Chromatic Fixed Point Theory

Nicholas J. Kuhn, University of Virginia

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Passcode: 998749

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. 

There have been two recent Inventiones papers (with two and six authors) studying the Balmer spectrum of the homotopy category of G-spectra.  Translated into less fancy terms, understanding the topology on the Balmer spectrum amounts to amounts to identifying "chromatic" variants of Smith's theorem, with mod p homology replaced by the Morava K-theories (at the prime p).  One such chromatic Smith theorem is proved in the six author paper: if G is a cyclic p-group and X is K(n)* acyclic, then X^G is K(n-1)* acyclic (and this answers questions like this for all abelian p-groups). 

In work with Chris Lloyd, we have been able to show that a chromatic analogue of Floyd's theorem is true whenever the chromatic Smith theorem holds.   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)*(X^G). 

This opens the door for many applications.  We have been able to resolve open questions involving the extraspecial 2-groups.  We can give constraints on the rational homology of the fixed point space of an involution acting on the 5-dimensional Wu manifold.  At the prime 2, we can show quick collapsing of the AHSS computing the Morava K-theory of some real Grassmanians: this is a non-equivariant result.

 In my talk, I'll try to give a coherent overview of some of this.