Summary: | For a based manifold M, *, the question of whether the surjective homomorphism Diff(M, *) → π0 Diff(M, *)$ admits a section is an example of a Nielsen realization problem. This question is related to a question about flat connections on M -bundles and is meaningful for M of any dimension. In dimension 2, Bestvina-Church-Souto showed a section does not exist when M is closed and has genus g ≥ 2. Their techniques are cohomological and certain aspects are specific to surfaces. We exhibit new cohomological obstructions to the existence of a section σ: π 0 Diff(M, *) → Diff(M, *). The main tools include Chern-Weil theory, Milnor-Wood inequalities, and Margulis superrigidity. For a large class of locally symmetric manifolds M = Γ\ G/K, we show that σ does not exist by showing that our obstructions are nonvanishing. For many M, this reduces to showing that M has some Pontryagin class pi ( M ) that is nonzero. We compute low degree Pontryagin classes for every closed locally symmetric manifold of noncompact type, building off an algorithm developed by Borel-Hirzebruch. As a result of our computation, we answer the question: Which closed locally symmetric M have at least one nonzero Pontryagin class?
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