Homotopy equivalences between p-subgroup categories

TL;DR

This paper extends Quillen's homotopy equivalence result from the Brown poset to the fusion category of nonidentity p-subgroups of a finite group G, exploring similar relations in related p-subgroup categories.

Contribution

It generalizes Quillen's homotopy equivalence to the fusion category and examines other p-subgroup categories, broadening understanding of their topological properties.

Findings

Homotopy equivalence between fusion category and elementary abelian subgroups

Extension of Quillen's result to additional p-subgroup categories

Broader framework for p-subgroup category homotopy types

Abstract

Let p be a prime number and G a finite group of order divisible by p. Quillen showed that the Brown poset of nonidentity p-subgroups of G is homotopy equivalent to its subposet of nonidentity elementary abelian subgroups. We show here that a similar statement holds for the fusion category of nonidentity p-subgroups of G. Other categories of p-subgroups of G are also considered.