A universal coefficient theorem for twisted K-theory

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.

💡 Research Summary

The paper presents a universal coefficient theorem (UCT) that dramatically simplifies the computation of twisted K‑theory for twists arising from integral three‑dimensional cohomology classes. After a concise review of the various definitions of twisted K‑theory—via the Freed‑Hopkins‑Teleman spectrum, Rosenberg’s C∗‑algebraic approach, and the classical bundle‑gerbe picture—the author focuses on a twist τ∈H³(X;ℤ). Such a class determines a principal ℂP^∞‑bundle P_τ→X, where ℂP^∞ serves as the classifying space for complex line bundles and, more importantly, as the representing space for the Bott element β in K‑theory.

The central observation is that the τ‑twisted K‑theory groups K_^{τ}(X) can be expressed entirely in terms of the ordinary K‑theory of the total space P_τ together with the module structure over K_(ℂP^∞). The K‑homology of ℂP^∞ is a polynomial ring ℤ