Published online by Cambridge University Press: 14 November 2011
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.