A second homotopy group for digital images (Q6635840)

A digital image \((X,c_k)\) is a subset \(X\) of \(\mathbb{Z}^n\) for \(n \geq 1\) together with an adjacency relation \(c_k\) induced by the integer lattice. In the paper under review, the authors define a second digital homotopy group \(\pi_2^d (X, x_0)\) of a digital image \((X, x_0)\), show that it is abelian, and construct a covariant functor \(\pi_2^d : \mathcal{D} \rightarrow \mathcal{AB}\) from the category \(\mathcal{D}\) of digital images and digitally continuous functions to the category \(\mathcal{AB}\) of abelian groups and group homomorphisms, which is closely related to the second homotopy group of a pointed space from classical homotopy theory. They also establish some basic properties such as independence of base points under the digitally same path-connected components, and take the general development enough to establish behavior with respect to products; that is, the second homotopy groups preserve the Cartesian products of digital images. The authors develop the so-called triangle-counting function for a map that represents a digital homotopy class of the homotopy group, and then show that the second homotopy group of a digital \(2\)-sphere is isomorphic to the group of integers by making use of elegant and interesting combinatorial ingredients as the main theorem in the article.