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

Abstract

In an exploration paper, {\it L. Chen, Algorithms for Deforming and Contracting Simply Connected Discrete Closed Manifolds (I)}, we designed algorithms for deforming and contracting a simply connected discrete closed manifold to a discrete sphere. However, the algorithms could not guarantee to be applicable to every case. This paper will be the continuation of the exploration. This paper contains two main procedures: (1) A shrinking procedure to contract a simply connected closed manifold. Unlike ones in the previous paper, we added a tree structure to support the process. (2) A more direct procedure for mapping a component from a separated simply connected closed manifold to a disk. We also discuss the practical use of these algorithms in topological data analysis. We think that we have an algorithmic solution, but careful detailed analysis should be done next.