Covering isotopies of 𝑀ⁿ⁻¹ in 𝑁ⁿ

Abstract

We show that a continuous family of locally flat separating embeddings of an ( n − 1 ) (n - 1) -manifold M n − 1 {M^{n - 1}} into an n-manifold N n {N^n} , where the family is parametrized by a locally compact finite-dimensional metric space B, can be covered locally and sometimes globally by a continuous family of homeomorphisms of N n {N^n} onto itself, provided n ≠ 4 n \ne 4 . Furthermore, the covering family can be chosen to extend a preassigned covering family corresponding to a compact connected subset of B. We derive a stronger result for embeddings of S n − 1 {S^{n - 1}} in S n {S^n} , and show that the natural map from the space of orientation preserving homeomorphisms of S n {S^n} to the space of locally flat embeddings of S n − 1 {S^{n - 1}} into S n , n ≠ 4 {S^n},n \ne 4 , is a Serre fibration and a weak homotopy equivalence.