Cyclic cohomology of etale groupoids.(English)Zbl 0812.19003

The authors indicate three necessary steps in a program to extend Bismut’s local index theorem for families to foliations (in the sense of Connes’ index theory). The first step which defines the curvature of the index in terms of cyclic homology is touched on in the present paper. The next two steps, extension of Quillen’s superconnection formalism to operators with nondiscrete spectrum and the local computations, are left to future work. Here they determine the Hochschild and cyclic homology and the cyclic cohomology of the algebra of functions on a separated smooth étale groupoid (which is Morita equivalent to the relevant foliation groupoid and is therefore supposed to have the same cyclic cohomology). Applications of the main results on cyclic (co)homology are obtained by specializing the groupoid, in particular, to the transformation groupoid obtained from a smooth action of a discrete group on a manifold. This contains as a special case the groupoid of an orbifold, and leads to an interpretation of the orbifold Euler characteristic in cyclic homology. Another application gives an expression of the Chern character of a vector bundle over a smooth manifold in terms of a defining cocycle of transition maps.

MSC:

 19D55 $$K$$-theory and homology; cyclic homology and cohomology
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 [1] Atiyah, M. and Segal, G., On equivariant Euler characteristics,J. Geom. Phys. 6 (1989), 671-677. · Zbl 0708.19004 [2] Atiyah, M. F. and Singer, I. M., The index of elliptic operators IV,Ann. of Math. 93 (1971), 119-138. · Zbl 0212.28603 [3] Bismut, J.-M., The index theorem for families of Dirac operators: two heat equation proofs,Invent. Math. 83 (1986), 91-151. · Zbl 0592.58047 [4] Blanc, Ph. and Brylinski, J.-L., Cyclic cohomology and the Selberg principle,J. Funct. Anal. 109 (1992), 289-330. · Zbl 0783.55004 [5] Block, J., Thesis, Harvard University, 1987. [6] Bott, R., Gelfand-Fuks cohomology and foliations, inProc. Eleventh Annual Holiday Symposium at New Mexico State University, 1973. [7] Bott, R., On the Chern-Weil homomorphism and the continuous cohomology of Lie groups,Adv. Math. 11 (1973), 289-303. · Zbl 0276.55011 [8] Bott, R. and Tu, L.,Differential forms in Algebraic Topology, Springer, New York, 1986. · Zbl 0496.55001 [9] Brylinski, J.-L., Algebras associated to group actions and their homology, preprint, 1987. [10] Brylinski, J.-L., Cyclic homology and equivariant theories,Ann. Inst. Fourier 37 (1987). · Zbl 0625.55003 [11] Burghelea, D., The cyclic cohomology of group rings,Comment. Math. Helv. 60 (1985), 354-365. · Zbl 0595.16022 [12] Connes, A., Sur la théorie noncommutative de l’intégration, inAlgébres d’Opérateurs, Lecture Notes in Math. 725, Springer, New York, 1982, pp. 19-143. [13] Connes, A., A survey of foliations and operator algebras, inOperator Algebras and Applications, Proc. Symp. Pure Math. 38, 1982, pp. 521-628. [14] Connes, A., Cohomology cyclique et foncteursExt n,C.R. Acad. Sci. Paris, I,296(23) (1983), 953-958. [15] Connes, A., Non-commutative differential geometry,Publ. Math. IHES,62 (1985), 41-144. · Zbl 0592.46056 [16] Connes, A., Cyclic cohomology and noncommutative differential geometry, inProc. Int. Cong. Math., Berkeley, volume 2, pp. 879-889, 1986. [17] Connes, A.,Géométrie non commutative, Inter Editions, Paris, 1990. [18] Dixon, L., Harvey, J., Vafa, C., and Witten, E., Strings on orbifolds.Nuclear Phys. B,261 (1985), 678. [19] Feigin, B. P. and Tsygan, B., AdditiveK-theory, inK-Theory, Arithmetic and Geometry, Lecture Notes in Math. 1289, Springer, Berlin, 1987, pp. 97-209. [20] Godement, R.,Topologie Algébrique et Théorie des Faisceaux, Hermann, Paris, 1958. · Zbl 0080.16201 [21] Griffiths, P. and Harris, J.,Principles of Algebraic Geometry, Wiley, New York, 1978. · Zbl 0408.14001 [22] Grothendieck, A., Produits tensoriels topologiques et espaces nucléaires,Mem. Amer. Math. Soc. 16, 1955. [23] Haefliger, A., Groupoïdes d’holonomie et espaces classifiants,Astérisque 116 (1984), 70-97. · Zbl 0562.57012 [24] Hilsum, M. and Skandalis, G., Stabilité des algèbres de feuilletages.Ann. Inst. Fourier Grenoble,33 (1983), 201-208. · Zbl 0505.46043 [25] Loday J.-L.,Cyclic Homology, Springer, Berlin, 1992. [26] Loday, J.-L. and Quillen, D., Cyclic homology and the Lie algebra homology of matrices,Comment. Math. Helv. 59 (1984), 565-591. · Zbl 0565.17006 [27] MacLane, S.,Homology, Springer, Berlin, 1963. [28] Moore, C. C. and Schochet, C.,Global Analysis on Foliated Spaces, Math. Sci. Res. Inst. 9, Springer, Berlin, 1988. · Zbl 0648.58034 [29] Nica, A. and Nistor, V., Work in progress. [30] Nistor, V., Cyclic cohomology of crossed products by algebraic groups,Invent. Math. 112 (1993), 615-638. · Zbl 0799.46078 [31] Nistor, V., Group cohomology and the cyclic cohomology of crossed products,Invent. Math. 99 (1990), 411-424. · Zbl 0692.46065 [32] Nistor, V., A bivariant Chern character forp-summable quasihomomorphisms,K-Theory 5 (1991), 193-211. · Zbl 0784.19001 [33] Nistor, V., A bivariant Chern-Connes character,Ann. of Math. 138 (1993), 555-590. · Zbl 0798.46051 [34] Renault, J., Muhly, P. S., and Williams, D., Equivalence and isomorphism for groupoidC*-algebras.J. Operator Theory 17 (1987), 3-22. · Zbl 0645.46040 [35] Quillen, D., Higher algebraicK-theory I, inAlgebraic K-theory I, Lecture Notes in Math. 341, Springer, New York, 1973, pp. 85-174. · Zbl 0292.18004 [36] Quillen, D., Superconnections and the Chern character,Topology 24 (1985), 89-95. · Zbl 0569.58030 [37] Renault, J.,A Groupoid Approach to C*-Algebras, Lecture Notes in Math. 793, Springer, Berlin, Heidelberg, New York, 1980. · Zbl 0433.46049 [38] Satake, I., The Gauss-Bonnet theorem forV-manifolds,J. Math. Soc. Japan (1957), 464-492. · Zbl 0080.37403 [39] Spanier, E. H.,Algebraic Topology, McGraw-Hill, New York, 1966. [40] Swan, R. G.,The Theory of Sheaves, University of Chicago Press, Chicago, 1964. · Zbl 0119.25801 [41] Tsygan, B. L., Homology of matrix Lie algebras over rings and Hochschild homology,Uspekhi Math. Nauk. 38 (1983), 217-218. · Zbl 0518.17002 [42] Wassermann, A., Cyclic cohomology I: Finite group actions and Schwarz algebras, preprint, 1987. [43] Whitehead, G. W.,Elements of Homotopy Theory, Springer, Berlin, 1978. · Zbl 0406.55001 [44] Winkelkemper, E., The graph of a foliation,Ann. Global Anal. Geom. 1 (1983), 53-73.
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