$^{\ast }$-taming sets for crumpled cubes. III. Horizontal sections in $2$-spheres

Abstract:

We prove that a 2-sphere S in ${E^3}$ is tame if each horizontal section of S has at most four components. Since there are wild spheres in ${E^3}$ whose horizontal sections have at most five components, this result is, in a sense, best possible. Much can nevertheless be said, however, even if certain sections have more than five components; and we show that the wildness of a 2-sphere S in ${E^3}$ is severely restricted by the requirement that each of the horizontal sections of S have at most finitely many components that separate S.