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

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 $tr:KR(A)\to THR(A)$ to the real topological Hochschild homology spectrum, which extends the $K$-theoretic trace of Bökstedt-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".

References in corpus (6)

• Semi-simplicial spaces
• Stable real K-theory and real topological Hochschild homology
• Real topological cyclic homology of spherical group rings
• Topological Hochschild homology and the cyclic bar construction in symmetric spectra
• A Grothendieck-Witt space for stable infinity categories with duality
• Tambara functors

Cited by in corpus (6)

• On the parametrized Tate construction and two theories of real $p$-cyclotomic spectra
• Real topological Hochschild homology of schemes
• Reflexive homology
• Real topological Hochschild homology via the norm and Real Witt vectors
• Reflexive homology and involutive Hochschild homology as equivariant Loday constructions
• An analogue of the Milnor conjecture for the de Rham-Witt complex in characteristic 2