Jeudi 31 Mars

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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.