Cross sectionally simple spheres can be wild

Abstract

The first part of this paper supplements earlier work of the first-named author by exhibiting an example of a wild n-sphere Σ in Eⁿ⁺¹ (n⩾4) for which each horizontal n-dimensional hyperplane of Eⁿ⁺¹ that does meet Σ intersects it either in a point or in an (n − 1)-sphere that is flatly embedded in the hyperplane. The second part sets forth an improved criterion for detecting the n-cell (n ≠ 4) based upon properties of slices determined as inverse sets associated with maps of a space to an interval.