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13
Dirac index classes and the noncommutative spectral flow
, 2003
"... We present a detailed proof of the existencetheorem for noncommutative spectral sections (see the noncommutative spectral flow, unpublished preprint, 1997). We apply this result to various indextheoretic situations, extending to the noncommutative context results of Booss– Wojciechowski, Melrose–P ..."
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Cited by 22 (5 self)
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We present a detailed proof of the existencetheorem for noncommutative spectral sections (see the noncommutative spectral flow, unpublished preprint, 1997). We apply this result to various indextheoretic situations, extending to the noncommutative context results of Booss– Wojciechowski, Melrose–Piazza and Dai–Zhang. In particular, we prove a variational formula, in K ðC n r ðGÞÞ; for the index classes associated to 1parameter family of Dirac operators on a Gcovering with boundary; this formula involves a noncommutative spectral flow for the boundary family. Next, we establish an additivity result, in K ðC * n r ðGÞÞ; for the index class defined by a Diractype operator associated to a closed manifold M and a map r: MBG when we assume that M is the union along a hypersurface F of two manifolds with boundary M Mþ,F M: Finally, we prove a defect formula for the signatureindex classes of two cutandpaste equivalent pairs ðM1; r1: M1BGÞ and ðM2; r2: M2BGÞ; where M1 Mþ, ðF;f1Þ M; M2 Mþ, ðF;f2Þ M and f jADiffðFÞ: The formula involves the noncommutative spectral flow of a suitable 1parameter family of twisted signature operators on F: We give applications to the problem of cutandpaste invariance of Novikov’s higher signatures on closed oriented manifolds.
Coarse geometry of foliations
"... We give a survey with many details of some of the recent work relating the coarse geometry of the leaves of foliations with their dynamics, index theory and spectral theory. ..."
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Cited by 6 (3 self)
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We give a survey with many details of some of the recent work relating the coarse geometry of the leaves of foliations with their dynamics, index theory and spectral theory.
THE TWISTED HIGHER HARMONIC SIGNATURE FOR FOLIATIONS
, 711
"... Abstract. We prove that the higher harmonic signature of an even dimensional oriented Riemannian foliation F of a compact Riemannian manifold M with coefficients in a leafwise U(p, q)flat complex bundle is a leafwise homotopy invariant. We also prove the leafwise homotopy invariance of the twisted ..."
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Cited by 6 (3 self)
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Abstract. We prove that the higher harmonic signature of an even dimensional oriented Riemannian foliation F of a compact Riemannian manifold M with coefficients in a leafwise U(p, q)flat complex bundle is a leafwise homotopy invariant. We also prove the leafwise homotopy invariance of the twisted higher Betti classes. Consequences for the Novikov conjecture for foliations and for groups are investigated. 1.
Available via WWW.ESI.AC.AT INDEX FORMULAS FOR GEOMETRIC DIRAC OPERATORS IN RIEMANNIAN FOLIATIONS
, 1994
"... Abstract. With regards to certain Riemannian foliations we consider Kasparov pairings of leafwise and transverse Dirac operators. Relative to a pairing with a transversal class we commence by establishing an index formula for foliations with leaves of non{positive sectional curvature. The underlyin ..."
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Abstract. With regards to certain Riemannian foliations we consider Kasparov pairings of leafwise and transverse Dirac operators. Relative to a pairing with a transversal class we commence by establishing an index formula for foliations with leaves of non{positive sectional curvature. The underlying ideas are then developed in a more general setting leading to pairings of images under the Baum{Connes map in geometric K{theory with transversal classes. Several ideas implicit in the work of Connes and Hilsum{Skandalis are formulated in the context of Riemannian foliations.>From these we establish the notion of a dual pairing in K{homology and a theorem of Grothendieck{Riemann{Roch type. 1991 Mathematics Subject Classication. Primary 58G10, 19K56; Secondary 19D55.