On $5$–manifolds with free fundamental group and simple boundary links in $S^5$

Abstract

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 a ${\pi }_1$–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.