A characterisation of punctured open $3$-cells

A proof is given using standard methods of the topology of three-dimensional manifolds of the following characterization of punctured cubes: A connected, open $3$-manifold $M$ is topological ${E^3}$ with $k$ points removed if and only if every polyhedral simple closed curve in $M$ lies in a topological cube in $M$ and the rank of ${\pi _2}(M)$ is $k$. An application is given.