How to Fold a Manifold

TL;DR

This paper presents methods for constructing and visualizing branched covers of spheres as folded embeddings or immersions, with explicit examples including the spun trefoil in 4D.

Contribution

It introduces techniques for embedding and immersing branched covers of spheres in higher dimensions, including explicit constructions for complex branch sets.

Findings

2-fold branched covers always embed in 3D and 4D.

3-fold branched covers of the 4-sphere can be explicitly embedded for certain knots.

The paper provides a method to visualize these branched covers as folded manifolds.

Abstract

Techniques for constructing codimension 2 embeddings and immersions of the 2 and 3-fold branched covers of the 3 and 4-dimensional spheres are presented. These covers are in braided form, and it is in this sense that they are folded. More precisely the composition of the embedding (or immersion) and the canonical projection induces the branched covering map. In the case of the 3-sphere, the branch locus is a knotted or linked curve in space, the 2-fold branched cover always embeds, and the 3-fold branch cover might be immersed. In the case of the 4-sphere, the branch locus is a knotted or linked orientable surface (surface knot or link), and the 2-fold branched cover is always embedded. We give an explicit embedding of the 3-fold branched cover of the 4-sphere when the branch set is the spun trefoil.