#### Volume 15, issue 5 (2015)

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Pontryagin classes of locally symmetric manifolds

### Bena Tshishiku

Algebraic & Geometric Topology 15 (2015) 2707–2754
##### 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?

##### Keywords
algebraic topology, differential geometry, characteristic classes
Primary: 57R20
Secondary: 06B15
##### Publication
Received: 9 April 2014
Revised: 14 December 2014
Accepted: 10 January 2015
Published: 10 December 2015
##### Authors
 Bena Tshishiku Department of Mathematics University of Chicago Chicago, IL 60615 USA