The Product Formula in Unitary Deformation K-theory

Abstract

For nitely generated groups G and H, we prove that there is a weak equivalence KG^ku KH ' K(G H) of ku-algebra spectra, where K denotes the \unitary deformation functor. Additionally, we give spectral sequences for computing the homotopy groups of KG and HZ^ku KG in terms of connective K-theory and homology of spaces of G-representations. The underlying goal of many programs in algebraic K-theory is to understand the algebraic K-groups of a eld F as being built from the K-groups of the algebraic closure of the eld, together with the action of the absolute Galois group. Specically , Carlsson's program (see (2)) is to construct a model for the algebraic K-theory spectrum using the Galois group and the K-theory spectrum of the algebraic closure F. In some specic instances, the absolute Galois group of the eld F is explicitly the pronite completion ^ G of a discrete group G. (For example, the absolute Galois group of the eld k(z) of rational func- tions, where k is an algebraically closed of characteristic zero, is the pronite completion of a free group.) In the case where F contains an algebraically closed subeld, the pronite completion of a \deformation spectrum KG is conjecturally equivalent to the pronite completion of the algebraic K-theory spectrum KF . Additionally, it would be advantageous for this description to be compatible with the motivic spectral sequence. This deformation K- theory spectrum has an Atiyah-Hirzebruch spectral sequence arising from a spectrum level ltration. The ltration quotients are spectra built from isomorphism classes of representations of the group. It is hoped that this ltration is related to the motivic spectral sequence, and that this relation would give a greater understanding of the rela- tionships between Milnor K-theory, Galois cohomology, and the repre- sentation theory of the Galois group. We now outline the construction of deformation K-theory. To a nitely generated group G one associates the categoryC of nite dimen- sional unitary representations of G, with morphisms being equivariant isometric isomorphisms. Elementary methods of representation theory