Twisted \(K\)-theory. (English) Zbl 1151.55301

Atiyah, Michael; Segal, Graeme

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)

Keywords:

algebraic topology; \(K\)-theory; equivariant cohomology; \(K\)-theory; operator algebras