Three manifolds as geometric branched coverings of the three sphere

TL;DR

This paper presents a geometric approach to representing all closed orientable 3-manifolds as branched coverings of the 3-sphere, emphasizing the significance of geometric methods in 3-manifold theory.

Contribution

It provides a geometric version of the classical result that every closed orientable 3-manifold can be obtained as a 3-sheeted branched covering of the 3-sphere.

Findings

Every closed orientable 3-manifold can be realized as a geometric branched covering of the 3-sphere.

The minimal number of sheets in such a covering is three, with a geometric perspective.

Highlights the importance of geometry in understanding 3-manifold structures.

Abstract

One method for obtaining every closed orientable 3-manifold is as branched covering of the 3-sphere over a link. There is a classical topological result showing that the minimun possible number of sheets in the covering is three. In this paper we obtain a geometric version of this result. The interest is given by the growing importance of geometry in 3-manifolds theory.