Amenable category of three–manifolds

Abstract

A closed topological n-manifold M n is of ame-category 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 1 .W / ! 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 D 3, M 3 has ame-category 4. We characterize all closed 3-manifolds of ame-category 1, 2 and 3. 55M30, 57M27, 57N10; 57N16