Generic Immersions of the Two-Sphere to R3 and Their Skeletons

Abstract

Let f:S2 ↬ R3 be a generic smooth immersion. The skeleton of f is the following triple (Γ, D, p): Γ ⊂ R3 is the 1-polyhedron of singular points of f, D = f−1(Γ) ⊂ S2 is also a 1-polyhedron, and p : D → Γ, x ↦ f(x), is the projection. For triples of the form (D,Γ,p), where Γ has at most four vertices, we give an iff-condition under which the triple is the skeleton of a smooth immersion f : S2 ↬ R3. Bibliography: 4 titles.