Homotopy relations for digital images
This work addresses a theoretical problem in digital topology for researchers, but it appears incremental as it extends existing homotopy concepts without broad practical impact.
The paper tackles the problem of expressing similarity between finite and infinite digital images with respect to homotopy by introducing three generalizations of homotopy equivalence. It shows that these generalizations are distinct from ordinary homotopy equivalence, imply isomorphism of fundamental groups, and are preserved under wedges and Cartesian products.
We introduce three generalizations of homotopy equivalence in digital images, to allow us to express whether a finite and an infinite digital image are similar with respect to homotopy. We show that these three generalizations are not equivalent to ordinary homotopy equivalence, and give several examples. We show that, like homotopy equivalence, our three generalizations imply isomorphism of fundamental groups, and are preserved under wedges and Cartesian products.