Twisted bundles and twisted K-theory

Many papers have been devoted recently to twisted K-theory as originally defined in [15] and [29]. See for instance the references [2], [23] and the very accessible paper [30]. We offer here a more direct approach based on the notion of “twisted vector bundles”. This is not an entirely new idea, since we find it in [4], [6], [7], [8] and [9] for instance, under different names and from various viewpoints. However, a careful look at this notion shows that we may interpret such bundles as modules over suitable algebra bundles. More precisely, the category of twisted vector bundles is equivalent to the category of vector bundles which are modules over algebra bundles with fibre End(V ), where V is a finite dimensional vector space. This notion was first explored in [15] in order to define twisted K-theory. In the same vein, twisted Hilbert bundles may be used to define extended twisted K-groups, following [14] and [29]. More generally, we also analyse the notion of “twisted principal bundles” with structural group G. Under favourable circumstances, we show that the associated category is equivalent to the category of locally trivial fibrations, with an action of a bundle of groups with fibre G, which is simply transitive on each fibre. Such bundles are classically called “torsors” in the literature. When the bundle of groups is trivial, we recover the usual notion of principal G-bundle.