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\DeclareMathOperator{\sgn}{sgn} \DeclareMathOperator{\Ext}{Ext} \DeclareMathOperator{\TropB}{TropB} \DeclareMathOperator{\weight}{wt} \DeclareMathOperator{\Span}{span} \DeclareMathOperator{\Coh}{Coh} \DeclareMathOperator{\Pic}{Pic} \DeclareMathOperator{\Fuk}{Fuk} \DeclareMathOperator{\str}{star} \DeclareMathOperator{\Ob}{Ob} \DeclareMathOperator{\Coh}{Coh} \DeclareMathOperator{\CritVal}{CritV} \DeclareMathOperator{\Sing}{Sing} \DeclareMathOperator{\FS}{FS} \DeclareMathOperator{\Vect}{Vect} \DeclareMathOperator{\grad}{grad} \DeclareMathOperator{\Supp}{Supp} \DeclareMathOperator{\Bl}{Bl} \DeclareMathOperator{\Spec}{Spec} \DeclareMathOperator{\Tw}{Tw} \DeclareMathOperator{\Int}{Int} \DeclareMathOperator{\Arg}{\mathbf{M}}\begin{filecontents}{references.bib} @article{ballard2012hochschild, title={Hochschild dimensions of tilting objects}, author={Ballard, Matthew and Favero, David}, journal={International Mathematics Research Notices}, volume={2012}, number={11}, pages={2607--2645}, 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There is a map \[H^\bullet(X)\to \SH(X).\] \end{proposition} \begin{proof} \label{prf:homologyIncludesIntoSH} Let $m_0$ be the minimal period of a Reeb orbit in $(\partial X, \alpha)$. Pick a slope $0< m< m_0$, and consider a linear Hamiltonian $H^m$ which is $C^2$ small on $X$. Then the only orbits for $H^m$ will be the constant orbits, and there is a quasi-isomorphism of chain complexes between $\CF(\hat X, H^m_t)$ and the Morse complex $CM^\bullet(\hat X, H^m_t)$ sending each constant orbit to its associated critical point of $H^m_t$. Since $H^m$ has gradient which points outward along the boundary of $X$, this is a valid Morse function for computing the Morse cohomology of $X$. \end{proof} \begin{application} We look at an application of \cref{prp:homologyIncludesIntoSH}. Let $X$ be a Liouville domain, and suppose that $(\partial X, \alpha)$ has no Reeb orbits. Then the map from \cref{prp:homologyIncludesIntoSH} is an isomorphism \[H^\bullet(X)\to \SH(X).\] Therefore, we can compute $\SH(X)$ to show that $(\partial X)$ has a Reeb orbit. \begin{example} \label{exm:SHofBall} A key example is the standard contact structure on the sphere. One can compute that the standard symplectic ball $B^{2n}:=\{(z_1, \ldots, z_n)\st \sum_{i=1}^n |z_i|^2=1 \}\subset \CC^n$ is a Liouville domain, and that $\SH(B^{2n})=0$. We can conclude that $(S^{2n-1}, \alpha)$ has a Reeb orbit.\end{example} A generalization of this result (due to \cite{oancea2003suite}) states that whenever $X$ is a subcritical \underline{\href{https://jeffhicks.net/snippets/index.php?tag=exm:steinDomain}{ Stein domain}} (so the Morse indexes of the critical points of $\phi: X\to \RR$ are all less than $n$) then $\SH(X)=0$. \end{application}\begin{definition} \label{def:liouvilleSubomain} Suppose that $(X, \lambda)$ is a Liouville domain. A \emph{Liouville subdomain} is a compact submanifold with boundary $X_0\subset X\setminus \partial X$ such that the Liouville vector $Z$ points outwards along $\partial X_0$. \label{def:liouvillesubDomain} \end{definition} \begin{theorem}\cite{viterbo1999functors} \label{thm:viterboRestriction} \label{thm:viterboRestriction} Let $X_0\subset X$ be a Liouville subdomain. Then there is a restriction map $\SH(X)\to \SH(X_0)$, which is a unital ring homomorphism. Furthermore, we have a commutative diagram \[ \begin{tikzpicture} \node (v3) at (-2,1) {$SH^\bullet(X)$}; \node (v4) at (1,1) {$SH^\bullet(X_0)$}; \node (v1) at (-2,-1) {$H^\bullet(X)$}; \node (v2) at (1,-1) {$H^\bullet(X_0)$}; \draw (v1) edge[->] (v2); \draw (v1) edge[->] (v3); \draw (v3) edge[->] (v4); \draw (v2) edge[->] (v4); \end{tikzpicture}\] where the horizontal maps are given by \cref{prp:homologyIncludesIntoSH}. \end{theorem} In the setting of subcritical Stein domains ( \cref{app:weinsteinConjectureInRn} ), the sublevel sets $X|_{\phi< t}$ form a nested sequence of Liouville subdomains. One way to prove the vanishing of $\SH(X)$ is to show that the map $\SH(X|_{\phi< t_{i+1}})\to \SH(X|_{\phi< t_{i}})$ is a an isomorphism for all $t_{i} < t_{i+1}$. When the only critical points of $\phi$ contained in $\SH(X|_{\phi < t_0})$ are minima, then $X|_{t_0}$ is a ball, which has vanishing symplectic cohomology. \printbibliography \end{document}