Fundamental groups and Euler characteristics of sphere-like digital images

Laurence Boxer, P. Christopher Staecker

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 dierent 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 the paper [15] 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.

Keywords

digital topology; digital image; fundamental group; Euler characteristic

Subject classification

68R10; 55Q40

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References

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1. A fundamental group for digital images
Gregory Lupton, John Oprea, Nicholas A. Scoville
Journal of Applied and Computational Topology  vol: 5  issue: 2  first page: 249  year: 2021  
doi: 10.1007/s41468-021-00067-1



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