Real Ktheory

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 Ktheory of Real Varieties, by Max Karoubi and Charles Weibel: K0473
[arXiv:1212.4310] Stable real Ktheory and real topological Hochschild homology from arXiv Front: math.AT by Emanuele Dotto The classical trace map is a highly nontrivial map from algebraic Ktheory to topological Hochschild homology (or topological cyclic homology) introduced by Bökstedt, Hsiang and Madsen. It led to many computations of algebraic Ktheory of rings. Hesselholt and Madsen recently introduced a Z/2equivariant version of Waldhausen Sconstruction for categories with duality. The output is a certain spectrum with involution, called the real Ktheory spectrum KR, and associated bigraded groups analogous to Atiyah's real (topological) Kgroups. This thesis develops a theory of topological Hochschild homology for categories with duality, and a Z/2equivariant trace map from real Ktheory 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 DundasMcCarthy for ordinary Ktheory.
arXiv:0910.0617 Higher real Ktheories 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 Ktheory EO_n a summand of the K(n)localization of a TAFspectrum associated to a unitary similitude group of type U(1,n1)? We answer this question in the affirmative for p in {2, 3, 5, 7} and n = (p1)p^{r1} 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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