Baum–Connes conjecture

Baum–Connes conjecture
Renate Schmid · CC BY-SA 2.0 de · source
Name	Baum–Connes conjecture
Field	Noncommutative geometry; Operator K-theory
Proposed by	Paul Baum; Alain Connes
Year	1982
Status	Partially proved; counterexamples with coefficients

Baum–Connes conjecture is a conjecture in noncommutative geometry and operator K-theory proposing an isomorphism between topological K-homology of a classifying space for proper actions of a locally compact group and the K-theory of the reduced group C*-algebra. It connects ideas from Paul Baum, Alain Connes, Gennadi Kasparov, Michael Atiyah and Isadore Singer with analytic techniques of C*-algebra theory and index theory on manifolds such as those studied by Stephen Smale and René Thom. The conjecture has driven research involving figures and institutions including George Mackey, Victor Kac, Jean-Louis Verdier, Max Karoubi, Nigel Higson, Gennady Lyubarskii, Andrei Suslin, and organizations like the Institute for Advanced Study, Mathematical Sciences Research Institute, Clay Mathematics Institute, and American Mathematical Society.

Statement of the conjecture

The conjecture asserts that for a second countable, locally compact group G the assembly map from the equivariant K-homology group K^G_*(\u{E}G) of the universal space for proper G-actions to the K-theory group K_*(C^*_r(G)) of the reduced group C*-algebra C^*_r(G) is an isomorphism. This statement intertwines constructions of Gennadi Kasparov such as KK-theory, the Dirac element and dual-Dirac element with index-theoretic invariants inspired by Michael Atiyah and Isadore Singer. The assembly map uses analytic induction procedures connected to representations studied by George Mackey and spectral considerations found in the work of Kiyoshi Itô and John von Neumann.

Historical background and motivation

Motivated by the index theorem of Michael Atiyah and Isadore Singer and by efforts in noncommutative topology led by Alain Connes, the conjecture was formulated to generalize classical index theory to arbitrary groups. Early precursor ideas appear in the study of group actions on manifolds by Hermann Weyl and representation-theoretic frameworks of Harish-Chandra and Eugène Wigner. The formulation leveraged KK-theory developed by Gennadi Kasparov and ideas from Paul Baum and Alain Connes to relate topology of classifying spaces studied by John Milnor and Raoul Bott to operator algebras investigated by Israel Gelfand and Mark Naimark. Seminal conferences where the conjecture gained attention include workshops at Institut des Hautes Études Scientifiques, Centre International de Rencontres Mathématiques, and Simons Foundation programs.

Known cases and results

The conjecture is known for large classes of groups: for a-T-menable groups (also called groups with the Haagerup property) due to work of Nigel Higson, Gennadi Kasparov, and Guoliang Yu; for word-hyperbolic groups following contributions by Marius Gromov, Vincent Lafforgue and Beverly Jones; for virtually cyclic groups and many crystallographic groups tied to theorems by John Milnor and Armand Borel; and for discrete subgroups of Lie groups with suitable properties via methods related to George Lusztig and David Vogan. Positive results include the proof for amenable groups from work of Hervé Oyono-Oyono and other contributors, and special cases handled using methods of Paul-Emile Paradan, Boris Tsygan and Ryszard Nest. Computations of K-theory for specific groups leverage ideas from the Baum–Connes program alongside techniques introduced by Jean-Pierre Serre, Atle Selberg, and Masayoshi Nagata.

Counterexamples and obstructions

Counterexamples with coefficients were constructed using expander graphs and property (T) groups, involving contributions by Mikhael Gromov, Willem de Laat, Tim de Laat, Piotr Nowak, and Gilles Pisier. These constructions draw on combinatorial objects studied by Jean Bourgain and analytic concepts related to Margulis expander families and the Kazhdan's property (T) work of David Kazhdan. Obstructions are linked to failures of local-to-global principles seen in examples from Boris Goldfarb and to rigidity phenomena investigated by Fisher Margulis and Grigori Margulis. Distinctions between the reduced and maximal group C*-algebras, illuminated by examples from Alain Connes and Nathanial Brown, clarify where the assembly map may fail.

Methods and approaches

Techniques include Kasparov's KK-theory and the Dirac-dual Dirac method of Gennadi Kasparov; coarse geometry and controlled K-theory methods of John Roe and Guoliang Yu; and Lafforgue's strong Banach property (T) framework introduced by Vincent Lafforgue. Analytical tools draw on cyclic cohomology of Alain Connes and index-theoretic methods developed by Michael Atiyah and Isadore Singer. Operator-algebraic deformation techniques relate to work by Elliott Classification Program contributors such as George Elliott and Andrew Toms, while geometric group theory strategies deploy ideas from Mikhail Gromov and Cornelia Drutu. Computational K-theory and assembly map analyses use spectral sequence approaches inspired by Jean Leray and categorical frameworks from Alexandre Grothendieck.

Applications and implications

When valid, the conjecture implies the Novikov conjecture on homotopy invariance of higher signatures proven for classes of groups by Shmuel Weinberger, Ferry Pedersen, and Markus Matschke. It impacts the classification of manifolds considered by William Thurston and the study of positive scalar curvature metrics developed by Rosenberg Jonathan and John Roe. Consequences extend to rigidity theorems in the style of Mostow Rigidity and to computations in algebraic topology influenced by Edwin Spanier and Samuel Eilenberg. The Baum–Connes framework informs quantum field theoretic models studied by Edward Witten and noncommutative space considerations of Alain Connes in mathematical physics contexts encountered at institutes like CERN and Perimeter Institute.

Related conjectures and developments

Related problems include the Farrell–Jones conjecture formulated by F. T. Farrell and L. E. Jones in algebraic K- and L-theory; the Novikov conjecture connected to work of Sergei Novikov; and the Kadison–Kaplansky conjecture associated with Irving Kaplansky and Richard Kadison. Developments in geometric group theory by Mikhail Gromov and operator algebra classification by George Elliott continue to influence progress, as do advances in coarse geometry by John Roe and controlled topology methods from William Browder. Ongoing research programs at Institute for Advanced Study, Mathematical Sciences Research Institute, and university groups led by Nigel Higson, Guoliang Yu, Gennadi Kasparov, and Vincent Lafforgue pursue refined formulations, counterexamples, and relations to conjectures from Alain Connes and Shmuel Weinberger.

Category:Conjectures in mathematics