Periodicity in the stable representation theory of crystallographic groups

TL;DR

This paper proves a periodicity theorem for the deformation K-theory spectra of crystallographic groups, revealing structured patterns in their stable representation spaces and connecting homotopy groups to cohomology.

Contribution

It establishes a periodicity theorem for crystallographic groups' deformation K-theory spectra and links homotopy groups to cohomology using algebraic geometry and representation theory.

Findings

Spectrum is 2-periodic above rational cohomological dimension.

Vanishing of homotopy groups for certain torsion-free groups.

Compactification of representation moduli space has bounded CW complex dimension.

Abstract

Deformation K-theory associates to each discrete group G a spectrum built from spaces of finite dimensional unitary representations of G. In all known examples, this spectrum is 2-periodic above the rational cohomological dimension of G (minus 2), in the sense that T. Lawson's Bott map is an isomorphism on homotopy in these dimensions. We establish a periodicity theorem for crystallographic subgroups of the isometries of k-dimensional Euclidean space. For a certain subclass of torsion-free crystallographic groups, we prove a vanishing result for the homotopy groups of the stable moduli space of representations, and we provide examples relating these homotopy groups to the cohomology of G. These results are established as corollaries of the fact that for each n > 0, the one-point compactification of the moduli space of irreducible n-dimensional representations of G is a CW complex of dimension at most k. This is proven using real algebraic geometry and projective representation theory.