Math Conjectures

Baum–Connes conjecture without coefficients

OPENLandmarkConjectureProposed 1982 · Standard version

Canonical statement

For every countable discrete group GG, the analytic assembly map
μG:K∗G(E‾G)⟶K∗ ⁣(Cr∗(G)) \mu_G:K_*^G(\underline EG)\longrightarrow K_*\!\left(C_r^*(G)\right)
is an isomorphism. Here E‾G=EFinG\underline EG=E_{\mathrm{Fin}}G is the terminal GG-CW complex whose HH-fixed-point space is contractible for finite subgroups H≤GH\le G and empty for infinite HH; K∗GK_*^G is equivariant topological KK-homology with GG-compact supports; and Cr∗(G)C_r^*(G) is the operator-norm closure of the left regular representation of CG\mathbb CG on ℓ2(G)\ell^2(G).
View source LaTeX
For every countable discrete group \(G\), the analytic assembly map \[ \mu_G:K_*^G(\underline EG)\longrightarrow K_*\!\left(C_r^*(G)\right) \] is an isomorphism. Here \(\underline EG=E_{\mathrm{Fin}}G\) is the terminal \(G\)-CW complex whose \(H\)-fixed-point space is contractible for finite subgroups \(H\le G\) and empty for infinite \(H\); \(K_*^G\) is equivariant topological \(K\)-homology with \(G\)-compact supports; and \(C_r^*(G)\) is the operator-norm closure of the left regular representation of \(\mathbb CG\) on \(\ell^2(G)\).
ANAL-FA-017:[BaumConnesHigson1994Classifying][HigsonKasparov2001ETheory][HigsonLafforgueSkandalis2002Counterexamples][GomezAparicioJulgValette2019BaumConnesSurvey]
Analysis and functional analysisGeometry and topologyAlgebra and representation theory

Notes

The assembly map is an isomorphism for amenable and Haagerup groups, hyperbolic groups, and many Lie groups and lattices, but not for every countable discrete group. Counterexamples to the stronger conjecture with arbitrary coefficient C∗C^*-algebras do not settle this coefficient-free statement.
The coefficient algebra here is only C\mathbb C. This record excludes the Baum--Connes conjecture with arbitrary coefficients and the coarse Baum--Connes conjecture.

References (4)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.