Chromatic fixed point theory and the Balmer spectrum for extraspecial 2-groups

Abstract

In the early 1940's, P.A.Smith showed that if a finite p-group G acts on a finite complex X that is mod $p$ acyclic, then its space of fixed points, X^G, will also be mod p acyclic. In their recent study of the Balmer spectrum of equivariant stable homotopy theory, Balmer and Sanders were led to study chromatic versions of this statement, with the question: given H<G and n, what is the smallest r such that if X^H is acyclic in the (n+r)th Morava K-theory, then X^G must be acyclic in the nth Morava K-theory? Barthel et.al. then answered this when G is abelian, by finding general lower and upper bounds for these `blue shift' numbers which agree in the abelian case. In our paper, we first prove that these potential chromatic versions of Smith's theorem are equivalent to chromatic versions of a 1952 theorem of E.E.Floyd, which replaces acyclicity by bounds on dimensions of homology, and thus applies to all finite G-spaces. This unlocks new techniques and applications in chromatic fixed point theory. In one direction, we are able to use classic constructions and representation theory to search for blue shift number lower bounds. We give a simple new proof of the known lower bound theorem, and then get the first results about nonabelian 2-groups that don't follow from previously known results. In particular, we are able to determine all blue shift numbers for extraspecial 2-groups. As samples of new applications, we offer a new result about involutions on the 5-dimensional Wu manifold, and a calculation of the mod 2 K-theory of a 100 dimensional real Grassmanian that uses a C_4 chromatic Floyd theorem.