The ring structure for equivariant twisted Ktheory
Abstract
We prove, under some mild conditions, that the equivariant twisted Ktheory group of a crossed module admits a ring structure if the twisting 2cocycle is 2multiplicative. We also give an explicit construction of the transgression map $T_1: H^*(\Gamma;A) \to H^{*1}((N\rtimes \Gamma;A)$ 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 Ktheory group $K^*_{e,\Gamma}(N)$ admits a ring structure. As an application, we prove that for a compact, connected and simply connected Lie group G, the equivariant twisted Ktheory group $K_{[c], G}^* (G)$ is endowed with a canonical ring structure $K^{i+d}_{[c],G}(G)\otimes K^{j+d}_{[c],G}(G)\to K^{i+j+d}_{[c], G}(G)$, where $d=dim G$ and $[c]\in H^2(G\rtimes G;S^1)$.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 April 2006
 arXiv:
 arXiv:math/0604160
 Bibcode:
 2006math......4160T
 Keywords:

 Mathematics  KTheory and Homology;
 High Energy Physics  Theory;
 Mathematical Physics;
 Mathematics  Algebraic Topology;
 Mathematics  Differential Geometry;
 Mathematics  Operator Algebras;
 19L47 (Primary);
 55N91;
 46L80;
 20L05 (Secondary)
 EPrint:
 47 pages. To appear in Crelle