Covering isotopies of $M^{n-1}$ in $N^{n}$

Abstract:

We show that a continuous family of locally flat separating embeddings of an $(n - 1)$-manifold ${M^{n - 1}}$ into an n-manifold ${N^n}$, 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 ${N^n}$ onto itself, provided $n \ne 4$. 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 ${S^{n - 1}}$ in ${S^n}$, and show that the natural map from the space of orientation preserving homeomorphisms of ${S^n}$ to the space of locally flat embeddings of ${S^{n - 1}}$ into ${S^n},n \ne 4$, is a Serre fibration and a weak homotopy equivalence.