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On Hochschild and cyclic homology of certain homogeneous spaces. (English) Zbl 0802.55010

The author gives explicit calculations of the Hochschild homology and the cyclic homology for certain homogeneous spaces. Mainly the Cartan pairs or the Riemannian symmetric spaces. These spaces are particular cases of formal spaces, therefore this paper is related to the paper of M. Vigué-Poirrier [J. Pure Appl. Algebra 91, No. 1-3, 347-354 (1994; see the review below)].

MSC:

55P62 Rational homotopy theory
55N91 Equivariant homology and cohomology in algebraic topology

Citations:

Zbl 0802.55011
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References:

[1] J. M. Boardman, R. M. Vogt: Homotopy invariant algebraic structures on topological spaces. LNM, no. 347, Berlin-Heidelberg-New York, 1973. · Zbl 0285.55012
[2] A. Borel: Sur la cohomologie des espaces fibres principaux et des espaces homogenes de groupes de Lie compacts. Annals of Math. 57 (1953), 115-207. · Zbl 0052.40001 · doi:10.2307/1969728
[3] A. Borel, F. Hirzebruch: Characteristic classes and homogeneous spaces 1. Amer. J. Math. 80 (1958), 459-538. · Zbl 0097.36401 · doi:10.2307/2372795
[4] A. K. Bousfield, V. K. A. M. Guggenheim: On PL de Rham theory and rational homotopy type. Memoirs Amer. Math. Soc. 8(179) (1976). · Zbl 0338.55008
[5] D. Burghelea: Cyclic homology and the algebraic \(K\)-theory of spaces 1. Contemp. Math. 55 (1986), 89-115. · Zbl 0615.55009
[6] D. Burghelea, Z. Fiedorowicz: Cyclic homology and algebraic \(K\)-theory of spaces 2. Topology 25 (1986), 303-317. · Zbl 0639.55003 · doi:10.1016/0040-9383(86)90046-7
[7] A. Connes: De Rham homology and non-commutative algebra. Publ. Math. IHES 62 (1985), 94-144.
[8] A. Connes: Cohomologie cyclique et founkteur Ext\(^n\). C. r. Acad. Sci. Paris, Ser. A 296 (1983), 953-958. · Zbl 0534.18009
[9] A. Connes, H. Moscovici: Cyclic homology, the Novikov conjecture and hyperbolic groups. Topology 29 (1990), 345-388. · Zbl 0759.58047 · doi:10.1016/0040-9383(90)90003-3
[10] K. Doan: Poicaré polynomials of compact homogeneous Riemannian spaces with irreducible isotropy subgroup. Trudy Semin. Vect. Tens. Anal. 14 (1968), 33-93.
[11] M. El-Hauari: Cohomologie de Hochschild et \(k\)-formalite intrinseque. C. r. Acad. Sci. Paris, Ser. 1 310 (1990), 731-734.
[12] T. Goodwillie: Cyclic homology, derivations and the free loop space. Topology 24 (1985), 187-215. · Zbl 0569.16021 · doi:10.1016/0040-9383(85)90055-2
[13] V. Greub, S. Halperin, R. Vanstone: Curvature, connections and cohomology, v. 3. Academic Press, New York, 1976.
[14] P. Griffiths, J. Morgan: Rational homotopy theory and differential forms. Birkhäuser, Boston, 1981. · Zbl 0474.55001
[15] S. Halperin, M. Vigué-Poirrier: The homology of a free loop space. Pacif. J. Math. 147 (1991), 311-324. · Zbl 0666.55011 · doi:10.2140/pjm.1991.147.311
[16] M.-C. Heidemann-Tcherkez, M. Vigué-Poirrier: Application de la theorie des polynomes de Hilbert-Samuel a l’etude de certaines algebres differentielles. C. r. Acad. Sci. Paris, Ser. A 278 (1974), 1607-1610. · Zbl 0321.13011
[17] S. Helgason: Differential Geometry, Lie Groups and Symmetric Spaces. Academic Press, New York, 1978. · Zbl 0451.53038
[18] Y. Kihoon: Almost complex homogeneous spaces and their submanifolds. World Scientific, Singapoor, 1987.
[19] R. Krasauskas: Certain topological applications of dyhedral homology. Ph.D. Thesis, Moscow University, 1987.
[20] D. Lehmann: Theorie homotopique des formes differentielles (d’apres D. Sullivan). Societe mathematique de France, Asterisque 45 (1977).
[21] P. Rashevskii: On real cohomology of homogeneous spaces. Uspekhi Mat. Nauk 24(3) (1969), 23-90.
[22] M. Vigué-Poirrier, D. Burghelea: A model for cyclic homology and algebraic \(K\)-theory of 1-connected topological spaces. J. Diff. Geom. 22 (1985), 243-253. · Zbl 0595.55009
[23] M. Vigué-Poirrier, D. Burghelea: Cyclic homology of commutative algebras. Publ. IRMA-Lille 8(1) (1987). · Zbl 0666.13007
[24] M. Vigué-Poirrier: Cyclic homology of algebraic hypersurfaces. Publ. IRMA-Lille 10(7) (1987). · Zbl 0732.16008
[25] M. Vigué-Poirrier: Cyclic homology and Quillen homology of a commutative algebra. Publ. IRMA-Lille 5(1) (1986). · Zbl 0675.13008
[26] M. Vigué-Poirrier: Homologie de Hochschild et homologie cyclique des algebres differentielles graduees. Publ. IRMA-Lille 17(1) (1989).
[27] E. Witten: The index of a Dirac operator in loop space. LNM, no. 1326, 1986, pp. 161-181.
[28] A. E. Tralle: Cyclic homology of certain topological spaces which are formal in the sense of Sullivan. Matem. Zametki 50(6) (1991), 131-141, · Zbl 0735.55009
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