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SUMMARY:Chromatic Smith theory for group actions on finite dimensional com
 plexes - Nicholas Kuhn (University of Virginia)
DTSTART:20250527T130000Z
DTEND:20250527T140000Z
UID:TALK231220@talks.cam.ac.uk
DESCRIPTION:Around 1940\, P. A. Smith showed that if finite p-group P acts
  on a finite dimensional complex X that is acyclic in mod p homology\, the
 n the space of fixed points\, X^P\, would also be acyclic in mod p homolog
 y.\n&nbsp\;\nThe more recent chromatic Smith theorem of Barthel et. al.\, 
 says that if a finite abelian p-group A of rank r acts on a finite complex
  X that is acyclic in K(n+r) homology then X^A will be acyclic in K(n) hom
 ology. (When stated this way\, it has been implicitly assumed that X^A is 
 nonempty.)\n&nbsp\;\nWith William Balderrama\, the speaker has given anoth
 er proof of this theorem\, in the spirit of standard proofs of Smith's ori
 ginal theorem. The hypothesis that X^A is nonempty is not needed: indeed t
 his is proved enroute. Much of our proof works for all finite dimensional 
 A-spaces X\, not just finite ones\, including proving the existence of a f
 ixed point. This opens the question of whether the chromatic Smith theorem
  might also hold under this weaker hypothesis. Examples show that there is
  an obvious problem when n=0\, but a nonequivariant theorem of Bousfield h
 ints that this might be the only problem.\n&nbsp\;\nIn my talk\, I will di
 scuss our proof\, and various open questions that it suggests.
LOCATION:Seminar Room 2\, Newton Institute
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