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

Abstract

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