Fundamental Groups and Euler Characteristics of Sphere-like Digital Images

TL;DR

This paper investigates the fundamental groups and Euler characteristics of digital models of the sphere, showing they are trivial and exploring how digital surface operations affect these properties, correcting previous errors and presenting new findings.

Contribution

It provides corrected and new results on the fundamental groups and Euler characteristics of digital sphere-like images, including effects of connected sums and different 'no holes' notions.

Findings

Fundamental groups of considered digital models are trivial.

Euler characteristics vary among models, not always equal.

Connected sum operations influence fundamental groups and Euler characteristics.

Abstract

The current paper focuses on fundamental groups and Euler characteristics of various digital models of the 2-dimensional sphere. For all models that we consider, we show that the fundamental groups are trivial, and compute the Euler characteristics (which are not always equal). We consider the connected sum of digital surfaces and investigate how this operation relates to the fundamental group and Euler characteristic. We also consider two related but different notions of a digital image having "no holes," and relate this to the triviality of the fundamental group. Many of our results have origins in a paper by S.-E. Han, which contains many errors. We correct these errors when possible, and leave some open questions. We also present some original results.