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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:On the homotopy type of p-subgroup posets - Kevin
Piterman (Philipps-Universität Marburg)
DTSTART;TZID=Europe/London:20220726T110000
DTEND;TZID=Europe/London:20220726T113000
UID:TALK176540AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/176540
DESCRIPTION:Let Ap(G) be the poset of non-trivial elementary a
belian p-subgroups of a finite group G at a given
prime p. Daniel Quillen established important conn
ections between intrinsic algebraic properties of
G and homotopical properties of Ap(G). For example
\, he showed that Ap(G) is disconnected if and onl
y if G contains a strongly p-embedded subgroup\, a
nd that Ap(G) is contractible if G contains a non-
trivial normal p-subgroup. He conjectured the conv
erse of the latter giving rise to the well-known Q
uillen's conjecture. Although there has been signi
ficant progress on the conjecture\, it is still op
en. One of the major advances was achieved by Mich
ael Aschbacher and Stephen D. Smith: they proved t
hat the conjecture holds for p>5\, under certain r
estrictions on finite unitary groups.In this talk\
, we will see some techniques to understand the ho
motopy type of the poset Ap(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-subgroup posets\, which leads to
new ways of understanding the homotopy type of th
e Ap-posets. As a consequence of these methods\, I
will present new developments on Quillen's conjec
ture: the extension of Aschbacher-Smith's theorem
to every odd prime p\, and also to p=2 (under cert
ain restrictions on some families of simple groups
of Lie type). These results were obtained in coll
aboration with Stephen D. Smith.
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:
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