$K$-theory of Hermitian Mackey functors, real traces, and assembly

Abstract

We define a $\mathbb{Z}∕2$-equivariant real algebraic $K$-theory spectrum $KR\left(A\right)$, for every $\mathbb{Z}∕2$-equivariant spectrum $A$ equipped with a compatible multiplicative structure. This construction extends the real algebraic $K$-theory of Hesselholt and Madsen for discrete rings, and the Hermitian $K$-theory of Burghelea and Fiedorowicz for simplicial rings. It supports a trace map of $\mathbb{Z}∕2$-spectra $tr:KR\left(A\right)\to THR\left(A\right)$ to the real topological Hochschild homology spectrum, which extends the $K$-theoretic trace of Bökstedt, Hsiang and Madsen.

We show that the trace provides a splitting of the real $K$-theory of the spherical group-ring. We use the splitting induced on the geometric fixed points of $KR$, which we regard as an $L$-theory of $\mathbb{Z}∕2$-equivariant ring spectra, to give a purely homotopy theoretic reformulation of the Novikov conjecture on the homotopy invariance of the higher signatures, in terms of the module structure of the rational $L$-theory of the “Burnside group-ring”.