Journal of the Mathematical Society of Japan

On the topology of stable maps

Nicolas DUTERTRE and Toshizumi FUKUI

Full-text: Open access


We investigate how Viro's integral calculus applies for the study of the topology of stable maps. We also discuss several applications to Morin maps and complex maps.

Article information

J. Math. Soc. Japan, Volume 66, Number 1 (2014), 161-203.

First available in Project Euclid: 24 January 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57R45: Singularities of differentiable mappings
Secondary: 57R20: Characteristic classes and numbers 57R70: Critical points and critical submanifolds 58C25: Differentiable maps

singularities of maps stable maps Euler integration Euler characteristic Morin maps


DUTERTRE, Nicolas; FUKUI, Toshizumi. On the topology of stable maps. J. Math. Soc. Japan 66 (2014), no. 1, 161--203. doi:10.2969/jmsj/06610161.

Export citation


  • V. I. Arnol'd, S. M. Guseĭn-Zade and A. N. Varchenko, Singularities of differentiable maps, Vol.,I, The classification of critical points, caustics and wave fronts, Monogr. Math., 82, Birkhäuser Boston, Inc., Boston, MA, 1985.
  • A. Borel and J. C. Moore, Homology theory for locally compact spaces, Michigan Math. J., 7 (1960), 137–159.
  • S. A. Broughton, On the topology of polynomial hypersurfaces, In: Singularities. Part 1, Arcata, Calif., 1981, (ed. P. Orlik), Proc. Sympos. Pure Math., 40, Amer. Math. Soc., Providence, RI, 1983, pp.,167–178.
  • R.-O. Buchweitz and G.-M. Greuel, The Milnor number and deformations of complex curve singularities, lnvent. Math., 58 (1980), 241–281.
  • S. K. Donaldson and P. B. Kronheimer, The Geometry of Four-Manifolds, Oxford Math. Monogr., Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1990.
  • A. H. Durfee, Neighborhoods of algebraic sets, Trans. Amer. Math. Soc., 276 (1983), 517–530.
  • J. M. Èlia\u sberg, On singularities of folding type, Math. USSR-Izv., 4 (1970), 1119–1134.
  • T. Fukuda, Local topological properties of differentiable mappings. I, Invent. Math., 65 (1981/82), 227–250.
  • T. Fukuda, Local topological properties of differentiable mappings. II, Tokyo J. Math., 8 (1985), 501–520.
  • T. Fukuda, Topology of folds, cusps and Morin singularities, In: A fête of topology, (eds. Y. Matsumoto, T. Mizutani and S. Morita), Academic Press, Boston, MA, 1988, pp.,331–353.
  • T. Fukuda and G. Ishikawa, On the number of cusps of stable perturbations of a plane-to-plane singularity, Tokyo J. Math., 10 (1987), 375–384.
  • T. Fukui and J. Weyman, Cohen-Macaulay properties of Thom-Boardman strata. II, The defining ideals of $\Sigma^{i,j}$, Proc. London Math. Soc., 87 (2003), 137–163.
  • T. Gaffney and D. M. Q. Mond, Cusps and double folds of germs of analytic maps $\mathbb{C}^2\to\mathbb{C}^2$, J. London Math. Soc. (2), 43 (1991), 185–192.
  • K. Ikegami and O. Saeki, Cobordism of Morse maps and its application to map germs, Math. Proc. Cambridge Philos. Soc., 147 (2009), 235–254.
  • G. M. Khimshiashvili, On the local degree of a smooth map, Soobshch. Akad. Nauk Gruz. SSR, 85 (1977), 309–311.
  • H. I. Levine, Mappings of manifolds into the plane, Amer. J. Math., 88 (1966), 357–365.
  • J. W. Milnor, Topology from the differentiable viewpoint, Based on notes by David W. Weaver, The University Press of Virginia, Charlottesville, Va., 1965.
  • I. Nakai, Characteristic classes and fiber products of smooth mappings, Preprint.
  • I. Nakai, Elementary topology of stratified mappings, In: Singularities-Sapporo 1998, (eds. J. P. Brasselet and T. Suwa), Adv. Stud. Pure Math., 29, Kinokuniya, Tokyo, 2000, pp.,221–243.
  • J. R. Quine, A global theorem for singularities of maps between oriented 2-manifolds, Trans. Amer. Math. Soc., 236 (1978), 307–314.
  • O. Saeki, Studying the topology of Morin singularities from a global viewpoint, Math. Proc. Cambridge Philos. Soc., 117 (1995), 223–235.
  • Z. Szafraniec, On the Euler characteristic of analytic and algebraic sets, Topology, 25 (1986), 411–414.
  • R. Thom, Les singularités des applications différentiables, Ann. Inst. Fourier (Grenoble), 6 (1955–1956), 43–87.
  • M. Tibăr and A. Zaharia, Asymptotic behaviour of families of real curves, Manuscripta Math., 99 (1999), 383–393.
  • O. Ya. Viro, Some integral calculus based on Euler characteristic, In: Topology and Geometry – Rohlin Seminar, (ed. O. Ya. Viro), Lecture Notes in Math., 1346, Springer-Verlag, Berlin, 1988, pp.,127–138.
  • O. Ya. Viro, Plane real algebraic curves: constructions with controlled topology, Leningrad Math. J., 1 (1990), 1059–1134.
  • Y. Yomdin, The structure of strata $\mu={const}$ in a critical set of a complete intersection singularity, In: Singularities. Part 2, Arcata, Calif., 1981, (ed. P. Orlik), Proc. Sympos. Pure Math., 40, Amer. Math. Soc., Providence, RI, 1983, pp.,663–665.
  • C. T. C. Wall, Topological invariance of the Milnor number mod 2, Topology, 22 (1983), 345–350.
  • C. T. C. Wall, Transversality in families of mappings, Proc. Lond. Math. Soc., 99 (2009), 67–99.