Skip to main content

On Kontsevich’s characteristic classes for higher dimensional sphere bundles I: the simplest class

Abstract

This paper studies the simplest one of the sequence of characteristic classes of framed smooth fiber bundles constructed by M. Kontsevich. By introducing a correction term to the characteristic number of the Kontsevich class, we obtain an invariant of unframed sphere bundles over a sphere. The correction term is given by a multiple of Hirzebruch’s signature defect. We observe that a reduction of our invariant modulo a certain integer agrees with a multiple of Milnor’s λ′-invariant of exotic spheres. Furthermore, our invariant is non-trivial for many fiber dimensions. Hence we can detect some ‘exotic’ non-trivial subspace of π i (Diff(S d)) ⊗ \({\mathbb {Q}}\) for some pairs (i, d) which are not in Igusa’s stable range.

This is a preview of subscription content, access via your institution.

References

  1. Antonelli P.L., Burghelea D., Kahn P.J.: Gromoll groups, Diff S n and bilinear constructions of exotic spheres. Bull. Am. Math. Soc. 76, 772–779 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  2. Antonelli P.L., Burghelea D., Kahn P.J.: The non-finite homotopy type of some diffeomorphism groups. Topology 11, 1–49 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  3. Atiyah M.F., Singer I.M.: The index of elliptic operators. III. Ann. Math. 87(2), 546–604 (1968)

    Article  MathSciNet  Google Scholar 

  4. Axelrod, S., Singer, I.M.: Chern–Simons perturbation theory. In: Catto S., Rocha A. (eds.) Proceedings of the XXth DGM Conference, pp. 3–45. World Scientific, Singapore (1992)

  5. Burghelea, D.: In: Problems Concerning Manifolds. Manifolds—Amsterdam 1970 (Proc. Nuffic Summer School) p. 223. Lecture Notes in Math., vol. 197. Springer, Berlin (1971)

  6. Burton D.M.: Elementary Number Theory. McGraw-Hill, New York (2002)

    Google Scholar 

  7. Bott R., Cattaneo A.: Integral Invariants of 3-Manifolds. J. Differ. Geom. 48, 91–133 (1998)

    MathSciNet  MATH  Google Scholar 

  8. Cerf J.: La stratification naturelle des espaces de fonctions différentiables réelles et le theóreḿe de la pseudo-isotopie. Inst. Hautes Et́udes Sci. Publ. Math. 39, 5–173 (1970)

    MathSciNet  MATH  Google Scholar 

  9. Chern S.S., Simons J.: Characteristic forms and geometric invariants. Ann. Math. 99, 48–69 (1974)

    Article  MathSciNet  Google Scholar 

  10. Eells J., Kuiper N.: An invariant for certain smooth manifolds. Ann. Math. 60, 93–110 (1962)

    MathSciNet  Google Scholar 

  11. Farrell F.T., Hsiang W.C.: On the rational homotopy groups of the diffeomorphism groups of discs, spheres and aspherical manifolds. Proc. Sympos. Pure Math. 32, 325–337 (1978)

    MathSciNet  Google Scholar 

  12. Fulton W., MacPherson R.: A compactification of configuration spaces. Ann. Math. 139, 183–225 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hatcher A.: A proof of a Smale conjecture, Diff(S 3)≃ O(4). Ann. Math. (2) 117(3), 553–607 (1983)

    Article  MathSciNet  Google Scholar 

  14. Hatcher A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)

    MATH  Google Scholar 

  15. Igusa, K.: Higher Franz–Reidemeister Torsion, Studies in Adv. Math. 31 AMS/IP (2002)

  16. Igusa K.: Axioms for higher torsion invariants of smooth bundles. J. Topol. 1, 159–186 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. Itõ, K. (ed.): Encyclopedic Dictionary of Mathematics, 2nd edn. The Mathematical Society of Japan, vol. IV. MIT Press, Cambridge, (1987)

  18. Kervaire M.: Relative characteristic classes. Am. J. Math. 79(3), 517–558 (1957)

    Article  MathSciNet  Google Scholar 

  19. Kervaire, M., Milnor, J.: Bernoulli numbers, homotopy groups and a theorem of Rohlin. In: Proceedings of the International Congress of Mathematicians, Edinburgh, or in [27, Part 1] Mil3 (1958)

  20. Kervaire, M., Milnor, J.: Groups of homotopy spheres: I. Ann. Math. 77, 504–537, (1963) or in [27, Part 1]Mil3

    Google Scholar 

  21. Kontsevich, M.: Feynman Diagrams and Low-dimensional Topology, First European Congress of Mathematics, vol. II (Paris, 1992), pp. 97–121. Birkhäuser, Basel (1994)

  22. Kuperberg, G., Thurston, D.: Perturbative 3-manifold invariants by cut-and-paste topology, preprint, math.GT/9912167

  23. Lescop, C.: On the Kontsevich-Kuperberg-Thurston construction of a configuration-space invariant for rational homology 3-spheres, math.GT/0411088, Prépublication de l’Institut Fourier 655 (2004)

  24. Lescop, C.: Splitting formulae for the Kontsevich–Kuperberg–Thurston invariant of rational homology 3-spheres, math.GT/0411431, Prépublication de l’Institut Fourier, p. 656 (2004)

  25. Milnor, J.: Differential structures on spheres. Am. J. Math. 81, 962–971 (1959) or in [27, Part 1]Mil3

    Google Scholar 

  26. Milnor, J.: Differentiable manifolds which are homotopy spheres (mimeographed). Princeton University, Princeton (1959) or in [27, Part 1]Mil3

  27. Milnor, J.: Collected papers of John Milnor. III. Differential Topology. American Mathematical Society, Providence (2007)

  28. Milnor J., Moore J.: On the structure of Hopf algebras. Ann. Math. 81(2), 211–264 (1965)

    Article  MathSciNet  Google Scholar 

  29. Milnor, J., Stasheff, J.: Characteristic Classes. Ann. of Math. Stud., No. 76. Princeton University Press, Princeton (1974)

  30. Morita S.: Casson invariant, signature defect of framed manifolds and the secondary characteristic classes of surface bundles. J. Differ. Geom. 47, 560–599 (1997)

    MATH  Google Scholar 

  31. Morita, S.: Geometry of Characteristic Classes, Transl. of Math. Monogr., vol. 199. American Mathematical Society, Providence (2001)

  32. Novikov, S.P.: Differentiable Sphere Bundles, Izv. Akad. Nauk SSSR Mat 29, 1–96 (Amer. Math. Soc. Transl, Ser. 2, vol. 63) (1965)

  33. Ohtsuki T.: Finite type invariants of integral homology 3-spheres. J. Knot Theory Ramifications 5, 101–115 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  34. Serre J.-P.: Homologie singulière des espaces fibrés. Appl. Ann. Math. 54(2), 425–505 (1951)

    MathSciNet  Google Scholar 

  35. Smale S.: Diffeomorphisms of the 2-sphere. Proc. Am. Math. Soc. 10, 621–626 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  36. Smale S.: Differentiable and combinatorial structures on manifolds. Ann. Math. 74, 498–502 (1961)

    Article  MathSciNet  Google Scholar 

  37. Smale S.: On the structure of manifolds. Am. J. Math. 84, 387–399 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  38. Switzer R.M.: Algebraic Topology–Homotopy and Homology. Springer, Berlin (1975)

    MATH  Google Scholar 

  39. Wall C.T.C.: Determination of the cobordism ring. Ann. Math. 72, 292–311 (1960)

    Article  Google Scholar 

  40. Witten E.: Quantum field theory and the Jones polynomial. Comm. Math. Phys. 121, 351–399 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  41. Witten, E.: Chern–Simons gauge theory as a string theory. In: The Floer Memorial Volume, Progr. Math., vol. 133, pp. 637–678. Birkhäuser, Basel (1995)

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Tadayuki Watanabe.

Additional information

Dedicated to Professor Akio Kawauchi for his 60th birthday.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Watanabe, T. On Kontsevich’s characteristic classes for higher dimensional sphere bundles I: the simplest class. Math. Z. 262, 683 (2009). https://doi.org/10.1007/s00209-008-0396-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00209-008-0396-4

Keywords

  • Cohomology Class
  • Homotopy Class
  • Signature Defect
  • Euler Class
  • Sphere Bundle