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

TL;DR

This paper investigates the homotopy type of the classifying space for commutativity in fundamental groups of orientable geometric 3-manifolds, revealing it is always a wedge of circles, thus deepening understanding of their algebraic topology.

Contribution

It establishes that for fundamental groups of closed orientable geometric 3-manifolds, the classifying space for commutativity is homotopy equivalent to a wedge of circles, providing new structural insights.

Findings

The space $E_{com}(G)$ is homotopy equivalent to a wedge of circles for these groups.

Structural results on the homotopy type of $E_{com}(G)$ are established.

The space relates to the order complex of cosets of abelian subgroups.

Abstract

For a topological group $G$ let $E_{\textsf{com}}(G)$ 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 $E_{\textsf{com}}(G)$ 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. On our way to prove this result we also establish some structural results on the homotopy type of $E_{\textsf{com}}(G)$.