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On the Boundary of the Milnor Fiber

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Singularities and Their Interaction with Geometry and Low Dimensional Topology

Part of the book series: Trends in Mathematics ((TM))

Abstract

In this work we study the topology of complex non-isolated hypersurface singularities. Inspired by work of Siersma and others, we compare the topology of the link L f with that of the boundary of the Milnor fiber, ∂F f. We review the three proofs in the literature showing that for functions \({\mathbb {C}}^3 \to {\mathbb {C}}\), the manifold ∂F f is Waldhausen: one by Némethi-Szilárd, another by Michel-Pichon and a more recent one by Fernández de Bobadilla-Menegon. We then consider an arbitrary real analytic space with an isolated singularity and maps on it with an isolated critical value. We study and define for these the concept of vanishing zone for the Milnor fiber, when this exists. We then introduce the concept of vanishing boundary cycles and compare the homology of L f and that of ∂F f. For holomorphic map germs with a one-dimensional critical set, we give a necessary and sufficient condition to have that ∂F f and L f are homologically equivalent.

To András, in celebration of his first 60th Birthday Anniversary!

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References

  1. Aguilar, M., Menegon, A., Seade, J.: On the topology of the boundary of the Milnor fiber of non-isolated complex singularities (2021). Preprint

    Google Scholar 

  2. Andreotti, A., Frankel, Th.: The Lefschetz theorem on hyperplane sections. Ann. Math. 69, 713–717 (1959)

    Article  MathSciNet  Google Scholar 

  3. Brauner, K.: Zur Geometrie der Funktionen zweier komplexen Verändlichen III, IV. Abh. Math. Sem. Hambg. 6, 8–54 (1928)

    Google Scholar 

  4. Cisneros-Molina, J.L., Seade, J., Snoussi, J.: Milnor fibrations and d-regularity for real analytic singularities. Int. J. Math. 21, 419–434 (2010)

    Article  MathSciNet  Google Scholar 

  5. Cisneros-Molina, J.L., Menegon, A., Seade, J., Snoussi, J.: Fibration theorems à la Milnor for differentiable maps with non-isolated singularities (2017). Preprint, arXiv:2002.07120

    Google Scholar 

  6. Colmez, P., Serre, J.-P. (eds.): Correspondance Grothendieck-Serre. Documents Mathématiques (Paris) 2. Société Mathématique de France, Paris (2001)

    Google Scholar 

  7. Curmi, O.: Topology of smoothings of non-isolated singularities of complex surfaces. Math. Ann. 377, 1711–1755 (2020)

    Article  MathSciNet  Google Scholar 

  8. Curmi, O.: Boundary of the Milnor fiber of a Newton non degenerate surface singularity. Adv. Math. 372, Article 107281 (2020)

    Article  MathSciNet  Google Scholar 

  9. Deligne, P., Katz, N.: Groupes de monodromie en géométrie algébrique, SGA7 t. II. LNM, vol. 340. Springer, Berlin (1973)

    Google Scholar 

  10. Dimca, A.: Singularities and Topology of Hypersurfaces. Universitext. Springer, New York (1992)

    Book  Google Scholar 

  11. du Val, P.: On absolute and non-absolute singularities of algebraic surfaces. Rev. Fac. Sci. Univ. Istanb. (A) 91, 159–215 (1944)

    Google Scholar 

  12. Durfee, A.: The signature of smoothings of complex surface singularities. Math. Ann. 232, 85–98 (1978)

    Article  MathSciNet  Google Scholar 

  13. Fernández de Bobadilla, J., Menegon, A.: The boundary of the Milnor fiber of complex and real analytic non-isolated singularities. Geom. Dedicata 173, 143–162 (2014)

    Google Scholar 

  14. Hironaka, H.: Stratification and flatness. In: Real and Complex Singularities (Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, 1976), pp. 199–265. Sijthoff and Noordhoff, Alphen aan den Rijn (1977)

    Google Scholar 

  15. Kato, M., Matsumoto, Y.: On the connectivity of the Milnor fiber of a holomorphic function at a critical point. In: Manifolds Tokyo 1973 (Proc. Internat. Conf., Tokyo, 1973), pp. 131–136. Univ. Tokyo Press, Tokyo (1975)

    Google Scholar 

  16. Laufer, H.B.: On μ for surface singularities. In: “Several Complex Variables”. Proc. Symp. Pure Math. AMS, XXX, Part 1, pp. 45–49 (1977)

    Google Scholar 

  17. Lê, D.T.: Some remarks on relative monodromy. In: Real and complex singularities (Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, 1976), pp. 397–403. Sijthoff and Noordhoff, Alphen aan den Rijn (1977)

    Google Scholar 

  18. Lê, D.T.: Introduction à la théorie des singularités t. 1 et t. 2. In: Lê, D.T. (ed.) Travaux en cours, pp. 36–37. Hermann Ed., Paris (1988)

    Google Scholar 

  19. Lê, D.T., Perron, O.: Sur la fibre de Milnor d’une singularité isolée en dimension complexe trois. C. R. Acad. Sci. Paris Sé r. A-B 289(2), A115–A118 (1979)

    MATH  Google Scholar 

  20. Lê, D.T., Nuño-Ballesteros, J.J., Seade, J.: The topology of the Milnor fiber. In: Cisneros, J.L. et al. (eds.) Handbook of geometry and topology of singularities I. Springer, Verlag (2020)

    Google Scholar 

  21. López de Medrano, S.: Topology of the intersection of quadrics in \({\mathbb {R}}^n\). In: Carlsson et al. (eds.) “Algebraic Topology” (Arcata, CA, 1986). Lecture Notes in Math., vol. 1370, pp. 280–292. Springer, Berlin (1989)

    Google Scholar 

  22. López de Medrano, S.: Singular intersections of quadrics I. In: Cisneros-Molina, J.L. et al. (eds.) Singularities in Geometry, Topology, Foliations and Dynamics. Trends in Mathematics. Birkhäuser, Basel (2017)

    Google Scholar 

  23. Massey, D.B.: Lê Cycles and Hypersurface Singularities. Lecture Notes in Mathematics, vol. 1615. Springer, Berlin (1995)

    Google Scholar 

  24. Massey, D.B.: Notes on Perverse Sheaves and Vanishing Cycles (2016). eprint arXiv:math/9908107, version 7

    Google Scholar 

  25. Maxim, L.: On the Milnor classes of complex hypersurfaces. In: G. Friedman et al. (eds.) Topology of Stratified Spaces, vol. 58. MSRI Publications, Camb. Univ. Press, Cambridge (2011)

    Google Scholar 

  26. Menegon Neto, A., Seade, J.: On the Lê-Milnor fibration for real analytic maps. Math. Nachr. 290(2–3), 382–392 (2017)

    Article  MathSciNet  Google Scholar 

  27. Menegon Neto, A., Seade, J.: Vanishing zones and the topology of non-isolated singularities. J. Geom Dedicata 202(1), 321–335 (2019). https://doi.org/10.1007/s10711-018-0415-5

    Article  MathSciNet  Google Scholar 

  28. Michel, F., Pichon, A.: On the boundary of the Milnor fiber of non-isolated singularities. Int. Math. Res. Notices 43, 2305–2311 (2003)

    Article  Google Scholar 

  29. Michel, F., Pichon, A.: On the boundary of the Milnor fiber of non-isolated singularities (Erratum). Int. Math. Res. Notices 6, 309–310 (2004)

    Article  Google Scholar 

  30. Michel, F., Pichon, A.: Carousel in family and non-isolated hypersurface singularities in \({\mathbb {C}}^3\). J. Reine Angew. Math. (Crelles J.) 720, 1–32 (2016)

    Google Scholar 

  31. Michel, F., Pichon, A., Weber, C.: The boundary of the Milnor fiber of Hirzebruch surface singularities. In: Cheniot, D. et al. (eds.) Singularity Theory, pp. 745–760. World Sci. Publ., Hackensack (2007)

    Chapter  Google Scholar 

  32. Michel, F., Pichon, A., Weber, C.: The boundary of the Milnor fiber for some non-isolated singularities of complex surfaces. Osaka J. Math. 46, 291–316 (2009)

    MathSciNet  MATH  Google Scholar 

  33. Milnor, J.W.: Differentiable structures on spheres. Am. J. Math. 81, 962–972 (1959)

    Article  MathSciNet  Google Scholar 

  34. Milnor, J.W.: Singular points of complex hypersurfaces. Ann. Math. Studies, vol. 61. Princeton University Press, Princeton (1968)

    Google Scholar 

  35. Mumford, D.: The topology of normal singularities of an algebraic surface and a criterion for simplicity. Publ. Math.. Inst. Hautes Études Sci. 9, 5–22 (1961)

    Article  MathSciNet  Google Scholar 

  36. Némethi, A.: Invariants of normal surface singularities. Contemp. Math. 354, 161–208 (2004)

    Article  MathSciNet  Google Scholar 

  37. Némethi, A.: Lattice cohomology of normal surface singularities. Publ. Res. Inst. Math. Sci. 44, 507–543 (2008)

    Article  MathSciNet  Google Scholar 

  38. Némethi, A., Szilárd, Á.: Resolution graphs of some surface singularities, II. (Generalized Iomdin series). Contemp. Math. 266, 129–164 (2000)

    Google Scholar 

  39. Némethi, A., Szilárd, Á.: Milnor Fiber Boundary of a Non-isolated Surface Singularity. Lecture Notes in Mathematics, vol. 2037. Springer, Berlin (2012)

    Google Scholar 

  40. Neumann, W.D.: A calculus for plumbing applied to the topology of complex surface singularities and degenerating complex curves. Trans. Am. Math. Soc. 268, 299–344 (1981)

    Article  MathSciNet  Google Scholar 

  41. Oka, M.: Topology of polar weighted homogeneous hypersurfaces. Kodai Math. J. 31(2), 163–182 (2008)

    Article  MathSciNet  Google Scholar 

  42. Pichon, A., Seade, J.: Fibered multilinks and singularities \(f \bar g\). Math. Ann. 342(3), 487–514 (2008)

    Google Scholar 

  43. Randell, R.: On the topology of non-isolated singularities. In: Proceedings 1977 Georgia Topology Conference, pp. 445–473 (1977)

    Google Scholar 

  44. Sabbah, C.: Vanishing cycles and their algebraic computation. http://www.cmls.polytechnique.fr/perso/sabbah/livres/sabbah_notredame1305.pdf

  45. Seade, J.: On the Topology of Isolated Singularities in Analytic Spaces. Progress in Mathematics, vol. 241. Birkhauser, Basel (2005)

    Google Scholar 

  46. Seade, J.: On Milnor’s Fibration theorem and its offspring after 50 years. Bull. Am. Math. Soc. 56, 281–348 (2019)

    Article  MathSciNet  Google Scholar 

  47. Siersma, D.: Variation mappings on singularities with a 1-dimensional critical locus. Topology 30, 445–469 (1991)

    Article  MathSciNet  Google Scholar 

  48. Siersma, D.: The vanishing topology of non isolated singularities. In: New Developments in Singularity Theory, Cambridge 2000, pp. 447–472. Kluwer, Dordrecht (2001)

    Google Scholar 

  49. Verdier, J.-L.: Stratifications de Whitney et théorème de Bertini-Sard. Invent. Math. 36, 295–312 (1976)

    Article  MathSciNet  Google Scholar 

  50. Verdier, J.-L.: Spécialisation des classes de Chern. Astérisque 82–83, pp. 149–159. Société mathématique de France, Marseille (1981)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Instituto de Matemáticas, Universidad Nacional Autónoma de México, Mexico City, Mexico

    Marcelo Aguilar & José Seade

  2. Universidade Federal da Paraiba, João Pessoa, Brazil

    Aurélio Menegon

Authors
  1. Marcelo Aguilar
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  2. Aurélio Menegon
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  3. José Seade
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Corresponding author

Correspondence to José Seade .

Editor information

Editors and Affiliations

  1. Basque Center for Applied Mathematics Ikerbasque (Basque Foundation for Science), Bilbao, Spain

    Javier Fernández de Bobadilla

  2. Faculty of Mathematics and Computer Science, Babeş-Bolyai University, Cluj-Napoca, Romania

    Tamás László

  3. Alfréd Rényi Institute of Mathematics, Budapest, Hungary

    András Stipsicz

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Aguilar, M., Menegon, A., Seade, J. (2021). On the Boundary of the Milnor Fiber. In: Fernández de Bobadilla, J., László, T., Stipsicz, A. (eds) Singularities and Their Interaction with Geometry and Low Dimensional Topology . Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-61958-9_14

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