Homotopy Equivalence in Finite Digital Images

Abstract

For digital images, there is an established homotopy equivalence relation which parallels that of classical topology. Many classical homotopy equivalence invariants, such as the Euler characteristic and the homology groups, do not remain invariants in the digital setting. This paper develops a numerical digital homotopy invariant and begins to catalog all possible connected digital images on a small number of points, up to homotopy equivalence.

Fixed poin sets in digital topology, 1

“…The following provides a negative answer to this question. (3,2), (3,4), (3,6), (0, 6)} (see Figure 5) is a minimal freezing set for (D, c 1 ). Note (2, 2) and (2, 4) are endpoints of maximal horizontal and vertical bounding segments of D and are not members of A.…”

Fundamental groups and Euler characteristics of sphere-like digital images

“…The difference between pointed and unpointed homotopy turns out to be complex, and must be carefully considered. This issue has been explored in [11] and [9], and we continue that work in this paper. In particular, Example 2.9 shows that contractibility does not imply pointed contractibility.…”

“…Let f be the 8 point loop in X which circles the deleted point (4, 2, 2). By Theorem 3.12 of [11], the only loops homotopic to f by loop-preserving homotopies are rotations of f . Thus f does not contract by a loop-preserving homotopy, and so X has a loophole.…”

“…Despite the existence of images that are homotopy equivalent but not pointed homotopy equivalent ( [11,9], Example 2.9), notice that we do not require the homotopy equivalence in the following theorem to be a pointed homotopy equivalence.…”

Fixed point sets in digital topology, 2

“…Rigid maps and digital images are discussed in [12,10]. Clearly, a rigid map is pointed rigid, and a rigid digital image is pointed rigid.…”