Two Fold Branched Covers

TL;DR

This paper explores the properties of three-dimensional manifolds as two-fold branched covers of the sphere, providing examples, classifications, and a census related to knots with fewer than eleven crossings.

Contribution

It offers a comprehensive exposition on two-fold branched covers, classifies various types of manifolds based on their covering properties, and analyzes exceptional symmetries in knot surgeries.

Findings

Some homology spheres do not cover any manifold via two-fold branched covers.

Certain manifolds only cover the sphere or non-trivial manifolds, but not both.

A census of knot behaviors with less than eleven crossings is included.

Abstract

Many three dimensional manifolds are two-fold branched covers of the three dimensional sphere. However, there are some that are not. This paper includes exposition about two-fold branched covers and many examples. It shows that there are three dimensional homology spheres that do not two-fold branched cover any manifold, ones that only two-fold branched cover the three dimensional sphere, ones that just two-fold branched cover a non-trivial manifold, and ones that two-fold branched cover the sphere and non-trivial manifolds. When a manifold is surgery on a knot, the possible quotients via involutions generically correspond to quotients of the knot. There can however be a finite number of surgeries for which there are exceptional additional symmetries. The included proof of this result follows the proof of Thurston's Dehn surgery theorem. The paper also includes examples of such exceptional symmetries. Since the quotients follow the behavior of knots a census of the behavior for knots with less than eleven crossings is included.