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.
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In [7], the transitions \(A_2^{\sigma ,+,-}\) and \(A_2^{\sigma ,-,-}\) are denoted by B and P, respectively.
References
Gibson, C.G.: Singular Points of Smooth Mappings. Reasearch Notes in Mathematics, Pitman, London (1978)
Goryunov, V.V.: Local invariants of maps between 3-manifolds. J. Topol. 6, 1–20 (2013)
Goryunov, V.V.: Local invariants of framed fronts in 3-manifolds. Arnold Math. J. 1, 211–232 (2015)
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)
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)
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)
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)
Kauffman, L. H.: Knots and physics. World Scientific Publishing Company. (2001)
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)
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)
Oset Sinha, R.: Topological invariants of stable maps from 3-manifolds to three-space. PhD Dissertation, Valencia, (2009)
Rolfsen, D.: Knots and links. Mathematics Lecture Series. Publish or Perish, Berkeley, CA, 346. (2003)
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)
Sinha, R.O., Romero Fuster, M.C.: Graphs of stable maps from 3-manifolds to 3-space. Mediterr. J. Math. 10, 1107–1126 (2013)
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).
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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.
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This work was partially supported by Vicerrectorado de investigación de la Universidad Nacional de San Cristóbal de Huamanga, VRI-UNSCH.
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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
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DOI: https://doi.org/10.1007/s00009-024-02644-x