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A higher Lefschetz formula for flat bundles

Author: Moulay-Tahar Benameur
Journal: Trans. Amer. Math. Soc. 355 (2003), 119-142
MSC (2000): Primary 19L47, 19M05, 19K56
Published electronically: September 5, 2002
MathSciNet review: 1928080
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Abstract: In this paper, we prove a fixed point formula for flat bundles. To this end, we use cyclic cocycles which are constructed out of closed invariant currents. We show that such cyclic cocycles are equivariant with respect to isometric longitudinal actions of compact Lie groups. This enables us to prove fixed point formulae in the cyclic homology of the smooth convolution algebra of the foliation.

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  • 1. M. F. Atiyah, Elliptic operators, discrete groups and von Neumann algebras, Colloque ``Analyse et Topologie" en l'Honneur de Henri Cartan (Orsay, 1974), pp. 43-72. Asterisque, No. 32-33, Soc. Math. France, Paris, 1976. MR 54:8741
  • 2. M. F. Atiyah and F. Hirzebruch, Spin-manifolds and group actions, 1970 Essays on Topology and Related Topics (Mémoires dédiés à Georges de Rham) pp. 18-28, Springer-Verlag, New York. MR 43:4064
  • 3. M. F. Atiyah, V. K. Patodi and I. M. Singer, Spectral asymmetry and Riemannian geometry. III, Math. Proc. Cambridge Philos. Soc. 79 (1976), no. 1, 71-99. MR 53:1655c
  • 4. M. F. Atiyah and I. M. Singer, The index of elliptic operators. III, Ann. of Math. (2) 87 (1968), 546-604. MR 38:5245
  • 5. M. F. Atiyah and G. Segal, The index of elliptic operators. II, Ann. of Math. (2) 87 (1968) 531-545. MR 38:5244
  • 6. M.-T. Benameur, A longitudinal Lefschetz theorem in $K$-theory, $K$-Theory 12 (1997), no. 3, 227-257. MR 98j:19007
  • 7. M.-T. Benameur, Cyclic cohomology and the family Lefschetz theorem, to appear in Math. Ann.
  • 8. M.-T. Benameur and V. Nistor, Homology of complete symbols and noncommutative geometry, ``Collected papers on Quantization of Singular Symplectic Quotients'', Progress in Math. series of Birkhäuser 198, (2001).
  • 9. M.-T. Benameur, On the equivariant Chern-Connes character in noncommutative geometry, preprint.
  • 10. M.-T. Benameur and H. Oyono-Oyono, Computation of the range of the trace for quasi-crystals, preprint.
  • 11. J.-M. Bismut and J. Cheeger, Families index for manifolds with boundary, superconnections, and cones, I. Families of manifolds with boundary and Dirac operators, J. Funct. Anal. 89 (1990), no. 2, 313-363. MR 91e:58180
  • 12. A. V. Brenner, M. A. Shubin, The Atiyah-Bott-Lefschetz formula for elliptic complexes on a manifold with boundary, (Russian) Translated in J. Soviet Math. 64 (1993), no. 4, 1069-1111. Itogi Nauki i Tekhniki, Current problems in mathematics. Newest results, Vol. 38 (Russian), 119-183, 186, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1990. MR 93k:58212
  • 13. A. Connes, A survey of foliations and operator algebras, Proc. Sympos. Pure Math. 38, Amer. Math. Soc., Providence, RI, 1982. MR 84m:58140
  • 14. A. Connes.
    Sur la théorie non commutative de l'integration (French), Algèbres d'opérateurs (Sém., Les Plans-sur-Bex, 1978), pp. 19-143, Lecture Notes in Math., 725, Springer-Verlag, Berlin, 1979. MR 81g:46090
  • 15. A. Connes, cours du collège de France, 94/95.
  • 16. A. Connes,
    Noncommutative Geometry,
    Academic Press, San Diego, CA, 1994. MR 95j:46063
  • 17. A. Connes, Noncommutative differential geometry, Inst. Hautes Études Sci. Publ. Math. 62 (1985), 257-360. MR 87i:58162
  • 18. A. Connes and H. Moscovici, The local index formula in noncommutative geometry, Geom. Funct. Anal. 5 (1995), no. 2, 174-243. MR 96e:58149
  • 19. A. Connes and H. Moscovici, Hopf algebras, cyclic cohomology and the transverse index theorem, Comm. Math. Phys. 198 (1998), no. 1, 199-246. MR 99m:58186
  • 20. A. Connes and G. Skandalis, The longitudinal index theorem for foliations, Publ. Res. Inst. Math. Sci. 20 (1984), no. 6, 1139-1183. MR 87h:58209
  • 21. M. Crainic and I. Moerdijk, Foliation groupoids and their cyclic homology, Adv. Math. 157 (2001), no. 2, 177-197. MR 2002a:22004
  • 22. R. Douglas, S. Hurder and J. Kaminker, Cyclic cocycles, renormalization and eta-invariants, Invent. Math. 103 (1991), no. 1, 101-179. MR 91m:58152
  • 23. A. Haefliger, Some remarks on foliations with minimal leaves, J. Differential Geom. 15 (1980), 269-284. MR 82j:57027
  • 24. J. L. Heitsch and C. Lazarov, A Lefschetz theorem for foliated manifolds, Topology 29 (1990), no. 2, 127-162. MR 91g:58281
  • 25. J. L. Heitsch and C. Lazarov, Rigidity theorems for foliations by surfaces and spin manifolds, Michigan Math. J. 38 (1991), no. 2, 285-297. MR 92b:58220
  • 26. J. L. Heitsch and C. Lazarov, A general families index theorem, $K$-Theory 18 (1999), no. 2, 181-202. MR 2000h:58046
  • 27. X. Jiang, An index theorem on foliated flat bundles, $K$-Theory 12 (1997), no. 4, 319-359. MR 99f:58197
  • 28. S. Klimek, W. Kondracki and A. Lesniewski, Equivariant entire cyclic cohomology. I. Finite groups, $K$-Theory 4 (1991), no. 3, 201-218. MR 92g:46091
  • 29. R. Lauter, B. Monthubert, and V. Nistor, Pseudodifferential operators on continuous family groupoids, Doc. Math. (electronic) 5 (2000) 625-655.
  • 30. J.-L. Loday, Cyclic homology. Appendix E by Maria O. Ronco. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 301, Springer-Verlag, Berlin, 1992. MR 94a:19004
  • 31. R. Melrose and V. Nistor, Homology of pseudodifferential operators I. Manifolds with boundary, to appear in Amer. J. Math.
  • 32. V. Nistor, The index of operators on foliated bundles, J. Funct. Anal. 141 (1996), no. 2, 421-434. MR 97j:19005
  • 33. V. Nistor, A. Weinstein and Ping Xu, Pseudodifferential operators on differential groupoids, Pacific J. Math. 189 (1999), no. 1, 117-152. MR 2000c:58036
  • 34. P. Tondeur, Geometry of foliations, Monographs in Mathematics, 90. Birkhäuser-Verlag, Basel, 1997. MR 98d:53037
  • 35. E. Witten, The index of the Dirac operator in loop space, Elliptic curves and modular forms in algebraic topology (Princeton, NJ, 1986), 161-181, Lecture Notes in Math. 1326, Springer-Verlag, Berlin, 1988.

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Additional Information

Moulay-Tahar Benameur
Affiliation: Institut Girard Desargues, Université Claude Bernard, Lyon 1, France

Keywords: $C^*$-algebra, $K$-theory, Lefschetz, foliations.
Received by editor(s): November 23, 2001
Received by editor(s) in revised form: March 12, 2002
Published electronically: September 5, 2002
Article copyright: © Copyright 2002 American Mathematical Society

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