Amenable category of three–manifolds

Gómez-Larrañaga, José Carlos; González-Acuña, Francisco; Heil, Wolfgang

doi:10.2140/agt.2013.13.905

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Amenable category of three–manifolds

José Carlos Gómez-Larrañaga, Francisco González-Acuña and Wolfgang Heil

Algebraic & Geometric Topology 13 (2013) 905–925

DOI: 10.2140/agt.2013.13.905

Abstract

A closed topological $n$ –manifold $M^{n}$ is of $ame$ –category $\leq k$ if it can be covered by $k$ open subsets such that for each path-component $W$ of the subsets the image of its fundamental group $π_{1} (W) \to π_{1} (M^{n})$ is an amenable group. ${cat}_{ame} (M^{n})$ is the smallest number $k$ such that $M^{n}$ admits such a covering. For $n = 3$ , $M^{3}$ has $ame$ –category $\leq 4$ . We characterize all closed $3$ –manifolds of $ame$ –category $1$ , $2$ and $3$ .

Keywords

coverings of $n$–manifolds with amenable subsets, amenable cover of 3–manifolds, Lusternik–Schnirelmann, virtually solvable 3–manifold groups

Mathematical Subject Classification 2010

Primary: 55M30, 57M27, 57N10

Secondary: 57N16

References

Publication

Received: 31 August 2011

Revised: 23 October 2012

Accepted: 2 November 2012

Published: 30 March 2013

Authors

José Carlos Gómez-Larrañaga
Centro de Investigación en Matemáticas A.P. 402 36000 Guanajuato Mexico

Francisco González-Acuña
Instituto de Matemáticas UNAM Morelos Campus 62210 Cuernavaca, Mexico	Centro de Investigación en Matemáticas A.P. 402 36000 Guanajuato Mexico

Wolfgang Heil
Department of Mathematics Florida State University Tallahassee, FL 32306-4510 USA
http://www.math.fsu.edu/People/faculty.php?id=16

Volume 13, issue 2 (2013)