Pontryagin classes of locally symmetric manifolds

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In this note we compute low degree rational Pontryagin classes for every closed locally symmetric manifold of noncompact type. In particular, we answer the question: Which locally symmetric M have at least one nonzero Pontryagin class? This computation was motivated by a Nielsen realization problem. Specifically, in my paper “Cohomological obstructions to Nielsen realization”, I show that Pontryagin classes are obstructions to realizing the point-pushing subgroup for nonpositively curved manifolds (and in particular locally symmetric manifolds of noncompact type).

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