Skip to main content
Log in

Rational model for the string coproduct of pure manifolds

  • Published:
Journal of Homotopy and Related Structures Aims and scope Submit manuscript

Abstract

The string coproduct is a coproduct on the homology with field coefficients of the free loop space of a closed oriented manifold introduced by Sullivan in string topology. The coproduct and the Chas-Sullivan loop product give an infinitesimal bialgebra structure on the homology if the Euler characteristic is zero. The aim of this paper is to study the string coproduct using Sullivan models in rational homotopy theory. In particular, we give a rational model for the string coproduct of pure manifolds. Moreover, we study the behavior of the string coproduct in terms of the Hodge decomposition of the rational cohomology of the free loop space. We also give computational examples of the coproduct rationally.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Basu, S.: Transversal string topology and invariants of manifolds. Diss. The Graduate School, Stony Brook University: Stony Brook, NY (2011)

  2. Buijs, U., Aniceto, M.: Basic constructions in rational homotopy theory of function spaces. Annales de l’institut Fourier. 56, 3 (2006)

    Article  MathSciNet  Google Scholar 

  3. Brown, E.H., Szczarba, R.H.: Continuous cohomology and real homotopy type. Trans. Am. Math. Soc. 311(1), 57–106 (1989)

    Article  MathSciNet  Google Scholar 

  4. Brown, E.H., Szczarba, R.H.: On the rational homotopy type of function spaces. Trans. Am. Math. Soc. 349(12), 4931–4951 (1997)

    Article  MathSciNet  Google Scholar 

  5. Chas, M., Sullivan, D.: String topology. arXiv:math/9911159 (1999)

  6. Cohen, R.L., Godin, V.: A polarized view of string topology. Topology, geometry and quantum field theory. Lond. Math. Soc. Lect. Note Ser. 308, 127–154 (2004)

    MATH  Google Scholar 

  7. Félix, Y., Halperin, S., Thomas, J.-C.: Rational Homotopy Theory. Vol. 205. Springer, New York (2012)

  8. Félix, Y., Oprea, J., Tanré, D.: Algebraic Models in Geometry, vol. 17. Oxford University Press, Oxford (2008)

    MATH  Google Scholar 

  9. Félix, Y., Thomas, J.-C.: Rational BV-algebra in string topology. Bull. Soc. Math. Fr. 136(2), 311–327 (2008)

    Article  MathSciNet  Google Scholar 

  10. Félix, Y., Thomas, J.-C.: String topology on Gorenstein spaces. Math. Ann. 345(2), 417–452 (2009)

    Article  MathSciNet  Google Scholar 

  11. Goresky, M., Hingston, N.: Loop products and closed geodesics. Duke Math. J. 150(1), 117–209 (2009)

    Article  MathSciNet  Google Scholar 

  12. Hingston, N., Wahl, N.: Product and coproduct in string topology. arXiv:1709.06839 (2017)

  13. Hingston, N., Wahl, N.: Homotopy invariance of the string topology coproduct. arXiv:1908.03857 (2019)

  14. Kuribayashi, K.: A rational model for the evaluation map. Georg Math. J. 13, 127–141 (2006)

    Article  MathSciNet  Google Scholar 

  15. Naito, T.: String Operations on Rational Gorenstein Spaces. arXiv:1301.1785 (2013)

  16. Sullivan, D.: Open and closed string field theory interpreted in classical algebraic topology. Topology, geometry and quantum field theory. Lond. Math. Soc. Lect. Note Ser. 308, 344–357 (2004)

  17. Tamanoi, H.: Loop coproducts in string topology and triviality of higher genus TQFT operations. J. Pure Appl. Algebra 214(5), 605–615 (2010)

    Article  MathSciNet  Google Scholar 

  18. Vigué-Poirrier, M.: Décompositions de l’homologie cyclique des algébres différentielles graduées commutatives. K-Theory 4(5), 399–410 (1991)

    Article  MathSciNet  Google Scholar 

  19. Vigué-Poirrier, M., Burghelea, D.: A model for cyclic homology and algebraic \(K\)-theory of \(1\)-connected topological spaces. J. Differ. Geom. 22(2), 243–253 (1985)

    Article  MathSciNet  Google Scholar 

  20. Wakatsuki, S.: Description and triviality of the loop products and coproducts for rational Gorenstein spaces. arXiv:1612.03563 (2016)

Download references

Acknowledgements

We would like to thank the referees for reading our work carefully and providing many valuable suggestions. This work was supported by JSPS KAKENHI Grant Number JP18K13403.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Takahito Naito.

Additional information

Communicated by Geoffrey Powell.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Naito, T. Rational model for the string coproduct of pure manifolds. J. Homotopy Relat. Struct. 16, 667–702 (2021). https://doi.org/10.1007/s40062-021-00293-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40062-021-00293-5

Keywords

Mathematics Subject Classification

Navigation