Twisted Ktheory

General Introduction
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Search results
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Online References
http://ncatlab.org/nlab/show/twisted+Ktheory
http://ncatlab.org/nlab/show/Karoubi+Ktheory
arXiv:1002.3004 Twists of Ktheory 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 Ktheory 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 fourstage 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 Ktheory from arXiv Front: math.AT by Mehdi Khorami 1 person liked this In this paper, we recall the definition of twisted Ktheory 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 Ktheory.
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Paper References
Basic bundle theory etc, in Ktheory 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 Ktheory from arXiv Front: math.AT by Markus Spitzweck, Paul Arne Østvær This paper sets out basic properties of motivic twisted Ktheory with respect to degree three motivic cohomology classes of weight one. Motivic twisted Ktheory is defined in terms of such motivic cohomology classes by taking pullbacks along the universal principal BGmbundle for the classifying space of the multiplicative group scheme. We show a Kuenneth isomorphism for homological motivic twisted Kgroups computing the latter as a tensor product of Kgroups over the Ktheory of BGm. The proof employs an Adams Hopf algebroid and a trigraded Torspectral sequence for motivic twisted Ktheory. By adopting the notion of an Einfinity ring spectrum to the motivic homotopy theoretic setting, we construct spectral sequences relating motivic (co)homology groups to twisted Kgroups. It generalizes various spectral sequences computing the algebraic Kgroups of schemes over fields. Moreover, we construct a Chern character between motivic twisted Ktheory 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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