An outstanding and readable, precise presentation of noncommutative geometry/non-commutative/non-Abelian algebraic topology application topics related to the Baum-Connes conjecture. Authors' description of these lecture notes as a verbatim quote: "The Baum-Connes conjecture has been an outstanding focal point of noncommutative geometry for over twenty years. The first part of this lecture course will show how to apply new ideas from homological algebra and homotopy theory to understand the conjecture more conceptually and to widen its scope towards locally compact quantum groups. The second part of the course will be devoted to applications of the Baum-Connes conjecture and its relations to classical topology." The following topics are covered in the free, online PDF download: Categorical aspects of Kasparov's KK-theory; Duality in Kasparov theory; Homological algebra in triangulated categories ; The Baum-Connes assembly map for locally compact groups ; Dirac-dual-Dirac method ; The Baum-Connes conjecture for discrete quantum groups that are duals of compact groups ; Towards the Baum-Connes conjecture for locally compact quantum groups ; Corollaries of the Baum-Connes conjecture ; Baum-Douglas model for K-homology ; Equivariant-bivariant Chern character; Geometric KK-theory ; 1 Noncommutative algebraic topology 3 1.1 What is noncommutative (algebraic) topology? 3 1.1.1 Kasparov KK-theory 5 1.1.2 Connection between abstract and concrete description . 6 1.1.3 Relation with K-theory p. 8 1.2 Equivariant theory p. 9 1.2.1 Tensor products p. 10 1.3 KK as triangulated category 12 1.4 Axioms of a triangulated categories p.14 1.5 Localisation of triangulated categories p.22 1.6 Index maps in K-theory and K-homology p.35 1.7 Mayer-Vietoris sequences p. 36 1.8 Localisation of functors p.39 1.9 Towards an analogue of the Baum-connes conjecture for quantum groups p.40 1.10 Quantum groups p.42 1.11 The Baum-Connes conjecture p. 44 Prerequisites: Basic knowledge of K-theory of C*-algebras and elementary category theory. (http://toknotes.mimuw.edu.pl/sem8/index.html) http://toknotes.mimuw.edu.pl/sem8/files/Meyer_bcclcqg.pdf
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Open acess with free PDF downloads at: http://toknotes.mimuw.edu.pl/sem8/index.html http://toknotes.mimuw.edu.pl/sem8/files/Meyer_bcclcqg.pdf
