On a class of $5$-manifolds with $\pi_1=\mathbb Z$ with applications to knottings in $S^5$

TL;DR

This paper classifies certain 5-manifolds with fundamental group Z using cohomological invariants and applies these results to study knottings in S^5 and fiber bundle structures over the circle.

Contribution

It provides a classification of 5-manifolds with fundamental group Z and finitely generated abelian second homotopy group based on cup product structures, with applications to knot theory and fiber bundles.

Findings

Classification of 5-manifolds with specified fundamental group and second homotopy group.

Application to simple knots in S^5 and fiber bundle characterization.

Insights into the topology of knots and fibered manifolds in higher dimensions.

Abstract

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.