Homotopy equivalence of finite digital images

TL;DR

This paper explores homotopy equivalence in finite digital images, introduces a new numerical invariant, and classifies small digital images up to homotopy, bridging digital and classical topology.

Contribution

It develops a digital homotopy invariant and catalogs all connected digital images with few points up to homotopy equivalence.

Findings

Introduces a numerical digital homotopy invariant.

Classifies small connected digital images up to homotopy.

Highlights differences between classical and digital homotopy invariants.

Abstract

For digital images, there is an established homotopy equivalence relation which parallels that of classical topology. Many classical homotopy equivalence invariants, such as the Euler characteristic and the homology groups, do not remain invariants in the digital setting. This paper develops a numerical digital homotopy invariant and begins to catalog all possible connected digital images on a small number of points, up to homotopy equivalence.