A Chen model for mapping spaces and the surface product
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 43 (2010) no. 5, p. 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 : https://doi.org/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},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 43},
     number = {5},
     year = {2010},
     pages = {811-881},
     doi = {10.24033/asens.2134},
     zbl = {1234.55009},
     mrnumber = {2721877},
     language = {en},
     url = {http://www.numdam.org/item/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/item/ASENS_2010_4_43_5_811_0/

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