About: Baum–Connes conjecture

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

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rdf:type	yago:WikicatConjectures yago:Abstraction100002137 yago:Cognition100023271 yago:Concept105835747 yago:Content105809192 yago:Hypothesis105888929 yago:Idea105833840 yago:PsychologicalFeature100023100 yago:Speculation105891783
rdfs:label	Baum–Connes conjecture (en) Conjecture de Baum-Connes (fr)
rdfs:comment	In mathematics, specifically in operator K-theory, the Baum–Connes conjecture suggests a link between the K-theory of the reduced C-algebra of a group and the K-homology of the classifying space of proper actions of that group. The conjecture sets up a correspondence between different areas of mathematics, with the K-homology of the classifying space being related to geometry, differential operator theory, and homotopy theory, while the K-theory of the group's reduced C-algebra is a purely analytical object. (en) En mathématiques, plus précisément en (en), la conjecture de Baum-Connes suggère un lien entre la K-théorie de la C*-algèbre d'un groupe et la (en) de l'espace classifiant les actions propres de ce groupe. Elle propose ainsi une correspondance entre deux objets mathématiques de nature différente, la K-homologie étant liée à la géométrie, à la théorie des opérateurs différentiels et à la théorie de l'homotopie, tandis que la K-théorie de la (en) est un objet purement analytique. (fr)
dcterms:subject	Conjectures Unsolved problems in mathematics Surgery theory K-theory C*-algebras
Wikipage page ID	11774498 (xsd:integer)
Wikipage revision ID	984441135 (xsd:integer)
Link from a Wikipage to another Wikipage	Presentation of a group Hyperbolic group Vincent Lafforgue Indefinite orthogonal group Conjectures Unsolved problems in mathematics Surgery theory Mathematics Novikov conjecture Second-countable space Classifying space Morphism Operator K-theory Antony Wassermann CAT(k) space K-theory Tree (graph theory) Haagerup property K-homology Linear algebraic group Local field Locally compact group Poincaré duality Alain Connes Amenable group Fourier transform P-adic number Differential operator Discrete group Isomorphism KK-theory Kaplansky's conjectures Kazhdan's property (T) Atiyah–Singer index theorem Coxeter group C-algebras Georges Skandalis Homotopy theory Assembly map Fredholm theory Group theory Michael Atiyah Paul Baum (mathematician) Nigel Higson Noncommutative geometry Index theory Proper action A-T-menability Reduced C-algebra dbr:Gennadi_Kasparov
Link from a Wikipage to an external page	https://web.archive.org/web/20080221181939/http:/www.math.ist.utl.pt/~matsnev/BCexpository.pdf
sameAs	Baum–Connes conjecture Baum–Connes conjecture Baum–Connes conjecture Baum–Connes conjecture
dbp:wikiPageUsesTemplate	dbt:Citation
has abstract	In mathematics, specifically in operator K-theory, the Baum–Connes conjecture suggests a link between the K-theory of the reduced C-algebra of a group and the K-homology of the classifying space of proper actions of that group. The conjecture sets up a correspondence between different areas of mathematics, with the K-homology of the classifying space being related to geometry, differential operator theory, and homotopy theory, while the K-theory of the group's reduced 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 discrete torsion-free groups, and the injectivity is closely related to the Novikov conjecture. The conjecture is also closely related to index theory, as the assembly map 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. (en) En mathématiques, plus précisément en (en), la conjecture de Baum-Connes suggère un lien entre la K-théorie de la C*-algèbre d'un groupe et la (en) de l'espace classifiant les actions propres de ce groupe. Elle propose ainsi une correspondance entre deux objets mathématiques de nature différente, la K-homologie étant liée à la géométrie, à la théorie des opérateurs différentiels et à la théorie de l'homotopie, tandis que la K-théorie de la (en) est un objet purement analytique. La conjecture, si elle était vraie, aurait pour conséquences quelques célèbres conjectures antérieures. Par exemple, la partie surjectivité implique la conjecture de Kadison-Kaplansky pour un groupe discret sans torsion et la partie injectivité est étroitement liée à la (en). La conjecture est aussi très liée à la (en) (car l'application d'assemblage µ est une sorte d'indice) et joue un rôle majeur dans le programme de géométrie non commutative d'Alain Connes. Les origines de la conjecture remontent à la (en), au théorème de l'indice d'Atiyah-Singer, et à l'interaction entre la géométrie et la K-théorie des opérateurs telle qu'elle est formulée dans les travaux de Brown, Douglas et Fillmore, parmi bien d'autres sujets motivants. (fr)
prov:wasDerivedFrom	wikipedia-en:Baum–Connes_conjecture?oldid=984441135&ns=0
page length (characters) of wiki page	7789 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Baum–Connes_conjecture
is Link from a Wikipage to another Wikipage of	Scalar curvature List of conjectures Vincent Lafforgue Indira Chatterji Geometric group theory Antony Wassermann Haagerup property Alain Connes Baum-Connes conjecture Farrell–Jones conjecture KK-theory Georges Skandalis Jean-Benoît Bost Thompson groups Paul Baum (mathematician) Unifying theories in mathematics Guoliang Yu List of unsolved problems in mathematics Random group

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