Mathematische Annalen

, Volume 328, Issue 1–2, pp 261–283 | Cite as

Order one invariants of immersions of surfaces into 3-space

Article

Abstract

We classify all order one invariants of immersions of a closed orientable surface F into ℝ3, with values in an arbitrary Abelian group . We show that for any F and and any regular homotopy class of immersions of F into ℝ3, the group of all order one invariants on is isomorphic to is the group of all functions from a set of cardinality . Our work includes foundations for the study of finite order invariants of immersions of a closed orientable surface into ℝ3, analogous to chord diagrams and the 1-term and 4-term relations of knot theory.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Banchoff, T.F.: Triple points and surgery of immersed surfaces. Proc. Am. Math. Soc. 46, 407–413 (1974)MATHGoogle Scholar
  2. 2.
    Goryunov, V.V.: Local Invariants of Mappings of Surfaces into Three-Space. Arnold-Gelfand Mathematical Seminars, Geometry and Singularity Theory. Birkhauser Boston Inc., 1997, pp. 223–255Google Scholar
  3. 3.
    Hirsch, M.W.: Immersions of manifolds. Trans. Am. Math. Soc. 93, 242–276 (1959)MATHGoogle Scholar
  4. 4.
    Hobbs, C.A., Kirk, N.P.: On the classification and bifurcation of multigerms of maps from surfaces to 3-space. Mathematica Scandinavica 89, 57–96 (2001)MathSciNetMATHGoogle Scholar
  5. 5.
    Kontsevich, M.: Vassiliev’s knot invariants. I.M. Gelfand Seminar, Advances in Soviet Mathematics 16, Part 2, Am. Math. Soc., Providence, RI, 1993, pp. 137–150Google Scholar
  6. 6.
    Max, N.: Turning a sphere inside out, a guide to the film. Computers in Mathematics, Marcel Dekker Inc., 1990, pp. 334–345Google Scholar
  7. 7.
    Max, N., Banchoff, T.: Every Sphere Eversion Has a Quadruple Point. Contributions to Analysis and Geometry, John Hopkins University Press, 1981, pp. 191–209Google Scholar
  8. 8.
    Nowik, T.: Quadruple Points of Regular Homotopies of Surfaces in 3-Manifolds. Topology 39, 1069–1088 (2000)CrossRefMathSciNetMATHGoogle Scholar
  9. 9.
    Nowik, T.: Finite order q-invariants of immersions of surfaces into 3-space. Mathematische Zeitschrift 236, 215–221 (2001)MathSciNetMATHGoogle Scholar
  10. 10.
    Nowik, T.: Automorphisms and embeddings of surfaces and quadruple points of regular homotopies. J. Diff. Geom. 58, 421–455 (2001)MathSciNetMATHGoogle Scholar
  11. 11.
    Smale, S.: A classification of immersions of the two-sphere. Trans. Am. Math. Soc. 90, 281–290 (1958)MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  1. 1.Department of MathematicsBar-Ilan UniversityIsrael

Personalised recommendations