Fundamental Groups and Euler Characteristics of Sphere-like Digital Images

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.