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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.
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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

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