Spherical posets from commuting elements

TL;DR

This paper investigates the homotopy type of the poset of cosets of abelian subgroups in extraspecial p-groups, revealing that its universal cover resembles a wedge of spheres, thus clarifying the structure of related classifying spaces.

Contribution

It establishes the homotopy equivalence of the universal cover of the nerve of cosets to a wedge of spheres, linking algebraic properties to topological structure.

Findings

Universal cover of the nerve is homotopy equivalent to a wedge of spheres.

Homotopy type determined by the rank of the Frattini quotient.

Results apply to classifying spaces of transitionally commutative bundles.

Abstract

In this paper we study the homotopy type of the partially ordered set of left cosets of abelian subgroups in an extraspecial $p$-group. We prove that the universal cover of its nerve is homotopy equivalent to a wedge of $r$-spheres where $2r \geq 4$ is the rank of its Frattini quotient. This determines the homotopy type of the universal cover of the classifying space of transitionally commutative bundles.