Jeudi 31 Mars

Heure:	10:15 - 12:00
Lieu:	Salle B405, bâtiment B, LAGA, Institut Galilée, Université Paris 13
Résumé:	Topologie algébrique - The p-subgroup posets and Quillen's conjecture -
Description:	Kevin Piterman The p-subgroup complexes have been largely studied in connection with group cohomology, finite group theory (including the Classification), representation theory, homotopy theory, etc. In this talk, we will mainly focus on understanding the homotopy type of the poset Ap(G) of non-trivial elementary abelian p-subgroups of a finite group G for a given prime p. This poset was introduced by D. Quillen, who established many connections between intrinsic algebraic properties of G and homotopical properties of Ap(G). In this context, Quillen proved that if Ap(G) has a fixed point by the action of G (that is, a non-trivial normal p-subgroup) then it is contractible. He conjectured the converse given rise to the well-known Quillen's conjecture. Although there have been important advances on the conjecture, it is still open. For example, one of the major advances was achieved by Aschbacher-Smith: they established the conjecture for p>5, under certain restrictions on the finite unitary groups. During the talk, I will present some new developments and results on the conjecture, obtained in collaboration with Stephen D. Smith. In particular, we will see that Aschbacher-Smith's theorem can be extended to every odd prime p, and also to p=2 (modulo further restrictions on some simple groups). These results rely on making suitable homotopical replacements of Ap(G) by some non-standard p-subgroup posets, which lead to new ways of understanding the homotopy type of these objects.