Algebra/Topology Seminar 08012021

Speaker: Kevin Piterman

Title: The topology of the complex of p-subgroups

Abstract: In the earlier seventies, Quillen and Brown studied the homotopical properties of the poset of nontrivial p-subgroups in connection with equivariant cohomology (mod p), group geometries, representation theory, and the intrinsic algebraic properties of the group. Quillen described the homotopy type of the poset Ap(G) of nontrivial elementary abelian p-subgroups for suitable families of finite groups and showed that it is contractible if G has a nontrivial normal p-subgroup. He conjectured the reciprocal, and proved the conjecture for solvable groups and for certain groups of Lie type. In fact, for these families, he established the stronger conclusion that Ap(G) has nontrivial rational homology. The most significant advance on the conjecture is due to Aschbacher-Smith, who established the rational homology version for p>5 under certain restrictions on the unitary components. However, the conjecture remains open so far.

In this talk, I will show some recent results on the topology of these complexes, some of them obtained in collaboration with G. Minian and with S.D. Smith. We proved that the fundamental group of these complexes is free in most cases, and I will show how this can be applied to attack Quillen's conjecture. For example, one can prove that the original conjecture (in terms of contractibility) is equivalent to the Z-acyclic version (that is, if Ap(G) is Z-acyclic, then G has a nontrivial normal p-subgroup). I will also discuss how to understand the homotopy type of Ap(G) from that of Ap(H), for suitable subgroups H, with the aim of propagating homology from Ap(H) to Ap(G). These tools recently led us to new cases of the conjecture and the extension of Aschbacher-Smith's theorem to every odd prime p.