# Symplectic homology of fiberwise convex sets and homology of loop spaces

## TL;DR

This paper establishes a deep connection between symplectic homology of fiberwise convex sets in cotangent bundles and the homology of loop spaces, providing new formulas and applications in symplectic capacity theory.

## Contribution

It proves an isomorphism between symplectic homology and loop space homology for fiberwise convex sets, and derives formulas for symplectic capacities using loop space homology.

## Key findings

- Symplectic homology capacity equals Ekeland-Hofer-Zehnder capacity for convex bodies.
- Established subadditivity of Hofer-Zehnder capacity.
- Connected symplectic invariants with loop space homology.

## Abstract

For any nonempty, compact and fiberwise convex set $K$ in $T^*\mathbb{R}^n$, 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.