Ukrainian Mathematical Bulletin
Volume 1, Issue 3, 2004 pp. 291-334.
Twisted K-theoryAuthors: Michael Atiyah and Graeme Segal
Author institution: Michael Atiyah, School of Mathematics, The University of Edinburgh, James Clerk Maxwell Building, Kings Buildings, Mayfield Road, Edinburgh, EH9 3JZ, United Kingdom Graeme Segal, All Souls College, Oxford OX1 4AL United Kingdom
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 $H^3(X;\Z)$. 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 $H^3_G(X;\Z)$. We also consider some basic examples of twisted $K$-theory classes, related to those appearing in the recent work of Freed-Hopkins-Teleman.
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