Advertisement

Toward Finite Models for the Stages of the Taylor Tower for Embeddings of the 2-Sphere

  • Adisa Bolić
  • Franjo Šarčević
  • Ismar VolićEmail author
Conference paper
Part of the Lecture Notes in Networks and Systems book series (LNNS, volume 83)

Abstract

We provide the beginning of the construction of a finite model for the stages of the Taylor tower for embeddings of the 2-sphere in a smooth manifold. We show how these stages can be described as iterated homotopy limits of punctured cubes of embedding spaces where the source manifolds are homotopy equivalent to unions of disks.

References

  1. [ALTV08]
    Arone, G., Lambrechts, P., Turchin, V., Volić, I.: Coformality and rational homotopy groups of spaces of long knots. Math. Res. Lett. 15(1), 1–14 (2008)MathSciNetCrossRefGoogle Scholar
  2. [BCKS14]
    Budney, R., Conant, J., Koytcheff, R., Sinha, D.: Embedding calculus knot invariants are of finite type. Algebr. Geom. Topol. 17(3), 1701–1742 (2017). arXiv:1411.1832MathSciNetCrossRefGoogle Scholar
  3. [BCSS05]
    Budney, R., Conant, J., Scannell, K.P., Sinha, D.: New perspectives on self-linking. Adv. Math. 191(1), 78–113 (2005)MathSciNetCrossRefGoogle Scholar
  4. [Goo92]
    Goodwillie, T.G.: Calculus II: analytic functors. K-Theory 5(4), 295–332 (1991/1992)MathSciNetCrossRefGoogle Scholar
  5. [GW99]
    Goodwillie, T.G., Weiss, M.: Embeddings from the point of view of immersion theory II. Geom. Topol. 3, 103–118 (1999)MathSciNetCrossRefGoogle Scholar
  6. [LTV10]
    Lambrechts, P., Turchin, V., Volić, I.: The rational homology of spaces of long knots in codimension \(>2\). Geom. Topol. 14, 2151–2187 (2010)MathSciNetCrossRefGoogle Scholar
  7. [Mun10]
    Munson, B.A.: Introduction to the manifold calculus of Goodwillie-Weiss. Morfismos 14(1), 1–50 (2010)MathSciNetGoogle Scholar
  8. [MV12]
    Munson, B.A., Volić, I.: Multivariable manifold calculus of functors. Forum Math. 24(5), 1023–1066 (2012)MathSciNetCrossRefGoogle Scholar
  9. [MV15]
    Munson, B.A., Volić, I.: Cubical Homotopy Theory. New Mathematical Monographs, vol. 25. Cambridge University Press, Cambridge (2015)CrossRefGoogle Scholar
  10. [Sin09]
    Sinha, D.P.: The topology of spaces of knots: cosimplicial models. Amer. J. Math. 131(4), 945–980 (2009)MathSciNetCrossRefGoogle Scholar
  11. [ŠV]
    Šarčević, F., Volić, I.: A streamlined proof of the convergence of the Taylor tower for embeddings in \(\mathbb{R}^n\). Colloq. Math. 156(1), 91–122 (2019)MathSciNetCrossRefGoogle Scholar
  12. [Vol06]
    Volić, I.: Finite type knot invariants and the calculus of functors. Compos. Math. 142(1), 222–250 (2006)MathSciNetCrossRefGoogle Scholar
  13. [Wei99]
    Weiss, M.: Embeddings from the point of view of immersion theory I. Geom. Topol. 3, 67–101 (1999)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Adisa Bolić
    • 1
  • Franjo Šarčević
    • 1
  • Ismar Volić
    • 2
    Email author
  1. 1.Department of MathematicsUniversity of SarajevoSarajevoBosnia and Herzegovina
  2. 2.Department of MathematicsWellesley CollegeWellesleyUSA

Personalised recommendations