Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-05T05:34:58.096Z Has data issue: false hasContentIssue false
  • English
  • Français

Tight stable surfaces, I

Published online by Cambridge University Press:  14 November 2011

Leslie Coghlan
Leslie Coghlan
Affiliation:
Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL 35294, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Synopsis

In this paper we begin a systematic study of smooth tight maps with singularities of surfaces into E3, particularly C∞-stable maps. The class of C∞-stable tight maps of surfaces into E3 is much larger and richer than the class of C∞ (or even topological) tight immersions. We describe the general structure of C∞-stable tight maps of surfaces into E3. We show that, given any integer n ≧ 2 and a compact surface X other than the sphere or the projective plane, there is a C∞ -stable tight map X → E3 with exactly n topcycles. This is very different from the situation for tight topological immersions, where Cecil and Ryan have shown that the number α(f) of topcycles of a map f: X → E3, X a compact surface other than S2, satisfies the bound 2 ≦ α(f) ≦ 2 − Euler number of X. We prove also an analogue of the Cecil–Ryan result for C∞-stable maps.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 107 , Issue 3-4 , 1987 , pp. 213 - 232
Copyright
Copyright © Royal Society of Edinburgh 1987

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

1Arnold, V. I.. Singularity Theory-Selected Papers. London Math. Soc. Lecture Notes 53 (Cambridge: Cambridge University Press, 1981).Google Scholar
2Banchoff, T. F.. The two-piece property and tight n-manifolds with boundaries in En. Trans. Amer. Math. Soc. 161 (1971), 259–267.Google Scholar
3Cecil, T. E. and Ryan, P. J.. On the number of Top-cycles of a Tight Surface in 3-space. J. London Math. Soc. (2) 30 (1984), 335–341.Google Scholar
4Cecil, T. E. and Ryan, P. J.. Tight and Taut Immersions of Manifolds. Research Notes in Mathematics 107 (New York: Pitman, 1985).Google Scholar
5Coghlan, L.. Tight Stable Surfaces, II (to appear).Google Scholar
6Coghlan, L.. Tight Stable Surfaces, III (to appear).Google Scholar
7Gibson, C. G.. Singular points of smooth mappings. Research Notes in Mathematics 25 (New York: Pitman, 1979).Google Scholar
8Golubitsky, M. and Guillemin, V.. Stable Mappings and their Singularities (Berlin: Springer, 1980).Google Scholar
9Kuiper, N. H.. On surfaces in euclidean three space. Bull. Soc. Math. Belg. 12 (1960), 5–12.Google Scholar
10Kuiper, N. H.. Convex immersions of closed surfaces in E 3. Comment. Math. Helv. 35 (1961), 85–92.CrossRefGoogle Scholar
11Kuiper, N. H.. Minimal total absolute curvature for immersions. Invent. Math. 10 (1970), 209–238.CrossRefGoogle Scholar
12Kuiper, N. H.. Stable surfaces in euclidean three space. Math. Scand. 36 (1975), 83–96.CrossRefGoogle Scholar
13Kuiper, N. H.. Tight Embeddings and Maps. Submanifolds of Geometrical Class Three in En. The Chern Symposium 1979, Berkeley, pp. 97–145 (Berlin: Springer, 1980).Google Scholar
14Lastufka, W. S.. Tight topological immersions of surfaces in Euclidean space. J. Differential Geom. 16 (1981), 373–400.CrossRefGoogle Scholar