Local systems on the free loop space and finiteness of the Hofer-Zehnder capacity

TL;DR

This paper investigates conditions under which symplectic homology vanishes for unit disk bundles, leading to finiteness results for the Hofer-Zehnder capacity and insights into the topology of cotangent bundles.

Contribution

It establishes new criteria for symplectic homology vanishing, proves finiteness of the Hofer-Zehnder capacity in these cases, and explores topological obstructions related to local systems.

Findings

Symplectic homology vanishes when the Hurewicz map is nonzero.

Finiteness of the Hofer-Zehnder capacity for certain cotangent bundles.

Obstructions to H-space structures via local systems.

Abstract

In this article we examine under which conditions symplectic homology with local coefficients of a unit disk bundle $D^*M$ vanishes. For instance this is the case if the Hurewicz map $\pi_2(M)\to H_2(M;\mathbb{Z})$ is nonzero. As an application we prove finiteness of the $\pi_1$-sensitive Hofer-Zehnder capacity of unit disk bundles in these cases. We also prove uniruledness for such cotangent bundles. Moreover, we find an obstruction to the existence of $H$-space structures on general topological spaces, formulated in terms of local systems.