{+})$ induced by a
continuation map on the homology level (this map is also called _continuation
map_) does not depend on the choice of continuation data from $(H^{-},J^{-})$
to $(H^{+},J^{+}).$

The groups $HF_{\ast}(H,J)$ together with the continuation maps form a
directed system of groups. As a consequence, the groups $HF_{\ast}(H,J)$ and
$HF_{\ast}(H^{\prime},J^{\prime})$ are canonically isomorphic whenever $(H,J)$
and $(H^{\prime},J^{\prime})$ are (regular) Floer data for the same admissible
contact Hamiltonian. Therefore, the Floer homology
$HF_{\ast}(h)=HF_{\ast}(W,h)$ for an admissible contact Hamiltonian
$h_{t}:\partial W\to\operatorname{\mathbb{R}}$ is well defined. The
continuation maps carry over to Floer homology for contact Hamiltonians. Due
to the “monotonicity”condition for the continuation data, the continuation map
$HF_{\ast}(h)\to HF_{\ast}(h^{\prime})$ is not well defined unless
$h_{t},h^{\prime}_{t}:\partial W\to\operatorname{\mathbb{R}}$ are admissible
contact Hamiltonians such that $h\leqslant h^{\prime},$ pointwise.

For a positive smooth function $f:\partial W\to\operatorname{\mathbb{R}}^{+}$,
the completions of the Liouville domains $W$ and $W^{f}$ can be naturally
identified. If a Hamiltonian
$H:\widehat{W}=\widehat{W^{f}}\to\operatorname{\mathbb{R}}$ has the slope
equal to $h$ with respect to the Liouville domain $W^{f}$, then it has the
slope equal to $f\cdot h$ with respect to the Liouville domain $W$. Therefore,
the groups $HF_{\ast}(W^{f},h)$ and $HF_{\ast}(W,f\cdot h)$ are canonically
isomorphic. Here, we tacitly identified $\partial W$ and $\partial W^{f}$ via
the contactomorphism furnished by the Liouville vector field, and regarded $h$
as both the function on $\partial W$ and $\partial W^{f}$.

##  3 Selective symplectic homology

This section defines formally the selective symplectic homology
$SH_{\ast}^{\Omega}(W)$. To this end, two sets of smooth functions on
$\partial W$ are introduced : $\mathcal{H}_{\Omega}(\partial W)$ and $\Pi(h)$.
The set $\mathcal{H}_{\Omega}(\partial W)$ consists of certain non-negative
smooth functions on $\partial W$, and $\Pi(h)$ is a set associated to
$h\in\mathcal{H}_{\Omega}(\partial W)$ that can be thought of as the set of
perturbations.

######  Definition 3.1.

Let $\Sigma$ be a closed contact manifold with a contact form $\alpha,$ and
let $\Omega\subset\Sigma$ be an open subset. Denote by
$\mathcal{H}_{\Omega}(\Sigma)=\mathcal{H}_{\Omega}(\Sigma,\alpha)$ the set of
smooth ($C^{\infty}$) autonomous contact Hamiltonians $h:\Sigma\to[0,+\infty)$
such that

  1. 1.

$h(p)\not=0\quad\Longleftrightarrow\quad p\in\Omega,$

  2. 2.

$dY^{h}(p)=0,$ for all $p\not\in\Omega,$

  3. 3.

the 1-periodic orbits of $h$ are constant.

In the definition above, the condition 2 follows from the condition 1 if the
set $\Omega$ is such that

$\overline{\operatorname{int}(\Sigma\setminus\Omega)}=\Sigma\setminus\Omega.$

However, if $\Sigma\setminus\Omega$ has an isolated point for instance, the
condition 2 is not implied by the condition 1.

######  Definition 3.2.

Let $\Sigma$ be a closed contact manifold with a contact form $\alpha,$ let
$\Omega\subset\Sigma$ be an open subset, and let
$h\in\mathcal{H}_{\Omega}(\Sigma).$ Denote by $\Pi(h)$ the set of smooth
positive functions $f:\Sigma\to\operatorname{\mathbb{R}}^{+}$ such that the
contact Hamiltonian $h+f$ has no 1-periodic orbits.

######  Definition 3.3.

Let $W$ be a Liouville domain, and let $\Omega\subset\partial W$ be an open
subset of the boundary $\Sigma:=\partial W.$ The _selective symplectic
homology_ with respect to $\Omega$ is defined to be

$SH_{\ast}^{\Omega}(W):=\underset{h\in\mathcal{H}_{\Omega}(\Sigma)}{\lim_{\longrightarrow}}\>\>\underset{f\in\Pi(h)}{\lim_{\longleftarrow}}\>HF_{\ast}(h+f).$

The limits are taken with respect to the continuation maps.

The condition 2 in Definition 3.1 is assumed for technical reasons. Given
$h\in\mathcal{H}_{\Omega}(\Sigma),$ this condition implies that for
$f:\Sigma\to\operatorname{\mathbb{R}}^{+}$ smooth and small enough (with
respect to the $C^{2}$ topology), the contact Hamiltonian $h+f$ does not have
any 1-periodic orbits (see Proposition 1.2 on page 1.2). As a consequence, the
groups $HF_{\ast}(h+f_{1})$ and $HF_{\ast}(h+f_{2})$ are canonically
isomorphic for $f_{1}$ and $f_{2}$ sufficiently small. In other words, the
inverse limit

$\underset{f\in\Pi(h)}{\lim_{\longleftarrow}}HF_{\ast}(h+f)$

stabilizes for $h\in\mathcal{H}_{\Omega}(W)$. This is proven in the next
lemma.

######  Lemma 3.4.

Let $W$ be a Liouville domain, let $\Omega\subset\partial W$ be an open
subset, and let $h\in\mathcal{H}_{\Omega}(W)$. Then, there exists an open
convex neighbourhood $U$ of 0 (seen as a constant function on $\partial W$) in
$C^{2}$ topology such that the natural map

$\underset{f\in\Pi(h)}{\lim_{\longleftarrow}}HF_{\ast}(h+f)\to HF_{\ast}(h+g)$

is an isomorphism for all $g\in C^{\infty}(\partial
W,\operatorname{\mathbb{R}}^{+})\cap U$.

###### Proof.

Proposition 1.2 on page 1.2 implies that there exists a convex $C^{2}$
neighbourhood $U$ of the constant function $\partial
W\to\operatorname{\mathbb{R}}:p\mapsto 0$ such that $h+f$ has no non-constant
1-periodic orbits if $f\in U$. Since $h+f$ is positive for a positive function
$f\in U$, it does not have any constant orbits (the corresponding vector field
is nowhere 0). Hence, $h+f$ has no 1-periodic orbits for all positive
functions $f:\partial W\to\operatorname{\mathbb{R}}^{+}$ from $U$. This, in
particular, implies

$\mathcal{O}:=C^{\infty}(\partial W,\operatorname{\mathbb{R}}^{+})\cap
U\subset\Pi(h).$

The set $\mathcal{O}$ is also convex. Therefore,

$(1-s)\cdot f_{a}+s\cdot f_{b}\in\mathcal{O}$

for all $f_{a},f_{b}\in\mathcal{O}$ and $s\in[0,1]$. If, additionally,
$f_{a}\leqslant f_{b}$, then $h+(1-s)\cdot f_{a}+s\cdot f_{b}$ is an
increasing family (in $s$-variable) of admissible contact Hamiltonians.
Theorem 1.3 from [25] asserts that the continuation map

$HF_{\ast}(h+f_{a})\to HF_{\ast}(h+f_{b})$

is an isomorphism in this case. This implies the claim of the lemma. ∎

The set $U$ from Lemma 3.4 is not unique. For technical reasons, it is useful
to choose one specific such set (we will denote it by $\mathcal{U}(h)$) for a
given contact Hamiltonian $h\in\mathcal{H}_{\Omega}(\partial W)$. The
construction of $\mathcal{U}(h)$ follows. Let $\psi_{j}:V_{j}\to\partial W$ be
charts on $\partial W$ and let $K_{j}\subset\psi(V_{j})$ be compact subsets,
$j\in\\{1,\ldots,m\\}$, such that $\bigcup_{j=1}^{m}K_{j}=\partial W$. Denote
by $\left\lVert\cdot\right\rVert_{C^{2}}$ the norm on $C^{\infty}(\partial
W,\operatorname{\mathbb{R}})$ defined by

$\left\lVert
f\right\rVert_{C^{2}}:=\underset{i\in\\{0,1,2\\}}{\max_{j\in\\{1,\ldots,m\\}}}\max_{K_{j}}\left\lVert
D^{i}(f\circ\psi_{j})\right\rVert.$

The norm $\left\lVert\cdot\right\rVert_{C^{2}}$ induces the $C^{2}$ topology
on $C^{\infty}(\partial W,\operatorname{\mathbb{R}})$. Denote by
$\mathcal{B}(\varrho)\subset C^{\infty}(\partial W,\operatorname{\mathbb{R}})$
the open ball with respect to $\left\lVert\cdot\right\rVert_{C^{2}}$ centered
at 0 of radius $\varrho$. Define $\mathcal{U}(h)$ as the union of the balls
$\mathcal{B}(\varrho)$ that have the following property: the contact
Hamiltonian $h+f$ has no non-constant 1-periodic orbits for all
$f\in\mathcal{B}(\varrho)$. The set $\mathcal{U}(h)$ is open as the union of
open subsets. It is convex as the union of nested convex sets. And, it is non-
empty by Lemma 3.4. The subset of $\mathcal{U}(h)$ consisting of strictly
positive functions is denoted by $\mathcal{O}(h)$, i.e.

$\mathcal{O}(h):=\mathcal{U}(h)\cap C^{\infty}(\partial
W,\operatorname{\mathbb{R}}).$

##  4 Behaviour under direct limits

The next theorem claims that the selective symplectic homology behaves well
with respect to direct limits.

######  Theorem 4.1.

Let $(W,\lambda)$ be a Liouville domain, and let
$\Omega_{1},\Omega_{2},\ldots$ be a sequence of open subsets of $\partial W$
such that $\Omega_{k}\subset\Omega_{k+1}$ for all $k\in\mathbb{N}.$ Denote
$\Omega:=\bigcup_{k}\Omega_{k}.$ Then, the map

$\displaystyle\mathfrak{P}:\lim_{k\to+\infty}SH_{\ast}^{\Omega_{k}}(W)\to
SH_{\ast}^{\Omega}(W),$

furnished by continuation maps, is an isomorphism.

###### Proof.

Let $\mathcal{U}(h)$ and $\mathcal{O}(h)$ be the sets introduced after Lemma
3.4 on page 3.4 and let $\left\lVert\cdot\right\rVert_{C^{2}}$ be the norm on
$C^{\infty}(\partial W,\operatorname{\mathbb{R}})$ used in the definition of
$\mathcal{U}(h)$. We first show that for an arbitrary
$h\in\mathcal{H}_{\Omega}(\partial W)$ there exist $k\in\mathbb{N},$
$g\in\mathcal{H}_{\Omega_{k}}(\partial W),$ and
$\varepsilon,\delta\in\operatorname{\mathbb{R}}^{+}$ such that

$h+\varepsilon\leqslant g+\delta,$

and such that $\varepsilon\in\mathcal{O}(h)$ and $\delta\in\mathcal{O}(g).$
There exists $a\in\operatorname{\mathbb{R}}^{+}$ such that
$f\in\mathcal{U}(h)$ for all smooth functions $f:\partial
W\to\operatorname{\mathbb{R}}$ that satisfy $\left\lVert
f\right\rVert_{C^{2}}\leqslant a.$ For $k$ large enough, there exists
$g\in\mathcal{H}_{\Omega_{k}}(\partial W)$ such that $\left\lVert
h-g\right\rVert_{C^{2}}<\frac{a}{4}.$ By the triangle inequality, it follows
that the contact Hamiltonian $f+g$ has no non-constant 1-periodic orbits for
all smooth $f:\partial W\to\operatorname{\mathbb{R}}$ such that $\left\lVert
f\right\rVert_{C^{2}}\leqslant\frac{3a}{4}.$ By the construction of
$\mathcal{U}(g)$, this means $f\in\mathcal{U}(g)$ for all smooth $f:\partial
W\to\operatorname{\mathbb{R}}$ with $\left\lVert
f\right\rVert_{C^{2}}\leqslant\frac{3a}{4}.$ Choose, for instance,
$\delta=\frac{3a}{4}$ and $\varepsilon=\frac{a}{4}.$

Now, we proceed to prove the theorem. The claim above implies that for every
$h\in\mathcal{H}_{\Omega}(W)$ there exists $k\in\mathbb{N}$ such that the
continuation maps furnish a map

$\underset{f\in\Pi(h)}{\lim_{\longleftarrow}}HF_{\ast}(h+f)\to
SH_{\ast}^{\Omega_{k}}(W).$

Therefore, continuation maps give rise to a map

$\mathfrak{Q}:SH_{\ast}^{\Omega}(W)=\underset{h\in\mathcal{H}_{\Omega}(W)}{\lim_{\longrightarrow}}\underset{f\in\Pi(h)}{\lim_{\longleftarrow}}HF_{\ast}(h+f)\to\lim_{k\to+\infty}SH_{\ast}^{\Omega_{k}}(W).$

Both maps $\mathfrak{P}$ and $\mathfrak{Q}$ are furnished by transfer
morphisms. Hence, their compositions

$\displaystyle\mathfrak{P}\circ\mathfrak{Q}:SH_{\ast}^{\Omega}(W)\to
SH_{\ast}^{\Omega}(W),$
$\displaystyle\mathfrak{Q}\circ\mathfrak{P}:\lim_{k\to+\infty}SH_{\ast}^{\Omega_{k}}(W)\to\lim_{k\to+\infty}SH_{\ast}^{\Omega_{k}}(W)$

are as well. Consequently, $\mathfrak{P}\circ\mathfrak{Q}=\operatorname{id}$
and $\mathfrak{Q}\circ\mathfrak{P}=\operatorname{id}$. In particular,
$\mathfrak{P}$ is an isomorphism. ∎

##  5 Conjugation isomorphisms

Let $(M,\lambda)$ be a Liouville domain of finite type. The group of
symplectomorphisms $\psi:M\to M$ that preserve the Liouville form outside of a
compact subset is denoted by $\operatorname{Symp}^{\ast}(M,\lambda)$. If
$M=\widehat{W}$ is the completion of a Liouville domain $(W,\lambda)$, then
for $\psi\in\operatorname{Symp}^{\ast}(M,\lambda)$ there exist a
contactomorphism $\varphi:\partial W\to\partial W$ and a positive smooth
function $f:\partial W\to\operatorname{\mathbb{R}}^{+}$ such that

$\psi(x,r)=(\varphi(x),r\cdot f(x)),$

for $x\in\partial W$ and $r\in\operatorname{\mathbb{R}}^{+}$ large enough. The
contactomorphism $\varphi$ is called the _ideal restriction_ of $\psi$. To an
element $\psi\in\operatorname{Symp}^{\ast}(M,\lambda)$, one can associate
isomorphisms, called _conjugation isomorphisms_ ,

$\displaystyle\mathcal{C}(\psi):HF_{\ast}(H,J)\to
HF_{\ast}(\psi^{\ast}H,\psi^{\ast}J),$

where $(H,J)$ is regular Floer data. The isomorphisms $\mathcal{C}(\psi)$ are
defined on the generators by

$\gamma\mapsto\psi^{\ast}\gamma=\psi^{-1}\circ\gamma.$

They are isomorphisms already on the chain level, and already on the chain
level, they commute with the continuation maps.

######  Proposition 5.1.

Let $(M,\lambda)$ be the completion of a Liouville domain $(W,\lambda)$, let
$\psi\in\operatorname{Symp}^{\ast}(M,\lambda)$, and let $\varphi:\partial
W\to\partial W$ be the ideal restriction of $\psi$. Then, the conjugation
isomorphisms with respect to $\psi$ give rise to isomorphisms (called the
same)

$\displaystyle\mathcal{C}(\psi):SH_{\ast}^{\Omega}(W)\to
SH_{\ast}^{\varphi^{-1}(\Omega)}(W),$

for every open subset $\Omega\subset\partial W$.

###### Proof.

Let $h\in\mathcal{H}_{\Omega}(\partial W)$, let $f\in\Pi(h)$, and let $(H,J)$
be Floer data for $W$ and for the contact Hamiltonian $h+f$. The Floer data
$(\psi^{\ast}H,\psi^{\ast}J)$ corresponds to the contact Hamiltonain
$g\cdot(h+f)\circ\varphi$, where $g:\partial
W\to\operatorname{\mathbb{R}}^{+}$ is a certain positive smooth function.
Moreover, $g\cdot h\circ\varphi\in\mathcal{H}_{\varphi^{-1}(\Omega)}(W)$ and
$g\cdot f\circ\varphi\in\Pi(g\cdot h\circ\varphi).$ Since the conjugation
isomorphisms commute with the continuation maps and since the relations above
hold, the conjugation isomorphisms give rise to an isomorphism

$\mathcal{C}(\psi):SH_{\ast}^{\Omega}(W)\to
SH_{\ast}^{\varphi^{-1}(\Omega)}(W).$

∎

Now, the proof of Theorem 1.6 from the introduction follows directly.

######  Theorem 1.6.

Let $W$ be a Liouville domain, let $\psi:\widehat{W}\to\widehat{W}$ be a
symplectomorphism that preserves the Liouville form outside of a compact set,
and let $\varphi:\partial W\to\partial W$ be the ideal restriction of $\psi$.
Let $\Omega_{a}\subset\Omega_{b}\subset\partial W$ be open subsets. Then, the
following diagram, consisting of conjugation isomorphisms and continuation
maps, commutes

${SH_{\ast}^{\Omega_{a}}(W)}$${SH_{\ast}^{\varphi^{-1}(\Omega_{a})}(W)}$${SH_{\ast}^{\Omega_{b}}(W)}$${SH_{\ast}^{\varphi^{-1}(\Omega_{b})}(W).}$$\scriptstyle{\mathcal{C}(\psi)}$$\scriptstyle{\mathcal{C}(\psi)}$

###### Proof.

The proof follows directly from the commutativity of the conjugation
isomorphisms and the continuation maps on the level of $HF_{\ast}(H,J)$. ∎

##  6 Selective symplectic homology for a Darboux chart

This section proves that sufficiently small open subsets on the boundary of a
Liouville domain have finite dimensional selective symplectic homology. Let
$a_{1},\ldots,a_{n},b\in\operatorname{\mathbb{R}}^{+}$. The contact polydisc
$P=P(a_{1},\ldots,a_{n},b)$ is a subset of the standard contact
$\operatorname{\mathbb{R}}^{2n+1}$ (endowed with the contact form
$dz+\sum_{j=1}^{n}(x_{j}dy_{j}-y_{j}dx_{j})$) that is given by

$P:=\left\\{(x,y,z)\in\operatorname{\mathbb{R}}^{n}\times\operatorname{\mathbb{R}}^{n}\times\operatorname{\mathbb{R}}\>|\>z^{2}\leqslant
b^{2}\>\&\>(\forall j\in\\{1,\ldots,n\\})\>x_{j}^{2}+y_{j}^{2}\leqslant
a_{j}^{2}\right\\}.$

######  Theorem 1.6.

Let $W$ be a Liouville domain and let $P\subset\partial W$ be a contact
polydisc in a Darboux chart. Then, the continuation map

$SH_{\ast}^{\varnothing}(W)\to SH_{\ast}^{P}(W)$

is an isomorphism.

The next lemma is used in the proof of Theorem 1.6.

######  Lemma 1.6.

Let $\alpha:=dz+\sum_{j=1}^{n}(x_{j}dy_{j}-y_{j}dx_{j})$ be the standard
contact form on $\operatorname{\mathbb{R}}^{2n+1}$. Denote by
$(r_{j},\theta_{j})$ polar coordinates in the $(x_{j},y_{j})$-plane,
$j=1,\ldots,n$. Let $h:\operatorname{\mathbb{R}}^{2n+1}\to[0,+\infty)$ be a
contact Hamiltonian of the form

$h(r,\theta,z):=\varepsilon+g(z)\cdot\prod_{j=1}^{n}f_{j}(r_{j}),$

where $\varepsilon\in\operatorname{\mathbb{R}}^{+}$,
$g:\operatorname{\mathbb{R}}\to[0,+\infty)$ is a smooth function, and
$f_{j}:[0,+\infty)\to[0,+\infty)$ is a (not necessarily strictly) decreasing
smooth function, $j=1,\ldots,n$. Then, the $z$-coordinate strictly decreases
along the trajectories of the contact Hamiltonian $h$ (with respect to the
contact form $\alpha$).

###### Proof.

Let $Y^{h}$ be the vector field of the contact Hamiltonian $h$, i.e. the
vector field that satisfies $\alpha(Y^{h})=-h$ and $d\alpha(Y^{h},\cdot)=dh-
dh(\partial_{z})\cdot\alpha$. Then,

$dz(Y^{h})=-\varepsilon+g(z)\cdot\left(-1+\frac{1}{2}\cdot\sum_{j=1}^{n}\left(r_{j}\cdot
f^{\prime}_{j}(r_{j})\cdot\prod_{k\not=j}f_{k}(r_{k})\right)\right).$

In particular, $dz(Y^{h}(p))\leqslant-\varepsilon$ for all
$p\in\operatorname{\mathbb{R}}^{2n+1}$. Let
$\gamma:I\to\operatorname{\mathbb{R}}^{2n+1}$ be a trajectory of the contact
Hamiltonian $h$. Then,

$\frac{d}{dt}\left(z(\gamma(t))\right)=dz(Y^{h}(\gamma(t)))\leqslant-\varepsilon.$

Consequently, the function $t\mapsto z(\gamma(t))$ is strictly decreasing. ∎

###### Proof of Theorem 1.6.

By assumptions, there exists a Darboux chart
$\psi:O\to\operatorname{\mathbb{R}}^{2n+1}$, $O\subset\partial W$, such that
$\psi(P)=P(a_{1},\ldots,a_{n},b)$ for some
$a_{1},\ldots,a_{n},b\in\operatorname{\mathbb{R}}^{+}$. Since
$P(a_{1},\ldots,a_{n},b)$ is compact and $\psi(O)$ open, there exist
$b^{\prime},a_{1}^{\prime},\ldots,a_{n}^{\prime}\in\operatorname{\mathbb{R}}^{+}$
such that

$P(a_{1},\ldots,a_{n},b)\subset\operatorname{int}P(a_{1}^{\prime},\ldots,a_{n}^{\prime},b^{\prime})\subset\psi(O).$

In particular, $b<b^{\prime}$. Denote $\varepsilon_{1}:=b^{\prime}-b$

Let $h\in\mathcal{H}_{P}(\partial W)$ be such that

$h\circ\psi^{-1}(r,\theta,z)=g(z)\cdot\prod_{j=1}^{n}f_{j}(r_{j})$

for some smooth function $g:\operatorname{\mathbb{R}}\to[0,+\infty)$ and some
smooth decreasing functions $f_{j}:[0,+\infty)\to[0,+\infty)$, $j=1,\ldots,n$.
Let $\varepsilon_{0}\in\operatorname{\mathbb{R}}^{+}$ be such that there are
no Reeb orbits on $\partial W$ of period less than or equal to
$\varepsilon_{0}$.

Now, we show that the contact Hamiltonian $h+\varepsilon$ has no 1-periodic
orbits if $0<\varepsilon<\min\\{\varepsilon_{0},\varepsilon_{1}\\}$. This
implies $\varepsilon\in\mathcal{O}(h)$ if
$0<\varepsilon<\min\\{\varepsilon_{0},\varepsilon_{1}\\}$. Let
$\gamma:\operatorname{\mathbb{R}}\to\partial W$ be a trajectory of the contact
Hamiltonian $h+\varepsilon$. If $\gamma$ does not intersect $P$, then $\gamma$
is also a trajectory of the reparametrized Reeb flow
$t\mapsto\varphi_{-\varepsilon\cdot t}$. Since $\varepsilon<\varepsilon_{0}$,
this implies that $\gamma$ is not 1-periodic. Assume, now, that $\gamma$ does
intersect $P$. If $\gamma$ is entirely contained in $O$, then Lemma 1.6
implies that $\gamma$ is not 1-periodic. If $\gamma$ is not entirely contained
in $O$, then (by Lemma 1.6) $\gamma$ intersects
$\psi^{-1}\left(\operatorname{\mathbb{R}}^{2n}\times[b,b^{\prime}]\right)$. On
$\psi^{-1}\left(\operatorname{\mathbb{R}}^{2n}\times[b,b^{\prime}]\right)$,
the contact Hamiltonian $h+\varepsilon$ is equal to $\varepsilon$ and
$\gamma(t)$ is equal to $\psi^{-1}(x,y,z-\varepsilon t)$ for some
$(x,y,z)\in\operatorname{\mathbb{R}}^{2n+1}$. In particular, $\gamma$ “spends”
at least $\frac{b^{\prime}-b}{\varepsilon}$ time passing through
$\psi^{-1}\left(\operatorname{\mathbb{R}}^{2n}\times[b,b^{\prime}]\right)$.
Since

$\frac{b^{\prime}-b}{\varepsilon}>\frac{b^{\prime}-b}{\varepsilon_{1}}=1,$

$\gamma$ cannot be 1-periodic.

The same argument shows that the contact Hamiltonian $h^{s}:=s\cdot
h+\varepsilon$ has no 1-periodic orbits for all $s\in[0,1]$. Additionally,
$\partial_{s}h^{s}\geqslant 0$. Therefore, the continuation map

$HF_{\ast}(\varepsilon)=HF_{\ast}(h^{0})\to HF_{\ast}(h^{1})=HF_{\ast}(h)$

is an isomorphism [25, Theorem 1.3]. ∎

##  7 Immaterial transverse circles and selective symplectic homology of their
complements

This section provides non-trivial examples where the selective symplectic
homology is “large”. We start by defining _immaterial_ subsets of contact
manifolds.

######  Definition 1.6.

A subset $A$ of a contact manifold $\Sigma$ is called _immaterial_ if there
exists a contractible loop $\varphi_{t}:\Sigma\to\Sigma$ of contactomorphisms
such that its contact Hamiltonian $h_{t}:\Sigma\to\operatorname{\mathbb{R}}$
(with respect to some contact form on $\Sigma$) is positive on $A$, i.e. such
that it satisfies

$(\forall x\in A)(\forall t\in\operatorname{\mathbb{R}})\quad h_{t}(x)>0.$

If a compact subset $A$ of a contact manifold $\Sigma$ is immaterial, then
there exists a contractible loop of contactomorphisms on $\Sigma$ whose
contact Hamiltonian is arbitrarily large on $A$. In fact, this property of a
compact subset $A$ is equivalent to $A$ being immaterial.

######  Lemma 1.6.

A compact subset $A$ of a contact manifold $\Sigma$ is immaterial if, and only
if, for every $a\in\operatorname{\mathbb{R}}^{+}$ there exists a contractible
loop of contactomorphisms on $\Sigma$ such that its contact Hamiltonian
$h_{t}:\Sigma\to\operatorname{\mathbb{R}}$ satisfies

$(\forall x\in A)(\forall t\in\operatorname{\mathbb{R}})\quad
h_{t}(x)\geqslant a.$

###### Proof.

Let $a\in\operatorname{\mathbb{R}}^{+}$ be an arbitrarily large positive
number and let $A$ be a compact immaterial subset of a contact manifold
$\Sigma$. Then, there exists a contractible loop $\varphi:\Sigma\to\Sigma$ of
contactomorphisms such that its contact Hamiltonian
$h_{t}:\Sigma\to\operatorname{\mathbb{R}}$ satisfies

$(\forall x\in A)(\forall t\in\operatorname{\mathbb{R}})\quad h_{t}(x)>0.$

Denote $m:=\min_{x\in A,t\in\operatorname{\mathbb{R}}}h_{t}(x)>0$. Let
$k\in\mathbb{N}$ be such that $k\cdot m>a$. Denote by
$h^{k}_{t}:\Sigma\to\operatorname{\mathbb{R}}$ the contact Hamiltonian defined
by $h^{k}_{t}(x):=k\cdot h_{kt}(x)$. The contact Hamiltonian $h^{k}$ furnishes
a loop of contactomorphisms that is obtained by concatenating $\varphi$ to
itself $k$ times. In particular, $h^{k}$ generates a contractible loop of
contactomorphisms. By construction

$(\forall x\in A)(\forall t\in\operatorname{\mathbb{R}})\quad
h^{k}_{t}(x)\geqslant k\cdot m>a.$

This proves one direction of the lemma. The other direction is obvious. ∎

The next lemma implies that a singleton (i.e. a set consisting of a single
point) is immaterial in every contact manifold of dimension greater than 3. By
continuity, every point in a contact manifold of dimension greater than 3 has
an immaterial neighbourhood.

######  Lemma 1.6.

Let $\Sigma$ be a contact manifold of dimension $2n+1\geqslant 3$. Then, there
exists a contractible loop $\varphi_{t}:\Sigma\to\Sigma$ of contactomorphisms
such that its contact Hamiltonian is positive at some point (for all times
$t$).

###### Proof.

Let $\mathbb{S}^{2n+1}$ be the standard contact sphere seen as the unit sphere
in $\mathbb{C}^{n+1}$ centered at the origin. The unitary matrices act on
$\mathbb{S}^{2n+1}$ as contactomorphisms. Let
$\psi_{t}:\mathbb{S}^{2n+1}\to\mathbb{S}^{2n+1}$ be the contact circle action
given by

$\psi_{t}(z):=\left(z_{1},\ldots,z_{n-1},e^{2\pi it}z_{n},e^{-2\pi
it}z_{n+1}\right).$

The loop

$t\mapsto\left[\begin{matrix}e^{2\pi it}&0\\\ 0&e^{-2\pi
it}\end{matrix}\right]$

is contractible in the unitary group $U(2)$. Hence, there exists a smooth
$s$-family $A^{s}$, $s\in[0,1]$, of loops in $U(2)$ such that

$A^{1}(t)=\left[\begin{matrix}e^{2\pi it}&0\\\ 0&e^{-2\pi
it}\end{matrix}\right]$

and such that $A^{0}(t)=\left[\begin{matrix}1&0\\\ 0&1\end{matrix}\right]$ for
all $t$. Denote $\psi^{s}_{t}(z):=\left[\begin{matrix}\mathbb{1}_{n-1}&\\\
&A^{s}(t)\end{matrix}\right]z$. For all $s\in[0,1]$, $\psi^{s}$ is a loop of
contactomorphisms of $\mathbb{S}^{2n+1}$ and $\psi_{t}^{0}=\operatorname{id}$,
$\psi_{t}^{1}=\psi_{t}$. Therefore, $\psi_{t}$ is a contractible loop of
contactomorphisms. Denote by $h^{s}_{t}:\mathbb{S}^{2n+1}\to\mathbb{R}$ the
contact Hamiltonian of $\psi^{s}_{t}$ and $h:=h^{1}$. Denote
$U(r):=\left\\{z\in\mathbb{S}^{2n+1}\>|\>\left\lvert
z_{1}\right\rvert>1-r\right\\}$ and let $\varepsilon\in(0,2)$. Let
$\mu:\mathbb{S}^{2n+1}\to[0,1]$ be a smooth cut-off function such that
$\mu(x)=0$ for $x$ in a neighbourhood of $p:=(1,0,\ldots,0)$ and such that
$\mu(x)=1$ for $x\in\mathbb{S}^{2n+1}\setminus U(\frac{\varepsilon}{2})$. Let
$f_{t}^{s}(x):=\mu(x)\cdot h^{s}_{t}(x)$. By the construction of $\mu$ and
since $U(r)$ is invariant under $\psi^{s}_{t}$ for all $r,s$, and $t$, the
contactomorphism $\varphi_{1}^{f^{s}}$ is compactly supported in
$U(\varepsilon)$ for all $s$. Let
$g^{s}_{t}:\mathbb{S}^{2n+1}\to\operatorname{\mathbb{R}}$, $s\in[0,1]$ be the
contact Hamiltonian that generates $t\mapsto\varphi_{1}^{f^{t\cdot s}}$, i.e.
$\varphi_{t}^{g^{s}}=\varphi_{1}^{f^{t\cdot s}}$. Denote $g:=g^{1}.$

The map $\varphi^{f^{1}}_{t}\circ(\varphi_{t}^{g})^{-1}$ is a loop of
contactomorphisms. Its contact Hamiltonian
$e_{t}:\mathbb{S}^{2n+1}\to\operatorname{\mathbb{R}}$ is equal to 0 in a
neighbourhood of $p$ and coincides with $f^{1}$ in $\mathbb{S}^{2n+1}\setminus
U(\varepsilon)$. Consequently (since $f^{1}$ and $h$ coincide in
$\mathbb{S}^{2n+1}\setminus U(\varepsilon)$), the contact Hamiltonians $e$ and
$h$ coincide in $\mathbb{S}^{2n+1}\setminus U(\varepsilon)$. This implies that
$\varphi^{f^{1}}_{t}\circ(\varphi_{t}^{g})^{-1}$ is a loop of
contactomorphisms of $\mathbb{S}^{2n+1}$ that are compactly supported in the
complement of a neighbourhood of $p$. Additionally, this implies that there
exists $q\in\mathbb{S}^{2n+1}\setminus U(\varepsilon)$ such that
$e_{t}(q)=h(q)>0$ for all $t$. The loop
$\varphi_{t}^{e}=\varphi^{f^{1}}_{t}\circ(\varphi_{t}^{g})^{-1}$ is
contractible via the homotopy
$\left\\{\varphi^{f^{s}}_{t}\circ(\varphi_{t}^{g^{s}})^{-1}\right\\}_{s\in[0,1]}$
that is also compactly supported in the complement of a neighbourhood of $p$.
Since $\mathbb{S}^{2n+1}\setminus\\{p\\}$ is contactomorphic to the standard
$\operatorname{\mathbb{R}}^{2n+1}$ and since every contact manifold has a
contact Darboux chart around each of its points, the lemma follows. ∎

The following theorem implies that the complement of an immaterial circle has
infinite dimensional selective symplectic homology under some additional
assumptions.

######  Theorem 1.6.

Let $W$ be a Liouville domain with $\dim SH_{\ast}(W)=\infty$, and let
$\Gamma\subset\partial W$ be an immaterial embedded circle that is transverse
to the contact distribution. Denote $\Omega:=\partial W\setminus\Gamma$. Then,
the continuation map $SH_{\ast}^{\Omega}(W)\to SH_{\ast}(W)$ has infinite
rank.

###### Proof.

For an admissible contact Hamiltonian $h_{t}:\partial
W\to\operatorname{\mathbb{R}}$, denote by $r(h)=r(W,h)$ the rank of the
canonical map $HF_{\ast}(h)\to SH_{\ast}(W)$. It is enough to prove that for
every admissible $\ell\in\operatorname{\mathbb{R}}$ there exists
$h\in\mathcal{H}_{\Omega}(\partial W)$ and $\varepsilon\in\mathcal{O}(h)$ such
that $r(\ell)\leqslant r(h+\varepsilon)$. Denote by $\alpha$ the contact form
on $\partial W$ (the restriction of the Liouville form). Without loss of
generality (see Theorem 2.5.15 and Example 2.5.16 in [13]), we may assume that
there exists an open neighbourhood $U\subset\partial W$ of $\Gamma$ and an
embedding $\psi:U\to\mathbb{C}^{n}\times\mathbb{S}^{1}$ such that
$\psi(\Gamma)=\\{0\\}\times\mathbb{S}^{1}$ and such that

$\alpha=\psi^{\ast}\left(d\theta+\frac{i}{2}\sum_{j=1}^{n}(z_{j}d\overline{z}_{j}-\overline{z}_{j}dz_{j})\right).$

Here, $z=(z_{1},\ldots,z_{n})\in\mathbb{C}^{n}$ and $\theta\in\mathbb{S}^{1}$.
Let $\ell\in\operatorname{\mathbb{R}}$ be an arbitrary admissible (constant)
slope. Since $\Gamma$ is immaterial, there exists a contractible loop of
contactomorphisms $\varphi^{f}_{t}:\partial W\to\partial W$ (which we see as a
1-periodic $\operatorname{\mathbb{R}}$-family of contactomorphisms) such that
its contact Hamiltonian $f_{t}:\partial W\to\operatorname{\mathbb{R}}$
satisfies $\min_{x\in\Gamma,t\in\operatorname{\mathbb{R}}}f_{t}(x)\geqslant
2\ell$. Denote $m:=\min_{x\in\partial
W,t\in\operatorname{\mathbb{R}}}f_{t}(x)$. Let
$h\in\mathcal{H}_{\Omega}(\partial W)$ be a strict contact Hamiltonian (i.e.
its flow preserves the contact form $\alpha$ ) such that $h(x)\geqslant\ell-m$
for $x$ in the set $\left\\{x\in\partial
W\>|\>\min_{t\in\operatorname{\mathbb{R}}}f_{t}(x)\leqslant\ell\right\\}.$ The
contact Hamiltonian $h$ can be constructed as follows. Since the function
$x\mapsto\min_{t\in\operatorname{\mathbb{R}}}f_{t}(x)$ is continuous, the set
$S:=\\{x\in\partial
W\>|\>\min_{t\in\operatorname{\mathbb{R}}}f_{t}(x)\leqslant\ell\\}$ is closed.
Therefore, there exists a ball $B(r)\subset\mathbb{C}^{n}$ centered at the
origin with sufficiently small radius $r$ such that
$\overline{B(r)}\times\mathbb{S}^{1}\subset\psi(\partial W\setminus S)$. Now,
we choose $h$ to be equal to a constant greater than $\ell-m$ on $\partial
W\setminus\psi^{-1}\left(\overline{B(r)}\times\mathbb{S}^{1}\right)$ and such
that $h\circ\psi^{-1}(z,\theta)=\overline{h}(z_{1}^{2}+\cdots+z_{n}^{2})$ for
$\left\lvert z\right\rvert<r$ and for some smooth function
$\overline{h}:[0,+\infty)\to[0,+\infty)$. Generically, $h$ has no non-constant
1-periodic orbits.

Let $\varepsilon\in\operatorname{\mathbb{R}}^{+}$ be a sufficiently small
positive number such that $\varepsilon\in\mathcal{O}(h)$ and denote
$h^{\varepsilon}:=h+\varepsilon.$ Let $g:=h^{\varepsilon}\\#f$ be the contact
Hamiltonian that generates the contact isotopy
$\varphi_{t}^{h^{\varepsilon}}\circ\varphi_{t}^{f}$, i.e.

$g_{t}(x):=h^{\varepsilon}(x)+f_{t}\circ\left(\varphi^{h^{\varepsilon}}_{t}\right)^{-1}(x).$

(In the last formula, we used that $h^{\varepsilon}$ is a strict contact
Hamiltonian.) If $h^{\varepsilon}(x)<\ell-m$, then (since $h^{\varepsilon}$ is
autonomous and strict)
$h^{\varepsilon}\circ\left(\varphi_{t}^{h^{\varepsilon}}\right)^{-1}(x)<\ell-m$
for all $t$. Consequently (by the choice of $h$),
$\min_{s\in\operatorname{\mathbb{R}}}f_{s}\circ\left(\varphi_{t}^{h^{\varepsilon}}\right)^{-1}(x)>\ell$.
This implies $g_{t}(x)\geqslant\ell$ for all $x\in\partial W$ and
$t\in\operatorname{\mathbb{R}}$.

Denote by $\overline{f}_{t}:\partial W\to\operatorname{\mathbb{R}}$ the
contact Hamiltonian that generates the loop
$\left(\varphi_{t}^{f}\right)^{-1}$. Since $\left(\varphi^{f}\right)^{-1}$ is
a contractible loop of contactomorphisms, there exists a Hamiltonian
$\overline{F}_{t}:\widehat{W}\to\operatorname{\mathbb{R}}$ such that
$\phi^{\overline{F}}_{1}=\operatorname{id}$ and such that the slope of
$\overline{F}$ is equal to $\overline{f}$. Let
$\mathcal{N}(\overline{F}):HF_{\ast}(h^{\varepsilon})\to HF_{\ast}(h\\#f)$ be
the isomorphism defined on generators by
$\gamma\mapsto\left(\phi^{\overline{F}}\right)^{\ast}\gamma$, where
$\left(\phi^{\overline{F}}\right)^{\ast}\gamma(t):=\left(\phi^{\overline{F}}_{t}\right)^{-1}\circ\gamma(t)$.
The map $\mathcal{N}(\overline{F})$ commutes with the continuation maps [24,
Lemma 2.29] and induces an isomorphism (denoted the same) $SH_{\ast}(W)\to
SH_{\ast}(W).$ Consequently, $r(h^{\varepsilon})=r(h^{\varepsilon}\\#f)=r(g)$.
Since $g\geqslant\ell$, we have $r(g)\geqslant r(\ell).$ This further implies
$r(h+\varepsilon)=r(h^{\varepsilon})\geqslant r(\ell)$ and the proof is
finished.

∎

##  8 Applications to contact non-squeezing

We start with a proof of Theorem 1.7 from the introduction.

######  Theorem 1.7.

Let $W$ be a Liouville domain and let $\Omega_{a},\Omega_{b}\subset\partial W$
be open subsets. If the rank of the continuation map
$SH_{\ast}^{\Omega_{b}}(W)\to SH_{\ast}(W)$ is (strictly) greater than the
rank of the continuation map $SH_{\ast}^{\Omega_{a}}(W)\to SH_{\ast}(W),$ then
$\Omega_{b}$ cannot be contactly squeezed into $\Omega_{a}$.

###### Proof.

Denote by $r(\Omega)\in\mathbb{N}\cup\\{0,\infty\\}$ the rank of the
continuation map $SH_{\ast}^{\Omega}(W)\to SH_{\ast}(W).$ Assume the contrary,
i.e. that there exist open subsets $\Omega_{a},\Omega_{b}\subset\partial W$
with $r(\Omega_{a})<r(\Omega_{b})$ and a contact isotopy $\varphi_{t}:\partial
W\to\partial W$, $t\in[0,1]$ such that $\varphi_{0}=\operatorname{id}$ and
such that $\varphi_{1}(\Omega_{b})\subset\Omega_{a}$. By Section 1.4,
$r(\varphi_{1}(\Omega_{b}))=r(\Omega_{b})$. Since
$\varphi_{1}(\Omega_{b})\subset\Omega_{a}$, the continuation map
$SH_{\ast}^{\varphi_{1}(\Omega_{b})}(W)\to SH_{\ast}(W)$ factors through the
continuation map $SH_{\ast}^{\Omega_{a}}(W)\to SH_{\ast}(W).$ Hence,
$r(\Omega_{b})=r(\varphi_{1}(\Omega_{b}))\leqslant r(\Omega_{a}).$ This
contradicts the assumption $r(\Omega_{a})<r(\Omega_{b})$. ∎

######  Theorem 1.4.

Let $n\geqslant 2$ be a natural number and let $W$ be a $2n$-dimensional
Liouville domain such that $SH_{\ast}(W)$ is infinite dimensional and such
that $\partial W$ is a homotopy sphere. Then, there exist two embedded closed
balls $B_{1},B_{2}\subset\partial W$ of dimension $2n-1$ such that $B_{1}$
cannot be contactly squeezed into $B_{2}$.

###### Proof.

Denote $\Sigma:=\partial W$, and denote by
$r(\Omega)\in\mathbb{N}\cup\\{0,\infty\\}$ the rank of the continuation map
$SH_{\ast}^{\Omega}(W)\to SH_{\ast}(W)$ for an open subset
$\Omega\subset\Sigma$.

Step 1 (A subset with a small rank). Since $SH_{\ast}^{\varnothing}(W)$ is
isomorphic to $H_{\ast}(W,\partial W;\mathbb{Z}_{2})$, Theorem 1.6 implies
that there exists a non-empty open subset $\Omega\subset\Sigma$ such that
$r(\Omega)$ is finite.

Step 2 (A subset with a large rank). This step proves that for every
$c\in\operatorname{\mathbb{R}}$, there exists an open non-dense subset
$U\subset\Sigma$ such that $r(U)>c$. Assume the contrary, i.e. that there
exists $c\in\operatorname{\mathbb{R}}$ such that $r(U)\leqslant c$ for every
open non-dense subset $U\subset\Sigma$. By Lemma 1.6, there exists a
sufficiently small contact Darboux chart on $\Sigma$ that is immaterial. Let
$\Gamma$ be an embedded circle in that chart that is transverse to the contact
distribution. Then, $\Gamma$ is immaterial as well. Consequently (by Theorem
1.6), the continuation map $\Phi:SH_{\ast}^{\Sigma\setminus\Gamma}(W)\to
SH_{\ast}(W)$ has infinite rank, i.e. $r(\Sigma\setminus\Gamma)=\infty$.
Hence, there exist $a_{1},\ldots,a_{k}\in
SH_{\ast}^{\Sigma\setminus\Gamma}(W)$, with $k>c$, such that
$\Phi(a_{1}),\ldots,\Phi(a_{k})$ are linearly independent. Let
$U_{1},U_{2},\ldots$ be an increasing family of open non-dense subsets of
$\Sigma$ such that $\bigcup_{j=1}^{\infty}U_{j}=\Sigma\setminus\Gamma$. Since,
by Theorem 4.1, continuation maps furnish an isomorphism

$\underset{j}{\lim_{\longrightarrow}}SH_{\ast}^{U_{j}}(W)\to
SH_{\ast}^{\Sigma\setminus\Gamma}(W),$

there exist $m\in\mathbb{N}$ and $b_{1},\ldots,b_{k}\in SH^{U_{m}}_{\ast}(W)$
such that $b_{1},\ldots,b_{k}$ are mapped to $a_{1},\ldots,a_{k}$ via the
continuation map $SH_{\ast}^{U_{m}}(W)\to
SH_{\ast}^{\Sigma\setminus\Gamma}(W).$ The images of $b_{1},\ldots,b_{k}$
under the continuation map $SH_{\ast}^{U_{m}}(W)\to SH_{\ast}(W)$ are equal to
$\Phi(a_{1}),\ldots,\Phi(a_{k})$, and therefore, are linearly independent.
Hence, $r(U_{m})\geqslant k>c.$ This contradicts the assumption and finishes
Step 2.

Step 3 (The final details). This step finishes the proof. Let
$\Omega,U\subset\Sigma$ be open non-empty subsets such that $U$ is non-dense,
and such that $r(\Omega)<r(U)$. Step 1 and Step 2 proved the existence of the
sets $\Omega$ and $U$. Let $a\in\Omega$ and
$b\in\Sigma\setminus\\{a\\}\setminus\overline{U}$ be two points. Since
$\Sigma$ is a homotopy sphere, there exists a Morse function
$f:\Sigma\to\operatorname{\mathbb{R}}$ that attains its minimum at $a$ and its
maximum at $b$ and that has no other critical points. The existence of such a
function $f$ is guaranteed by the results of Smale [21, 23, 22] and Cerf [3]
(see also [18, Proposition 2.2]). The Morse theory implies that
$\Sigma_{t}:=f^{-1}\big{(}(-\infty,t]\big{)}$ is the standard
$(2n-1)$-dimensional closed ball smoothly embedded into $\Sigma$ for all
$t\in(f(a),f(b))$ (see, for instance, [1]). For $s\in(f(a),f(b))$ sufficiently
close to $f(a)$, $\Sigma_{s}\subset\Omega$. Similarly, for
$\ell\in(f(a),f(b))$ sufficiently close to $f(b)$,
$\operatorname{int}\Sigma_{\ell}\supset U$. Since $r(\Omega)<r(U)$, by Theorem
1.7, $U$ cannot be contactly squeezed into $\Omega$. Hence, $\Sigma_{\ell}$
cannot be contactly squeezed into $\Sigma_{s}$ and the proof is finished (one
can take $B_{2}:=\Sigma_{s}$ and $B_{1}:=\Sigma_{\ell}$). ∎

Now, we prove the contact non-squeezing for the Ustilovsky spheres.

######  Theorem 1.2.

Let $\Sigma$ be an Ustilovsky sphere. Then, there exist two embedded closed
balls $B_{1},B_{2}\subset\Sigma$ of dimension equal to $\dim\Sigma$ such that
$B_{1}$ cannot be contactly squeezed into $B_{2}$.

###### Proof.

In the view of Theorem 1.4, it is enough to show that the symplectic homology
of the Brieskorn variety

$W:=\left\\{z=(z_{0},\ldots,z_{2m+1})\in\mathbb{C}^{2m+2}\>|\>z_{0}^{p}+z_{1}^{2}+\cdots+z_{2m+1}^{2}=\varepsilon\>\&\>\left\lvert
z\right\rvert\leqslant 1\right\\}$

is infinite demensional. Here, $m,p\in\mathbb{N}$ are natural numbers with
$p\equiv\pm 1\pmod{8}$ and $\varepsilon\in\operatorname{\mathbb{R}}^{+}$ is
sufficiently small. The Brieskorn variety $W$ is a Liouville domain whose
boundary is contactomorphic to an Ustilovsky sphere, and every Ustilovsky
sphere is contactomorphic to $\partial W$ for some choice of $m$ and $p$.

Let $\Sigma_{k}$ be the sequence of manifolds such that $\Sigma_{k}=\partial
W$ if $p\mid k$ (we use this notation for “$k$ is divisible by $p$”) and such
that

$\Sigma_{k}:=\left\\{z=(z_{1},\ldots,z_{2m+1})\in\mathbb{C}^{2m+1}\>|\>z_{1}^{2}+\cdots
z_{2m+1}^{2}=0\>\&\>\left\lvert z\right\rvert=1\right\\}$

if $p\nmid k$. Denote

$s_{k}:=\frac{k}{p}\cdot\left((4m-2)\cdot p+4\right)-2m$

if $p\mid k$, and

$s_{k}:=(4m-2)\cdot k+2\cdot\left\lfloor\frac{2k}{p}\right\rfloor-2m+2$

otherwise. Theorem 8.11 from [15] implies that there exists a spectral
sequence $E_{k,\ell}^{r}$ that converges to $SH_{\ast}(W)$ such that its first
page is given by

$E^{1}_{k,\ell}=\left\\{\begin{matrix}H_{k+\ell-
s_{k}}(\Sigma_{k};\mathbb{Z}_{2})&\text{if }k>0,\\\ H_{\ell+2m+1}(W,\partial
W;\mathbb{Z}_{2})&\text{if }k=0,\\\ 0&\text{if }k<0.\end{matrix}\right.$

The terms of $E^{1}$ can be explicitly computed. If $p\nmid k$, then
$\Sigma_{k}$ is diffeomorphic to the unit cotangent bundle
$S^{\ast}\mathbb{S}^{2m}$ of the sphere. Otherwise, $\Sigma_{k}$ is
diffeomorphic to $\mathbb{S}^{4m+1}$, because $p\equiv\pm 1\pmod{8}$ [2].
Therefore,

$H_{j}(\Sigma_{k};\mathbb{Z}_{2})\cong\left\\{\begin{matrix}\mathbb{Z}_{2}&\text{if
}j\in\\{0,2m-1,2m,4m-1\\},\\\ 0&\text{otherwise}\end{matrix}\right.$

if $p\nmid k$, and

$H_{j}(\Sigma_{k};\mathbb{Z}_{2})\cong\left\\{\begin{matrix}\mathbb{Z}_{2}&\text{if
}j\in\\{0,4m+1\\},\\\ 0&\text{otherwise}\end{matrix}\right.$

if $p\mid k$. The Brieskorn variety $W$ is homotopy equivalent to the bouquet
of $p-1$ spheres of dimension $2m+1$ [17, Theorem 6.5]. Therefore,

$H_{j}(W,\partial
W;\mathbb{Z}_{2})\cong\left\\{\begin{matrix}\mathbb{Z}_{2}&\text{if
}j=4m+2,\\\ \mathbb{Z}_{2}^{p-1}&\text{if }j=2m+1,\\\
0&\text{otherwise.}\end{matrix}\right.$

Figure 1 on page 1 shows the page $E^{1}_{k,\ell}$ for $p=7$ and $m=1.$ For
$k\in\mathbb{N}\cup\\{0\\}$, denote by $Q(k)$ the unique number in
$\mathbb{Z}$ such that $E^{1}_{k,Q(k)}\not=0$ and such that $E^{1}_{k,\ell}=0$
for all $\ell>Q(k)$. Similarly, for $k\in\mathbb{N}\cup\\{0\\}$, denote by
$q(k)$ the unique number in $\mathbb{Z}$ such that $E^{1}_{k,q(k)}\not=0$ and
such that $E^{1}_{k,\ell}=0$ for all $\ell<Q(k)$. Explicitly,

$\displaystyle Q(k)=\left\\{\begin{matrix}2m+1&\text{if }k=0,\\\
4m-1+s_{k}-k&\text{if }p\nmid k,\\\ 4m+1+s_{k}-k&\text{if }p\mid
k,\end{matrix}\right.$ $\displaystyle q(k)=\left\\{\begin{matrix}0&\text{if
}k=0,\\\ s_{k}-k&\text{for }k\in\mathbb{N}.\end{matrix}\right.$

We will show that the element in $E^{1}_{ap-1,Q(ap-1)}$ “survives” in
$SH_{\ast}(W)$ for $a\in\mathbb{N}$. (In Figure 1, the fields
$(ap-1,Q(ap-1))$, $a=1,2$ are emphasized by thicker edges.) Both sequences
$Q(k)$ and $q(k)$ are strictly increasing in $k$. Since $Q(k)$ is strictly
increasing, the element in $E^{1}_{ap-1,Q(ap-1)}$ cannot be “killed” by an
element from $E^{1}_{k,\ell}$ if $k\leqslant ap-1$. Since

$q(ap+1)-Q(ap-1)=4m-3\geqslant 1,$

the element in $E^{1}_{ap-1,Q(ap-1)}$ cannot be “killed” by an element from
$E^{1}_{k,\ell}$ if $k\geqslant ap+1$. Finally, since $Q(ap-1)-q(ap)=2$, the
non-zero groups $E_{ap,\ell}^{1}$ are the ones for $\ell=Q(ap-1)-2$ and
$\ell=Q(ap-1)+4m-1$. In particular, $E^{1}_{ap,Q(ap-1)}=0$. Therefore,
$E^{\infty}_{ap-1,Q(ap-1)}\not=0$ for all $a\in\mathbb{N}$. This implies $\dim
SH_{\ast}(W)=\infty$ and the proof is finished. ∎

$0$$7$$14$$0$$10$$20$6111111111111111111111111111111111111111111111111111111111
Figure 1: The first page of the spectral sequence from the proof of Theorem
1.2 for $p=7$ and $m=1$. The number in the field $(k,\ell)$ represents $\dim
E^{1}_{k,\ell}$. Empty fields are assumed to contain zeros.

Finally, we prove Corollary 1.3.

######  Corollary 1.2.

Let $m\in\mathbb{N}$. Then, there exist a contact structure $\xi$ on
$\operatorname{\mathbb{R}}^{4m+1}$ and an embedded closed ball
$B\subset\operatorname{\mathbb{R}}^{4m+1}$ of dimension $4m+1$ such that $B$
cannot be contactly squeezed into every non-empty open subset by a compactly
supported contact isotopy of
$\left(\operatorname{\mathbb{R}}^{4m+1},\xi\right)$.

###### Proof.

Let $S$ be a $(4m+1)$-dimensional Ustilovsky sphere and let
$B_{1},B_{2}\subset S$ be two embedded closed balls such that $B_{1}\cup
B_{2}\not=S$ and such that $B_{2}$ cannot be contactly squeezed into $B_{1}.$
Let $p\in S\setminus(B_{1}\cup B_{2})$. The Ustilovsky sphere $S$ is
diffeomerphic to the standard sphere. Hence, there exists a diffeomorphism
$\psi:\operatorname{\mathbb{R}}^{4m+1}\to S\setminus\\{p\\}.$ Let $\xi$ be the
pull back of the contact structure on $S$ via $\psi$. Now, we prove that $\xi$
and $B:=\psi^{-1}(B_{2})$ satisfy the conditions of the corollary. Assume the
contrary. Then, there exists a compactly supported contact isotopy
$\varphi_{t}:\operatorname{\mathbb{R}}^{4m+1}\to\operatorname{\mathbb{R}}^{4m+1}$
such that $\varphi_{0}=\operatorname{id}$ and such that
$\varphi_{1}(B)\subset\operatorname{int}\left(\psi^{-1}(B_{1})\right)$. Since
$\varphi$ is a compactly supported isotopy,
$\psi\circ\varphi_{t}\circ\psi^{-1}$ extends to a contact isotopy on $S$. This
contact isotopy squeezes $B_{2}$ into $B_{1}$. This is a contradiction that
finishes the proof. ∎

##  9  Well-definedness of the selective symplectic homology

This section proves some details that are required for a proper definition of
the selective symplectic homology. More precisely, it shows that the set
$\mathcal{H}_{\Omega}(\Sigma)$ (introduced on page 3.1) is non-empty for a
closed contact manifold $\Sigma$ with a contact form and for an open subset
$\Omega\subset\Sigma$. It also shows that the sets $\Pi(h)$, $\mathcal{U}(h)$
(and, consequently, $\mathcal{O}(h)$) are non-empty for
$h\in\mathcal{H}_{\Omega}(\Sigma)$. We first introduce some notation.

Let $\Sigma$ be a contact manifold with a contact form, and let
$\Omega\subset\Sigma$ be an open subset. Denote by
$\tilde{\mathcal{H}}_{\Omega}(\Sigma)$ the set of smooth ($C^{\infty}$)
autonomous contact Hamiltonians $h:\Sigma\to[0,+\infty)$ such that

  1. 1.

$h(p)\not=0\quad\Longleftrightarrow\quad p\in\Omega,$

  2. 2.

$dY^{h}(p)=0,$ for all $p\not\in\Omega.$

Hence, the set $\mathcal{H}_{\Omega}(\Sigma)$ is equal to the subset of
$\tilde{\mathcal{H}}_{\Omega}(\Sigma)$ consisting of those contact
Hamiltonians that have no non-constant 1-periodic orbits.

Non-emptiness of the set $\mathcal{H}_{\Omega}(\Sigma)$ is a direct
consequence of the next proposition. The proposition implies that, for a
generic $h\in\tilde{\mathcal{H}}_{\Omega}(\Sigma)$ and for almost every
$c\in\operatorname{\mathbb{R}}$, the contact Hamiltonian $c\cdot h$ has no
non-constant 1-periodic orbits.

######  Proposition 1.2.

Let $\Sigma$ be a closed contact manifold with a contact form and let
$\Omega\subset\Sigma$ be an open subset . Then, the periods of non-constant
orbits of a generic (in the $C^{\infty}$ sense) contact Hamiltonian
$h\in\tilde{\mathcal{H}}_{\Omega}(\Sigma)$ form a countable set.

###### Proof.

Step 1. (Transversality of the graph.) This step proves that the set of
periods of non-constant closed orbits of a contact Hamiltonian
$h\in\tilde{\mathcal{H}}_{\Omega}(\Sigma)$ is countable if the map

$\Gamma_{h}:\Sigma\times\operatorname{\mathbb{R}}\to\Sigma\times\operatorname{\mathbb{R}}\times\Sigma\quad:\quad(x,t)\mapsto\left(x,t,\varphi_{t}^{h}(x)\right)$

is transverse to the submanifold

$\Delta:=\left\\{(x,t,x)\>|\>x\in\Sigma,t\in\operatorname{\mathbb{R}}\right\\}\subset\Sigma\times\operatorname{\mathbb{R}}\times\Sigma$

along $\Omega\times(\operatorname{\mathbb{R}}\setminus\\{0\\})$. If
$\Gamma_{h}$ is transverse to $\Delta$ along
$\Omega\times(\operatorname{\mathbb{R}}\setminus\\{0\\})$, then

$A:=\Gamma_{h}^{-1}(\Delta)\cap\big{(}\Omega\times(\operatorname{\mathbb{R}}\setminus\\{0\\})\big{)}$

is a submanifold in $\Omega\times(\operatorname{\mathbb{R}}\setminus\\{0\\})$
of dimension 1. If $(x,t)\in A$, then $\left(\varphi^{h}_{s}(x),t\right)\in A$
for all $s\in\operatorname{\mathbb{R}}$. Therefore, the 1-dimensional
submanifold $A$ consists of embedded circles of the form
$\left\\{\left(\varphi_{s}^{h}(x),t\right)\>|\>s\in\operatorname{\mathbb{R}}\right\\}.$
(To conclude this, we also used $Y^{h}(x)\not=0$ for $x\in\Omega$.) Since $A$
is a submanifold, these circles are isolated from each other, and therefore,
there are countable many of them. To each non-zero period of a non-constant
orbit of $h$ corresponds at least one of these embedded circles. Hence, the
set of periods of closed transverse orbits of $h$ is countable.

Step 2. (Construction of a perturbation map.) Let
$h\in\tilde{\mathcal{H}}_{\Omega}(\Sigma)$. Let $\Omega^{\prime}\subset\Omega$
and $J\subset\operatorname{\mathbb{R}}\setminus\\{0\\}$ be open subsets such
that $\overline{\Omega^{\prime}}\subset\Omega$ and such that $\overline{J}$ is
a compact subset of $\operatorname{\mathbb{R}}\setminus\\{0\\}$. This step
shows that there exists arbitrarily $C^{\infty}$ small
$g\in\tilde{\mathcal{H}}_{\Omega}(\Sigma)$ such that $\Gamma_{h+g}$ is
transverse to $\Delta$ along $\Omega^{\prime}\times J$. First, we construct a
smooth map
$F:\Sigma\times\operatorname{\mathbb{R}}\times\operatorname{\mathbb{R}}^{m}\to\Sigma\times\operatorname{\mathbb{R}}\times\Sigma$
such that

$F(x,t,a_{1},\ldots,a_{m})=\left(x,t,\varphi_{t}^{h+f}(x)\right)$

for some smooth $f:\Sigma\to[0,+\infty)$ that is supported in $\Omega$ (and
that depends on $a_{1},\ldots a_{m}$) and such that $DF(p)$ is surjective for
$p\in\overline{\Omega^{\prime}}\times(\operatorname{\mathbb{R}}\setminus\\{0\\})\times\\{0\\}$.
To this map, the parametric transversality theorem [14, Theorem 2.7] will be
applied afterwards.

Since $\overline{\Omega^{\prime}}$ is compact and contained in $\Omega$, Lemma
1.2 below implies that there exists a finite number of contact Hamiltonians
$h_{1},\ldots,h_{m}:\Sigma\to[0,+\infty)$, each of which is supported in
$\Omega$, such that the following holds. For every
$x\in\overline{\Omega^{\prime}}$, there exist
$g_{1},\ldots,g_{2n+1}\in\\{h_{1},\ldots,h_{m}\\}$ such that

  1. 1.

$Y^{g_{1}}(x),\ldots,Y^{g_{2n+1}}(x)$ form a basis on $T_{x}\Sigma$,

  2. 2.

$\left[Y^{h},Y^{g_{j}}\right](y)=0$ for $j\in\\{1,\ldots,2n+1\\}$ and for $y$
in a neighbourhood of $x$.

Denote by
$F:\Sigma\times\operatorname{\mathbb{R}}\times\operatorname{\mathbb{R}}^{m}\to\Sigma\times\operatorname{\mathbb{R}}\times\Sigma$
the map defined by

$F(x,t,a_{1},\ldots,a_{m}):=\left(x,t,\varphi_{t}^{h+a_{1}h_{1}+\cdots+a_{m}h_{m}}(x)\right).$

We will prove that $DF(p)$ is surjective for
$p\in\Omega^{\prime}\times\operatorname{\mathbb{R}}\times\\{0\\}$. Indeed, if
$h$ and $f$ commute (i.e. if $\left[Y^{h},Y^{f}\right]=0$), then

$\varphi_{t}^{h+sf}=\varphi_{t}^{sf}\circ\varphi_{t}^{h}=\varphi^{f}_{st}\circ\varphi_{t}^{h}.$

Hence,

$\frac{d}{ds}\left(\varphi_{t}^{h+sf}(x)\right)=\frac{d}{ds}\left(\varphi_{st}^{f}\circ\varphi_{t}^{h}(x)\right)=t\cdot
Y^{f}\left(\varphi_{st}^{f}\circ\varphi_{t}^{h}(x)\right).$

For a fixed $x\in\overline{\Omega^{\prime}}$, pick
$g_{1},\ldots,g_{2n+1}\in\\{h_{1},\ldots,h_{m}\\}$ such that $g_{j}$ commutes
with $h$ in a neighbourhood of $x$ for all $j\in\\{1,\ldots,2n+1\\}$ and such
that $Y^{g_{1}}(x),\ldots,Y^{g_{2n+1}}(x)$ are linearly independent. Let
$b_{1},\ldots,b_{2n+1}$ be the elements of $\\{a_{1},\ldots,a_{m}\\}$ that
correspond to $g_{1},\ldots,g_{2n+1}$, respectively (e.g. if $g_{1}=h_{j}$,
then $b_{1}=a_{j}$). Then,

$\partial_{x_{1}}F(x,t,0),\ldots,\partial_{x_{2n+1}}F(x,t,0),\partial_{t}F(x,t,0),\partial_{b_{1}}F(x,t,0),\ldots,\partial_{b_{2n+1}}F(x,t,0)$

span $T_{F(x,t,0)}(\Sigma\times\operatorname{\mathbb{R}}\times\Sigma),$ for
$t\not=0.$ In particular, $DF(p)$ is surjective for
$p\in\overline{\Omega^{\prime}}\times(\operatorname{\mathbb{R}}\setminus\\{0\\})\times\\{0\\}.$
Since the surjectivity of $DF(p)$ is an open property and since
$\overline{\Omega^{\prime}}\times\overline{J}$ is a compact set, there exists
$\varepsilon>0$ such that $DF(p)$ is surjective for
$p\in\overline{\Omega^{\prime}}\times\overline{J}\times(-\varepsilon,\varepsilon)^{m}$.
Hence, the parametric transversality theorem [14, Theorem 2.7 on page 79]
implies that the map

$\Omega^{\prime}\times
J\to\Sigma\times\operatorname{\mathbb{R}}\times\Sigma\quad:\quad(x,t)\mapsto
F\left(x,t,a_{1},\ldots,a_{m}\right)$

is transverse to $\Delta$ for $(a_{1},\ldots,a_{m})$ that belongs to a dense
subset of $(-\varepsilon,\varepsilon)^{m}$. Since $\Omega^{\prime}\times J$ is
a codimension-0 submanifold of $\Sigma\times\operatorname{\mathbb{R}}$, the
map

$\Omega^{\prime}\times
J\to\Sigma\times\operatorname{\mathbb{R}}\times\Sigma\quad:\quad(x,t)\mapsto
F\left(x,t,a_{1},\ldots,a_{m}\right)$

is transverse to $\Delta$ if, and only if, the map

$\Sigma\times\operatorname{\mathbb{R}}\to\Sigma\times\operatorname{\mathbb{R}}\times\Sigma\quad:\quad(x,t)\mapsto\left(x,t,a_{1},\ldots,a_{m}\right)$

is transverse to $\Delta$ along the subset $\Omega^{\prime}\times
J\subset\Sigma\times\operatorname{\mathbb{R}}$. Therefore, there exists an
arbitrarily small $g\in\tilde{\mathcal{H}}_{\Omega}(\Sigma)$ (in the
$C^{\infty}$ sense) such that $\Gamma_{h+g}$ is transverse to $\Delta$ along
$\Omega^{\prime}\times J$.

Step 3. (The final details.) This step finishes the proof. Let
$\Omega_{1},\Omega_{2},\ldots\subset\Omega$ and
$J_{1},J_{2},\ldots\subset\operatorname{\mathbb{R}}\setminus\\{0\\}$ be
sequences of open subsets such that

  1. 1.

$\overline{\Omega_{k}}\subset\Omega_{k+1}$ for all $k\in\mathbb{N}$,

  2. 2.

$\bigcup_{k=1}^{\infty}\Omega_{k}=\Omega$,

  3. 3.

$\overline{J_{k}}$ is a compact subset of $J_{k+1}$ for all $k\in\mathbb{N}$,

  4. 4.

$\bigcup_{k=1}^{\infty}J_{k}=\operatorname{\mathbb{R}}\setminus\\{0\\}$.

Denote by $\mathcal{F}_{j}\subset\tilde{\mathcal{H}}_{\Omega}(\Sigma)$ the set
of contact Hamiltonians $f$ such that $\Gamma_{f}$ is transverse to $\Delta$
along $\overline{\Omega}_{j}\times\overline{J}_{j}$. By Step 2, the set
$\mathcal{F}_{j}$ is dense in $\tilde{\mathcal{H}}_{\Omega}(\Sigma)$. Theorem
2.1 on page 74 in [14] asserts that the set $\mathcal{F}_{j}$ is open as well.
Hence, the set

$\mathcal{F}:=\bigcap_{j=1}^{\infty}\mathcal{F}_{j}$

is residual (by definition, as a countable intersection of open and dense
subsets). On the other hand, $\Gamma_{f}$ is transverse to $\Delta$ along
$\Omega\times(\operatorname{\mathbb{R}}\setminus\\{0\\})$ for all
$f\in\mathcal{F}$ because

$\bigcup_{j=1}^{\infty}\overline{\Omega}_{j}\times\overline{J}_{j}=\Omega\times(\operatorname{\mathbb{R}}\setminus\\{0\\}).$

This finishes the proof. ∎

The next lemma is used in Step 2 of the proof of Proposition 1.2 to show that
$DF(p)$ is surjective for
$p\in\Omega^{\prime}\times\operatorname{\mathbb{R}}\times\\{0\\}$.

######  Lemma 1.2.

Let $\Sigma$ be a $(2n+1)$-dimensional contact manifold with a contact form
and let $h:\Sigma\to\operatorname{\mathbb{R}}$ be a contact Hamiltonian. Then,
for every $x\in\Sigma\setminus h^{-1}(0)$, there exist open subsets
$U,V\subset\Sigma$, and contact Hamiltonians
$h_{1},\ldots,h_{2n+1}:\Sigma\to[0,+\infty)$ such that the following holds.

  1. 1.

$x\in U.$

  2. 2.

$\overline{U}\subset V.$

  3. 3.

The vectors $Y^{h_{1}}(y),\ldots,Y^{h_{2n+1}}(y)$ form a basis on
$T_{y}\Sigma$ for all $y\in U.$

  4. 4.

$\left[Y^{h},Y^{h_{j}}\right](y)=0$ for all $j\in\\{1,\ldots,2n+1\\}$ and all
$y\in U$.

  5. 5.

$h_{j}(y)=0$ for all $j\in\\{1,\ldots,2n+1\\}$ and all $y\in\Sigma\setminus
V$.

###### Proof.

Since $h(x)\not=0$, the 1-form $\frac{\alpha}{h}$ is a well defined contact
form in a neighbourhood of $x$. By the Darboux theorem concerning 1-forms,
there exist local coordinates $x_{1},\ldots,x_{n},y_{1},\ldots,y_{n},z$ around
the point $p$ in which the 1-form $\frac{\alpha}{h}$ is equal to
$dz-\sum_{j=1}^{n}y_{j}dx_{j}$. In these coordinates, the vector field $Y^{h}$
is equal to $-\partial_{z}$ (because $-Y^{h}$ is the Reeb vector field with
respect to the contact form $\frac{\alpha}{h}$). Let
$a_{i},b_{i}:\operatorname{\mathbb{R}}^{2n+1}\to\operatorname{\mathbb{R}}$,
$i\in\\{1,\ldots,n\\}$, be the contact Hamiltonians given by

$a_{i}(x,y,z):=e^{x_{i}},\quad b_{i}(x,y,z):=e^{y_{i}}.$

Their vector fields are given by

$\displaystyle
Y^{a_{i}}(x,y,z)=-e^{x_{i}}\cdot\left(\partial_{y_{i}}+\partial_{z}\right),$
$\displaystyle
Y^{b_{i}}(x,y,z)=e^{y_{i}}\cdot(\partial_{x_{i}}+(y_{i}-1)\cdot\partial_{z})$

for $i\in\\{1,\ldots,n\\}$. The vector fields
$Y^{a_{1}},\ldots,Y^{a_{n}},Y^{b_{1}},\ldots,Y^{b_{n}},\partial_{z}$ are
linearly independent at every point in $\operatorname{\mathbb{R}}^{2n+1}$,
and, therefore, form a basis. In addition,

$\left[\partial_{z},Y^{a_{i}}\right]=\left[\partial_{z},Y^{b_{i}}\right]=\left[\partial_{z},\partial_{z}\right]=0,$

for all $i\in\\{1,\ldots,n\\}$. By cutting off the contact Hamiltonians
$a_{1},\ldots,a_{n},b_{1},\ldots,b_{n}$, and 1 appropriately, one obtains the
contact Hamiltonians $h_{1},\ldots,h_{2n+1}$ that satisfy the conditions of
the lemma. ∎

The next proposition implies that $\Pi(h)$ is non-empty for
$h\in\mathcal{H}_{\Omega}(\Sigma)$. It is also used in the proof of Lemma 3.4
which asserts that the inverse limit in the definition of the selective
symplectic homology stabilises and which implies the non-emptiness of the set
$\mathcal{U}(h)$.

######  Proposition 1.2.

Let $\Sigma$ be a closed contact manifold with a contact form. Let
$h:\Sigma\to\operatorname{\mathbb{R}}$ be a contact Hamiltonian such that $h$
has no non-constant 1-periodic orbits, and such that $dY^{h}(p)=0$ for all
$p\in\Sigma$ at which the vector field $Y^{h}$ vanishes. Then, there exists a
$C^{2}$ neighbourhood of $h$ in $C^{\infty}(\Sigma)$ such that the flow of $g$
has no non-constant 1-periodic orbits for all $g$ in that neighbourhood.

###### Proof.

Assume the contrary. Then, there exist a sequence of contact Hamiltonians
$h_{k}$ and a sequence $x_{k}\in\Sigma$ such that $h_{k}\to h$ in $C^{2}$
topology, such that $x_{k}\to x_{0},$ and such that
$t\mapsto\varphi_{t}^{h_{k}}(x_{k})$ is a non-constant 1-periodic orbit of
$h_{k}.$ This implies that $t\mapsto\varphi_{t}^{h}(x_{0})$ is a 1-periodic
orbit of $h,$ and therefore, has to be constant. By assumptions,
$dY^{h}(x_{0})=0.$ The map $C^{\infty}(\Sigma)\to\mathfrak{X}(\Sigma)$ that
assigns the contact vector field to a contact Hamiltonian is continuous with
respect to $C^{2}$ topology on $C^{\infty}(\Sigma)$ and $C^{1}$ topology on
$\mathfrak{X}(\Sigma)$. Consequently (since $h_{k}\to h$ in $C^{2}$ topology),
$Y^{h_{k}}\to Y^{h}$ in $C^{1}$ topology. Therefore, for each $L>0,$ there
exists a neighbourhood $U\subset\Sigma$ of $x_{0}$ and $N\in\mathbb{N}$ such
that $\left.Y^{h_{k}}\right|_{U}$ is Lipschitz with Lipschitz constant $L$ for
all $k\geqslant N.$ For $k$ big enough, the loop
$t\mapsto\varphi_{t}^{h_{k}}(x_{k})$ is contained in the neighbourhood $U.$
This contradicts [30] because for $L$ small enough there are no non-constant
1-periodic orbits of $h_{k}$ in $U.$ ∎

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