Algebraic & Geometric Topology

Pontryagin classes of locally symmetric manifolds

Bena Tshishiku

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Pontryagin classes pi(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 pi(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?

Article information

Algebr. Geom. Topol., Volume 15, Number 5 (2015), 2707-2754.

Received: 9 April 2014
Revised: 14 December 2014
Accepted: 10 January 2015
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57R20: Characteristic classes and numbers
Secondary: 06B15: Representation theory

algebraic topology differential geometry characteristic classes


Tshishiku, Bena. Pontryagin classes of locally symmetric manifolds. Algebr. Geom. Topol. 15 (2015), no. 5, 2707--2754. doi:10.2140/agt.2015.15.2709.

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