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15 September 2004 Homotopy type and volume of locally symmetric manifolds
Tsachik Gelander
Duke Math. J. 124(3): 459-515 (15 September 2004). DOI: 10.1215/S0012-7094-04-12432-7

Abstract

We consider locally symmetric manifolds with a fixed universal covering, and we construct for each such manifold M a simplicial complex ℝ whose size is proportional to the volume of M. When M is noncompact, $\mathcal{R}$ is homotopically equivalent to M, while when M is compact, $\mathcal{R}$ is homotopically equivalent to M\N, where N is a finite union of submanifolds of relatively small dimension. This reflects how the volume controls the topological structure of M, and yields concrete bounds for various finiteness statements that previously had no quantitative proofs. For example, it gives an explicit upper bound for the possible number of locally symmetric manifolds of volume bounded by v>0, and it yields an estimate for the size of a minimal presentation for the fundamental group of a manifold in terms of its volume. It also yields a number of new finiteness results.

Citation

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Tsachik Gelander. "Homotopy type and volume of locally symmetric manifolds." Duke Math. J. 124 (3) 459 - 515, 15 September 2004. https://doi.org/10.1215/S0012-7094-04-12432-7

Information

Published: 15 September 2004
First available in Project Euclid: 31 August 2004

zbMATH: 1076.53040
MathSciNet: MR2084613
Digital Object Identifier: 10.1215/S0012-7094-04-12432-7

Subjects:
Primary: 53C20 22E40

Rights: Copyright © 2004 Duke University Press

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Vol.124 • No. 3 • 15 September 2004
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