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

abstract:

In the early 1940s, P. A. Smith showed that if a finite $p$-group $G$ acts on a finite dimensional 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 a question that can be shown to be equivalent to the following: if a $G$-space $X$ is a equivariant homotopy retract of the $p$-localization of a based finite $G$-C.W. complex, 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 $n$th 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 mod $p$ homology, and thus applies to all finite dimensional $G$-spaces. This unlocks new techniques and applications in chromatic fixed point theory.

Applied to the problem of understanding blue shift numbers, we are able to use classic constructions and representation theory to search for 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 do not follow from previously known results. In particular, we are able to determine all blue shift numbers for extraspecial 2-groups.

Applied in ways analogous to Smith's original applications, we prove new fixed point theorems for $K(n)_*$-homology disks and spheres.

Finally, our methods offer a new way of using equivariant results to show the collapsing of certain Atiyah-Hirzebruch spectral sequences in certain cases. Our criterion appears to apply to the calculation of all 2-primary Morava $K$-theories of all real Grassmanians.