The classifying space for commutativity of geometric orientable 3-manifold groups

Abstract

For a topological group G let $E_{\mathrm{com}}\left(G\right)$ be the total space of the universal transitionally commutative principal G-bundle as defined by Adem–Cohen–Torres-Giese. So far this space has been most studied in the case of compact Lie groups; but in this paper we focus on the case of infinite discrete groups.

For a discrete group G, the space $E_{\mathrm{com}}\left(G\right)$ is homotopy equivalent to the geometric realization of the order complex of the poset of cosets of abelian subgroups of G. We show that for fundamental groups of closed orientable geometric 3-manifolds, this space is always homotopy equivalent to a wedge of circles.