Nicholas Kuhn: Chromatic Smith Fixed Point Theory

Abstract.

The study of the action of a finite p-group 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 acyclic, then so is X^G, 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 X^G.    The study of thick tensored categories in the category of G-spectra has led to the problem of identifying "chromatic" variants of these theorems, with mod p homology replaced by the Morava K-theories (at the prime p). An example of a new chromatic Floyd theorem is the following: 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)*(X^G).   These chromatic fixed point theorems open the door for various new applications. For example, one can deduce that a C_2 action on the 5 dimensional Wu manifold will have fixed points that have the rational homology of a sphere.     There are still open problems in this area about general `blue shift' numbers and about the truth of variants of the theorems for actions on complexes that are finite dimensional, but not necessarily finite.   My own contributions have included joint work with Chris Lloyd and William Balderrama.   In my talk, I'll try to give an overview of some of this.