Twisted bundles and twisted K-theory, by Max Karoubi

We offer here a more direct approach to twisted K-theory, based on the notion of twisted vector bundles (of finite or infinite dimension). This is also related to the classical notion ot torsor.

Roughly speaking, twisted K-theory is defined here as the Grothendieck group of twisted vector bundles (with a given twist). The usual operations on vector bundles (exterior powers, Adams operations) are easily extended to this twisted framework.

With this viewpoint, the usual Chern character ia also extended with a target which is "twisted cohomology". Already defined by many authors by other means, this Chern character is here given with the Chern-Weil method and provides explicit formulas in terms of Steenrod's coordinate bundles.

Max Karoubi <>