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Does anyone know of any detailed proof of the Atiyah-Singer Index Theorem using KK-theory/ Kasparov products? References to any papers and textbooks are greatly appreciated. Thanks!

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    $\begingroup$ A standard reference is A. Connes and G. Skandalis paper alainconnes.org/docs/longitudinal.pdf. The paper does the index theorem for foliations as well but you can only read the first 3 chapters. $\endgroup$
    – Omar
    Commented Jan 16, 2018 at 10:31

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I find the proof in the recent paper of Kasparov

G. Kasparov. Elliptic and transversally elliptic index theory from the viewpoint of KK-theory. J. Noncommut. Geom., 10(4):1303–1378, 2016

difficult but understandable. There are certainly more details than in Kasparov's Inventiones paper.

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