The Baum–Connes conjecture via localisation of categories

Abstract

We redefine the Baum-Connes assembly map using simplicial approximation in the equivariant Kasparov category. This new interpretation is ideal for studying functorial properties and gives analogues of the Baum-Connes assembly map for other equivariant homology theories. We extend many of the known techniques for proving the Baum-Connes conjecture to this more general setting.

Homological algebra in bivariant K-theory and other triangulated categories. I

“…In the latter case, you must replace KK (G) by an equivalent category (see [17]); since this is not important here, we do not bother about this issue.…”

“…It is shown in [17] that KK and KK G for a locally compact group G are triangulated categories with this extra structure. The same holds for the equivariant Kasparov theory KK S with respect to any C * -bialgebra S; this theory was defined by Baaj and Skandalis in [2].…”

“…The triangulated category structure encodes basic information about manipulations with long exact sequences and (total) derived functors. The main point of [17] is that the domain of the Baum-Connes assembly map is the total left derived functor of the functor that maps a G-C * -algebra A to K * (G ⋉ r A).…”

“…A basic example is the ideal in the Kasparov category KK defined by If F is the family of compact subgroups, then VC F is related to the Baum-Connes assembly map ( [17]). Of course, there are analogous ideals in more classical categories of (spectra of) G-CW-complexes.…”

“…This ideal arises if we try to compute G-equivariant homology theories in terms of H-equivariant homology theories for H ∈ F. The ideal VC is closely related to the approach to the Baum-Connes assembly map in [17].…”