Multiplicative structures and the twisted Baum-Connes assembly map

Abstract

Using a combination of Atiyah-Segal ideas on one side and of Connes and Baum-Connes ideas on the other, we prove that the twisted geometric K-homology groups of a Lie groupoid have an external multiplicative structure extending hence the external product structures for proper cases considered by Adem-Ruan in 2003 or by Tu, Xu and Laurent-Gengoux in 2004. These twisted geometric K-homology groups are the left-hand sides of the twisted geometric Baum-Connes assembly maps recently constructed by Carrillo Rouse and Wang (2016), and hence one can transfer the multiplicative structure via the Baum-Connes map to the twisted K-theory groups whenever these assembly maps are isomorphisms.