Amenable category of three–manifolds

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 $3$.