A Classical Construction for the Digital Fundamental Group

Abstract

A version of topology's fundamental group is developed for digital images in dimension at most 3 in [7] and [8]. In the latter paper, it is shown that such a digital image X ⊂ $\mathcal{Z}^k$, k ≤ 3, has a continuous analog C(X) ⊂ Rk such that X has digital fundamental group isomorphic to Π1(C(X)). However, the construction of the digital fundamental group in [7] and [8] does not greatly resemble the classical construction of the fundamental group of a topological space. In the current paper, we show how classical methods of algebraic topology may be used to construct the digital fundamental group. We construct the digital fundamental group based on the notions of digitally continuous functions presented in [10] and digital homotopy [3]. Our methods are very similar to those of [6], which uses different notions of digital topology. We show that the resulting theory of digital fundamental groups is related to that of [7] and [8] in that it yields isomorphic fundamental groups for the digital images considered in the latter papers (for certain connectedness types).