Vol. 4, No. 2, 2019

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$K\mkern-2mu$-theory of Hermitian Mackey functors, real traces, and assembly

Emanuele Dotto and Crichton Ogle

Vol. 4 (2019), No. 2, 243–316
Abstract

We define a $ℤ∕2$-equivariant real algebraic $K$-theory spectrum $KR\left(A\right)$, for every $ℤ∕2$-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 $ℤ∕2$-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 $ℤ∕2$-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”.

Keywords
Hermitian, $K\mkern-2mu$-theory, forms, $L$-theory, trace, Novikov
Mathematical Subject Classification 2010
Primary: 11E81, 19D55, 19G24, 19G38