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.Comment: 33 page