Skip to main content
SpringerLink
Log in

A geometric classification of immersions of 3-manifolds into 5-space

  • Published:
manuscripta mathematica Aims and scope Submit manuscript

Abstract.

In this paper we define two regular homotopy invariants c and i for immersions of oriented 3-manifolds into ℝ5 in a geometric manner. The pair (c(f), i(f)) completely describes the regular homotopy class of the immersion f. The invariant i corresponds to the 3-dimensional obstruction that arises from Hirsch-Smale theory and extends the one defined in [10] for immersions with trivial normal bundle.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Banghe, Li.: On classification of immersions of n-manifolds in (2n-1)-manifolds. Comment. Math. Helvetici 57, 135–144 (1982)

    Google Scholar 

  2. Ekholm, T.: Regular homotopy and Vassiliev invariants of generic immersions Sk → ℝ2k-1, k ≥4. J. Knoth Theory Ramifications 7 (8), 1041–1064 (1998)

    Article  Google Scholar 

  3. Ekholm, T., Szucs, A.: Geometric formulas for Smale invariants of codimension two immersions. Topology 42 (1), 171–196 (2003)

    Article  Google Scholar 

  4. Gompf, R. E.: Handlebody construction of Stein surfaces. Ann. Math. 148, 619–693 (1998)

    Google Scholar 

  5. Hirsch, M. W.: Immersions of manifolds. Trans. Am. Math. Soc. 93, 242–276 (1959)

    Google Scholar 

  6. Hughes, J. F., Melvin, P. M.: The Smale invariant of a knot. Comment. Math. Helv. 60, 615–627 (1985)

    Google Scholar 

  7. Juhász, A.: Regular homotopy classes of singular maps. Proc. London Math. Soc. (to appear)

  8. Paechter, G. F.: The groups π r (V n , m ). Quart. J. Math. Oxford Ser. 7 (2), 249–268 (1956)

    Google Scholar 

  9. Pontrjagin, L.: A classification of mappings of the three-dimensional complex into the two dimensional sphere. Matematicheskii Sbornik 9 (2), 331–363 (1941)

    Google Scholar 

  10. Saeki, O., Szucs, A., Takase, M.: Regular homotopy classes of immersions of 3-manifolds into 5-space. Manuscripta Math. 108, 13–32 (2002)

    Article  Google Scholar 

  11. Smale, S.: Classification of immersions of spheres in Euclidean space. Ann. Math. 69, 327–344 (1959)

    Google Scholar 

  12. Szucs, A.: Note on double points of immersions. Manuscripta Math. 76, 251–256 (1992)

    Google Scholar 

  13. Szucs, A.: On the singularities of hyperplane projections of immersions. Bull. London Math. Soc. 32, 364–374 (2000)

    Article  Google Scholar 

  14. Wu, Wen-Tsün,: On the immersion of C∞-3-manifolds in a Euclidean space. Sci. Sinica 13, 335–336 (1964)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Analysis, Eötvös Loránd University, Pázmány Péter sétány 1/C, 1117, Budapest, Hungary

    András Juhász

Authors
  1. András Juhász
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to András Juhász.

Additional information

Mathematics Subject Classification (2000): 57N35, 57R45, 57R42

Rights and permissions

Reprints and permissions

About this article

Cite this article

Juhász, A. A geometric classification of immersions of 3-manifolds into 5-space. manuscripta math. 117, 65–83 (2005). https://doi.org/10.1007/s00229-005-0543-8

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00229-005-0543-8

Keywords

Search

Navigation