On digital H-spaces

TL;DR

This paper explores algebraic properties of digital H-spaces within digital topology, classifies NP2-digital H-spaces, and completes the classification of digital topological groups, highlighting the impact of product adjacencies.

Contribution

It introduces a general construction for digital H-spaces not homotopy-equivalent to digital topological groups and classifies all NP2-digital H-spaces.

Findings

Digital H-spaces' properties depend on product adjacencies.

A construction produces NP2-digital H-spaces not homotopy-equivalent to groups.

Complete classification of digital topological groups achieved.

Abstract

In this article, we investigate properties of digital H-spaces in the graph theoretic model of digital topology. As in prior work, the results obtained often depend fundamentally on the choice between NP$_1$ and NP$_2$ product adjacencies. We explore algebraic properties of digital H-spaces preserved under digital homotopy equivalence, and we give a general construction that produces examples of digital H-spaces which are not homotopy-equivalent to digital topological groups in both categories. Further, we show that this construction essentially classifies all NP$_2$-digital H-spaces. In a short appendix, we resolve a question that was left unresolved in [17], and complete the full classification of digital topological groups.