# nLab Baum-Connes conjecture

Contents

### Context

#### Index theory

index theory, KK-theory

noncommutative stable homotopy theory

partition function

genus, orientation in generalized cohomology

## Definitions

operator K-theory

K-homology

# Contents

## Idea

The Baum-Connes conjecture asserts that under suitable technical conditions, the operator K-theory/KK-theory of the groupoid convolution algebra of an action topological groupoid $X//G$ is equivalent to the $G$-equivariant topological K-theory/equivariant operator K-theory/equivariant KK-theory of $X$. Moreover, it says that this equivalences is exhibited by the analytic assembly map. (Just the injectivity of this map is related to the Novikov conjecture.)

The original version of the Baum-Connes conjecture (Baum-Connes) stated for a suitable topological group $G$ that with $X = E G \simeq \ast$ the point incarnated as the $G$-universal principal bundle with its free $G$-action the analytic assembly map (the $G$-equivariant index map)

$K_\bullet^G(E G) \stackrel{}{\to} K_\bullet(C(\mathbf{B}G)) \,,$

from the $G$-equivariant K-theory of $E G$ to the operator K-theory of the group algebra, hence the groupoid convolution algebra of the delooping groupoid $\mathbf{B}G \simeq \ast // G$, is an isomorphism.

This Baum-Connes conjecture is known to be true for

It is not known if the conjecture is true for all discrete groups.

Later the statement was generalized (Tu 99) to more general groupoids.

In (Kasparov 88) the refinement of the analytic assembly map to equivariant KK-theory is given, and called the descent map. This is of the form (recalled as Blackadar, theorem 20.6.2)

$KK^G(A,B) \to KK(G \ltimes A, G \ltimes B)$

where on the left we have $G$-equivariant KK-theory and on the right ordinary KK-theory of crossed product C*-algebras (which by the discussion there are models for the groupoid convolution algebras of $G$-action groupoids).

This is an isomorphism at least for $G$ a compact topological group and restricted to operator K-theory (hence to the first argument being $\mathbb{C}$) and for $G$ a discrete group and restricted to K-homology (hence ot the second argument being $\mathbb{C}$). In this form this is the Green-Julg theorem, see below.

## Theorems

### Green-Rosenberg-Julg theorem

The Green-Rosenberg-Julg theorem identifies equivariant K-theory with the operator K-theory of crossed product algebras.

###### Theorem

(Green-Julg theorem)

Let $G$ be a topological group acting on a C*-algebra $A$.

1. If $G$ is a compact topological group then the descent map

$KK^G(\mathbb{C}, A) \to KK(\mathbb{C},G\ltimes A)$

is an isomorphism, identifying the equivariant operator K-theory of $A$ with the ordinary operator K-theory of the crossed product C*-algebra $G \ltimes A$.

2. if $G$ is a discrete group then the descent map

$KK^G(A, \mathbb{C}) \to KK(G \ltimes A, \mathbb{C})$

is an isomorphism, identifying the equivariant K-homology of $A$ with the ordinary K-homology of the crossed product C*-algebra $G \ltimes A$.

This goes back to (Green 82), (Julg 81). A KK-theory-proof is in (Echterhoff, theorem 0.2); a textbook account is in (Blackadar, 11.7, 20.2.7). See also around (Land 13, prop. 41).

(equivariant) cohomologyrepresenting
spectrum
equivariant cohomology
of the point $\ast$
cohomology
of classifying space $B G$
(equivariant)
ordinary cohomology
HZBorel equivariance
$H^\bullet_G(\ast) \simeq H^\bullet(B G, \mathbb{Z})$
(equivariant)
complex K-theory
KUrepresentation ring
$KU_G(\ast) \simeq R_{\mathbb{C}}(G)$
Atiyah-Segal completion theorem
$R(G) \simeq KU_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {KU_G(\ast)} \simeq KU(B G)$
(equivariant)
complex cobordism cohomology
MU$MU_G(\ast)$completion theorem for complex cobordism cohomology
$MU_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {MU_G(\ast)} \simeq MU(B G)$
(equivariant)
algebraic K-theory
$K \mathbb{F}_p$representation ring
$(K \mathbb{F}_p)_G(\ast) \simeq R_p(G)$
Rector completion theorem
$R_{\mathbb{F}_p}(G) \simeq K (\mathbb{F}_p)_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {(K \mathbb{F}_p)_G(\ast)} \!\! \overset{\text{Rector 73}}{\simeq} \!\!\!\!\!\! K \mathbb{F}_p(B G)$
(equivariant)
stable cohomotopy
$K \mathbb{F}_1 \overset{\text{Segal 74}}{\simeq}$ SBurnside ring
$\mathbb{S}_G(\ast) \simeq A(G)$
Segal-Carlsson completion theorem
$A(G) \overset{\text{Segal 71}}{\simeq} \mathbb{S}_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {\mathbb{S}_G(\ast)} \!\! \overset{\text{Carlsson 84}}{\simeq} \!\!\!\!\!\! \mathbb{S}(B G)$

## References

### Introductions and surveys

Introductions and surveys include

Textbook discussion:

### Original articles

The original article is

Proof of the conjecture for hyperbolic groups is in

• Vincent Lafforgue, La conjecture de Baum-Connes à coefficients pour les groupes hyperboliques, Journal of Noncommutative Geometry, Volume 6, Issue 1, 2012, pages 1-197 (arXiv:1201.4653)

Technical subtleties are discussed in

The generalization to Lie groupoids is due to

Proofs for some cases are in

KK-theory tools and the descent map are introduced in

• Gennady Kasparov, Equivariant KK-theory and the Novikov conjecture, Inventiones Mathematicae, vol. 91, p.147, 1988(web)

The “Green-Julg theorem” for commutative algebra and finite group is due to Michael Atiyah, for commutative algebra and general group due to

• P. Green, Equivariant if-theory and crossed product C- algebras_, pp. 337-338 in Operator algebras and applications (Kingston, Ont., 1980), vol. 1, edited by R. V. Kadison, Proc. Sympos. Pure Math. 38, Amer. Math. Soc, Providence, 1982.

and the general case is due to an unpublished result by Green and Jonathan Rosenberg and independently due to

• P. Julg, K-theorie equivariante et produits croises, C. R. Acad. Sci. Paris Ser. I Math. 292:13 1981, 629-632.

Further discussion is in

• Siegfried Echterhoff, The Green-Julg theorem pdf
• Walther Paravicini, A generalised Green-Julg theorem for proper groupoids and Banach algebras, (arXiv:0902.4365)

• V. Lafforgue, pdf

A modification of the Baum-Connes conjecture with coefficient where many counterexamples (to the conjecture with coefficients) are eliminated is in

• Paul Baum, Erik Guentner, Rufus Willett, Expanders, exact crossed products, and the Baum-Connes conjecture, arxiv/1311.2343

Discussion in terms of localization/homotopy theory is in

• Ralf Meyer, Ryszard Nest, The Baum-Connes conjecture via localisation of categories, Topology 45 (2006), no. 2, 209–259.

Last revised on March 14, 2021 at 11:32:28. See the history of this page for a list of all contributions to it.