Periodicity in the stable representation theory of crystallographic groups

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 , 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.