  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.

Symplectic Geometry;Algebraic Topology;Dynamical Systems