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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)      .

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