2004/12/18 by Hela Bettaieb, Bettaieb, Hela, Michel Matthey +3 · 1 citation
Mathematics · #FOS: Mathematics #Geometric Topology (math.GT) #K-Theory and Homology (math.KT) #Operator Algebras (math.OA) #math.GT #math.KT #math.OA
paper · pdf · doi:10.48550/arxiv.math/0412364
42 pages, 3 figures
arxiv created 2004/12/18 · arxiv updated 2009/12/01
Using the unbounded picture of analytical K-homology, we associate a well-defined K-homology class to an unbounded symmetric operator satisfying certain mild technical conditions. We also establish an ``addition formula'' for the Dirac operator on the circle and for the Dolbeault operator on closed surfaces. Two proofs are provided, one using topology and the other one, surprisingly involved, sticking to analysis, on the basis of the previous result. As a second application, we construct, in a purely analytical language, various homomorphisms linking the homology of a group in low degree, the K-homology of its classifying space and the analytic K-theory of its C^*-algebra, in close connection with the Baum-Connes assembly map. For groups classified by a 2-complex, this allows to reformulate the Baum-Connes Conjecture.