On digital simply connected spaces and manifolds: a digital simply connected 3-manifold is the digital 3-sphere

TL;DR

This paper explores digital topology, proving that digitally simply connected 3-manifolds are equivalent to digital 3-spheres, thus providing a digital analogue of the Poincaré conjecture for three-dimensional digital spaces.

Contribution

It introduces the concept of simple connectedness in digital spaces and establishes that simply connected digital 3-manifolds are digital 3-spheres, extending classical topological results to digital topology.

Findings

A digitally simply connected 2-manifold is the digital 2-sphere.

A digitally simply connected 3-manifold is the digital 3-sphere.

Digital simple connectedness parallels classical topological properties.

Abstract

In the framework of digital topology, we study structural and topological properties of digital n-dimensional manifolds. We introduce the notion of simple connectedness of a digital space and prove that if M and N are homotopy equivalent digital spaces and M is simply connected, then so is N. We show that a simply connected digital 2-manifold is the digital 2-sphere and a simply connected digital 3-manifold is the digital 3-sphere. This property can be considered as a digital form of the Poincar\'e conjecture for continuous three-manifolds.