The ring structure for equivariant twisted K-theory

Abstract

We prove, under some mild conditions, that the equivariant twisted K-theory group of a crossed module admits a ring structure if the twisting 2-cocycle is 2-multiplicative. We also give an explicit construction of the transgression map $T_1:H^{\ast }(\Gamma ;A)\to H^{\ast -1}((N⋊\Gamma ;A)$ for any crossed module $N\to \Gamma$ and prove that any element in the image is $\infty$-multiplicative. As a consequence, we prove that, under some mild conditions, for a crossed module $N \to \gm$ and any $e\in {\check{Z}}^3(\Gamma ;S^1)$, that the equivariant twisted K-theory group $K_{e,\Gamma }^{\ast }(N)$ admits a ring structure. As an application, we prove that for a compact, connected and simply connected Lie group G, the equivariant twisted K-theory group $K_{[c],G}^{\ast }(G)$ is endowed with a canonical ring structure $K_{[c],G}^{i+d}(G)\otimes K_{[c],G}^{j+d}(G)\to K_{[c],G}^{i+j+d}(G)$, where $d=dimG$ and $[c]\in H^2(G⋊G;S^1)$.

47 pages. To appear in Crelle

References in corpus (2)

• Twisted K-theory and loop group representations
• A Stringy Product on Twisted Orbifold K-theory