The Bott cofiber sequence in deformation K-theory and simultaneous similarity in U(n)

We show that there is a homotopy cofiber sequence of spectra relating Carlsson’s deformation K-theory of a group G to its “deformation representation ring,” analogous to the Bott periodicity sequence relating connective K-theory to ordinary homology. We then apply this to study simultaneous similarity of unitary matrices. The algebraic K-theory of a category uses the machinery of infinite loop space theory to associate spectra to symmetric monoidal categories. The homotopy groups of these spectra give information about the structure of the category itself. However, some symmetric monoidal categories arise with natural topologies on their objects and morphisms that give information about how objects in the category can behave in families. For example, given a group G, we can consider the category of its finitedimensional complex representations or unitary representations, each of which comes with a natural topology. Carlsson’s “deformation K-theory,” or the associated unitary variant, produces a K-theory spectrum which depends on both the symmetric monoidal structure and the behavior in families. The purpose of this article is to identify the cofiber of the Bott map on unitary deformation K-theory ([2], [8]) of a finitely generated group G. For a finite group G, this cofiber can be identified with the Eilenberg-MacLane spectrum associated to the complex representation ring R[G]. More generally one obtains a “unitary deformation representation ring,” also denoted by R[G], which is a commutative HZ-algebra spectrum. This deformation representation ring was considered in a previous paper [7]. Results of Park and Suh [9] will be applied to show that this deformation representation ring admits a cellular construction as an HZ-module spectrum. Supported in part by NSF grant DMS-0402950.