2023/07/11 by Antolín-Camarena, Omar, García-Hernández, Luis Eduardo, Saldaña, Luis Jorge Sánchez · 1 citation
#Algebraic Topology (math.AT) #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR)
paper · doi:10.48550/arxiv.2307.04997
For a topological group G let E_\textsfcom(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_\textsfcom(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_\textsfcom(G).