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

In this article we examine under which conditions symplectic homology with local coefficients of a unit disk bundle $D^{\ast }M$ 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.