The author proves that there is a homotopy cofiber sequence of \(ku\)-modules \(\Sigma^2{\mathcal K}G @>{\beta}>> {\mathcal K}G \to R[G]\) induced by the Bott map \(\beta : S^2 \to ku\) in connective \(K\)-theory, where \(G\) is a finitely generated group (Corollary 4). Here \({\mathcal K}G\) denotes the unitary deformation \(K\)-theory of \(G\) defined by G. Carlsson [“Structured stable homotopy theory and the descent problem for the algebraic \(K\)-theory of fields”, Preprint, http://math.stanford.edu/~gunnar/ (2003)] and \(R[G]\) denotes the unitary deformation representation ring introduced by the author [T. Lawson, Algebr. Geom. Topol. 6, 253–286 (2006; Zbl 1118.55009)]. There is also a natural map of spectra \({\mathcal K}G \to \text{Rep}[G]\). Then the main theorem of this paper is in fact that this map induces a weak equivalence \(\text{H}\mathbb{Z}\wedge_{ku}{\mathcal K}G \to R[G]\). Considering the smash product of the Bott periodicity sequence \(\Sigma^2ku @>{\beta}>> ku \to \text{H}{\mathbb Z}\) with \({\mathcal K}G\) over \(ku\), one finds that the assertion above is an immediate consequence of this theorem. The proof of the theorem is done based on results of the author [T. Lawson, \(K\)-Theory 37, No. 4, 395-422 (2006; Zbl 1110.19004)], using another type of spectrum associated to a certain category of representations of \(G\) instead of using \({\mathcal K}G\).
It is also shown that this cofiber sequence yields an Atiyah-Hirzebruch type spectral sequence converging to \(\pi_{p+q}({\mathcal K}G)\), whose \(E_2\)-term is \(\pi_p(R[G])\otimes\pi_q(ku)\). Actually this follows by considering the tower of spectra \(\cdots \to \Sigma^4\text{K}G @>{\Sigma^2\beta}>> \Sigma^2{\mathcal K}G @>{\beta}>> {\mathcal K}G\). The fourth section is devoted to example computations for several groups. For example, for \(G=\mathbb{Z}\rtimes\mathbb{Z}/2\) one has \(\pi_*({\mathcal K}G)=\mathbb{Z}^3\) if \(*\) is even, \(* \geq 0\), and 0 otherwise.
Finally as an application the author considers the case when \(G\) is free. This allows one to obtain information about simultaneous similarity of unitary matrices. Taking the eigenvalues of a matrix gives a map from \(\text{U}(n)^{\text{Ad}}/\text{U}(n)\) to \(\text{Sym}^{n}(S^1)\). This map can be generalized to a map \(X(n, k)=[\text{U}(n)^{\text{Ad}}]^k/\text{U}(n) \to [\text{Sym}^n(S^1)]\). It is proved here that the stabilization of this map \(X(\infty, k) \to [\text{Sym}^\infty(S^1)]\) is a homotopy equivalence. Moreover in the last section there is given another proof which amounts to showing that this stabilization map is a quasifibration.