A universal coefficient theorem for twisted K-theory

TL;DR

This paper establishes a universal coefficient theorem for twisted K-theory, linking twisted K-groups to untwisted K-theory of associated principal bundles via a tensor product isomorphism.

Contribution

It proves a universal coefficient isomorphism for twisted K-theory with three-dimensional integral cohomology class twists, extending classical K-theory results.

Findings

Established a universal coefficient isomorphism for twisted K-theory.

Connected twisted K-theory groups to untwisted K-theory of principal bundles.

Provided a framework for computations in twisted K-theory using classical tools.

Abstract

In this paper, we recall the definition of twisted K-theory in various settings. We prove that for a twist $\tau$ corresponding to a three dimensional integral cohomology class of a space X, there exist a "universal coefficient" isomorphism K_{*}^{\tau}(X)\cong K_{*}(P_{\tau})\otimes_{K_{*}(\mathbb{C}P^{\infty})} \hat{K}_{*} where $P_\tau$ is the total space of the principal $\mathbb{C}P^{\infty}$-bundle induced over X by $\tau$ and $\hat K_*$ is obtained form the action of $\mathbb{C}P^{\infty}$ on K-theory.