Multiplicative Structures and the Twisted Baum-Connes Assembly map

TL;DR

This paper establishes a multiplicative structure on twisted geometric K-homology groups of Lie groupoids, extending known products and enabling transfer of these structures to twisted K-theory via the Baum-Connes assembly map.

Contribution

It introduces a new multiplicative structure on twisted geometric K-homology groups of Lie groupoids and connects it to twisted K-theory through the Baum-Connes assembly map.

Findings

Defined an external multiplicative structure for twisted geometric K-homology groups.

Extended previous product structures to a broader class of Lie groupoids.

Provided a framework to transfer multiplicative structures to twisted K-theory.

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 [1] or by Tu,Xu and Laurent-Gengoux in [24]. These Twisted geometric K-homology groups are the left hand sides of the twisted geometric Baum-Connes assembly maps recently constructed in [9] and hence one can transfer the multiplicative structure via the Baum-Connes map to the Twisted K-theory groups whenever this assembly maps are isomorphisms.