Real K-theory
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General Introduction
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Search results
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Online References
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Paper References
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Definition
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Properties
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Standard theorems
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Open Problems
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Connections to Number Theory
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Computations and Examples
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History and Applications
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Some Research Articles
Algebraic and Real K-theory of Real Varieties, by Max Karoubi and Charles Weibel: K0473
[arXiv:1212.4310] Stable real K-theory and real topological Hochschild homology from arXiv Front: math.AT by Emanuele Dotto The classical trace map is a highly non-trivial map from algebraic K-theory to topological Hochschild homology (or topological cyclic homology) introduced by Bökstedt, Hsiang and Madsen. It led to many computations of algebraic K-theory of rings. Hesselholt and Madsen recently introduced a Z/2-equivariant version of Waldhausen S-construction for categories with duality. The output is a certain spectrum with involution, called the real K-theory spectrum KR, and associated bigraded groups analogous to Atiyah's real (topological) K-groups. This thesis develops a theory of topological Hochschild homology for categories with duality, and a Z/2-equivariant trace map from real K-theory to it. The main result of the thesis is that stable KR of the category of projective modules over a split square zero extension of a ring is equivalent to the real topological Hochschild homology of the ring with appropriate coefficients. This is the real version a theorem of Dundas-McCarthy for ordinary K-theory.
arXiv:0910.0617 Higher real K-theories and topological automorphic forms from arXiv Front: math.AT by Mark Behrens, Michael J. Hopkins Given a maximal finite subgroup G of the nth Morava stabilizer group at a prime p, we address the question: is the associated higher real K-theory EO_n a summand of the K(n)-localization of a TAF-spectrum associated to a unitary similitude group of type U(1,n-1)? We answer this question in the affirmative for p in {2, 3, 5, 7} and n = (p-1)p^{r-1} for a maximal finite subgroup containing an element of order p^r. We answer the question in the negative for all other odd primary cases. In all odd primary cases, we to give an explicit presentation of a global division algebra with involution in which the group G embeds unitarily.
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