Advertisement

Mathematische Annalen

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

Order one invariants of immersions of surfaces into 3-space

  • Tahl NowikEmail author
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.

Keywords

Abelian Group Homotopy Class Finite Order Orientable Surface Chord Diagram 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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)zbMATHGoogle 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)zbMATHGoogle 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)MathSciNetzbMATHGoogle 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)CrossRefMathSciNetzbMATHGoogle Scholar
  9. 9.
    Nowik, T.: Finite order q-invariants of immersions of surfaces into 3-space. Mathematische Zeitschrift 236, 215–221 (2001)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Nowik, T.: Automorphisms and embeddings of surfaces and quadruple points of regular homotopies. J. Diff. Geom. 58, 421–455 (2001)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Smale, S.: A classification of immersions of the two-sphere. Trans. Am. Math. Soc. 90, 281–290 (1958)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  1. 1.Department of MathematicsBar-Ilan UniversityIsrael

Personalised recommendations