On weakly flat $1$-spheres

Abstract:

With his cellularity criterion, McMillan has provided a characterization of those complementary domains of an $(n - 1)$-sphere in the $n$-sphere ${S^n}$ that are open $n$-cells $(n \ne 4)$. Paul F. Duvall has used the cellularity criterion to characterize those $k$-spheres $\Sigma$ in ${S^n}(2 \leqq k \leqq n - 3)$ such that ${S^n} - \Sigma$ is homeomorphic to the complement of the standard $k$-sphere in ${S^n}$. The purpose of this note is to indicate how McMillan’s methods can be reapplied to handle the case $k = 1(n \ne 4)$ as well.