Pontryagin classes of locally symmetric manifolds

Tshishiku, Bena

doi:10.2140/agt.2015.15.2709

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Pontryagin classes of locally symmetric manifolds

Bena Tshishiku

Algebraic & Geometric Topology 15 (2015) 2707–2754

DOI: 10.2140/agt.2015.15.2709

Abstract

Pontryagin classes $p_{i} (M)$ 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 $p_{i} (M)$ 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

Mathematical Subject Classification 2000

Primary: 57R20

Secondary: 06B15

References

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

Volume 15, issue 5 (2015)