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2019 $K$-theory of Hermitian Mackey functors, real traces, and assembly
Emanuele Dotto, Crichton Ogle
Ann. K-Theory 4(2): 243-316 (2019). DOI: 10.2140/akt.2019.4.243

Abstract

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”.

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Emanuele Dotto. Crichton Ogle. "$K$-theory of Hermitian Mackey functors, real traces, and assembly." Ann. K-Theory 4 (2) 243 - 316, 2019. https://doi.org/10.2140/akt.2019.4.243

Information

Received: 21 June 2018; Revised: 3 January 2019; Accepted: 18 January 2019; Published: 2019
First available in Project Euclid: 13 August 2019

zbMATH: 07102034
MathSciNet: MR3990786
Digital Object Identifier: 10.2140/akt.2019.4.243

Subjects:
Primary: 11E81, 19D55, 19G24, 19G38

Rights: Copyright © 2019 Mathematical Sciences Publishers

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Vol.4 • No. 2 • 2019
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