Stable real K-theory and real topological Hochschild homology

The classical trace map is a highly non-trivial map from algebraic K-theory to topological Hochschild homology (or topological cyclic homology) introduced by B\"okstedt, Hsiang and Madsen. It led to many computations of algebraic K-theory of rings. Hesselholt and Madsen recently introduced a Z/2-equivariant version of Waldhausen S-construction for categories with duality. The output is a certain spectrum with involution, called the real K-theory spectrum KR, and associated bigraded groups analogous to Atiyah's real (topological) K-groups. This thesis develops a theory of topological Hochschild homology for categories with duality, and a Z/2-equivariant trace map from real K-theory to it. The main result of the thesis is that stable KR of the category of projective modules over a split square zero extension of a ring is equivalent to the real topological Hochschild homology of the ring with appropriate coefficients. This is the real version a theorem of Dundas-McCarthy for ordinary K-theory.