Connection between continuous and digital n-manifolds and the Poincare conjecture

TL;DR

This paper explores the relationship between continuous n-manifolds and digital n-manifolds using LCL covers, providing new insights into their structures and implications for the Poincaré conjecture.

Contribution

It introduces LCL covers of closed n-manifolds, characterizes when a collection forms a digital n-sphere, and links these concepts to the classification of 3-manifolds.

Findings

LCL covers of spheres can be minimized to 2n+2 disks.

A collection is a cover of a continuous n-sphere iff its intersection graph is a digital n-sphere.

Conditions are identified under which a 3-manifold is a 3-sphere.

Abstract

We introduce LCL covers of closed n-dimensional manifolds by n-dimensional disks and study their properties. We show that any LCL cover of an n-dimensional sphere can be converted to the minimal LCL cover, which consists of 2n+2 disks. We prove that an LCL collection of n-disks is a cover of a continuous n-sphere if and only if the intersection graph of this collection is a digital n-sphere. Using a link between LCL covers of closed continuous n-manifolds and digital n-manifolds, we find conditions where a continuous closed three-dimensional manifold is the three-dimensional sphere. We discuss a connection between the classification problems for closed continuous three-dimensional manifolds and digital three-manifolds.