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Department of Applied Mathematics and Theoretical Physics

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, then the space of fixed points, X^P, would also be acyclic in mod p homology.
 
The 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) homology. (When stated this way, it has been implicitly assumed that X^A is nonempty.)
 
With William Balderrama, the speaker has given another proof of this theorem, in the spirit of standard proofs of Smith's original theorem. The hypothesis that X^A is nonempty is not needed: indeed this 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 fixed 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 hints that this might be the only problem.
 
In my talk, I will discuss our proof, and various open questions that it suggests.

Further information

Time:

27May
May 27th 2025
14:00 to 15:00

Venue:

Seminar Room 2, Newton Institute

Speaker:

Nicholas Kuhn (University of Virginia)

Series:

Isaac Newton Institute Seminar Series