A Chen model for mapping spaces and the surface product

Grégory Ginot; Thomas Tradler; Mahmoud Zeinalian

Annales scientifiques de l'École Normale Supérieure (2010)

  • Volume: 43, Issue: 5, page 811-881
  • ISSN: 0012-9593

Abstract

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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.

How to cite

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Ginot, Grégory, Tradler, Thomas, and Zeinalian, Mahmoud. "A Chen model for mapping spaces and the surface product." Annales scientifiques de l'École Normale Supérieure 43.5 (2010): 811-881. <http://eudml.org/doc/272175>.

@article{Ginot2010,
abstract = {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.},
author = {Ginot, Grégory, Tradler, Thomas, Zeinalian, Mahmoud},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {string topology; (higher) Hochschild homology; Hochschild cohomology; Chen integrals; mapping spaces; surface product},
language = {eng},
number = {5},
pages = {811-881},
publisher = {Société mathématique de France},
title = {A Chen model for mapping spaces and the surface product},
url = {http://eudml.org/doc/272175},
volume = {43},
year = {2010},
}

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
PB - Société mathématique de France
VL - 43
IS - 5
SP - 811
EP - 881
AB - 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.
LA - eng
KW - string topology; (higher) Hochschild homology; Hochschild cohomology; Chen integrals; mapping spaces; surface product
UR - http://eudml.org/doc/272175
ER -

References

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