A universal coefficient theorem for twisted K-theory

Understand

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.