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

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