Baum-Connes Conjecture

General Idea

There are two versions of the conjecture: without and with coefficient, the analytic assembly maps being, respectively $$ \mu^G:RK^G_\ast(\mathcal{E}G)\to K_\ast(C^\ast_r(G)) $$ and $$ \mu^{G,A}: R\operatorname{KK}_\ast^G(\mathcal{E}G,A)\to K_\ast(A\rtimes_r G). $$ where $G$ is a second-countable locally compact group and $A$ is a $C^\ast$-algebra on which $G$ acts by automorphisms. The conjecture with coefficient is in general false, but still interesting for classes of groups.

Rough introduction

Group $C^\ast$-algebras

Given a locally compact group $G$, consider the convolution algebra $L^1(G)$ of integrable complex-valued functions on $G$. There is a natural involution on $L^1(G)$ that makes it into a Banach $\ast$-algebra. The non-degenerate $\ast$-representations of $L^1(G)$ on Hilbert space are in 1-to-1 correspondence with the unitary representations of $G$. The group $C^\ast$-algebra $C^\ast(G)$ is the enveloping $C^\ast$-algebra of $L^1(G)$. Its representations are also in 1-to-1 correspondence with the unitary representations of $G$.

When the unitary dual $\hat{G}$ is viewed as a topological space, the group $C^\ast$-algebra $C^\ast(G)$ is a good object to study. When $G$ is abelian, the Fourier transform provides an isomorphism from $C^\ast(G)$ to the commutative $C^\ast$-algebra of continuous functions on the Pontrjagin dual vanishing at infinity, so $C^\ast(G)$ captures precisely the topological structure of $\hat{G}$. For nonabelian $G$, in the view point of noncommutative geometry, the noncommutative $C^\ast(G)$ is thought of as the algebra of continuous functions on the noncommutative space $\hat{G}$ of unitary representations of $G$.

K-theory and index theory

The $K$-theory groups of a $C^\ast$-algebra $A$ are defined in such a way that if $A = C_0(X)$, the $C^\ast$-algebra of continuous complex-valued functions vanishing at infinity on a locally compact space $X$, then $K_j(A) = K^{-j}(X)$, the Atiyah-Hirzebruch topological $K$-theory group. For a locally compact group $G$ abelian, the Fourier isomorphism guarentees that $K_j(C^\ast(G))\cong K^{-j}(\hat{G})$, and for noncommutative spaces $\hat{G}$ (where $G$ is nonabelian), $K_j(C^\ast(G))$ is seen as a substitute for $K^{-j}(\hat{G})$.

Let $M$ be a smooth closed manifold and $D$ an elliptic partial differential operator on $M$. The operator $D$ has an integer-valued Fredholm index. A more refined index, the equivariant index, valued in $K_0(C^\ast(G))$ can be defined if $\pi_1(M)$ is provided with a homomorphism into a discrete group $G$. Let $\tilde{M}$ be the universal covering. The quotient of $\tilde{M}\times C^\ast(G)$ by the diagonal action of $\pi_1(M)$ is a flat bundle over $M$ whose fibers are finitely-generated projective modules over $C^\ast(G)$. Let $D_G$ denote the canonical lifting of $D$ to the flat bundle, then $\operatorname{Ker}(D_G)$ and $\operatorname{Coker}(D_G)$ are both $C^\ast(G)$-modules, and usually they are finitely generated projective modules (if not under some perturbation they can be made to become finitely generated and projective). Now define $$ \operatorname{Index}_G(D) = [\operatorname{Ker}(D_G)] - [\operatorname{Coker}(D_G)] \in K_0(C^\ast(G)). $$ Associated to the regular representation of $G$ the algebra $C^\ast(G)$ has a natural trace $\tau$, and it can be shown that $\tau(\operatorname{Index}_G(D)) = \operatorname{Index}(D)$.

For $G$ finite $\operatorname{Index}_G(D)$ doesn’t contain more information aside for those contained in the Fredholm index $\operatorname{Index}(D)$, while for $G$ infinite it can contain a good deal more. These is a proposition:

Proposition. The $G$-index of the Dirac operator on a closed spin manifold vanishes if the manifold ahs positive scalar curvature. The $G$-index of the signature operator on an oriented manifold is an oriented homotopy invariant.

The assembly map

A Dirac operator on a closed manifold $M^n$ and a map $M^n\to X$ determines a class in the $K$-homology group $K_n(X)$. Baum realized $K_n(X)$ as equivalence classes of triples $(M,E,f)$ where $M$ is a closed $\operatorname{Spin}^c$-$n$-manifold, $E$ a complex vector bundle over $M$, and $f:M\to X$ a continuous map, and the equivalence relation involves direct sum, bordism, relation related to the multiplicativity of the index of elliptic operators on fiber bundles, etc.

If $X=BG$, then a trile $(M^{2n},E,f)$ has an index in $K_0(C^\ast(G))$. Form the Dirac operator on $M$ with coefficients in $E$ and take its $G$-index along the map $\pi_1(M)\to G$ induced from $f$. The index depends only on the equivalence class of $(M,E,f)$ and with a related construction for odd-dimensional manifolds it defines a map $$ \mu:K_\ast(BG)\to K_\ast(C^\ast(G)) $$ called the *assembly map$, so-called because of its connection with the assembly map of surgery theory. There is a conjecture

Conjecture(Strong Novikov conjecture) . The assembly map $\mu:K_\ast BG \to K_\ast(C^\ast(G))$ is rationally injective.

The Baum-Connes conjecture

The Baum-Connes conjecture expresses the idea that every class in the $K$-theory of a group $C^\ast$-algebra is an index, and that the only relations among elements are the natural relations (such as bordism) among index theory problems.

The reduced $C^\ast(G)$-algebra $C^*_r(G)$ of $G$ is the completion of $L^1(G)$ in its regular representation as bounded operators on $L^2(G)$. $C^\ast(G)\cong C^\ast_r(G)$ iff $G$ is amenable, and $\hat{G}_r \subset \hat{G}$ is the closed subset compriesd of those unitary representations which are weakly contained in the regular representation. For a finite subgroup $H\subseteq G$ of any discrete group $G$, its contribution to $C^\ast(G)$ is a projection $\frac{1}{|H|}\sum_{h\in H}[h]$, whose $K$-theory class is not in the image of the assembly, hence upon restriction to torsion-free groups

Conjecture(Baum-Connes for torsion-free groups) . Let $G$ be a discrete, torsion-free group, then the reduced assembly $$ \mu_r:K_\ast(BG)\to K_\ast(C^\ast_r(G)) $$ obtained from the assembly map $\mu$ using the regular representation $C^\ast(G)\to C^\ast_r(G)$ is an isomorphism.

FOr general second-countable locally compact groups, the statement of the conjecture uses Kasparov’s equivariant $KK$-theory. Associated to any $G$ there is a universal proper $G$-space $\mathcal{E}G$ such that any other proper $G$-space maps uniquely, up to equivariant homotopy, into $\mathcal{E}G$. For $G$ discrete and torsion free $\mathcal{E}G = EG$ the universal principal space, and the equivariant $K$-homology $RK^G_\ast(\mathcal{E}G) = K_\ast(BG)$, and for general $G$ there is an assembly $$ \mu_r : RK^G_\ast(\mathcal{E}G)\to K_\ast(C^\ast_r(G)). $$ Where $RK_\ast = \lim_{X \text{ compact in }\mathcal{E}G} K_i(X)$. A cycle for $RK^G_ast(\mathcal{E}G)$ is an ‘abstract’ elliptic operator $D$ on a proper $G$-space, and $\mu_r$ associates to $D$ its equivariant index.

Conjecture(Baum-Connes) . For $G$ a second countable locally compact group, the assembly map $\mu_r$ is an isomorphism.

Remarks

Higson:

A major obstacle to further progress is the lack of a full understanding of the relationship between harmonic analysis and the Baum-Connes conjecture. It seems likely that underly- ing the conjecture is an as yet unknown governing principle of harmonic analysis. But the conjecture has not drawn the attention of harmonic analysts the way it has the topologists, and this issue remains largely unexamined.

Relation to group representation theory

Let $G$ be a connected Lie group and $H$ its maximal compact subgroup. The homogeneous space $G/H$ is the universal proper $G$-space, $EG$.

[Under Construction]

Fourier-Mukai transform

For free abelian groups the conjecture amounts to a $K$-theoretic form of Fourier-Mukai duality.

See

Recall that crossed products by $\mathbb{R}^n$ are strict quantization in the sense of Rieffel. Quantization as a $G$-equivariant index also suggests that Fourier-Mukai and Baum-Connes Might be related to quantization through

Relation to foliations

Let $\mathcal{F}$ be a $C^\infty$-foliation on a manifold $M$. $M/\mathcal{F}$ is a bad quotient generally. To obtain information concerning the transverse structure of the foliation, other types of objects rather than the quotient should be used. Some of them are

  1. The dynamics of $\mathcal{F}$ is described by its holonomy groupoid $\mathcal{G}$. The groupoid can be considered as a desingularization of the leaf space $M/\mathcal{F}$. To $\mathcal{G}$ assocaite a $C^\ast$-algebra $C^\ast_r(\mathcal)$ of functions, interpreted as the “space of continuous functions vanishing at infinity” on $M/\mathcal{F}$. The analytic $K$-theory of the leaf space $K_a(M/\mathcal{F}$ is defined as $K_\ast(C^\ast_r(\mathcal{G})$.
  2. Construct a classifying space $B\mathcal{G}$ for $\mathcal{G}$, which is in general not a manifold, where $\mathcal{G}$ acts freely and properly. The space $B\mathcal{G}$ can be seen as the leaf space modulo homotopy. The $\mathcal{G}$-equivariant $K$-theory $K_{\ast,\tau}(B\mathcal{G})$ associated with this object is called the topological $K$-theory $K_{\text{top}}(M/\mathcal{F})$.

Elliptic operators provide a map between these $K$-theory groups, $$ \mu: K_{\text{top}}(M/\mathcal{F}) \to K_a(M/\mathcal{F}) $$ and the Baum-Connes conjecture asserts that $\mu$ is a group isomorphism if the holonomy groups are torsion-free.

More

Index theoretic interpretation,

Construction of the geometric Baum-Connes assembly map for twisted Lie groupoids, and the proof of the Morita invariance of the assembly map, giving thus a precise meaning to the geometric assembly map for twisted differentiable stacks.