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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  1. [1] K. Behrend, G. Ginot, B. Noohi & P. Xu, String topology for stacks, preprint arXiv:math/0712.3857. Zbl1253.55007MR2977576
  2. [2] M. Bökstedt, W. C. Hsiang & I. Madsen, The cyclotomic trace and algebraic K -theory of spaces, Invent. Math.111 (1993), 465–539. Zbl0804.55004MR1202133
  3. [3] E. H. J. Brown & R. H. Szczarba, On the rational homotopy type of function spaces, Trans. Amer. Math. Soc.349 (1997), 4931–4951. Zbl0927.55012MR1407482
  4. [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. Zbl0666.13007MR952571
  5. [5] M. Chas & D. Sullivan, String topology, preprint arXiv:9911159. 
  6. [6] K. T. Chen, Iterated integrals of differential forms and loop space homology, Ann. of Math.97 (1973), 217–246. Zbl0227.58003
  7. [7] K. T. Chen, Iterated path integrals, Bull. Amer. Math. Soc.83 (1977), 831–879. Zbl0389.58001
  8. [8] R. L. Cohen & J. D. S. Jones, A homotopy theoretic realization of string topology, Math. Ann.324 (2002), 773–798. Zbl1025.55005
  9. [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. Zbl1089.57002
  10. [10] Y. Félix, S. Halperin & J.-C. Thomas, Rational homotopy theory, Graduate Texts in Math. 205, Springer, 2001. Zbl0961.55002
  11. [11] Y. Félix & J.-C. Thomas, Rational BV-algebra in string topology, Bull. Soc. Math. France136 (2008), 311–327. Zbl1160.55006
  12. [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. Zbl1060.57019
  13. [13] Y. Félix, J.-C. Thomas & M. Vigué-Poirrier, Rational string topology, J. Eur. Math. Soc.9 (2007), 123–156. Zbl1200.55015
  14. [14] E. Getzler, J. D. S. Jones & S. Petrack, Differential forms on loop spaces and the cyclic bar complex, Topology30 (1991), 339–371. Zbl0729.58004
  15. [15] G. Ginot, Higher order Hochschild cohomology, C. R. Math. Acad. Sci. Paris346 (2008), 5–10. Zbl1157.53042
  16. [16] P. G. Goerss & J. F. Jardine, Simplicial homotopy theory, Progress in Math. 174, Birkhäuser, 1999. Zbl0949.55001MR1711612
  17. [17] T. G. Goodwillie, Relative algebraic K -theory and cyclic homology, Ann. of Math.124 (1986), 347–402. Zbl0627.18004MR855300
  18. [18] A. Haefliger, Rational homotopy of the space of sections of a nilpotent bundle, Trans. Amer. Math. Soc.273 (1982), 609–620. Zbl0508.55019MR667163
  19. [19] P. Hu, Higher string topology on general spaces, Proc. London Math. Soc.93 (2006), 515–544. Zbl1103.55008MR2251161
  20. [20] P. Lambrechts & D. Stanley, Poincaré duality and commutative differential graded algebras, Ann. Sci. Éc. Norm. Supér. 41 (2008), 495–509. Zbl1172.13009MR2489632
  21. [21] J.-L. Loday, Cyclic homology, Grund. Math. Wiss. 301, Springer, 1992. Zbl0780.18009MR1217970
  22. [22] S. MacLane, Homology, first éd., Grundl. der math. Wiss. 114, Springer, 1967. Zbl0133.26502MR349792
  23. [23] R. McCarthy, On operations for Hochschild homology, Comm. Algebra21 (1993), 2947–2965. Zbl0809.18009MR1222750
  24. [24] F. Patras & J.-C. Thomas, Cochain algebras of mapping spaces and finite group actions, Topology Appl.128 (2003), 189–207. Zbl1027.55017MR1956614
  25. [25] T. Pirashvili, Hodge decomposition for higher order Hochschild homology, Ann. Sci. École Norm. Sup.33 (2000), 151–179. Zbl0957.18004MR1755114
  26. [26] D. Quillen, Rational homotopy theory, Ann. of Math.90 (1969), 205–295. Zbl0191.53702MR258031
  27. [27] D. Sullivan, Infinitesimal computations in topology, Publ. Math. I.H.É.S. 47 (1977), 269–331. Zbl0374.57002MR646078
  28. [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. Zbl1080.53085MR2131009

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