\(K\)-theory, Hermitian \(K\)-theory and the Karoubi tower
Let \((R,\alpha,\varepsilon)\) be a Hermitian ring; thus \(R\) is a ring with unit, \(\alpha\) is an anti-involution, and \(\varepsilon\) is a central unit such that \(\alpha(\varepsilon)=\varepsilon^{-1}\). Beside the usual \(K\)-theory infinite loop space \(KR\) based on finitely generated projective modules, there is a Hermitian \(K\)-theory infinite loop space \(K\text{Herm}(R,\alpha,\varepsilon)\) based on finitely generated projective modules with Hermitian forms. It is homotopy equivalent to the fixed subspace \(KR^{{\mathbb Z}/2}\) under the involution induced by \(\alpha\). The Karoubi tower is a doubly infinite sequence \[ \ldots \to \Omega^2 KH^{(2)} \to \Omega KH^{(1)} \to K\text{Herm}(R,\alpha,\varepsilon) \to KH^{(-1)} \to KH^{(-2)} \to \ldots, \] where each space \(KH^{(n)}\) is the Hermitian \(K\)-theory of a Hermitian ring constructed from \((R,\alpha,\varepsilon)\). The homotopy fibre of each map is an iterated looping or delooping of \(KR\). If \(2\) is invertible in \(R\), then the sequence is \(4\)-periodic. There are two main results in this paper. The first produces homotopy fibration sequences \[ \Omega^{n+1}KH^{(n+1)} \to K\text{Herm}(R,\alpha,\varepsilon) \to A_n \] for \(n\geq 0\), where the \(A_n\) form a natural tower with homotopy limit equal to the homotopy fixed point set \(KR^{h{\mathbb Z}/2}\). As a corollary, the homotopy limit of the Karoubi tower is the homotopy fibre of \[ K\text{Herm}(R,\alpha,\varepsilon)=KR^{{\mathbb Z}/2} \to KR^{h{\mathbb Z}/2}. \] Dually, the second result produces homotopy cofibration sequences \[ B_n \to K\text{Herm}(R,\alpha,\varepsilon) \to KH^{(-n-1)} \] for \(n\geq 0\), where the \(B_n\) form a sequence with homotopy colimit equal to the homotopy orbit space \(KR_{h{\mathbb Z}/2}\). As a corollary, the homotopy colimit of the Karoubi tower is the homotopy cofibre of \[ KR_{h{\mathbb Z}/2} \to KR^{{\mathbb Z}/2}=K\text{Herm}(R,\alpha,\varepsilon). \] There are calculations for the real numbers, complex numbers and quaternions, which include Bott periodicity; there are also calculations for finite fields. See also the review below (Zbl 0931.19004).
scientific article; zbMATH DE number 1332922