Twisted cohomology

General Introduction
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Online References
[CDATA[http://ncatlab.org/nlab/show/twisted+cohomology
http://ncatlab.org/nlab/show/history+of+cohomology+with+local+coefficients
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 PoincarĂ© duality and umkehr maps in this setting.
arXiv:1406.3231 Twisted differential cohomology arXiv Front: math.AT by Ulrich Bunke, Thomas Nikolaus The main goal of the present paper is the construction of twisted generalized differential cohomology theories and the comprehensive statement of its basic functorial properties. Technically it combines the homotopy theoretic approach to (untwisted) generalized differential cohomology developed by HopkinsSinger and later by the first author and D. Gepner with the oocategorical treatement of twisted cohomology by AndoBlumbergGepner.We introduce the notion of a differential twist for a given generalized cohomology theory and construct twisted differential cohomology groups (resp. spectra). The main technical results of the paper are existence and uniqueness statements for differential twists. These results will be applied in a variety of examples, including Ktheory, topological modular forms and other cohomology theories.
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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
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Other Information
Stasheff sent and email to ALGTOP on Aug 15, 2009, asking for info so that he could recover the history of twisted cohomology.
http://mathoverflow.net/questions/27567/twistingsforothercohomologytheories
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