2 results
The Baum–Connes conjecture localised at the unit element of a discrete group
- Part of
-
- Journal:
- Compositio Mathematica / Volume 156 / Issue 12 / December 2020
- Published online by Cambridge University Press:
- 14 January 2021, pp. 2536-2559
- Print publication:
- December 2020
-
- Article
- Export citation
-
We construct a Baum–Connes assembly map localised at the unit element of a discrete group
$\Gamma$. This morphism, called 
$\mu _\tau$, is defined in 
$KK$-theory with coefficients in 
$\mathbb {R}$ by means of the action of the idempotent 
$[\tau ]\in KK_{\mathbin {{\mathbb {R}}}}^\Gamma (\mathbb {C},\mathbb {C})$ canonically associated to the group trace of 
$\Gamma$. We show that the corresponding 
$\tau$-Baum–Connes conjecture is weaker than the classical version, but still implies the strong Novikov conjecture. The right-hand side of 
$\mu _\tau$ is functorial with respect to the group 
$\Gamma$.
Comparing Assembly Maps in Algebraic K-Theory
-
- Journal:
- Journal of K-Theory / Volume 7 / Issue 1 / February 2011
- Published online by Cambridge University Press:
- 05 February 2010, pp. 145-168
- Print publication:
- February 2011
-
- Article
- Export citation
-
Given a group G and a ring R, Loday [Lod, 1976] described an assembly map αG : hn(BG;L(R)) → Kn(RG) where L(R) is a spectrum with nth space K0(SnR) × BGL(SnR)+ for n ≥ 0 and Kn(RG) = πn(BGL(RG)+ × K0(RG)). Hambleton and Pederson, [HP, 2004], indicate a proof that this map is isomorphic to the map on homotopy groups from the assembly map
. We will complete the proof of this.
. We will complete the proof of this.