Symplectic homology of fiberwise convex sets and homology of loop spaces

Abstract

For any nonempty, compact and fiberwise convex set $K$ in $T^{\ast }{\mathbb{R}}^n$, we prove an isomorphism between symplectic homology of $K$ and a certain relative homology of loop spaces of ${\mathbb{R}}^n$. We also prove a formula which computes symplectic homology capacity (which is a symplectic capacity defined from symplectic homology) of $K$ using homology of loop spaces. As applications, we prove (i) symplectic homology capacity of any convex body is equal to its Ekeland-Hofer-Zehnder capacity, (ii) a certain subadditivity property of the Hofer-Zehnder capacity, which is a generalization of a result previously proved by Haim-Kislev.

arXiv comment40 pages. Revised following comments by the referee. Proposition 1.11 added