Symplectic homology of fiberwise convex sets and homology of loop spaces

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.

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Received 6 February 2020

Accepted 26 June 2021

Published 23 December 2022