Fundamental groups and Euler characteristics of sphere-like digital images

Laurence Boxer

United States

Niagara University

Laurence Boxer received his Ph.D. in Mathematics from the University of Illinois at Urbana-Champaign, in 1976. He is Professor and past Chair of Computer and Information Sciences at Niagara University, and Research Professor of Computer Science and Engineering at SUNY-Bualo.

P. Christopher Staecker

United States

Fairfield University

P. Christopher Staecker received his Ph.D. in Mathematics from the University of California, Los Angeles, in 2005. He is now an Associate Professor in the Department of Mathematics at Faireld University in Faireld, Connecticut, USA.

Submitted: 2016-02-01

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Accepted: 2016-04-25

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Published: 2016-10-03

DOI: https://doi.org/10.4995/agt.2016.4624
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Keywords:

digital topology, digital image, fundamental group, Euler characteristic

Supporting agencies:

This research was not funded

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.
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