A Chen model for mapping spaces and the surface product
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 43 (2010) no. 5, pp. 811-881.

We develop a machinery of Chen iterated integrals for higher Hochschild complexes. These are complexes whose differentials are modeled on an arbitrary simplicial set much in the same way the ordinary Hochschild differential is modeled on the circle. We use these to give algebraic models for general mapping spaces and define and study the surface product operation on the homology of mapping spaces of surfaces of all genera into a manifold. This is an analogue of the loop product in string topology. As an application, we show this product is homotopy invariant. We prove Hochschild-Kostant-Rosenberg type theorems and use them to give explicit formulae for the surface product of odd spheres and Lie groups.

Dans cet article, on étend le formalisme des intégrales itérées de Chen aux complexes de Hochschild supérieurs. Ces derniers sont des complexes de (co)chaînes modelés sur un espace (simplicial) de la même manière que le complexe de Hochschild classique est modelé sur le cercle. On en déduit des modèles algébriques pour les espaces fonctionnels que l'on utilise pour étudier le produit surfacique. Ce produit, défini sur l'homologie des espaces de fonctions continues de surfaces (de genre quelconque) dans une variété, est un analogue du produit de Chas-Sullivan sur les espaces de lacets en topologie des cordes. En particulier, on en déduit que le produit surfacique est un invariant homotopique. On démontre également un théorème du type Hochschild-Kostant-Rosenberg pour les complexes de Hochschild modelés sur les surfaces qui permet d'obtenir des formules explicites pour le produit surfacique des sphères de dimension impaire ainsi que pour les groupes de Lie.

DOI: 10.24033/asens.2134
Classification: 18G60,  55P50,  18G30,  55P62
Keywords: string topology, (higher) Hochschild homology, Hochschild cohomology, Chen integrals, mapping spaces, surface product
@article{ASENS_2010_4_43_5_811_0,
     author = {Ginot, Gr\'egory and Tradler, Thomas and Zeinalian, Mahmoud},
     title = {A {Chen} model for mapping spaces and the surface product},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {811--881},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 43},
     number = {5},
     year = {2010},
     doi = {10.24033/asens.2134},
     zbl = {1234.55009},
     mrnumber = {2721877},
     language = {en},
     url = {http://www.numdam.org/articles/10.24033/asens.2134/}
}
TY  - JOUR
AU  - Ginot, Grégory
AU  - Tradler, Thomas
AU  - Zeinalian, Mahmoud
TI  - A Chen model for mapping spaces and the surface product
JO  - Annales scientifiques de l'École Normale Supérieure
PY  - 2010
DA  - 2010///
SP  - 811
EP  - 881
VL  - Ser. 4, 43
IS  - 5
PB  - Société mathématique de France
UR  - http://www.numdam.org/articles/10.24033/asens.2134/
UR  - https://zbmath.org/?q=an%3A1234.55009
UR  - https://www.ams.org/mathscinet-getitem?mr=2721877
UR  - https://doi.org/10.24033/asens.2134
DO  - 10.24033/asens.2134
LA  - en
ID  - ASENS_2010_4_43_5_811_0
ER  - 
%0 Journal Article
%A Ginot, Grégory
%A Tradler, Thomas
%A Zeinalian, Mahmoud
%T A Chen model for mapping spaces and the surface product
%J Annales scientifiques de l'École Normale Supérieure
%D 2010
%P 811-881
%V Ser. 4, 43
%N 5
%I Société mathématique de France
%U https://doi.org/10.24033/asens.2134
%R 10.24033/asens.2134
%G en
%F ASENS_2010_4_43_5_811_0
Ginot, Grégory; Tradler, Thomas; Zeinalian, Mahmoud. A Chen model for mapping spaces and the surface product. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 43 (2010) no. 5, pp. 811-881. doi : 10.24033/asens.2134. http://www.numdam.org/articles/10.24033/asens.2134/

[1] K. Behrend, G. Ginot, B. Noohi & P. Xu, String topology for stacks, preprint arXiv:math/0712.3857. | MR | Zbl

[2] M. Bökstedt, W. C. Hsiang & I. Madsen, The cyclotomic trace and algebraic K-theory of spaces, Invent. Math. 111 (1993), 465-539. | MR | Zbl

[3] E. H. J. Brown & R. H. Szczarba, On the rational homotopy type of function spaces, Trans. Amer. Math. Soc. 349 (1997), 4931-4951. | MR | Zbl

[4] D. Burghelea & M. Vigué-Poirrier, Cyclic homology of commutative algebras. I, in Algebraic topology-rational homotopy (Louvain-la-Neuve, 1986), Lecture Notes in Math. 1318, Springer, 1988, 51-72. | MR | Zbl

[5] M. Chas & D. Sullivan, String topology, preprint arXiv:9911159.

[6] K. T. Chen, Iterated integrals of differential forms and loop space homology, Ann. of Math. 97 (1973), 217-246. | Zbl

[7] K. T. Chen, Iterated path integrals, Bull. Amer. Math. Soc. 83 (1977), 831-879. | Zbl

[8] R. L. Cohen & J. D. S. Jones, A homotopy theoretic realization of string topology, Math. Ann. 324 (2002), 773-798. | Zbl

[9] R. L. Cohen & A. A. Voronov, Notes on string topology, in String topology and cyclic homology, Adv. Courses Math. CRM Barcelona, Birkhäuser, 2006, 1-95. | Zbl

[10] Y. Félix, S. Halperin & J.-C. Thomas, Rational homotopy theory, Graduate Texts in Math. 205, Springer, 2001. | Zbl

[11] Y. Félix & J.-C. Thomas, Rational BV-algebra in string topology, Bull. Soc. Math. France 136 (2008), 311-327. | EuDML | Numdam | Zbl

[12] Y. Felix, J.-C. Thomas & M. Vigué-Poirrier, The Hochschild cohomology of a closed manifold, Publ. Math. Inst. Hautes Études Sci. 99 (2004), 235-252. | EuDML | Numdam | Zbl

[13] Y. Félix, J.-C. Thomas & M. Vigué-Poirrier, Rational string topology, J. Eur. Math. Soc. 9 (2007), 123-156. | EuDML | Zbl

[14] E. Getzler, J. D. S. Jones & S. Petrack, Differential forms on loop spaces and the cyclic bar complex, Topology 30 (1991), 339-371. | Zbl

[15] G. Ginot, Higher order Hochschild cohomology, C. R. Math. Acad. Sci. Paris 346 (2008), 5-10. | Zbl

[16] P. G. Goerss & J. F. Jardine, Simplicial homotopy theory, Progress in Math. 174, Birkhäuser, 1999. | MR | Zbl

[17] T. G. Goodwillie, Relative algebraic K-theory and cyclic homology, Ann. of Math. 124 (1986), 347-402. | MR | Zbl

[18] A. Haefliger, Rational homotopy of the space of sections of a nilpotent bundle, Trans. Amer. Math. Soc. 273 (1982), 609-620. | MR | Zbl

[19] P. Hu, Higher string topology on general spaces, Proc. London Math. Soc. 93 (2006), 515-544. | MR | Zbl

[20] P. Lambrechts & D. Stanley, Poincaré duality and commutative differential graded algebras, Ann. Sci. Éc. Norm. Supér. 41 (2008), 495-509. | EuDML | Numdam | MR | Zbl

[21] J.-L. Loday, Cyclic homology, Grund. Math. Wiss. 301, Springer, 1992. | MR | Zbl

[22] S. Maclane, Homology, first éd., Grundl. der math. Wiss. 114, Springer, 1967. | MR | Zbl

[23] R. Mccarthy, On operations for Hochschild homology, Comm. Algebra 21 (1993), 2947-2965. | MR | Zbl

[24] F. Patras & J.-C. Thomas, Cochain algebras of mapping spaces and finite group actions, Topology Appl. 128 (2003), 189-207. | MR | Zbl

[25] T. Pirashvili, Hodge decomposition for higher order Hochschild homology, Ann. Sci. École Norm. Sup. 33 (2000), 151-179. | EuDML | Numdam | MR | Zbl

[26] D. Quillen, Rational homotopy theory, Ann. of Math. 90 (1969), 205-295. | MR | Zbl

[27] D. Sullivan, Infinitesimal computations in topology, Publ. Math. I.H.É.S. 47 (1977), 269-331. | EuDML | Numdam | MR | Zbl

[28] D. Sullivan, Sigma models and string topology, in Graphs and patterns in mathematics and theoretical physics, Proc. Sympos. Pure Math. 73, Amer. Math. Soc., 2005, 1-11. | MR | Zbl

Cited by Sources: