## Algebraic & Geometric Topology

### Pontryagin classes of locally symmetric manifolds

Bena Tshishiku

#### Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.agt/1510841029

Digital Object Identifier
doi:10.2140/agt.2015.15.2709

Mathematical Reviews number (MathSciNet)
MR3426690

Zentralblatt MATH identifier
1336.57036

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

#### Citation

Tshishiku, Bena. Pontryagin classes of locally symmetric manifolds. Algebr. Geom. Topol. 15 (2015), no. 5, 2707--2754. doi:10.2140/agt.2015.15.2709. https://projecteuclid.org/euclid.agt/1510841029

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