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4 - How Can It Be True?
Shmuel Weinberger, University of Chicago
Book:

Variations on a Theme of Borel

Published online:

24 November 2022

Print publication:

08 December 2022, pp 79-128
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Summary

The Novikov conjecture arises as a piece of the Borel conjecture, although generalized beyond the setting of aspherical manifolds. This chapted gives several methods, from spltting theorems to index theory to bounded topology, for verifying this conjecture.

The Baum–Connes conjecture localised at the unit element of a discrete group
Part of
Paolo Antonini, Sara Azzali, Georges Skandalis
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
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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 $\Gamma$.

9 - Products and Poincaré Duality
from Part Three - Differential Operators
Rufus Willett, University of Hawaii, Manoa, Guoliang Yu, Texas A & M University
Book:

Higher Index Theory

Published online:

11 June 2020

Print publication:

02 July 2020, pp 353-392
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Summary

Applications of assembly maps to the Kadison-Kaplansky conjecture on the existence of idempotents in group algebras, to the existence and study of positive scalar curvature metrics, and to the Novikov conjecture in manifold topology.Historical motivation and some overview of the literature.

DUALIZING THE COARSE ASSEMBLY MAP
Heath Emerson, Ralf Meyer
Journal:

Journal of the Institute of Mathematics of Jussieu / Volume 5 / Issue 2 / April 2006

Published online by Cambridge University Press:

07 December 2005, pp. 161-186

Print publication:

April 2006
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We formulate and study a new coarse (co-)assembly map. It involves a modification of the Higson corona construction and produces a map dual in an appropriate sense to the standard coarse assembly map. The new assembly map is shown to be an isomorphism in many cases. For the underlying metric space of a group, the coarse co-assembly map is closely related to the existence of a dual Dirac morphism and thus to the Dirac dual Dirac method of attacking the Novikov conjecture.

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