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SUMMARY:Nick Kuhn (University of Virginia)
DTSTART:20221003T203000Z
DTEND:20221003T213000Z
DTSTAMP:20260423T004758Z
UID:MITTop/52
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/MITTop/52/">
 Chromatic Fixed Point Theory</a>\nby Nick Kuhn (University of Virginia) as
  part of MIT topology seminar\n\nLecture held in Room 2-131 in the Simons 
 building.\n\nAbstract\n\\noindent The study of the action of a finite p-gr
 oup G on a finite G-CW complex X is one of the oldest topics in algebraic 
 topology. In the late 1930's\, P. A. Smith proved that if X is mod p acycl
 ic\, then so is XG\, its subspace of fixed points. A related theorem of Ed
  Floyd from the early 1950's says that the dimension of the mod p homology
  of X will bound the dimension of the mod p homology of XG.\n\n\\smallskip
 \n\nThe study of the Balmer spectrum of the homotopy category of G-spectra
  has lead to the problem of identifying "chromatic" variants of Smith's th
 eorem\, with mod p homology replaced by the Morava K-theories (at the prim
 e p). One such chromatic Smith theorem is proved by Barthel et.al.: if G i
 s a cyclic p-group and X is K(n) acyclic\, then XG is K(n−1) acyclic (an
 d this answers questions like this for all abelian p-groups).\n\n\\smallsk
 ip\n\nIn work with Chris Lloyd\, we have been able to show that a chromati
 c analogue of Floyd's theorem is true whenever a chromatic Smith theorem h
 olds. For example\, if G is a cyclic p-group\, then the dimension over K(n
 )∗ of K(n)∗(X) will bound the dimension over K(n−1)∗ of K(n−1)
 ∗(XG).\n\n\\smallskip\n\nThe proof that chromatic Smith theorems imply t
 he stronger chromatic Floyd theorems uses the representation theory of the
  symmetric groups.\n\n\\smallskip\n\nThese chromatic Floyd theorems open t
 he door for many applications. We have been able to resolve open questions
  involving the Balmer spectrum for the extraspecial 2-groups. In a differe
 nt direction\, at the prime 2\, we can show quick collapsing of the AHSS c
 omputing the Morava K-theory of some real Grassmanians: this is a non-equi
 variant result.\n\n\\smallskip\n\nIn my talk\, I'll try to give an overvie
 w of some of this.\n
LOCATION:https://researchseminars.org/talk/MITTop/52/
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