Slicing theorems for $n$-spheres in Euclidean $(n+1)$-space

Abstract:

This paper describes conditions on the intersection of an n-sphere $\Sigma$ in Euclidean $(n + 1)$-space ${E^{n + 1}}$ with the horizontal hyperplanes of ${E^{n + 1}}$ sufficient to determine that the sphere be nicely embedded. The results generally are pointed towards showing that the complement of $\Sigma$ is 1-ULC (uniformly locally 1-connected) rather than towards establishing the stronger property that $\Sigma$ is locally flat. For instance, the main theorem indicates that ${E^{n + 1}} - \Sigma$ is 1-ULC provided each non-degenerate intersection of $\Sigma$ and a horizontal hyperplane be an $(n - 1)$-sphere bicollared both in that hyperplane and in $\Sigma$ itself $(n \ne 4)$.