Simplicial volume and fillings of hyperbolic manifolds

Fujiwara, Koji; Manning, Jason

doi:10.2140/agt.2011.11.2237

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Simplicial volume and fillings of hyperbolic manifolds

Koji Fujiwara and Jason Fox Manning

Algebraic & Geometric Topology 11 (2011) 2237–2264

DOI: 10.2140/agt.2011.11.2237

Abstract

Let $M$ be a hyperbolic $n$ –manifold whose cusps have torus cross-sections. In an earlier paper, the authors constructed a variety of nonpositively and negatively curved spaces as “ $2 π$ –fillings” of $M$ by replacing the cusps of $M$ with compact “partial cones” of their boundaries. These $2 π$ –fillings are closed pseudomanifolds, and so have a fundamental class. We show that the simplicial volume of any such $2 π$ –filling is positive, and bounded above by $\frac{Vol (M)}{v_{n}}$ , where $v_{n}$ is the volume of a regular ideal hyperbolic $n$ –simplex. This result generalizes the fact that hyperbolic Dehn filling of a $3$ –manifold does not increase hyperbolic volume.

In particular, we obtain information about the simplicial volumes of some $4$ –dimensional homology spheres described by Ratcliffe and Tschantz, answering a question of Belegradek and establishing the existence of $4$ –dimensional homology spheres with positive simplicial volume.

Keywords

simplicial volume, pseudomanifold, Dehn filling

Mathematical Subject Classification 2010

Primary: 20F65, 53C23

References

Publication

Received: 2 February 2011

Accepted: 3 June 2011

Published: 25 August 2011

Authors

Koji Fujiwara
Graduate School of Information Sciences Tohoku University Sendai 980-8579 Japan

Jason Fox Manning
Department of Mathematics University at Buffalo, SUNY Buffalo NY 14260 USA

Volume 11, issue 4 (2011)