On a class of $5$-manifolds with ${\pi }_1=\mathbb{Z}$ with applications to knottings in $S^5$

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 $k\text{\hspace{0.17em}⁣}:S^3\subset S^5$ 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.