Advertisement

Embedding Calculus and the Little Discs Operads

  • Victor Turchin
Chapter
Part of the MATRIX Book Series book series (MXBS, volume 1)

Abstract

This note describes recent development in the study of embedding spaces from the manifold calculus viewpoint. An important progress that has been done was the discovery and application of the connection to the theory of operads. This allows one to describe embedding spaces as certain derived operadic module maps and to produce their explicit deloopings.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

The author/speaker is grateful to Gabriel C. Drummond-Cole for his amazing ability of simultaneous tex-typing during the lectures. The final version of this note is a slight improvement of his. The author/speaker is also grateful to P. Hackney and M. Robertson for organizing the conference and the MATRIX institute for providing support and base for this conference.

References

  1. 1.
    Arone, G., Turchin, V.: On the rational homology of high dimensional analogues of spaces of long knots. Geom. Topol. 18, 1261–1322 (2014)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Boavida de Brito, P., Weiss, M.: Manifold calculus and homotopy sheaves. Homol. Homot. Appl. 15(2), 361–383 (2013)Google Scholar
  3. 3.
    Boavida de Brito, P., Weiss, M.: Spaces of smooth embeddings and configuration categories. To appear in J. Topology (2018)Google Scholar
  4. 4.
    Budney, R.: Little cubes and long knots. Topology 46(1), 1–27 (2007)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Ducoulombier, J., Turchin, V.: Delooping manifold calculus tower on a closed disc (2017, preprint). arXiv:1708.02203Google Scholar
  6. 6.
    Dwyer, W., Hess, K.: Long knots and maps between operads. Geom. Topol. 16(2), 919–955 (2012)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Dwyer, W., Hess, K.: Delooping the space of long embeddings (paper to appear)Google Scholar
  8. 8.
    Fiedorowicz, Z., Vogt, R.M.: An additivity theorem for the interchange of E n structures. Adv. Math. 273, 421–484 (2015)Google Scholar
  9. 9.
    Fresse, B., Willwacher, T.: The intrinsic formality of E n operads (2015, preprint). arXiv:1503.08699Google Scholar
  10. 10.
    Fresee, B., Turchin, V., Willwacher, T.: The rational homotopy of mapping spaces of E n operads (2017, preprint). arXiv:1703.06123Google Scholar
  11. 11.
    Goodwillie, T.G., Klein, J.: Multiple disjunction for spaces of smooth embeddings. J. Topol. 8(3), 651–674 (2015)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Goodwillie, T.G., Weiss, M.: Embeddings from the point of view of immersion theory: part II. Geom. Topol. 3, 103–118 (1999)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Haefliger, A.: Knotted (4k − 1)-spheres in 6k-space. Ann. Math. (2) 75, 452–466 (1962)Google Scholar
  14. 14.
    Haefliger, A.: Enlacements de sphères en codimension supérieure à 2. Comm. Math. Helv. 41, 51–72 (1966–1967)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Kontsevich, M.: Operads and motives in deformation quantization. Lett. Math. Phys. 48(1), 35–72 (1999). Moshé Flato (1937–1998)Google Scholar
  16. 16.
    Lambrechts, P., Volić, I.: Formality of the little N-disks operad. Mem. Am. Math. Soc. 230(1079), viii+116 (2014)Google Scholar
  17. 17.
    McClure, J.E., Smith, J.H.: Cosimplicial objects and little n-cubes. I. Am. J. Math. 126(5), 1109–1153 (2004)Google Scholar
  18. 18.
    Pirashvili, T.: Hodge decomposition for higher order Hochschild homology. Ann. Sci. Ecole Norm. Sup (4) 33(2), 151–179 (2000)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Rezk, C.: A model for the homotopy theory of homotopy theory. Trans. Am. Math. Soc. 353(3), 973–1007 (2001)Google Scholar
  20. 20.
    Robertson, M.: Spaces of Operad Structures. Preprint. arXiv:1111.3904Google Scholar
  21. 21.
    Sinha, D.: Operads and knot spaces. J. Am. Math. Soc. 19(2), 461–486 (2006)Google Scholar
  22. 22.
    Tamarkin, D.E.: Formality of chain operad of little discs. Lett. Math. Phys. 66(1–2), 65–72 (2003)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Turchin, V.: Context-free manifold calculus and the Fulton-MacPherson operad. Algebr. Geom. Topol. 13(3), 1243–1271 (2013)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Turchin, V.: Delooping totalization of a multiplicative operad. J. Homotopy Relat. Struct. 9(2), 349–418 (2014)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Turchin V., Willwacher, T.: Relative (non-)formality of the little cubes operads and the algebraic Cerf lemma. To appear in Amer. J. Math. (2018)Google Scholar
  26. 26.
    Weiss, M.: Embeddings from the point of view of immersion theory. I. Geom. Topol. 3, 67–101 (1999)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Math DepartmentKansas State UniversityManhattanUSA

Personalised recommendations