  We classify compact oriented $5$-manifolds with free fundamental group and
$\pi_{2}$ a torsion free abelian group in terms of the second homotopy group
considered as $\pi_1$-module, the cup product on the second cohomology of the
universal covering, and the second Stiefel-Whitney class of the universal
covering. We apply this to the classification of simple boundary links of
$3$-spheres in $S^5$. Using this we give a complete algebraic picture of closed
$5$-manifolds with free fundamental group and trivial second homology group.
