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Abstract
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We define a
-equivariant
real algebraic
-theory
spectrum
, for
every
-equivariant
spectrum
equipped with a compatible multiplicative structure. This construction extends the real
algebraic
-theory
of Hesselholt and Madsen for discrete rings, and the Hermitian
-theory of
Burghelea and Fiedorowicz for simplicial rings. It supports a trace map of
-spectra
to the real topological Hochschild homology spectrum, which extends the
-theoretic
trace of Bökstedt, Hsiang and Madsen.
We show that the trace provides a splitting of the real
-theory of
the spherical group-ring. We use the splitting induced on the geometric fixed points of
, which we regard
as an
-theory
of
-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
-theory
of the “Burnside group-ring”.
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Keywords
Hermitian, $K\mkern-2mu$-theory, forms, $L$-theory, trace,
Novikov
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Mathematical Subject Classification 2010
Primary: 11E81, 19D55, 19G24, 19G38
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Milestones
Received: 21 June 2018
Revised: 3 January 2019
Accepted: 18 January 2019
Published: 16 June 2019
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