Emanuele Dotto: Real topological Hochschild homology and the Hermitian K-theory...

Emanuele Dotto: Real topological Hochschild homology and the Hermitian K theory of Z2 equivariant rin

The lecture was held within the framework of the Hausdorff Trimester Program: K-Theory and Related Fields.

Real topological Hochschild homology (THR) is a Z/2-equivariant spectrum introduced by Hesselholt and Madsen as the recipient of a trace map from real algebraic K-theory of discrete rings with anti-involution.

In joint work with Moi and Patchkoria we interpret THR as a derived smash product of modules over the Hill-Hopkins-Ravenel norm, and carry out calculations for Fp, group-algebras and in π0.

In joint work with Ogle we extend the construction of real K-theory to ring spectra, and use the trace to THR to show that the restricted assembly map of the spherical group-ring splits. One can then reformulate the Novikov conjecture in terms of the vanishing of the trace on the kernel of a certain linearization map in rational Hermitian K-theory.