Tag - Subgroup complexes

K. Brown introduced in 1975 the p-subgroup complex of a finite group G. It is the simplicial complex whose vertices are the nontrivial p-subgroups of G, where a collection of subgroups spans a simplex if it is a chain. This complex was further studied by Quillen, who observed that for a finite group of Lie type G with defining characteristic p, this complex is homotopy equivalent to the building of G. He also conjectured that the p-subgroup complex is contractible if and only if G contains a nontrivial normal p-subgroup and proved his conjecture for solvable groups. The Quillen conjecture remains open but was proved for almost simple groups by Aschbacher and Kleidman, and strong reduction theorem was obtained by Aschbacher and Smith.

The group G acts on its p-subgroups by conjugation and hence acts simplicially on the p-subgroup complex. Webb conjectured in 1987 that the orbit space of the p-subgroup complex is always contractible. He proved that its mod-p homology vanishes using methods from group cohomology. Webb's conjecture was first proved by Symonds in 1998, and a number of other proofs have since appeared. All the proofs I am aware of go through Robinson's subcomplex, which is G-homotopy equivalent to Brown's. None of the proofs are explicit. Symonds computes the fundamental group and integral homology and uses the Hurewicz and Whitehead theorems. Bux gave an inductive  proof using a variant of Bestvina-Brady style discrete Morse theory. In this talk, I will use Brown's theory of collapsing schemes to give an explicit sequence of elementary collapses that collapses the orbit space of Robinson's subcomplex to the vertex corresponding to the conjugacy class of Sylow p-subgroups.