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Topological classification of corank 1 map germs from \(\mathbb {R}^{3}\) to \(\mathbb {R}^{3}\)

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Abstract

The link of a real analytic map germ \(f: (\mathbb {R}^{3}, 0) \rightarrow (\mathbb {R}^{3}, 0)\) is obtained by taking the intersection of the image with a small enough sphere \(S^2_\epsilon \) centered at the origin in \(\mathbb {R}^3\). If \(f\) is finitely determined, this link becomes a stable map from \(S^2\) to \(S^2\). In a previous work, we defined the Gauss paragraph which contains all the topological information of the link when the singular set \(S(\gamma )\) is connected. Now, starting from this point, we give a classification of some finitely determined weighted homogeneous map germs with two-jet equivalent to \((x,y,xz)\). In particular, we classify all 2-ruled map germs from \(\mathbb {R}^3\) to \(\mathbb {R}^3\).

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References

  1. Demoto, S.: Stable maps between 2-spheres with a connected fold curve. Hiroshima Math. J. 35, 93–113 (2005)

    MATH  MathSciNet  Google Scholar 

  2. Fukuda, T.: Local topological properties of differentiable mappings I. Invent. Math. 65, 227–250 (1981/82)

    Google Scholar 

  3. Hacon, D., Mendes de Jesus, C., Romero Fuster, M.C.: Fold maps from the sphere to the plane. Exp. Math. 15(4), 492–497 (2006)

    Article  MathSciNet  Google Scholar 

  4. Kuiper, N.H.: \(C^1\)-equivalence of functions near isolated critical points. Ann. Math. Stud. 69, 199–218 (1972)

    MathSciNet  Google Scholar 

  5. Marar, W.L., Nuño-Ballesteros, J.J.: The doodle of a finitely determined map germ from \(\mathbb{R}^{2}\) to \(\mathbb{R}^{3}\). Adv. Math. 221, 1281–1301 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  6. Marar, W.L., Tari, F.: On the geometry of simple germs of corank \(1\) maps from \(\mathbb{R}^{3}\) to \(\mathbb{R}^{3}\). Math. Proc. Camb. Philos. Soc. 119(3), 469–481 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  7. Martins, R., Nuño-Ballesteros, J.J.: Finitely determied singularities of ruled surfaces in \(\mathbb{R}^{3}\). Math. Proc. Camb. Philos. Soc. 147, 701–733 (2009)

    Article  MATH  Google Scholar 

  8. Milnor, J.: Singular points of complex hypersurfaces. In: Annals of Mathematics Studies, vol. 61. Princeton University Press, NJ (1968)

  9. Moya-Pérez, J.A., Nuño-Ballesteros, J.J.: The link of a finitely determined map germ from \(\mathbb{R}^{2}\) to \(\mathbb{R}^{2}\). J. Math. Soc. Jpn. 62(4), 1069–1092 (2010)

    Article  MATH  Google Scholar 

  10. Moya-Pérez, J.A., Nuño-Ballesteros, J.J.: Gauss words and the topology of map germs from \(\mathbb{R}^3\) to \(\mathbb{R}^3\). http://www.uv.es/nuno (preprint)

  11. Quine, J.R.: A global theorem for singularities of maps between oriented 2-manifolds. Trans. Am. Math. Soc. 236, 307–314 (1978)

    MATH  MathSciNet  Google Scholar 

  12. Scott Carter, J.: Classifying immersed curves. Proc. Am. Math. Soc. 111(1), 281–287 (1991)

    Article  MATH  Google Scholar 

  13. Wall, C.T.C.: Finite determinacy of smooth map germs. Bull. Lond. Math. Soc. 13, 481–539 (1981)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Departament de Geometria i Topologia, Universitat de València, Campus de Burjassot, 46100 , Burjassot, Spain

    J. A. Moya-Pérez & J. J. Nuño-Ballesteros

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  1. J. A. Moya-Pérez
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  2. J. J. Nuño-Ballesteros
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Correspondence to J. A. Moya-Pérez.

Additional information

Work partially supported by DGICYT Grant MTM2009-08933. The second author is partially supported by MEC-FPU Grant no. AP2009–2646.

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Moya-Pérez, J.A., Nuño-Ballesteros, J.J. Topological classification of corank 1 map germs from \(\mathbb {R}^{3}\) to \(\mathbb {R}^{3}\) . Rev Mat Complut 27, 421–445 (2014). https://doi.org/10.1007/s13163-013-0137-z

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