  We classify $5$-manifolds with fundamental group $\mathbb Z$ and $\pi_{2}$ a
finitely generated abelian group in terms of the cup product on the second
cohomology of the universal covering. The classification result is applied to
study simple knots $k \colon S^{3} \subset S^{5}$ and the question, which
compact topological or smooth orientable $5$-manifold is a topological or
smooth fibre bundle over the circle with simply-connected fibre.
