The Gottschalk Conjecture

TL;DR

This paper proves the Gottschalk Conjecture by demonstrating that a minimal flow on a 3-sphere leads to a contradiction via foliation and Novikov's theorem, confirming the conjecture's validity.

Contribution

It provides a novel proof of the Gottschalk Conjecture using foliation theory and an induction process on 3-manifolds, which was previously unresolved.

Findings

Minimal flow on 3-sphere implies existence of a transverse foliation.

Application of Novikov's theorem leads to contradiction.

Established the conjecture's validity for 3-spheres.

Abstract

The central idea of the proof is to show that a minimal flow v on a compact 3-manifold M implies the existence of a codimension one foliation F on it, which is transverse to the flow. If M is the 3-sphere, Novikov's theorem applies to show that one of the leaves of F is a compact surface X. It is now easy to derive a contradiction. The foliation is achieved by an induction procedure in which the manifold M is partitioned into spaces called 'tubes', each of which is homeomorphic to a closed hollow cylinder. These cylinders are, in turn, 'filled in' with closed discs, each of which has its boundary on the tube. The first two sections are devoted to showing how a surface is chosen, which serves as a base for the first step of the induction procedure. The remainder of the paper describes the foliation procedure in detail.