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. 
Tyler Lawson