TOPOLOGICAL INVARIANTS OF ELLIPTIC OPERATORS. I:K-HOMOLOGY

Noncommutative geometry of foliations

“…This notion is a functional-analytic abstraction of the notion of elliptic pseudodifferential operator on a compact manifold. The idea that abstract elliptic operators can be naturally considered as elements of the K-homology groups, suggested by Atiyah [2], was realized by Kasparov (see [110]). …”

A history and survey of the Novikov conjecture

“…By analyzing the Khomology class of the signature operator on a non-simply connected manifold, one could hope to redo what Lusztig had done, but in a more powerful setting. Kasparov's earliest results in this direction, as well as the first announcements of his results on the Novikov Conjecture, appeared in [Kas1] and [Kas2], although the power of his methods did not become clear until the development of the "KK calculus" in [Kas3]. (For more informal expositions, see also [Black] and [Fack1].)…”

On the quantization of conjugacy classes

“…The element (µ, l) ∈ H 3 G (Ψ) is realized by the Morita trivialization (22). Now let C be the conjugacy class of exp(ξ), and Φ : C → G the inclusion.…”

“…In this Section we review Kasparov's definition of K-homology [23,22] for C * -algebras. Excellent references for this material are the books by Higson-Roe [19] and Blackadar [5].…”