Abstract

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.