## Twisted $$K$$-theory.(English)Zbl 1151.55301

Ukr. Mat. Visn. 1, No. 3, 287-300 (2004) and in Ukr. Math. Bull. 1, No. 3, 291-334 (2004).
Summary: Twisted complex $$K$$-theory can be defined for a space $$X$$ equipped with a bundle of complex projective spaces, or, equivalently, with a bundle of $$C^*$$-algebras. Up to equivalence, the twisting corresponds to an element of $$H^3(X;\mathbb Z)$$. We give a systematic account of the definition and basic properties of the twisted theory, emphasizing some points where it behaves differently from ordinary $$K$$-theory. (We omit, however, its relations to classical cohomology, which we shall treat in a sequel.) We develop an equivariant version of the theory for the action of a compact Lie group, proving that then the twistings are classified by the equivariant cohomology group $$H^3_G(X;\mathbb Z)$$. We also consider some basic examples of twisted $$K$$-theory classes, related to those appearing in the recent work of Freed-Hopkins-Teleman.

### MSC:

 55N15 Topological $$K$$-theory 55N91 Equivariant homology and cohomology in algebraic topology 19K99 $$K$$-theory and operator algebras 19L47 Equivariant $$K$$-theory 46L80 $$K$$-theory and operator algebras (including cyclic theory)
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