#### Volume 13, issue 2 (2013)

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 The Journal About the Journal Subscriptions Editorial Board Editorial Interests Editorial Procedure Submission Guidelines Submission Page Author Index To Appear ISSN (electronic): 1472-2739 ISSN (print): 1472-2747
Amenable category of three–manifolds

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

Algebraic & Geometric Topology 13 (2013) 905–925
##### Abstract

A closed topological $n$–manifold ${M}^{n}$ is of $ame$–category $\le k$ 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 ${\pi }_{1}\left(W\right)\to {\pi }_{1}\left({M}^{n}\right)$ is an amenable group. ${cat}_{ame}\left({M}^{n}\right)$ is the smallest number $k$ such that ${M}^{n}$ admits such a covering. For $n=3$, ${M}^{3}$ has $ame$–category $\le 4$. 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
##### Publication
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