Multiplicative structures and the twisted Baum-Connes assembly map

Abstract

Using a combination of Atiyah-Segal ideas on one side and of Connes and BaumConnes 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.

Twisted geometric K-homology for proper actions of discrete groups

“…In section 2, we recall definitions, methods, and results related to twisted equivariant K-Theory for proper actions [5], [4], [3].…”

“…Date: October 20, 2018. In section 4, the construction is compared to equivariant Kasparov KK-groups via the definition of an index map, which is proved in Theorem 4.6 to give an isomorphism between both constructions in the case of a proper, finite G-CW complex. On the way, several results, including versions of KK-Theoretical Poincaré Duality [14], [13] and consequences of the Thom isomorphisms and Pushforward Maps for Twisted Equivariant K-Theory of [3] are discussed.…”

“…Twisted Equivariant K-Theory for proper actions of discrete groups was introduced in [5]. We will recall definitions, results and methods from [5] and [3].…”

“…We recall one of the main results from [3], obtained using a version of the Thom Isomorphism and suitable deformation index maps in Twisted Equivariant K-theory. It is stated there (Proposition 5.18) in the more general context of Metalinear structures, but, as observed there, the proof readily applies to the Spin(c)-case.…”

“…The main methods in this note are a combination of consequences of the pushforward homomorphism constructed in [3], and ideas ultimately originated in geometric pictures of K-homology [9], [8].…”