Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-30T20:16:10.906Z Has data issue: false hasContentIssue false
  • English
  • Français

Studying the topology of Morin singularities from a global viewpoint

Published online by Cambridge University Press:  24 October 2008

Osamu Saeki
Osamu Saeki
Affiliation:
Department of Mathematics, Faculty of Science, Hiroshima University, Higashi-Hiroshima 724, Japan
Get access Rights & Permissions [Opens in a new window]

Abstract

Let f: M → N be a smooth map of a closed n-manifold into a p-manifold (n ≥ p) having only Morin singularities [17]. We study the topology of such a map and obtain a modulo 2 congruence formula involving the Euler characteristics of M, N, the singular sets and the regular fibres of f. We also consider applications of this formula to the existence problem of maps having only fold singular points. Stable maps into 3-manifolds are also studied and we obtain a modulo 2 congruence formula involving the swallow tails and the number of triple points of the discriminant set.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 117 , Issue 2 , March 1995 , pp. 223 - 235
Copyright
Copyright © Cambridge Philosophical Society 1995

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Bott, R.. Nondegenerate critical manifolds. Ann. Math. 60 (1954), 248–261.CrossRefGoogle Scholar
[2]Brown, R. L. W.. A note on immersions up to cobordism. Illinois J. Math. 21 (1977), 240–241.CrossRefGoogle Scholar
[3]Chess, D. S.. A note on the classes . Proc. Sympos. Pure Math., 40, Part 1, Amer. Math. Soc., Providence, RI, 1983, 221–224.CrossRefGoogle Scholar
[4]Donaldson, S. K. and Kronheimer, P. B.. The geometry of four-manifolds (Oxford, 1990).CrossRefGoogle Scholar
[5]Èliašberg, J. M.. On singularities of folding type. Math. USSR-Izv. 4 (1970), 1119–1134.CrossRefGoogle Scholar
[6]Èliašberg, J. M.. Surgery of singularities of smooth mappings. Math. USSE-Izv. 6 (1972), 1302–1326.CrossRefGoogle Scholar
[7]Fukuda, T.. Topology of folds, cusps and Morin singularities. In A Fete of Topology, eds. Matsumoto, Y., Mizutani, T. and Morita, S. (Academic Press, 1987) pp. 331–353.Google Scholar
[8]Izumiya, S. and Marar, W. L.. The Euler number of a topologically stable singular surface in a 3-manifold (preprint (1992)).Google Scholar
[9]Kjkuchi, S. and Saeki, O.. Remarks on the topology of folds. Proc. Amer. Math. Soc., to appear.Google Scholar
[10]Kobayashi, M.. Simplifying certain mappings from simply connected 4-manifolds into the plane. Tokyo J. Math. 15 (1992), 327–349.CrossRefGoogle Scholar
[11]Levine, H.. The singularities, . Illinois J. Math. 8 (1964), 152–168.Google Scholar
[12]Levine, H.. Elimination of cusps. Topology 3 (suppl. 2) (1965), 263–296.CrossRefGoogle Scholar
[13]Mather, J.. Stability of C ∞ mappings, IV: Classification of stable map germs by R-algebras. Publ. Math. I.H.E.S. 37 (1969), 223–248.CrossRefGoogle Scholar
[14]Mather, J.. Stability of C ∞ mappings: V: Transversality. Adv. Math. 4 (1970), 301–336.CrossRefGoogle Scholar
[15]Mather, J.. Generic projections. Ann. Math. 98 (1973), 226–245.CrossRefGoogle Scholar
[16]Milnor, J. and Stasheff, J.. Characteristic classes. Ann. Math. Stud. no. 76 (Princeton Univ. Press, 1974).CrossRefGoogle Scholar
[17]Morin, B.. Formes canoniques des singularités d'une application différentiable. C.E. Acad. Sci. Paris 260 (1965), 5662–5665, 6503–6506.Google Scholar
[18]Ballesteros, J. J. Nuño and Saeki, O.. On the number of singularities of a stable surface with boundary in a 3-manifold (preprint (1994)).Google Scholar
[19]Phillips, A.. Submersions of open manifolds. Topology 6 (1966), 171–206.CrossRefGoogle Scholar
[20]Saeki, O.. Notes on the topology of folds. J. Math. Soc. Japan 44 (1992), 551–566.CrossRefGoogle Scholar
[21]Sakuma, K.. On special generic maps of simply connected 2n-manifolds into R3. Topology Appl. 50 (1993), 249–261.CrossRefGoogle Scholar
[22]Sakuma, K.. On the topology of simple fold maps. Tokyo J. Math. 17 (1994), 21–31.CrossRefGoogle Scholar
[23]Szücs, A.. Surfaces in R3. Bull. London Math. Soc. 18 (1986), 60–66.CrossRefGoogle Scholar
[24]Thom, R.. Les singularités des applications différentiables. Ann. Inst. Fourier (Grenoble) 6 (1955–1956), 43–87.CrossRefGoogle Scholar