Characterization of knot complements in the n-sphere

Abstract

Knot complements in the n-sphere are characterized. A connected open subset W of $S^n$ is homeomorphic with the complement of a locally flat (n-2)-sphere in $S^n$, n ≥ 4, if and only if the first homology group of W is infinite cyclic, W has one end, and the homotopy groups of the end of W are isomorphic to those of $S^1$ in dimensions less than n/2. This result generalizes earlier theorems of Daverman, Liem, and Liem and Venema