Twisted K-theory
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General Introduction
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Search results
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Online References
http://ncatlab.org/nlab/show/twisted+K-theory
http://ncatlab.org/nlab/show/Karoubi+K-theory
arXiv:1002.3004 Twists of K-theory and TMF from arXiv Front: math.AT by Matthew Ando, Andrew J. Blumberg, David Gepner We explore an approach to twisted generalized cohomology from the point of view of stable homotopy theory and quasicategory theory provided by arXiv:0810.4535. We explain the relationship to the twisted K-theory provided by Fredholm bundles. We show how our approach allows us to twist elliptic cohomology by degree four classes, and more generally by maps to the four-stage Postnikov system BO<0...4>. We also discuss Poincaré duality and umkehr maps in this setting.
arXiv:1001.4790 A universal coefficient theorem for twisted K-theory from arXiv Front: math.AT by Mehdi Khorami 1 person liked this In this paper, we recall the definition of twisted K-theory in various settings. We prove that for a twist $\tau$ corresponding to a three dimensional integral cohomology class of a space X, there exist a "universal coefficient" isomorphism K{*}^{\tau}(X)\cong K{}(P{\tau})\otimes{K_{}(\mathbb{C}P^{\infty})} \hat{K}{*} where $P\tau$ is the total space of the principal $\mathbb{C}P^{\infty}$-bundle induced over X by $\tau$ and $\hat K_*$ is obtained form the action of $\mathbb{C}P^{\infty}$ on K-theory.
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Paper References
Basic bundle theory etc, in K-theory folder.
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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
arXiv:1008.4915 Motivic twisted K-theory from arXiv Front: math.AT by Markus Spitzweck, Paul Arne Østvær This paper sets out basic properties of motivic twisted K-theory with respect to degree three motivic cohomology classes of weight one. Motivic twisted K-theory is defined in terms of such motivic cohomology classes by taking pullbacks along the universal principal BGm-bundle for the classifying space of the multiplicative group scheme. We show a Kuenneth isomorphism for homological motivic twisted K-groups computing the latter as a tensor product of K-groups over the K-theory of BGm. The proof employs an Adams Hopf algebroid and a tri-graded Tor-spectral sequence for motivic twisted K-theory. By adopting the notion of an E-infinity ring spectrum to the motivic homotopy theoretic setting, we construct spectral sequences relating motivic (co)homology groups to twisted K-groups. It generalizes various spectral sequences computing the algebraic K-groups of schemes over fields. Moreover, we construct a Chern character between motivic twisted K-theory and twisted periodized rational motivic cohomology, and show that it is a rational isomorphism. The paper includes a discussion of some open problems.
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Other Information
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