The Baum–Connes Conjecture and Assembly¶

Job: state the Baum–Connes assembly map $\mu:K^\bullet_G(\underline{EG})\to K_\bullet(C^*_r(G))$ from equivariant $K$-homology of the classifying space for proper actions to the $K$-theory of the reduced group $C^*$-algebra, the conjecture that $\mu$ is an isomorphism, the proven cases (a-T-menable / Haagerup, hyperbolic, $\mathrm{SL}(n,\mathbb{R})$), the Higson–Lafforgue–Skandalis counterexamples with coefficients, and the descent to higher indices (Novikov, Gromov–Lawson). Worked example: $G=\mathbb{Z}$ as the Fourier-dual $S^1$ picture.

Core¶

The assembly map¶

Let $G$ be a locally compact second countable group (the construction extends to groupoids and quantum groups via KK-theory). The classifying space for proper $G$-actions $\underline{EG}$ is a proper $G$-CW-complex, terminal among proper $G$-spaces (for discrete $G$, a $G$-CW complex with finite stabilizers; for $G=\mathbb{R}^n$, a point; for $G=\mathbb{Z}$, $\mathbb{R}$). It is not $BG$ (the classifying space for free $G$-actions) — they differ for groups with torsion or for non-discrete $G$.

The equivariant $K$-homology $K^\bullet_G(\underline{EG})$ is the geometric / topological side: cycles are $G$-equivariant generalized elliptic operators on proper $G$-manifolds (more precisely: $G$-equivariant $K$-homology with compact supports via Kasparov's $KK$-theory, $K^\bullet_G(\underline{EG})=\varinjlim KK_G^\bullet(C_0(\underline{EG}),\mathbb{C})$).

The analytic side is $K_\bullet(C^*_r(G))$, the $K$-theory of the reduced group $C^*$-algebra $C^*_r(G)$ (closure of the left-regular representation $\lambda$ of $G$ on $L^2(G)$).

The assembly map (Baum–Connes; Kasparov descent). There is a natural homomorphism $$ \mu:K^\bullet_G(\underline{EG})\longrightarrow K_\bullet(C^*_r(G)) $$ constructed via Kasparov's equivariant $KK$-theory: a $G$-equivariant $K$-homology class $[D]$ on a proper $G$-manifold $\tilde M$ descends (via the descent homomorphism $j^G:KK_G^\bullet(C_0(\tilde M),\mathbb{C})\to KK^\bullet(C_0(\tilde M)\rtimes_r G,C^*_r(G))$ and the Mishenko–Fomenko compression $\tilde M\to\underline{EG}$) to a class in $KK^\bullet(\mathbb{C},C^*_r(G))=K_\bullet(C^*_r(G))$.

For a compact group $G$, $\underline{EG}=\mathrm{pt}$ and $\mu$ is the usual index map $R(G)\to K_\bullet(C^*(G))$ (Frobenius reciprocity). For a discrete torsion-free $G$, $\underline{EG}=EG=BG$, and the domain is $K^\bullet_G(EG)=K^\bullet(BG)$.

The conjecture¶

Baum–Connes conjecture (without coefficients). For every locally compact second countable group $G$, the assembly map $\mu$ is an isomorphism.

Proven cases and obstructions¶

Class of groups	Status	Reference / method
Amenable / a-T-menable (Haagerup property)	$\mu$ iso (Higson–Kasparov)	Hilbert-space compression of $G$ into $\underline{EG}$
Connected Lie groups (e.g. $\mathrm{SL}(n,\mathbb{R})$)	$\mu$ iso (Lafforgue)	Banach $KK$-theory, "strong property T" circumvented
Gromov-hyperbolic groups	$\mu$ iso (Lafforgue)	Banach $KK$, bounded cohomology controls
One-relator groups, Coxeter groups, braid groups, mapping class groups	$\mu$ iso	via geometry (Higson, Mislin, Kasprowski, Bartels–Lück–Reich)
Groups acting on buildings, $\mathrm{CAT}(0)$-cube complexes	$\mu$ iso	Bartels–Lück–Reich, Farrell–Jones via controlled topology
Property (T) groups (general)	open (but $\mu$ known for some, e.g. $\mathrm{SL}(3,\mathbb{R})$, lattices)	Lafforgue for special cases; BC open in general
Groups acting on expanders	$\mu$ with coefficients fails (Higson–Lafforgue–Skandalis)	counterexamples to the stronger version

Obstructions to extending proofs. The geometric methods (a-T-menability) require a proper affine isometric action on Hilbert space; property (T) groups cannot admit such an action (Bekka–de la Harpe–Valette). Lafforgue's method (Banach $KK$-theory) bypasses this for hyperbolic and some property-T groups but requires control of rapid decay or Banach-KK boundedness; the general property-T case remains open.

The descent and the higher index theorem¶

Let $\tilde M\to M$ be a Galois covering with deck group $G$, $\tilde M$ a proper cocompact $G$-manifold (so $M$ compact). A $G$-equivariant elliptic operator $D$ on $\tilde M$ defines a class $[D]\in K^\bullet_G(\tilde M)$ pushed to $K^\bullet_G(\underline{EG})$, and its image under $\mu$ is the higher index $\mathrm{Ind}_G(D)\in K_\bullet(C^*_r(G))$ — the NC higher index.

Theorem (NC higher index / Mishenko–Fomenko–Kasparov). Pairing $\mathrm{Ind}_G(D)$ with the canonical trace $\tau$ on $C^*_r(G)$ (or with more refined traces, e.g. on group von Neumann algebras) recovers the ordinary $L^2$-index: $$ \langle\mathrm{Ind}G(D),\tau\rangle=\mathrm{Index} $$ (Atiyah's }(D)\in\mathbb{R$L^2$-index). For the leafwise Dirac on the universal cover $\tilde M\to M$, this is the $\hat A$-genus of $M$ when $M$ spin and $\dim\equiv0\pmod4$.

Novikov conjecture. The higher signatures $\langle L(M)\cup f^*\alpha,[M]\rangle$ for $\alpha\in H^*(BG)$ are homotopy invariants. They are computed as pairings of the higher signature operator with classes in $K^\bullet(BG)$; the injectivity of the rational assembly map $\mu\otimes\mathbb{Q}$ injective implies the Novikov conjecture (Mishenko–Kasparov).

Gromov–Lawson conjecture. For a closed spin manifold $M$ with positive scalar curvature, $\hat A(M)=0$. The assembly for the universal cover $\tilde M\to M$ (deck group $G=\pi_1 M$) gives $\hat A(M)=\langle\mathrm{Ind}_G(D\!\!\!/),\tau\rangle$, which vanishes by Roe's argument using the positivity (a positive scalar curvature estimate forces the class to vanish in $K_0(C^*_r(G))\otimes\mathbb{Q}$).

Baum–Connes with coefficients¶

For a $G$-C*-algebra $A$ there is an assembly map with coefficients $$ \mu_{A,G}:KK^\bullet_G(A,\underline{EG}\otimes A)\longrightarrow K_\bullet(A\rtimes_r G). $$

Counterexample (Higson–Lafforgue–Skandalis). There exist groups $G$ (e.g. groups acting on expander graphs, Gromov's random groups) and $G$-C*-algebras $A$ such that $\mu_{A,G}$ is not injective. Hence the Baum–Connes conjecture with coefficients is false.

The bare assembly map $\mu$ (without coefficients) is conjectured to be an isomorphism for all $G$ and has no known counterexample. State it as "the conjecture without coefficients" only; with coefficients needs the HLS caveat.

See it¶

Assembly for $G=\mathbb{Z}$. The universal proper $\mathbb{Z}$-space is $\underline{E\mathbb{Z}}=\mathbb{R}$ (the universal cover of $S^1$, with $\mathbb{Z}$ acting by translation; any proper $\mathbb{Z}$-space maps equivariantly to $\mathbb{R}$ via a Busemann-type function). Equivariant $K$-homology of $\mathbb{R}$: $$ K^0_{\mathbb{Z}}(\mathbb{R})\cong\mathbb{Z},\qquad K^1_{\mathbb{Z}}(\mathbb{R})\cong\mathbb{Z}. $$ The $K^0$ class is represented by the (lift of the) Dirac on $\mathbb{R}$ (the Bott class); the $K^1$ class by the orbit (the fundamental class of the $\mathbb{Z}$-action, realized by a shift operator / the generator $z\mapsto z$ in $C(S^1)$).

The reduced group C*-algebra is, by Pontryagin duality, $$ C^*_r(\mathbb{Z})\cong C(S^1),\qquad\lambda(n)(z)=z^n, $$ so $K_0(C(S^1))\cong\mathbb{Z}$ (rank of vector bundles over $S^1$, detected by the trace / winding at $z=1$) and $K_1(C(S^1))\cong\mathbb{Z}$ (winding number of invertible functions, detected by $\frac{1}{2\pi i}\oint u^{-1}du$).

The assembly map is the isomorphism $$ \mu:\mathbb{Z}\oplus\mathbb{Z}\xrightarrow{\ \sim\ }\mathbb{Z}\oplus\mathbb{Z},\qquad K^0_{\mathbb{Z}}(\mathbb{R})\oplus K^1_{\mathbb{Z}}(\mathbb{R})\xrightarrow{\sim}K_0(C(S^1))\oplus K_1(C(S^1)), $$ sending the Bott class to the unit $1\in C(S^1)$ (the rank-1 projection) and the orbit class to the generator $z\in C(S^1)$ (the unitary of winding number 1). This is the prototypical assembly: the $\mathbb{Z}$ case is the Fourier-dual $S^1$ picture, the base case for the Novikov conjecture for $G=\mathbb{Z}$ (where higher signatures reduce to ordinary signatures since $BG=B\mathbb{Z}=S^1$ has trivial cohomology in degrees $\ge2$). The Bott periodicity equivalence $K^\bullet(\mathrm{pt})\cong K^\bullet(\mathbb{R})$ underlies the isomorphism.

Check¶

Why does injectivity (not surjectivity) of the rational assembly map suffice for the Novikov conjecture?

The higher signatures are pairings $\langle L(M)\cup f^*\alpha,[M]\rangle=\langle\mathrm{Ind}_G(D_{\mathrm{sign}}),\,f^*\alpha\rangle$ with $\alpha\in H^\bullet(BG)$, and the index class lies in the image of $\mu\otimes\mathbb{Q}$. Homotopy invariance of $\alpha$ (it is a class on $BG$, which is homotopy invariant) plus injectivity of $\mu\otimes\mathbb{Q}$ means the higher signature is determined by the image in $K_\bullet(C^*_r(G))\otimes\mathbb{Q}$, which is functorial and hence homotopy invariant. Surjectivity is never needed.

Construct an explicit $\mathbb{Z}$-equivariant $K^0$-cycle on $\mathbb{R}$ and identify its image in $K_0(C(S^1))$.

The Dirac on $\mathbb{R}$ with the $\mathbb{Z}$-action by translation gives a $\mathbb{Z}$-equivariant elliptic operator $D=-i\,d/dx$ on $L^2(\mathbb{R})$; its $KK_\mathbb{Z}$-class $[D]\in K^0_\mathbb{Z}(\mathbb{R})$ is the Bott generator. The descent $j^\mathbb{Z}:KK_\mathbb{Z}(\mathbb{R})\to KK(C(S^1),\mathbb{C})=K^0(C(S^1))=K_0(C(S^1))$ sends $[D]$ to the class of the Mishenko bundle / the unit $1\in C(S^1)$ (the rank-1 trivial projection), which generates $K_0(C(S^1))\cong\mathbb{Z}$.

Pitfalls¶

The assembly map WITHOUT coefficients is the safe conjecture (no counterexamples); the version WITH coefficients is false (Higson–Lafforgue–Skandalis). Do not state the stronger version without the HLS caveat.
Injectivity (not surjectivity) suffices for the Novikov conjecture — do not require the full isomorphism.
$\underline{EG}$ is the classifying space for PROPER actions, not $BG$ (free actions). They differ for groups with torsion (where $BG$ has cells with finite stabilizers $\ne 1$) and for non-discrete $G$ (e.g. $G=\mathbb{R}$, where $\underline{EG}=\mathrm{pt}$ but $BG$ is not a point classifying).
a-T-menability (Haagerup property) is sufficient but not necessary — Lafforgue proves BC for some property-T groups (e.g. $\mathrm{SL}(3,\mathbb{R})$) using Banach $KK$-theory; do not conflate a-T-menability with BC.
The reduced $C^*$-algebra $C^*_r(G)$, not the full $C^*(G)$ — the assembly lands in $K_\bullet(C^*_r(G))$; the surjection $C^*(G)\twoheadrightarrow C^*_r(G)$ is generally not iso (it is iff $G$ amenable), and only the reduced version is conjectured to be iso.
For non-discrete $G$ (e.g. Lie groups), the construction uses locally compact groupoids and continuous fields; the discrete-group picture of $BG$ does not directly apply.

Refs¶

Baum–Connes, "K-theory for Lie groups and foliations" (L'Enseign. Math. 2000); Mislin–Valette, Proper Group Actions and the Baum–Connes Conjecture (MSRI 2003)
Higson–Kasparov, "E-theoretic and KK-theoretic approaches to the Baum–Connes conjecture" (2001)
Lafforgue, "K-théorie bivariante pour les algèbres de Banach et conjecture de Baum–Connes" (Invent. 2002)
Higson–Lafforgue–Skandalis, "Counterexamples to the Baum–Connes conjecture" (2002)
Bartels–Lück–Reich, "On the Farrell–Jones conjecture and its applications" (J. Topol. 2008)

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