Links and submersions to the plane on an open 3-manifold

TL;DR

This paper investigates when an oriented link in an open 3-manifold can be realized as a fiber of a submersion to the plane, providing a corrected and refined criterion based on the first homology group with mod 2 coefficients.

Contribution

It offers a necessary and sufficient condition for link realization as a fiber of a submersion, correcting previous errors and using homological invariants.

Findings

Established a criterion based on the first homology group with mod 2 coefficients.

Corrected earlier results with an improved understanding of the realization problem.

Provided a complete characterization for the realization of links as fibers in open 3-manifolds.

Abstract

We study the realization problem which asks if a given oriented link in an open 3-manifold can be realized as a fiber of a submersion to the Euclidean plane. We correct the results obtained before by the author which contains an error and certain imperfection, and investigate a necessary and sufficient condition for the realization in the words of well-known invariants. We obtain the condition expressed by the first homology group with mod 2 coefficient.