The two digital homology theories

TL;DR

This paper compares four homology theories for digital images, establishing isomorphisms between some and highlighting differences, thereby clarifying their relationships and properties in digital topology.

Contribution

It proves isomorphisms between certain digital homology theories and introduces a new cubical homology for $c_1$-adjacent digital images, expanding the understanding of their relationships.

Findings

Simplicial homology theories are isomorphic.

Cubical homology theories are isomorphic.

Simplicial and cubical theories differ in higher dimensions.

Abstract

In this paper we prove results relating to four homology theories developed in the topology of digital images: a simplicial homology theory by Arslan et al which is the homology of the clique complex, a singular simplicial homology theory by Lee, a cubical homology theory by Jamil and Ali, and a new kind of cubical homology for digital images with $c_1$-adjacency which is easily computed, and generalizes a construction by Karaca & Ege. We show that the two simplicial homology theories are isomorphic, and that the two cubical theories are isomorphic, but that the simplicial theory is not isomorphic to the cubical theory. Thus we obtain, up to isomorphism, two different homology theories: simplicial, and cubical. We discuss briefly the relationship between the two theories. Notably, the two theories always agree in dimension zero, and in dimension 1 there is always a surjection from the cubical homology to the simplicial homology. We give examples to show that there need be no similar relationships in higher dimensions.