The canonical Schoenflies theorem

Abstract:

M. Brown has shown that if $f:{S^{n - 1}} \times I’ \to {S^n}$, where $I’ = [ - 1,1]$, is an embedding, then the closure of either complementary domain of $f({S^{n - 1}} \times 0)$ is homeomorphic to ${B^n}$; in fact there is an embedding $g : {B^n} \to {S^n}$ satisfying $g|{S^{n - 1}} = f|{S^{n - 1}} \times 0$. This paper shows that the choice of embedding $g$ can be made to be “canonical,” i.e. if $f’$ is an embedding near $f$ in the compact-open sense, then the embedding $g’$ corresponding to $f’$ is near $g$.