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

We define a 2-equivariant real algebraic K-theory spectrum KR(A), for every 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 2-spectra tr : KR(A) THR(A) 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 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”.

Hermitian, $K\mkern-2mu$-theory, forms, $L$-theory, trace, Novikov
Mathematical Subject Classification 2010
Primary: 11E81, 19D55, 19G24, 19G38
Received: 21 June 2018
Revised: 3 January 2019
Accepted: 18 January 2019
Published: 16 June 2019
Emanuele Dotto
Mathematical Institute
University of Bonn
Crichton Ogle
Department of Mathematics
The Ohio State University
Columbus, OH
United States