# Baum–Connes conjecture

In mathematics, specifically in operator K-theory, the Baum–Connes conjecture suggests a link between the K-theory of the C*-algebra of a group and the K-homology of the corresponding classifying space of proper actions of that group. The conjecture sets up a correspondence between different areas of mathematics, with the K-homology being related to geometry, differential operator theory, and homotopy theory, while the K-theory of the reduced ${\displaystyle C^{*}}$-algebra is a purely analytical object.

The conjecture, if true, would have some older famous conjectures as consequences. For instance, the surjectivity part implies the Kadison–Kaplansky conjecture for a discrete torsion-free group, and the injectivity is closely related to the Novikov conjecture.

The conjecture is also closely related to index theory, as the assembly map ${\displaystyle \mu }$ is a sort of index, and it plays a major role in Alain Connes' noncommutative geometry program.

The origins of the conjecture go back to Fredholm theory, the Atiyah–Singer index theorem and the interplay of geometry with operator K-theory as expressed in the works of Brown, Douglas and Fillmore, among many other motivating subjects.

## Formulation

Let Γ be a second countable locally compact group (for instance a countable discrete group). One can define a morphism

${\displaystyle \mu \colon RK_{*}^{\Gamma }({\underline {E\Gamma }})\to K_{*}(C_{\lambda }^{*}(\Gamma )),}$

called the assembly map, from the equivariant K-homology with ${\displaystyle \Gamma }$-compact supports of the classifying space of proper actions ${\displaystyle {\underline {E\Gamma }}}$ to the K-theory of the reduced C*-algebra of Γ. The index * can be 0 or 1.

The assembly map μ is an isomorphism.

As the left hand side tends to be more easily accessible than the right hand side, because there are hardly any general structure theorems of the ${\displaystyle C^{*}}$-algebra, one usually views the conjecture as an "explanation" of the right hand side.

The original formulation of the conjecture was somewhat different, as the notion of equivariant K-homology was not yet common in 1982.

In case ${\displaystyle \Gamma }$ is discrete and torsion-free, the left hand side reduces to the non-equivariant K-homology with compact supports of the ordinary classifying space ${\displaystyle B\Gamma }$ of ${\displaystyle \Gamma }$.

There is also more general form of the conjecture, known as Baum–Connes conjecture with coefficients, where both sides have coefficients in the form of a ${\displaystyle C^{*}}$-algebra ${\displaystyle A}$ on which ${\displaystyle \Gamma }$ acts by ${\displaystyle C^{*}}$-automorphisms. It says in KK-language that the assembly map

${\displaystyle \mu _{A,\Gamma }\colon RKK_{*}^{\Gamma }({\underline {E\Gamma }},A)\to K_{*}(A\rtimes _{\lambda }\Gamma ),}$

is an isomorphism, containing the case without coefficients as the case ${\displaystyle A=\mathbb {C} }$.

However, counterexamples to the conjecture with coefficients were found in 2002 by Nigel Higson, Vincent Lafforgue and Georges Skandalis, basing on not universally accepted, as of 2008, results of Gromov on expanders in Cayley graphs. Even provided validity of Higson, Lafforgue & Skandalis, conjecture with coefficients remains an active area of research, since it is, not unlike the classical conjecture, often seen as a statement concerning particular groups or class of groups.

## Examples

Let ${\displaystyle \Gamma }$ be the integers ${\displaystyle \mathbb {Z} }$. Then the left hand side is the K-homology of ${\displaystyle B\mathbb {Z} }$ which is the circle. The ${\displaystyle C^{*}}$-algebra of the integers is by the commutative Gelfand–Naimark transform, which reduces to the Fourier transform in this case, isomorphic to the algebra of continuous functions on the circle. So the right hand side is the topological K-theory of the circle. One can then show that the assembly map is KK-theoretic Poincaré duality as defined by Gennadi Kasparov, which is an isomorphism.

Another simple example is given by compact groups. In this case, both sides identify naturally with the complex representation ring ${\displaystyle R(K)}$ of the group in such a way that the assembly map becomes the identity.

## Results

The conjecture without coefficients is still open, although the field has received great attention since 1982. The conjecture is proved for the following classes of groups:

Injectivity is known for a much larger class of groups thanks to the Dirac-dual-Dirac method. This goes back to ideas of Michael Atiyah and was developed in great generality by Gennadi Kasparov in 1987. Injectivity is known for the following classes:

• Discrete subgroups of connected Lie groups or virtually connected Lie groups.
• Discrete subgroups of p-adic groups.
• Bolic groups (a certain generalization of hyperbolic groups).
• Groups which admit an amenable action on some compact space.

The simplest example of a group for which it is not known whether it satisfies the conjecture is ${\displaystyle SL_{3}(\mathbb {Z} )}$.

## References

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