Skip to main content
SpringerLink
Log in

Minimal models, GT-action and formality of the little disk operad

  • Published:
Selecta Mathematica Aims and scope Submit manuscript

Abstract

We give a new proof of formality of the operad of little disks. The proof makes use of an operadic version of a simple formality criterion for commutative differential graded algebras due to Sullivan. We see that formality is a direct consequence of the fact that the Grothendieck–Teichmüller group operates on the chain operad of little disks.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  1. Bar-Natan, D.: On associators and the Grothendieck-Teichmuller group. I. Sel. Math. (N.S.) 4(2), 183–212 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  2. Deligne, P.: La conjecture de Weil. II. Inst. Hautes Études Sci. Publ. Math. 52, 137–252 (1980)

    Google Scholar 

  3. Deligne, P.: Poids dans la cohomologie des variétés algébriques. In: Proceedings of the International Congress of Mathematicians (Vancouver, BC, 1974), vol. 1, pp. 79–85. Canad. Math. Congress, Montreal, Que., (1975)

  4. Deligne, P., Griffiths, P., Morgan, J., Sullivan, D.: Real homotopy theory of Kähler manifolds. Invent. Math. 29(3), 245–274 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  5. Drinfel’d, V.: On quasitriangular quasi-Hopf algebras and on a group that is closely connected with \({{\rm Gal}}({\overline{{{\bf Q}}}/{{\bf Q}}})\). Algebra i Analiz 2(4), 149–181 (1990)

  6. Fresse, B.: Homotopy of operads and Grothendieck-Teichmüller groups. Book in preparation. http://math.univ-lille1.fr/~fresse/OperadHomotopyBook/ (2013)

  7. Granåker, J.: Strong homotopy properads. Int. Math. Res. Not. IMRN, (14):Art. ID rnm044, 26, (2007)

  8. Ihara, Y.: On the embedding of \({{\rm Gal}}({\overline{{\bf Q}}/{{\bf Q}}})\) into \({\widehat{{\rm GT}}}\). In: The Grothendieck Theory of Dessins d’Enfants (Luminy, 1993), volume 200 of London Math. Soc. Lecture Note Ser., pp. 289–321. Cambridge Univ. Press, Cambridge (1994)

  9. Kim, M.: Weights in cohomology groups arising from hyperplane arrangements. Proc. Am. Math. Soc. 120(3), 697–703 (1994)

    Article  MATH  Google Scholar 

  10. Kontsevich, M.: Operads and motives in deformation quantization. Lett. Math. Phys. 48(1), 35–72 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  11. Lambrechts, P., Volic, I.: Formality of the little \(N\)-disks operad. Preprint. arXiv:0808.0457 (2008)

  12. Markl, M.: Models for operads. Commun. Algebra 24(4), 1471–1500 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  13. Merkulov, S.: Grothendieck-Teichmüller group in algebra, geometry and quantization: a survey. Notes from a seminar at Stockholm University organized by Torsten Ekedahl and Sergei Merkulov. http://www2.math.su.se/~sm/ (2011)

  14. Morava, J.: The motivic Thom isomorphism. In: Elliptic Cohomology, volume 342 of London Math. Soc. Lecture Note Ser., pp. 265–285. Cambridge Univ. Press, Cambridge (2007)

  15. Santos, F.G., Navarro, V., Pascual, P., Roig, A.: Moduli spaces and formal operads. Duke Math. J. 129(2), 291–335 (2005)

    Google Scholar 

  16. Sullivan, D.: Infinitesimal computations in topology. Inst. Hautes Études Sci. Publ. Math., (47), 269–331 (1978)

  17. Tamarkin, D.E.: Formality of chain operad of little discs. Lett. Math. Phys. 66(1–2), 65–72 (2003)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, KTH Royal Institute of Technology, 100 44 , Stockholm, Sweden

    Dan Petersen

Authors
  1. Dan Petersen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dan Petersen.

Additional information

Supported by the Göran Gustafsson foundation for scientific and medical research.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Petersen, D. Minimal models, GT-action and formality of the little disk operad. Sel. Math. New Ser. 20, 817–822 (2014). https://doi.org/10.1007/s00029-013-0135-5

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00029-013-0135-5

Keywords

Mathematics Subject Classification (2010)

Search

Navigation