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

TL;DR

This paper presents polynomial-time algorithms for deforming and contracting simply connected discrete closed manifolds into discrete spheres within a higher-dimensional ambient space, with plans for further analysis of special cases.

Contribution

It introduces a new polynomial-time algorithm for deforming simply connected discrete manifolds into spheres, expanding computational topology methods.

Findings

Algorithm works for most cases

Operates in polynomial time

Further analysis needed for special cases

Abstract

In this exploration paper, we design algorithms for deforming and contracting a simply connected discrete closed manifold to a discrete sphere. Such a contraction is a kind of shrinking or reducing process. In our algorithms, we need to assume an ambient space for the discrete manifold, and this ambient space also a simply connected discrete space in higher dimensions. Our algorithm would work for most of cases. For some special cases, we will make detailed analysis in the next paper. In other words, this paper has not provided a complete proof for each case. The algorithm designed in this paper is in polynomial time.