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SUMMARY:On the homotopy type of p-subgroup posets - Kevin Piterman (Philip
 ps-Universität Marburg)
DTSTART:20220726T100000Z
DTEND:20220726T103000Z
UID:TALK176540@talks.cam.ac.uk
DESCRIPTION:Let Ap(G) be the poset of non-trivial elementary abelian p-sub
 groups of a finite group G at a given prime p. Daniel Quillen established 
 important connections between intrinsic algebraic properties of G and homo
 topical properties of Ap(G). For example\, he showed that Ap(G) is disconn
 ected if and only if G contains a strongly p-embedded subgroup\, and that 
 Ap(G) is contractible if G contains a non-trivial normal p-subgroup. He co
 njectured the converse of the latter giving rise to the well-known Quillen
 's conjecture. Although there has been significant progress on the conject
 ure\, it is still open. One of the major advances was achieved by Michael 
 Aschbacher and Stephen D. Smith: they proved that the conjecture holds for
  p>5\, under certain restrictions on finite unitary groups.In this talk\, 
 we will see some techniques to understand the homotopy type of the poset A
 p(G) from a subposet Ap(H)\, where H is some subgroup of G. This will allo
 w us to perform homotopical-replacements of Ap(G) by non-standard p-subgro
 up posets\, which leads to new ways of understanding the homotopy type of 
 the Ap-posets. As a consequence of these methods\, I will present new deve
 lopments on Quillen's conjecture: the extension of Aschbacher-Smith's theo
 rem to every odd prime p\, and also to p=2 (under certain restrictions on 
 some families of simple groups of Lie type). These results were obtained i
 n collaboration with Stephen D. Smith.
LOCATION:Seminar Room 1\, Newton Institute
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