Models of Twisted K-theory.

Abstract

This thesis concerns geometrical models in complete generality of twistings in complex K-theory, in particular of higher twistings. In the first three chapters we treat the non-equivariant situation. In the rest of the chapters, we treat the equivariant situation with some restrictions, namely, we treat higher twistings on the Borel cohomology theory associated to equivariant K-theory. This amounts to considering the completion of equivariant K-theory with respect to the augmentation ideal. We only treat the particular case of a point.

Our model is based on the construction of a suitable classifying space for K-theory that has the structure of a semigroup with respect to two different operations that correspond to the tensor product and Whitney sum of vector bundles. We build such a space based on a category whose objects are Fredholm operators.

During the construction, there arises a problem of how to replace actions of topological semigroups with units up to homotopy on spaces, by topological group actions for groups of the same weak homotopy type. We give a positive answer to this problem under some conditions that hold in our particular application.

In the last chapter we show that when working with a compact Lie group the higher twistings of the completion of equivariant K-theory over a point vanish. This means that the only nontrivial twistings are those already known; those which correspond to twisted representations. Working in the Borel cohomology theory associated with equivariant K-theory allows us to consider the theory over general topological groups. For non compact groups we give examples where the Borel cohomology K-theory higher twistings do not vanish. We also give examples where the corresponding analogue of the Atiyah-Hirzebruch spectral sequence has nontrivial higher differentials in arbitrarily large dimension.