2017/03/31 by Emanuele Dotto, Crichton Ogle
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Combinatorics #Equivariant map #Functor #Hermitian matrix #Homotopy #Homotopy and Cohomology in Algebraic Topology #Mathematics #Physics #Pure mathematics #Ring (chemistry) #Spectrum (functional analysis) #TRACE (psycholinguistics) #math.AT #math.KT
paper · pdf · doi:10.2140/akt.2019.4.243
published as Ann. K-Th. 4 (2019) 243-316
arxiv created 2017/10/25 · openalex publication_date 2019/06/16 · arxiv updated 2019/08/14 · openalex created_date 2020/11/23 · openalex updated_date 2026/06/11
We define a [math] -equivariant real algebraic [math] -theory spectrum [math] , for every [math] -equivariant spectrum [math] equipped with a compatible multiplicative structure. This construction extends the real algebraic [math] -theory of Hesselholt and Madsen for discrete rings, and the Hermitian [math] -theory of Burghelea and Fiedorowicz for simplicial rings. It supports a trace map of [math] -spectra [math] to the real topological Hochschild homology spectrum, which extends the [math] -theoretic trace of Bökstedt, Hsiang and Madsen.\n¶ We show that the trace provides a splitting of the real [math] -theory of the spherical group-ring. We use the splitting induced on the geometric fixed points of [math] , which we regard as an [math] -theory of [math] -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 [math] -theory of the “Burnside group-ring”.