Algorithms for Deforming and Contracting Simply Connected Discrete Closed Manifolds (III)

TL;DR

This paper extends previous algorithms for deforming simply connected discrete manifolds to standard PL complexes, proving topological properties and homeomorphism to a 3-sphere using triangulation techniques.

Contribution

It demonstrates the applicability of contraction algorithms to PL complexes on smooth manifolds and proves a key topological theorem for 3-manifolds.

Findings

A 2-cycle separating a 3-manifold implies the components are simply-connected.

The manifold can be algorithmically transformed into a 3-sphere.

The method relates to Jordan separation and Schoenflies theorems.

Abstract

In a recent paper, {\it Algorithms for Deforming and Contracting Simply Connected Discrete Closed Manifolds (II)}, we discussed two algorithms for deforming and contracting a simply connected discrete closed manifold into a discrete sphere. The first algorithm was a continuation of work that began in {\it Algorithms for Deforming and Contracting Simply Connected Discrete Closed Manifolds (I)}, the second algorithm contained a more direct treatment of contraction for discrete manifolds. In this paper, we clarify that we can use this same method on standard piecewise linear (PL) complexes on the triangulation of general smooth manifolds. Our discussion is based on triangulation techniques invented by Cairns, Whitehead, and Whitney more than half of a century ago. In this paper, we use PL or simplicial complexes to replace certain concepts of discrete manifolds in previous papers. Note that some details in the original papers related to discrete manifolds may also need to be slightly modified for this purpose. In this paper, we use the algorithmic procedure (a contraction process) to prove the following theorem: For a finite triangulation of a simply-connected closed and orientable 3-manifold $M$ in Euclidean space, if a (simply-connected closed) 2-cycle which was made by 2-cells of this triangulation separates $M$ into two connected components, then each of the components will also be simply-connected. In addition, we can algorithmically make $M$ to be homeomorphic to a 3-sphere. The relationship of the theorem to a generalized Jordan separation problem, the general Jordan-Schoenflies theorem, and other important problems are also discussed. This revision adds more figures and explanations as well as two more special cases. We will post the detailed algorithm/procedure for practical triangulations in the following papers.