K-Theory, Hermitian K-Theory and the Karoubi Tower

The goal of this paper is to present a new view on the link between the well known K-theory of finitely generated projective modules and much less understood K-theory of Hermitian forms on finitely generated projective modules.

For a given Hermitian ring (R, α, ε), we obtain the K-theory space KR equipped with an involution and the Hermitian K-theory space K Herm(R, α, ε). The fixed subspace KR^Z₂ is homotopy equivalent to the Hermitian K-theory space K Herm(R, α, ε). We construct the Karoubi Tower diagram, which is obtained by iterating Karoubi's construction of the homotopy fibers of the forgetful and hyperbolic maps. Using an interesting factorization of these maps, we prove various homotopy properties of the Karoubi Tower. The homotopy inverse limit of the Karoubi Tower is homotopy equivalent to the homotopy fibre of the inclusion of the fixed set KR^Z₂ into the homotopy fixed set KR^hZ₂. Considering Karoubi's fundamental periodicity theorem, the Karoubi Tower generalizes the low dimensional connections between Hermitian K-theory and K-theory groups. Illustrative examples of the Karoubi Tower are given by the finite field case and the classical Hermitian rings over and ℝ, ℂ, and H. Considering topological K-theory for these cases, the Karoubi Tower comprises the classical Bott periodicity.

Another important application of the Karoubi Tower is an elegant and comprehensive generalization of the classical invariants of quadratic forms.