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

TL;DR

This paper classifies certain 5-manifolds with free fundamental group and applies the results to classify simple boundary links in 5-spheres, providing a comprehensive algebraic understanding of these manifolds.

Contribution

It offers a new classification framework for 5-manifolds with specific fundamental group and homotopy properties, and applies it to boundary links in spheres.

Findings

Complete classification of 5-manifolds with free fundamental group and torsion-free second homotopy group.

Algebraic description of simple boundary links in S^5.

Characterization of closed 5-manifolds with free fundamental group and trivial second homology.

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 $\pi_1$-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.