Skip to main content
SpringerLink
Log in

Stable Maps from \(\#^n(S^1\times S^2)\) to the Euclidean 3-Space

  • Published:
Mediterranean Journal of Mathematics Aims and scope Submit manuscript

Abstract

We introduce a graph associated with any stable map defined from the connected sum \(\#^n(S^1\times S^2)\) of n copies of the product \(S^1\times S^2\) to the Euclidean 3-space. This graph has a weight on each vertex and a pair of weights on each edge, and its properties provide a necessary and sufficient condition to be the graph of a stable map defined from \(\#^n(S^1\times S^2)\) to the Euclidean 3-space.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

Data availability

Not applicable.

Notes

  1. In [7], the transitions \(A_2^{\sigma ,+,-}\) and \(A_2^{\sigma ,-,-}\) are denoted by B and P, respectively.

References

  1. Gibson, C.G.: Singular Points of Smooth Mappings. Reasearch Notes in Mathematics, Pitman, London (1978)

  2. Goryunov, V.V.: Local invariants of maps between 3-manifolds. J. Topol. 6, 1–20 (2013)

    Article  MathSciNet  Google Scholar 

  3. Goryunov, V.V.: Local invariants of framed fronts in 3-manifolds. Arnold Math. J. 1, 211–232 (2015)

    Article  MathSciNet  Google Scholar 

  4. Hacon, D., Mendes de Jesus, C., Romero Fuster, M. C.: Topological Invariants of Stable Maps from a Surface to the Plane from a Global Viewpoint. Real and Complex Singularities, Informa UK Limited, (2003)

  5. Hacon, D., Mendes de Jesus, C., Romero Fuster, M.C.: Fold maps from the sphere to the plane. Exp. Math. 15(4), 491–497 (2006)

    Article  MathSciNet  Google Scholar 

  6. Hacon, D., Mendes de Jesus, C., Romero Fuster, M.C.: Stable maps from surfaces to the plane with prescribed branching data. Topol. Appl. 154, 166–175 (2007)

    Article  MathSciNet  Google Scholar 

  7. Huamaní, N.B., Mendes de Jesus, C., Palacios, J.: Invariants of stable maps from the \(3\)-sphere to the Euclidean \(3\)-space. Bull. Braz. Math. Soc. New Ser. 50, 913–932 (2019)

    Article  MathSciNet  Google Scholar 

  8. Kauffman, L. H.: Knots and physics. World Scientific Publishing Company. (2001)

  9. Mather, J. N.: Stability of \(C^{\infty }\) mappings VI: the nice dimensions. In: Proc. Liverpool Singularities-Sympos., I (1969/70), Lecture notes in Math, (vol. 192, pp. 207–253), Springer-Verlag, Berlin and New York, (1971)

  10. Mendes de Jesus, C., Sinha, R.O., Romero Fuster, M.C.: Global topological invariants of stable maps from 3-manifolds to \(R^{3}\). Proc. Steklov. Inst. Math. 267, 205–216 (2009)

    Article  MathSciNet  Google Scholar 

  11. Oset Sinha, R.: Topological invariants of stable maps from 3-manifolds to three-space. PhD Dissertation, Valencia, (2009)

  12. Rolfsen, D.: Knots and links. Mathematics Lecture Series. Publish or Perish, Berkeley, CA, 346. (2003)

  13. Sinha, R.O., Romero Fuster, M.C.: First order semi-local invariants for stable maps from 3-manifolds to \({\mathbb{R} }^3\). Michigan Math. J. 61, 385–414 (2012)

    MathSciNet  Google Scholar 

  14. Sinha, R.O., Romero Fuster, M.C.: Graphs of stable maps from 3-manifolds to 3-space. Mediterr. J. Math. 10, 1107–1126 (2013)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work was partially supported by Vicerrectorado de investigación de la Universidad Nacional de San Cristóbal de Huamanga, VRI-UNSCH and the Instituto de Matemática y Ciencias Afines (Imca-Uni).

Author information

Authors and Affiliations

  1. Departamento de Matemática y Física, Universidad Nacional de San Cristóbal de Huamanga, Portal Independencia No. 57, Ayacucho, Perú

    N. B. Huamaní

  2. Departamento de Matemática, Universidade Federal de Juiz de Fora, CEP: 36036-900, Juiz de Fora, MG, Brazil

    C. Mendes de Jesus

  3. Instituto de Matemática y Ciencias Afines, (Imca-Uni), Calle Los Biólogos 245, 15012, Lima, Perú

    J. Palacios

Authors
  1. N. B. Huamaní
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. C. Mendes de Jesus
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. J. Palacios
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Nelson Berrocal Huamaní and Catarina Mendes de Jesus Sánchez wrote the main manuscript text. Nelson Berrocal Huamaní prepared all the figures. Joe Albino Palacios Baldeón rewrote the proof of some theorems and made the translation into English. All authors reviewed the manuscript.

Corresponding author

Correspondence to N. B. Huamaní.

Ethics declarations

Conflict of interest

The authors declare no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work was partially supported by Vicerrectorado de investigación de la Universidad Nacional de San Cristóbal de Huamanga, VRI-UNSCH.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Huamaní, N.B., de Jesus, C.M. & Palacios, J. Stable Maps from \(\#^n(S^1\times S^2)\) to the Euclidean 3-Space. Mediterr. J. Math. 21, 100 (2024). https://doi.org/10.1007/s00009-024-02644-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00009-024-02644-x

Keywords

Mathematics Subject Classification

Search

Navigation