Pontryagin classes of locally symmetric manifolds

Abstract

Pontryagin classes $p_i\left(M\right)$ are basic invariants of a smooth manifold $M$, and many topological problems can be reduced to computing these classes. For a locally symmetric manifold, Borel and Hirzebruch gave an algorithm to determine if $p_i\left(M\right)$ is nonzero. In addition they implemented their algorithm for a few well-known $M$ and for $i=1$, $2$. Nevertheless, there remained several $M$ for which their algorithm was not implemented. In this note we compute low-degree Pontryagin classes for every closed, locally symmetric manifold of noncompact type. As a result of this computation, we answer the question: Which closed locally symmetric $M$ have at least one nonzero Pontryagin class?