Computability of digital cubical singular homology of $c_1$-digital images

TL;DR

This paper investigates the computability of discrete cubical homology for $c_1$-digital images, comparing it with simpler homology theories to facilitate calculations and establish invariance properties.

Contribution

It introduces a chain map linking $c_1$-digital homology with $c_1$-cubical homology, proving functoriality and homotopy invariance for the latter.

Findings

Constructed a chain map between $c_1$-digital and $c_1$-cubical homology.

Proved functoriality of $c_1$-homology.

Established homotopy invariance of the $c_1$-homology.

Abstract

Discrete cubical homology arose as the homology theory associated with discrete cubical homotopy theory. Despite the combinatorial nature of this homology, its computation has posed a significant challenge to the researchers in the field. This paper focuses on determining the discrete cubical homology of $c_1$-digital images, which are subgraphs of the integer lattice. We compare the discrete cubical homology of $c_1$-digital images with the computationally simpler $c_1$-cubical homology as a possible route to simplifying these computations. This comparison is motivated by the classical equivalence between simplicial and singular homology theories, but the construction and proof of the chain map was found to be unexpectedly difficult. Furthermore, via the chain map constructed in this work, the $c_1$-homology, developed by the second author, is shown to be functorial and homotopy-type invariant.