The Bott cofiber sequence in deformation K - theory and simultaneous similarity in

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 finite-dimensional complex representations or unitary representations, each of which comes with a natural topology. Carlsson’s “deformationK-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. There is a resulting first quadrant Atiyah-Hirzebruch style spectral sequence converging to the homotopy groups of deformation K-theory, as follows. E 2 = E p,q 3 = πp(R[G])⊗ πq(ku) ⇒ πp+qKG. As a side effect of this identification of R[G] with the cofiber of the Bott map, we obtain results about the homotopy type of spaces parameterizing representations of the group G. In particular, when G is free, we obtain information about simultaneous similarity. The spectral theorem in linear algebra implies that a unitary matrix A is determined, up to similarity, by its set of eigenvalues {z1, . . . , zn}, counted with multiplicity. Taking 2 Tyler Lawson the eigenvalues of a matrix gives a map from U(n) to the n-fold symmetric product Sym(S1), inducing a bijection U(n)/U(n) → Sym(S). In fact, both sides have natural topologies that make this map a homeomorphism. The simultaneous similarity problem in U(n) is to classify the orbits of k-tuples of matrices (A1, . . . , Ak) under unitary change of basis, or simultaneous conjugation. There is an analogous classification in GL(n) due to Friedland [4], which generalizes the Jordan canonical form but is much more involved. The simplest invariant that can be extracted from this situation is the collection of eigenvalues. This gives a continuous eigenvalue map φn,k : X(n, k) = [ U(n) ]k /U(n) → [ Sym(S) k . In addition, there are stabilization maps X(n, k) ֒→ X(n+ 1, k), given by (Ai) 7→ ([