# $K$-theory of Hermitian Mackey functors and a reformulation of the Novikov Conjecture

## TL;DR

This paper develops a new genuine $Z/2$-equivariant real algebraic $K$-theory spectrum extending previous theories, introduces a trace map to real topological Hochschild homology, and reformulates the Novikov conjecture using equivariant $L$-theory.

## Contribution

It defines a genuine $Z/2$-equivariant real algebraic $K$-theory spectrum and uses it to reformulate the Novikov conjecture in terms of equivariant $L$-theory.

## Key findings

- Constructed a genuine $Z/2$-equivariant real algebraic $K$-theory spectrum.
- Established a trace map extending classical $K$-theoretic trace.
- Reformulated the Novikov conjecture using equivariant $L$-theory.

## Abstract

We define a genuine $\mathbb{Z}/2$-equivariant real algebraic $K$-theory spectrum $KR(A)$, for every genuine $\mathbb{Z}/2$-equivariant spectrum $A$ equipped with a compatible multiplicative structure. This construction extends the real $K$-theory of Hesselholt-Madsen for discrete rings and the Hermitian $K$-theory of Burghelea-Fiedorowicz for simplicial rings. We construct a natural trace map of $\mathbb{Z}/2$-spectra $tr\colon KR(A)\to THR(A)$ to the real topological Hochschild homology spectrum, which extends the $K$-theoretic trace of B\"okstedt-Hsiang-Madsen. The trace provides a splitting of the real $K$-theory of the spherical group-ring. We use this splitting on the geometric fixed points of $KR$, which we regard as an $L$-theory of genuinely equivariant ring spectra, to reformulate the Novikov conjecture on the homotopy invariance of the higher signatures purely in terms of the module structure of the rational $L$-theory of the "Burnside group-ring".