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Maps with finitely many critical points into high dimensional manifolds

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Abstract

Assume that there exists a smooth map between two closed manifolds \(M^m\rightarrow N^k\), where \(2\le k\le m\le 2k-1\), with only finitely many singular points, all of which are cone-like. If \((m,k)\not \in \{(2,2), (4,3), (5,3), (8,5), (16,9)\}\), then \(M^m\) admits a locally trivial topological fibration over \(N^k\) and there exists a smooth map \(M^m\rightarrow N^k\) with at most one critical point.

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References

  1. Andrica, D., Funar, L.: On Smooth maps with finitely many critical points, J. London Math. Soc. 69, 783-800 (2004), Addendum 73 (2006), 231–236

  2. Andrica, D., Funar, L., Kudryavtseva, E.: On the minimal number of critical points of maps between closed manifolds, Russian Journal of Mathematical Physics, special issue Conference for the 65-th birthday of Nicolae Teleman, Brasselet, J.-P., Legrand, A., Longo, R., Mishchenko, A. eds. 16 (2009), 363–370

  3. Araújo Dos Santos, R.N., Hohlenwerger, M.A.B., Saeki, O., Souza, T.O.: New examples of Neuwirth–Stallings pairs and non-trivial real Milnor fibrations. Ann. Institut Fourier 66, 83–104 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  4. Church, P.T., Lamotke, K.: Non-trivial polynomial isolated singularities. Indag. Math. 37, 149–154 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  5. Church, P.T., Timourian, J.G.: Differentiable maps with 0-dimensional critical set I. Pacific J. Math. 41, 615–630 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  6. Church, P.T., Timourian, J.G.: Differentiable maps with 0-dimensional critical set II. Indiana Univ. Math. J. 24, 17–28 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  7. Curtis, M.L.: Cartesian products with intervals. Proc. Am. Math. Soc. 12, 819–820 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  8. Durfee, A.H.: Fibered knots and algebraic singularities. Topology 13, 47–59 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  9. Funar, L.: Global classification of isolated singularities in dimensions (4,3) and (8,5). Ann. Scuola Norm. Sup. Pisa Cl. Sci. X, 819–861 (2011)

    MathSciNet  MATH  Google Scholar 

  10. Funar, L., Pintea, C., Zhang, P.: Smooth maps with finitely many critical points in dimensions \((4,3)\) and \((8,5)\). Proc. Am. Math. Soc. 138, 355–365 (2010)

    Article  MathSciNet  Google Scholar 

  11. Funar, L., Pintea, C.: Manifolds which admit maps with finitely many critical points into spheres of small dimensions. Michigan Math. J. 67, 585–615 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  12. Haefliger, A.: Plongements différentiables des variétés dans le domaine stable. Sem. Bourbaki exp.245, 63–77 (1964)

    MATH  Google Scholar 

  13. Kato, M.: A classification of simple spinnable structures on a 1-connected Alexander manifold. J. Math. Soc. Jpn 26, 454–463 (1974)

    MathSciNet  MATH  Google Scholar 

  14. Kauffman, L., Neumann, W.: Products of knots, branched fibrations and sums of singularities. Topology 16, 369–393 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  15. King, H.C.: Topological type of isolated singularities. Ann. Math. 107, 385–397 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  16. King, H.C.: Topology of isolated critical points of functions on singular spaces, Stratifications, singularities and differential equations, II (Marseille, 1990; Honolulu, HI, 1990), 63–72, Trotman, D., Wilson, L.C. eds., Travaux en Cours, 55, Hermann, Paris, (1997)

  17. Kneser, H., Kneser, M.: Reell-analytische Strukturen der Alexandroff–Halbgeraden und der Alexandroff–Geraden. Arch. Math. 11, 104–106 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  18. Looijenga, E.: A note on polynomial isolated singularities. Indag. Math. 33, 418–421 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  19. Mazur, B.: A note on some contractible 4-manifolds. Ann. Math. 73, 221–228 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  20. Milnor, J.: Singular Points of Complex Hypersurfaces. Princeton University Press, Princeton (1968)

    MATH  Google Scholar 

  21. Poenaru, V.: La décomposition de l’hypercube en produit topologique. Bull. Soc. Math. France 88, 113–129 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  22. Smale, S.: On the structure of manifolds. Am. J. Math. 83, 387–399 (1962)

    Article  MATH  Google Scholar 

  23. Stallings, J.: Polyhedral homotopy spheres. Bull. Am. Math. Soc. 66, 485–488 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  24. Takens, F.: Isolated critical points of \(\cal{C}^{\infty }\) and \(\cal{C}^{\omega }\) functions. Indag. Math. 28, 238–243 (1967)

    Article  MathSciNet  Google Scholar 

  25. Weinberger, S.: Homology manifolds, Handbook of geometric topology, 1085–1102. North-Holland, Amsterdam (2002)

    Google Scholar 

  26. Zeeman, E.C.: The generalized Poincaré conjecture. Bull. Am. Math. Soc. 67, 270 (1961)

    Article  MATH  Google Scholar 

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Acknowledgements

The author is grateful to Norbert A’Campo, Cornel Pintea, Nicolas Dutertre, Valentin Poenaru and Mihai Tibǎr for useful discussions and to the referees for corrections and helpful remarks. The author was partially supported by the GDRI Eco-Math.

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  1. Institut Fourier BP 74, UMR 5582, Laboratoire de Mathématiques, Université Grenoble Alpes, CS 40700, 38058, Grenoble cedex 9, France

    Louis Funar

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Funar, L. Maps with finitely many critical points into high dimensional manifolds. Rev Mat Complut 34, 585–595 (2021). https://doi.org/10.1007/s13163-020-00362-y

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