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Index Theory and Non-Commutative Geometry II. Dirac Operators and Index Bundles

Published online by Cambridge University Press:  30 November 2007

Moulay-Tahar Benameur  and
James L. Heitsch
Moulay-Tahar Benameur
Affiliation:
[email protected] UMR 7122 du CNRS, Université de Metz, Ile du Saulcy, Metz, France
James L. Heitsch
Affiliation:
[email protected], [email protected] Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Mathematics Northwestern University
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Abstract

When the index bundle of a longitudinal Dirac type operator is transversely smooth, we define its Chern character in Haefliger cohomology and relate it to the Chern character of the K—theory index. This result gives a concrete connection between the topology of the foliation and the longitudinal index formula. Moreover, the usual spectral assumption on the Novikov-Shubin invariants of the operator is improved.

Keywords

Non-commutative geometryfoliationindex theoryDirac operatorindex bundle
Type
Research Article
Information
Journal of K-Theory , Volume 1 , Issue 2 , April 2008 , pp. 305 - 356
Copyright
Copyright © ISOPP 2008

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References

A75. Atiyah, M. F., Elliptic operators, discrete groups and von Neumann algebras, Asterisque 32/33 (1976), 43–72Google Scholar
BH04. Benameur, M-T. and Heitsch, J. L., Corrigendum, K-Theory 36 (2005), 397–402.CrossRefGoogle Scholar
BH04. Benameur, M-T. and Heitsch, J. L., Index theory and Non-Commutative Geometry I. Higher Families Index Theory, K-Theory 33 (2004), 151–183.CrossRefGoogle Scholar
BH06. Benameur, M-T. and Heitsch, J. L., The Higher Harmonic Signatures for Foliations I: the Untwisted Case, submittedGoogle Scholar
BV87. Berline, N. and Vergne, M., A proof of Bismut local index theorem for a family of Dirac operators. Topology 26 (1987), 435–463CrossRefGoogle Scholar
B86. Bismut, J.-M., The Atiyah-Singer index theorem for families of Dirac operators: two heat equation proofs. Invent. Math. 83 (1986), 91–151CrossRefGoogle Scholar
Con79. Connes, A., Sur la théorie de l'intégration non commutative. Lect. Notes in Math. 725, 1979CrossRefGoogle Scholar
Con81. Connes, A.. A survey of foliations and operator algebras. Operator algebras and applications, Part I, Proc. Sympos. Pure Math 38, Amer. Math. Soc., (1982), 521–628CrossRefGoogle Scholar
Con85. Connes, A.. Noncommutative differential geometry. Inst. Hautes Etudes Sci. Publ. Math. No. 62 (1985), 257–360Google Scholar
Con86. Connes, A.. Cyclic cohomology and the transverse fundamental class of a foliation. Geometric methods in operator algebras (Kyoto, 1983), 52–144, Pitman Res. Notes Math. Ser., 123, Longman Sci. Tech., Harlow, 1986Google Scholar
Con94. Connes, A., Noncommutative Geometry, Academic Press, New York, 1994Google Scholar
CM91. Connes, A. and Moscovici, H., Cyclic cohomology and the Novikov conjecture for hyperbolic groups, Topology 29 (1990), 345–388CrossRefGoogle Scholar
CS84. Connes, A., and Skandalis, G.. The longitudinal index theorem for foliations, Publ. RIMS Kyoto 20 (1984), 1139–1183CrossRefGoogle Scholar
CQ97. Cuntz, J. and Quillen, D., Excision in bivariant periodic cyclic cohomology. Invent. Math. 127 (1997), no. 1, 67–98CrossRefGoogle Scholar
GL03. Gorokhovsky, A., and Lott, J.. Local index theory over étale groupoids, J. Reine Angew. Math. 560 (2003), 151–198Google Scholar
GL05. Gorokhovsky, A., and Lott, J.. Local index theory over foliation groupoids, Adv. Math. 204 (2006), 413–447CrossRefGoogle Scholar
H80. Haefliger, A.. Some remarks on foliations with minimal leaves, J. Diff. Geo. 15 (1980), 269–284Google Scholar
He95. Heitsch, J. L.. Bismut superconnections and the Chern character for Dirac operators on foliated manifolds, K-Theory 9 (1995), 507–528CrossRefGoogle Scholar
HL90. Heitsch, J. L. and Lazarov, C.. A Lefschetz theorem for foliated manifolds, Topology 29 (1990), 127–162CrossRefGoogle Scholar
HL99. Heitsch, J. L. and Lazarov, C.. A general families index theorem, K-Theory 18 (1999), 181–202CrossRefGoogle Scholar
HL02. Heitsch, J. L. and Lazarov, C.. Riemann-Roch-Grothendieck and torsion for foliations J. Geo. Anal. 12 (2002), 437–468CrossRefGoogle Scholar
K91. Kordyukov, Yu. Lp-theory of elliptic differential operators on manifolds of bounded geometry Acta Appl. Math. 23 (1991), no. 3, 223–260CrossRefGoogle Scholar
K95. Kordyukov, Yu. Functional calculus for tangentially elliptic operators on foliated manifolds. Analysis and Geometry in Foliated Manifolds (Santiago de Compostela, 1994), 113–136, World Sci. Publishing, River Edge, NJ, 1995Google Scholar
LM89. Lawson, H. B. and Michelson, M.-L., Spin geometry, Princeton Math. Series 38, Princeton, 1989Google Scholar
MN96. Melrose, R. and Nistor, V., Homology of pseudodifferential operators I (manifolds with boundary), to appear in Amer. J. Math.Google Scholar
Nis93. Nistor, V., A bivariant Chern-Connes character. Ann. of Math. (2) 138 (1993), no. 3, 555–590CrossRefGoogle Scholar
NWX96. Nistor, V., Weinstein, A. and Xu, P., Pseudodifferential operators on differential groupoids, Pacific J. Math. 189 (1999), no. 1, 117–152CrossRefGoogle Scholar
Ph87. Phillips, J., The holonomic imperative and the homotopy groupoid of a foliation, Rocky Mountain J. of Math. 17 (1987) no. 1, 151–165CrossRefGoogle Scholar
RS80. Reed, M., and Simon, B.. Methods of Modern Mathematical Physics I: Functional Analysis, Academic Press, New York, 1980Google Scholar
S92. Shubin, M. A., Spectral theory of elliptic operators on noncompact manifolds. Methodes semi-classiques, Vol. 1 (Nantes, 1991). Asterisque No. 207 (1992), 35–108Google Scholar