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Northeastern Brazilian Meeting on Singularities

Brazil-Mexico Meeting on Singularities

NBMS 2015, BMMS 2015: Singularities and Foliations. Geometry, Topology and Applications pp 3–49Cite as

Combinatorial Models in the Topological Classification of Singularities of Mappings

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Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 222))

Abstract

The topological classification of finitely determined map germs \(f:(\mathbb R^n,0)\rightarrow (\mathbb R^p,0)\) is discrete (by a theorem due to R. Thom), hence we want to obtain combinatorial models which codify all the topological information of the map germ f. According to Fukuda’s work, the topology of such germs is determined by the link, which is obtained by taking the intersection of the image of f with a small enough sphere centered at the origin. If \(f^{-1}(0)=\{0\}\), then the link is a topologically stable map \(\gamma :S^{n-1}\rightarrow S^{p-1}\) (or stable if (n, p) are nice dimensions) and f is topologically equivalent to the cone of \(\gamma \). When \(f^{-1}(0)\ne \{0\}\), the situation is more complicated. The link is a topologically stable map \(\gamma :N\rightarrow S^{p-1}\), where N is a manifold with boundary of dimension \(n-1\). However, in this case, we have to consider a generalized version of the cone, so that f is again topologically equivalent to the cone of the link diagram. We analyze some particular cases in low dimensions, where the combinatorial models are provided by objects which are well known in Computational Geometry, for instance, the Gauss word or the Reeb graph.

This work has been partially supported by DGICYT Grant MTM2015–64013–P and by PVE - CAPES Grant 88881.062217/2014–01.

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References

  1. Arnold, V.I.: Topological classification of Morse functions and generalisations of Hilbert’s 16-th problem. Math. Phys. Anal. Geom. 10, 227–236 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  2. Batista, E.B., Costa, J.C.F., Nuño-Ballesteros, J.J.: The Reeb graph of a map germ from \({\mathbb{R}}^{3}\) to \({\mathbb{R}}^{2}\) with non isolated zeros. Bull. Braz. Math. Soc. https://doi.org/10.1007/s00574-017-0058-4

  3. Batista, E.B., Costa, J.C.F., Nuño-Ballesteros, J.J.: The cone structure theorem for map germs with non isolated zeros. http://www.uv.es/nuno/Preprints/ConeTheorem.pdf

  4. Batista, E.B., Costa, J.C.F., Nuño-Ballesteros, J.J.: The Reeb graph of a map germ from \({\mathbb{R}}^{3}\) to \({\mathbb{R}}^{2}\) with isolated zeros. Proc. Edinb. Math. Soc. 60(2), 319–348 (2017)

    Google Scholar 

  5. Costa, J.C.F., Nuño-Ballesteros, J.J.: Topological \(\cal{K}\)-classification of finitely determined map germs. Geom. Dedicata 166, 147–162 (2013)

    Google Scholar 

  6. Dehn, M.: Über kombinatorische topologie. Acta Math. 67(1), 123–168 (1936)

    Google Scholar 

  7. de Jong, T., Pfister, G.: Local analytic geometry. Basic Theory and Applications. Advanced Lectures in Mathematics. Friedr. Vieweg & Sohn, Braunschweig (2000)

    Google Scholar 

  8. Ehresmann, C.: Les connexions infinitésimales dans un espace fibré différentiable, Colloque de topologie (espaces fibrés), Bruxelles, pp. 29–55 (1950); Georges Thone, Liége; Masson et Cie., Paris, 1951

    Google Scholar 

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

    Google Scholar 

  10. Fukuda, T.: Local topological properties of differentiable mappings II. Tokyo J. Math. 8(2), 501–520 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  11. Gauss, C.F.: Werke VIII, pp. 271–286. Teubner, Leipzig (1900)

    Google Scholar 

  12. Gibson, C.G., Wirthmüller, K., du Plessis, A.A., Looijenga, E.J.N.: Topological stability of smooth mappings. Lecture Notes in Mathematics, vol. 552. Springer, Berlin (1976)

    Google Scholar 

  13. Golubitsky, M., Guillemin, V.: Stable mappings and their singularities. Graduate Texts in Mathematics, vol. 14. Springer, New York (1973)

    Google Scholar 

  14. Greuel, G.M., Pfister, G.: A Singular Introduction to Commutative Algebra. Springer, Berlin (2008)

    MATH  Google Scholar 

  15. Izar, S.A.: Funções de Morse e topologia das superfícies II: Classificação das funções de Morse estáveis sobre superfícies, Métrica no. 35, Estudo e Pesquisas em Matemática, IBILCE, Brazil (1989). http://www.ibilce.unesp.br/Home/Departamentos/Matematica/metrica-35.pdf

  16. Mather, J.: Stability of \(C^\infty \) mappings I: the division theorem. Ann. Math. 87(2), 89–104 (1968)

    Google Scholar 

  17. Mather, J.: Stability of \(C^\infty \) mappings III: finitely determined mapgerms. Publ. Math. Inst. Hautes Études Sci. 35, 279–308 (1968)

    Article  MathSciNet  Google Scholar 

  18. Mather, J.: Stability of \(C^\infty \) mappings II: infinitesimal stability implies stability. Ann. Math. 89(2), 254–291 (1969)

    Google Scholar 

  19. Mather, J.: Stability of \(C^\infty \) mappings IV: classification of stable germs by \(\mathbb{R}\)-algebras. Publ. Math. Inst. Hautes Études Sci. 37, 223–248 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  20. Mather, J.: Stability of \(C^\infty \) mappings V: transversality. Adv. Math. 4, 301–336 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  21. Mather, J.: Stability of \(C^\infty \) mappings VI: the nice dimensions. In: Proceedings of Liverpool Singularities-Symposium, I (1969/70). Lecture Notes in Mathematics, vol. 192, pp. 207–253. Springer, Berlin (1971)

    Google Scholar 

  22. 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(4), 1281–1301 (2009)

    Google Scholar 

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

    Google Scholar 

  24. Martins, R., Nuño-Ballesteros, J.J.: The link of a ruled frontal surface singularity. Real and Complex Singularities. Contemporary Mathematics, vol. 675, pp. 181–195. American Mathematical Society, Providence (2016)

    Google Scholar 

  25. Mendes, R., Nuño-Ballesteros, J.J.: Knots and the topology of singular surfaces in \({\mathbb{R}}^{4}\). Real and Complex Singularities. Contemporary Mathematics, vol. 675, pp. 229–239. American Mathematical Society, Providence (2016)

    Google Scholar 

  26. Milnor, J.: Morse theory. Annals of Mathematics Studies, vol. 51. Princeton University Press, Princeton (1963)

    Google Scholar 

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

    Google Scholar 

  28. Mond, D.: On the classification of germs of maps from \(\mathbb{R}^2\) to \(\mathbb{R}^3\). Proc. Lond. Math. Soc. 50, 333–369 (1985)

    Google Scholar 

  29. Montesinos-Amilibia, A.: SphereXSurface, computer program. http://www.uv.es/montesin

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

    Google Scholar 

  31. Moya-Pérez, J.A., Nuño-Ballesteros, J.J.: Topological triviality of families of map germs from \(\mathbb{R}^2\) to \(\mathbb{R}^2\). J. Singul. 6, 112–123 (2012)

    MathSciNet  MATH  Google Scholar 

  32. 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(2), 421–445 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  33. Moya-Pérez, J.A., Nuño-Ballesteros, J.J.: Some remarks about the topology of corank 2 map germs from \(\mathbb{R}^2\) to \(\mathbb{R}^2\). J. Singul. 10, 200–224 (2014)

    MathSciNet  MATH  Google Scholar 

  34. Moya-Pérez, J.A., Nuño-Ballesteros, J.J.: Gauss words and the topology of map germs from \(\mathbb{R}^3\) to \(\mathbb{R}^3\). Rev. Mat. Iberoam. 31(3), 977–988 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  35. Moya-Pérez, J.A., Nuño-Ballesteros, J.J.: Topological triviality of families of map germs from \(\mathbb{R}^3\) to \(\mathbb{R}^3\). Rocky Mountain J. Math. 46(5), 1643–1664 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  36. Reeb, G.: Sur les points singuliers d’une forme de Pfaff completement intégrable ou d’une fonction numérique. C. R. Math. Acad. Sci. Paris 222, 847–849 (1946)

    MATH  Google Scholar 

  37. Sharko, V.V.: Smooth and topological equivalence of functions on surfaces. Ukr. Math. J. 55, 832–846 (2003)

    Article  MathSciNet  Google Scholar 

  38. Thom, R.: La stabilité topologique des applications polynomiales. Enseign. Math. 8(2), 24–33 (1962)

    Google Scholar 

  39. Thom, R.: Local topological properties of differentiable mappings. Colloquium on Differential Analysis (Tata Institute), pp. 191–202. Oxford University Press, Oxford (1964)

    Google Scholar 

  40. Wall, C.T.C.: Finite determinacy of smooth map-germs. Bull. Lond. Math. Soc. 13(6), 481–539 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  41. Wall, C.T.C.: Classification and Stability of Singularities of Smooth Maps Singularity Theory (Trieste, 1991), pp. 920–952. World Scientific Publishing, River Edge (1995)

    Google Scholar 

  42. Whitney, H.: The singularities of a smooth \(n\)-manifold in \((2n-1)\)-space. Ann. Math. 45(2), 247–293 (1944)

    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. J. Nuño-Ballesteros

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  1. J. J. Nuño-Ballesteros
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Correspondence to J. J. Nuño-Ballesteros .

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Editors and Affiliations

  1. ICMC, University of São Paulo, São Carlos, São Paulo, Brazil

    Raimundo Nonato Araújo dos Santos

  2. Department of Mathematics, Federal University of Paraíba, João Pessoa, Paraíba, Brazil

    Aurélio Menegon Neto

  3. Mathematics Institute, University of Warwick, Coventry, United Kingdom

    David Mond

  4. ICMC, University of São Paulo, São Carlos, São Paulo, Brazil

    Marcelo J. Saia

  5. Instituto de Matemáticas, Unidad Cuernavaca, Universidad Nacional Autónoma de México, Cuernavaca, Morelos, Mexico

    Jawad Snoussi

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Nuño-Ballesteros, J.J. (2018). Combinatorial Models in the Topological Classification of Singularities of Mappings. In: Araújo dos Santos, R., Menegon Neto, A., Mond, D., Saia, M., Snoussi, J. (eds) Singularities and Foliations. Geometry, Topology and Applications. NBMS BMMS 2015 2015. Springer Proceedings in Mathematics & Statistics, vol 222. Springer, Cham. https://doi.org/10.1007/978-3-319-73639-6_1

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