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

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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
##### 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