In the late 1980’s, Krasauskas and Fiedorowicz-Loday independently developed the theory of crossed simplicial groups, which generalize Connes’ cyclic category. Of particular interest is the Dihedral category, which has recently been used to develop the theory of Real topological Hochschild homology, a first approximation to Grothendieck-Witt groups.

In the first part of my talk, I will discuss ongoing joint work with Mona Merling and Maximilien Péroux on a topological analogue of the homology of crossed simplicial groups. As a special case, we recover the theory of Real topological Hochschild homology.

In the second part of my talk, I will discuss joint work with Teena Gerhardt and Mike Hill. We provide a norm model for Real topological Hochschild homology, prove a multiplicative double coset formula for Real topological Hochschild homology, and we construct the Real Witt vectors of rings with anti-involution.