A higher Lefschetz formula for flat bundles

Benameur, Moulay-Tahar

doi:10.1090/S0002-9947-02-03111-2

A higher Lefschetz formula for flat bundles
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Trans. Amer. Math. Soc. 355 (2003), 119-142 Request permission

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.

References

M. F. Atiyah, Elliptic operators, discrete groups and von Neumann algebras, Colloque “Analyse et Topologie” en l’Honneur de Henri Cartan (Orsay, 1974) Astérisque, No. 32-33, Soc. Math. France, Paris, 1976, pp. 43–72. MR 0420729
Michael Atiyah and Friedrich Hirzebruch, Spin-manifolds and group actions, Essays on Topology and Related Topics (Mémoires dédiés à Georges de Rham), Springer, New York, 1970, pp. 18–28. MR 0278334
M. F. Atiyah, V. K. Patodi, and I. M. Singer, Spectral asymmetry and Riemannian geometry. I, Math. Proc. Cambridge Philos. Soc. 77 (1975), 43–69. MR 397797, DOI 10.1017/S0305004100049410
M. F. Atiyah and I. M. Singer, The index of elliptic operators. III, Ann. of Math. (2) 87 (1968), 546–604. MR 236952, DOI 10.2307/1970717
M. F. Atiyah and G. B. Segal, The index of elliptic operators. II, Ann. of Math. (2) 87 (1968), 531–545. MR 236951, DOI 10.2307/1970716
Moulay-Tahar Benameur, A longitudinal Lefschetz theorem in $K$-theory, $K$-Theory 12 (1997), no. 3, 227–257. MR 1480591, DOI 10.1023/A:1007747731149
M.-T. Benameur, Cyclic cohomology and the family Lefschetz theorem, to appear in Math. Ann.
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).
M.-T. Benameur, On the equivariant Chern-Connes character in noncommutative geometry, preprint.
M.-T. Benameur and H. Oyono-Oyono, Computation of the range of the trace for quasi-crystals, preprint.
Jean-Michel Bismut and Jeff 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 1042214, DOI 10.1016/0022-1236(90)90098-6
A. V. Brenner and M. A. Shubin, The Atiyah-Bott-Lefschetz formula for elliptic complexes on a manifold with boundary, Current problems in mathematics. Newest results, Vol. 38 (Russian), Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1990, pp. 119–183, 186 (Russian). Translated in J. Soviet Math. 64 (1993), no. 4, 1069–1111. MR 1156775, DOI 10.1016/0022-328X(90)87204-Q
A. Connes, A survey of foliations and operator algebras, Operator algebras and applications, Part 1 (Kingston, Ont., 1980) Proc. Sympos. Pure Math., vol. 38, Amer. Math. Soc., Providence, R.I., 1982, pp. 521–628. MR 679730
Alain Connes, Sur la théorie non commutative de l’intégration, Algèbres d’opérateurs (Sém., Les Plans-sur-Bex, 1978) Lecture Notes in Math., vol. 725, Springer, Berlin, 1979, pp. 19–143 (French). MR 548112
A. Connes, cours du collège de France, 94/95.
Alain Connes, Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994. MR 1303779
Alain Connes, Noncommutative differential geometry, Inst. Hautes Études Sci. Publ. Math. 62 (1985), 257–360. MR 823176
A. Connes and H. Moscovici, The local index formula in noncommutative geometry, Geom. Funct. Anal. 5 (1995), no. 2, 174–243. MR 1334867, DOI 10.1007/BF01895667
A. Connes and H. Moscovici, Hopf algebras, cyclic cohomology and the transverse index theorem, Comm. Math. Phys. 198 (1998), no. 1, 199–246. MR 1657389, DOI 10.1007/s002200050477
A. Connes and G. Skandalis, The longitudinal index theorem for foliations, Publ. Res. Inst. Math. Sci. 20 (1984), no. 6, 1139–1183. MR 775126, DOI 10.2977/prims/1195180375
Marius Crainic and Ieke Moerdijk, Foliation groupoids and their cyclic homology, Adv. Math. 157 (2001), no. 2, 177–197. MR 1813430, DOI 10.1006/aima.2000.1944
R. G. Douglas, S. Hurder, and J. Kaminker, Cyclic cocycles, renormalization and eta-invariants, Invent. Math. 103 (1991), no. 1, 101–179. MR 1079841, DOI 10.1007/BF01239510
André Haefliger, Some remarks on foliations with minimal leaves, J. Differential Geometry 15 (1980), no. 2, 269–284 (1981). MR 614370
James L. Heitsch and Connor Lazarov, A Lefschetz theorem for foliated manifolds, Topology 29 (1990), no. 2, 127–162. MR 1056266, DOI 10.1016/0040-9383(90)90004-4
James L. Heitsch and Connor Lazarov, Rigidity theorems for foliations by surfaces and spin manifolds, Michigan Math. J. 38 (1991), no. 2, 285–297. MR 1098862, DOI 10.1307/mmj/1029004334
James L. Heitsch and Connor Lazarov, A general families index theorem, $K$-Theory 18 (1999), no. 2, 181–202. MR 1711704, DOI 10.1023/A:1007886112487
Xinhui Jiang, An index theorem on foliated flat bundles, $K$-Theory 12 (1997), no. 4, 319–359. MR 1485433, DOI 10.1023/A:1007775309863
Sławomir Klimek, Witold Kondracki, and Andrzej Lesniewski, Equivariant entire cyclic cohomology. I. Finite groups, $K$-Theory 4 (1991), no. 3, 201–218. MR 1106952, DOI 10.1007/BF00569446
R. Lauter, B. Monthubert, and V. Nistor, Pseudodifferential operators on continuous family groupoids, Doc. Math. (electronic) 5 (2000) 625–655.
Jean-Louis Loday, Cyclic homology, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 301, Springer-Verlag, Berlin, 1992. Appendix E by María O. Ronco. MR 1217970, DOI 10.1007/978-3-662-21739-9
R. Melrose and V. Nistor, Homology of pseudodifferential operators I. Manifolds with boundary, to appear in Amer. J. Math.
Victor Nistor, The index of operators on foliated bundles, J. Funct. Anal. 141 (1996), no. 2, 421–434. MR 1418513, DOI 10.1006/jfan.1996.0135
Victor Nistor, Alan Weinstein, and Ping Xu, Pseudodifferential operators on differential groupoids, Pacific J. Math. 189 (1999), no. 1, 117–152. MR 1687747, DOI 10.2140/pjm.1999.189.117
Philippe Tondeur, Geometry of foliations, Monographs in Mathematics, vol. 90, Birkhäuser Verlag, Basel, 1997. MR 1456994, DOI 10.1007/978-3-0348-8914-8
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.