P. Christopher Staecker

Dae-Woong Lee, P. Christopher Staecker

In this article, we develop the basic theory of digital topological groups. The basic definitions directly lead to two separate categories, based on the details of the continuity required of the group multiplication. We define $\NP_1$- and $\NP_2$-digital topological groups, and investigate their properties and algebraic structure. The $\NP_2$ category is very restrictive, and we give a complete classification of $\NP_2$-digital topological groups. We also give many examples of $\NP_1$-digital topological groups. We define digital topological group homomorphisms, and describe the digital counterpart of the first isomorphism theorem.

P. Christopher Staecker

In this paper we prove results relating to two homotopy relations and four homology theories developed in the topology of digital images. We introduce a new type of homotopy relation for digitally continuous functions which we call "strong homotopy." Both digital homotopy and strong homotopy are natural digitizations of classical topological homotopy: the difference between them is analogous to the difference between digital 4-adjacency and 8-adjacency in the plane. We also consider four different digital homology theories: a simplicial homology theory by Arslan et al which is the homology of the clique complex, a singular simplicial homology theory by D. W. Lee, a cubical homology theory by Jamil and Ali, and a new kind of cubical homology for digital images with $c_1$-adjacency which is easily computed, and generalizes a construction by Karaca \& Ege. We show that the two simplicial homology theories are isomorphic to each other, but distinct from the two cubical theories. We also show that homotopic maps have the same induced homomorphisms in the cubical homology theory, and strong homotopic maps additionally have the same induced homomorphisms in the simplicial theory.

P. Christopher Staecker

We introduce a new type of homotopy relation for digitally continuous functions which we call ``strong homotopy.'' Both digital homotopy and strong homotopy are natural digitizations of classical topological homotopy: the difference between them is analogous to the difference between digital 4-adjacency and 8-adjacency in the plane. We explore basic properties of strong homotopy, and give some equivalent characterizations. In particular we show that strong homotopy is related to ``punctuated homotopy,'' in which the function changes by only one point in each homotopy time step. We also show that strongly homotopic maps always have the same induced homomorphisms in the digital homology theory. This is not generally true for digitally homotopic maps, though we do show that it is true for any homotopic selfmaps on the digital cycle $C_n$ with $n\ge 4$. We also define and consider strong homotopy equivalence of digital images. Using some computer assistance, we produce a catalog of all small digital images up to strong homotopy equivalence. We also briefly consider pointed strong homotopy equivalence, and give an example of a pointed contractible image which is not pointed strongly contractible.

Laurence Boxer, P. Christopher Staecker

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.

Laurence Boxer, P. Christopher Staecker

We study connectivity preserving multivalued functions between digital images. This notion generalizes that of continuous multivalued functions studied mostly in the setting of the digital plane $Z^2$. We show that connectivity preserving multivalued functions, like continuous multivalued functions, are appropriate models for digital morpholological operations. Connectivity preservation, unlike continuity, is preserved by compositions, and generalizes easily to higher dimensions and arbitrary adjacency relations.

Laurence Boxer, P. Christopher Staecker

We present and explore in detail a pair of digital images with $c_u$-adjacencies that are homotopic but not pointed homotopic. For two digital loops $f,g: [0,m]_Z \rightarrow X$ with the same basepoint, we introduce the notion of {\em tight at the basepoint (TAB)} pointed homotopy, which is more restrictive than ordinary pointed homotopy and yields some different results. We present a variant form of the digital fundamental group. Based on what we call {\em eventually constant} loops, this version of the fundamental group is equivalent to that of Boxer (1999), but offers the advantage that eventually constant maps are often easier to work with than the trivial extensions that are key to the development of the fundamental group in Boxer (1999) and many subsequent papers. We show that homotopy equivalent digital images have isomorphic fundamental groups, even when the homotopy equivalence does not preserve the basepoint. This assertion appeared in Boxer (2005), but there was an error in the proof; here, we correct the error.

P. Christopher Staecker

The topology of digital images has been studied much in recent years, but no attempt has been made to exhaustively catalog the structure of binary images of small numbers of points. We produce enumerations of several classes of digital images up to isomorphism and decide which among them are homotopy equivalent to one another. Noting some patterns in the results, we make some conjectures about digital images which are irreducible but not rigid.

Jason Haarmann, Meg P. Murphy, Casey S. Peters et al.

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.