Transcription of The Longitudinal Index Theorem for Foliations
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The Longitudinal Index Theorem for Foliations
The Longitudinal Index Theorem for FoliationsA. Connes and G. Skandalis Institut des Hautes Etudes Scientifiques, 35 route de Chartres, 91440 Bures-sur-Yvette, France Universit e Pierre et Marie Curie, Tour 45-46, HUER 48, 4 place Jussieu, Paris, FranceIntroductionIn this paper1, we use the bivariantKtheory of Kasparov ([19]) as a basic tool to prove theK-theoreticalversion of the Index Theorem for Longitudinal elliptic differential operators for Foliations which is stated asa problem in [10], Section 10. When the foliation is by the fibers of a fibration, this Theorem reduces tothe Atiyah-Singer Index Theorem for families ([2], ). It implies the Index Theorem for measuredfoliations ([9], Theorem , p. 136) and unlike the latter makes sense for arbitrary Foliations , not necessarilygifted with a holonomy invariant transverse Index in the Atiyah Singer Theorem for families ([2]) is an element of theK-theoryK0(B) of the basespace of the fibration.
of an immersion, one gets that in all cases the map K(X) →K(Y) given by f! coincides with the classical wrong way map in K-theory. This statement is …
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