Laurence Boxer and Ismet KARACA

SOME PROPERTIES OF DIGITAL COVERING SPACES Laurence Boxer and Ismet KARACA Abstract. In this paper we study digital versions of some properties of covering spaces from algebraic topology. We correct and improve upon the presentation of assertions in Han s papers [10,11]. 1. Introduction Digital Topology has arisen for the study of geometric and topological properties of digital images. In our digital technology, such questions arise in a range of applications, including computer graphics, computer tomography, pattern analysis and robotic design. Knowledge of the digital fundamental group is a very important tool for Image Analysis. The digital fundamental group of a discrete object has been developed by Kong [15]. Boxer [2] shows how classical methods of Algebraic Topology may be used to construct the digital fundamental group. The digital covering space is an important tool for computing fundamental groups of digital images. A digital covering space has been introduced by Han [8]. Boxer [5] develops further the topic of digital covering space by deriving digital analogs of classical results of Algebraic Topology concerning the existence and properties of digital universal covering spaces. Boxer and Karaca [7] show the digital covering spaces are classified by the conjugacy class corresponding to a digital covering space. This paper is organized as follows. Section 2 provides some basic notions. In Section 3, we study an action of the fundamental group of the digital image of a digital covering map, on the fiber, the set of points mapping to the base point in the base digital image. In section 4, we investigate maps between digital coverings. 2. Preliminaries Let Z be the set of integers. Then Z n is the set of lattice points in the n-dimensional Euclidean space. Let X Z n and let κ be some adjacency relation for the members of X. Then the pair (X, κ) is said to be a (binary) digital image. A variety of adjacency 1991 Mathematics Subject Classification. Primary 55N35, 68R10, 68U05, 68U10. Key words and phrases. digital image, digital topology, homotopy, fundamental group, covering space, normal subgroup. 1 Typeset by AMS-TEX

2 LAURENCE BOXER AND ISMET KARACA relations are used in the study of digital images. Some of the better-known adjacencies are the following. For a positive integer l with 1 l n and two distinct points p and q are c l -adjacent [4] if p = (p 1, p 2,..., p n ), q = (q 1, q 2,..., q n ) Z n, (1) there are at most l indices i such that p i q i = 1 and (2) for all other indices j such that p j q j 1, p j = q j. Note that the notation c l represents the number of points q Z n which are adjacent to a given point p Z n. Thus, in Z we have c 1 = 2; in Z 2 we have c 1 = 4 and c 2 = 8; in Z 3 we have c 1 = 6, c 2 = 18, and c 3 = 26. A k-neighbor of a lattice point p is k-adjacent to p. More general adjacency relations are studied in [12]. Let κ be an adjacency relation defined on Z n. A digital image X Z n is κ-connected [12] if and only if for every pair of different points x, y X, there is a set {x 0, x 1,..., x r } of points of a digital image X such that x = x 0, y = x r and x i and x i+1 are κ-neighbors where i = 0, 1,..., r 1. A κ-component of a digital image X is a maximal κ-connected subset of X. Let a, b Z with a < b. A digital interval [1] is a set of the form [a, b] Z = {z Z a z b}. Let X Z n 0 and Y Z n 1 be digital images with κ 0 -adjacency and κ 1 -adjacency respectively. A function f : X Y is said to be (κ 0, κ 1 )-continuous ([2], [18]) if for every κ 0 -connected subset U of X, f(u) is a κ 1 -connected subset of Y. We say that such a function is digitally continuous. Proposition 2.1 [2], [18]. Let X Z n 0 and Y Z n 1 be digital images with κ 0 -adjacency and κ 1 -adjacency respectively. Then the function f : X Y is (κ 0, κ 1 )-continuous if and only if for every κ 0 -adjacent points {x 0, x 1 } of X, either f(x 0 ) = f(x 1 ) or f(x 0 ) and f(x 1 ) are κ 1 -adjacent in Y. Let X Z n 0 and Y Z n 1 be digital images with κ 0 -adjacency and κ 1 -adjacency respectively. A function f : X Y is a (κ 0, κ 1 )-isomorphism [5] if f is (κ 0, κ 1 )-continuous and bijective and further f 1 : Y X is (κ 1, κ 0 )-continuous. By a digital κ-path from x to y in a digital image X, we mean a (2, κ)-continuous function f : [0, m] Z X such that f(0) = x and f(m) = y. If f(0) = f(m), we called f a digital κ- loop, and the point f(0) is the base point of the loop f. A digital loop f is said to be a trivial loop if it is a constant function. A simple closed κ-curve of m 4 points in a digital image X is a sequence {f(0), f(1),..., f(m 1)} of images of the κ-path f : [0, m 1] Z X such that f(i) and f(j) are κ-adjacent if and only if j = i ± 1 mod m. Let X Z n 0 and Y Z n 1 be digital images with κ 0 -adjacency and κ 1 -adjacency respectively. Two (κ 0, κ 1 )-continuous functions f, g : X Y are said to be digitally (κ 0, κ 1 )- homotopic in Y [2] if there is a positive integer m and a function H : X [0, m] Z Y such that (1) for all x X, H(x, 0) = f(x) and H(x, m) = g(x),

SOME PROPERTIES OF DIGITAL COVERING SPACES 3 (2) for all x X, the induced function H x : [0, m] Z Y defined by H x (t) = H(x, t) for all t [0, m] Z, is (2, κ 1 )-continuous, and (3) for all t [0, m] Z, the induced function H t : X Y defined by is (κ 0, κ 1 )-continuous. H t (x) = H(x, t) for all x X, We say that the function H is a digital (κ 0, κ 1 )-homotopy between f and g. In [2] Boxer shows that the digital (κ 0, κ 1 )-homotopy relation is an equivalence relation among digitally continuous functions f : (X, κ 0 ) (Y, κ 1 ). A digital image (X, κ) is said to be κ-contractible [1] if its identity map is (κ, κ)-homotopic to a constant function c for some c X where the constant function c : X X is defined by c(x) = c for all x X. A pointed digital image is a pair (X, x 0 ) where (X, κ) is a digital image and x 0 X. If f : [0, m 1 ] Z X and g : [0, m 2 ] Z X are digital κ-paths with f(m 1 ) = g(0), then define the product (f g) : [0, m 1 + m 2 ] Z X [13] by { f(t) if t [0, m1 ] Z ; (f g)(t) = g(t m 1 ) if t [0, m 1 + m 2 ] Z. The restriction of loop classes to loops defined on the same digital interval is undesirable. We have the following notion of trivial extension which allows a loop to stretch and remain in the same digital class. Let f and f be κ-loops in a pointed digital image (X, x 0 ). We say f is a trivial extension of f [2] if there are sets of κ-paths {f 1, f 2,..., f r } and {F 1, F 2,..., F p } in X such that (1) r p; (2) f = f 1 f 2 f r ; (3) f = F 1 F 2 F p ; (4) There are indices 1 i 1 < i 2 < < i r p such that F ij = f j, 1 j r and i / {i 1, i 2,..., i r } implies F i is a trivial loop. Let f, g : [0, m] Z (X, x 0 ) be digital loops with base point x 0. If there is a digital homotopy H : [0, m] Z [0, M] Z X between f and g such that for all t [0, M] Z we have H(0, t) = H(m, t), then we say that H is loop-preserving [6]. More generally, if f, g : [0, m] Z (X, κ) are κ-paths such that f(0) = g(0) and f(m) = g(m), then we say that a homotopy H : [0, m] Z [0, M] Z X between f and g such that for all t [0, M] Z, H(0, t) = x 0 and H(m, t) = f(m), holds the endpoints fixed (this generalizes a definition of [3]). Two loops f 0, f 1 with the same base point x 0 X belong to the same loop class [f] X [3] if they have trivial extensions that can be joined by a homotopy that keeps the endpoints fixed. Define π1 κ (X, x 0 ) to be the set of digital homotopy classes of κ-loops [f] X in X with base point x 0. π1 κ (X, x 0 ) is a group under the product operation defined by [f] X [g] X = [f g] X (see [2]).

4 LAURENCE BOXER AND ISMET KARACA Let (E, κ) be a digital image and let ε N. The κ-neighborhood [8] of e 0 E with radius ε is the set N κ (e 0, ε) = {e E l κ (e 0, e) ε} {e 0 }, where l κ (e 0, e) is the length of a shortest κ-path from e 0 to e in E. The definition of digital covering maps in [8] was simplified in [5] as follows. Proposition 2.2. [5] Let (E, κ 0 ) and (B, κ 1 ) be digital images. Let p : E B be a (κ 0, κ 1 )-continuous surjection. Then the map p is a (κ 0, κ 1 )-covering map if and only if for each b B there exists an index set M such that (1) p 1 (N κ1 (b, 1)) = i M N κ0 (e i, 1) with e i p 1 (b); (2) if i, j M, i j, then N κ0 (e i, 1) N κ0 (e j, 1) = φ; and (3) the restriction map p Nκ0 (e i,1) : N κ0 (e i, 1) N κ1 (b, 1) is a (κ 0, κ 1 )-isomorphism for all i M. Definition 2.3. [9] For n N, a (κ 0, κ 1 )-covering (E, p, B) is a radius n local isomorphism if the restriction map p Nκ0 (e i,n) : N κ0 (e i, n) N κ1 (b, n) is a (κ 0, κ 1 )-isomorphism for all i M, b B. Let (E, κ 0 ), (B, κ 1 ), and (X, κ 2 ) be digital images, let p : E B be a (κ 0, κ 1 )-covering map, and let f : X B be (κ 2, κ 1 )-continuous. A lifting of f with respect to p is a (κ 2, κ 0 )-continuous function f : X E such that p f = f (see [8]). Theorem 2.4. [8] Let (E, κ 0 ) be a digital image and e 0 E. Let (B, κ 1 ) be a digital image and b 0 B. Let p : E B be a (κ 0, κ 1 )-covering map such that p(e 0 ) = b 0. Then any κ-path f : [0, m] Z B beginning at b 0 has a unique lifting to a path f in E beginning at e 0. Theorem 2.5. [9] Let (E, κ 0 ) be a digital image and e 0 E. Let (B, κ 1 ) be a digital image and b 0 B. Let p : E B be a (κ 0, κ 1 )-covering map such that p(e 0 ) = b 0. Suppose that p is a radius 2 local isomorphism. For κ 0 -paths g 0, g 1 : [0, m] Z E starting at e 0, if there is a κ 1 -homotopy in B from p g 0 to p g 1 that holds the endpoints fixed, then g 0 (m) = g 1 (m), and there is a κ 0 -homotopy in E from g 0 to g 1 that holds the endpoints fixed. A digital pointed image (X, x 0 ) is said to be simply κ-connected [8] if π1 κ (X, x 0 ) is a trivial group. In [2] Boxer proves that if f : (X, x 0 ) (Y, y 0 ) is a (κ 0, κ 1 )-continuous map of pointed digital images, then f : π κ 0 1 (X, x 0) π κ 1 1 (Y, y 0), defined by f ([g]) = [f g], is a group homomorphism. From Theorem 2.5, we immediately have the following result; Corollary 2.6. [5] Let (E, κ 0 ) be a digital image and e 0 E. Let (B, κ 1 ) be a digital image and b 0 B. Let p : E B be a (κ 0, κ 1 )-covering map such that p(e 0 ) = b 0. Suppose that p is a radius 2 local isomorphism. Then the induced homomorphism p : π κ 0 1 (E, e 0) π κ 1 1 (B, b 0) is a monomorphism. The following result describes an algebraic condition that is sufficient for existence of a lifting of a function. Practically all of the applications we have encountered have involved necessary algebraic conditions for the existence of certain digital topological features.

SOME PROPERTIES OF DIGITAL COVERING SPACES 5 Theorem 2.7. [5] Let ((E, e 0 ), κ 0 ) and ((B, b 0 ), κ 1 ) be pointed digital images. Let p : (E, e 0 ) (B, b 0 ) be a pointed (κ 0, κ 1 )-covering map. Let X be a κ 2 -connected digital image, x 0 X. Let φ : (X, x 0 ) (B, b 0 ) be a (κ 2, κ 1 )-continuous map of pointed digital images. Consider the following statements. (1) There exists a lifting φ : (X, x 0 ) (E, e 0 ) of φ with respect to p. (2) φ (π κ 2 1 (X, x 0)) p (π κ 0 1 (E, e 0)). Then (1) implies (2). Further, if p is a radius 2 local isomorphism, then (2) implies (1). 3. The action of the group π κ 1 1 (B, b 0) on the set p 1 (b 0 ) Many of our results are based on analogs in the algebraic topology of Euclidean spaces, as presented, for example, in the section 5.7 of [16]. In this section we study an action of the digital fundamental group of the base digital image of a digital covering map, on the fiber, the set of points mapping to the base point in the base digital image. This will play an important role in the study of the classification problem. Let p : (E, e 0 ) (B, b 0 ) be a (κ 0, κ 1 )-covering map that is a radius 2 local isomorphism. Let b 0 B be a fixed base point. To simplify notation, we define G = π κ 1 1 (B, b 0) and F = p 1 (b 0 ). We are going to describe an action of the group G on the set F as a group of permutations. For convenience the group will act on the right of the set. Let e F and α G. Represent α by a κ 1 -path f : [0, m f ] B. Lift f to get a κ 0 -path g in E with g(0) = e. Then define the dot operator [16,10] as a function from F G to F by e α = g(m f ). By Theorem 2.5, this does not depend on the choice of f α and so this function is well-defined. Now we shall derive some properties of this function. For e F and α, β G, we have the following equations (1) and (2), whose proofs are given below (note these properties appear as (4.2) of [10], but proofs are not given there). (1) e 1 = e, (2) (e α) β = e (αβ). These imply that π κ 1 1 (B, b 0) acts as a group of permutations of p 1 (b 0 ). Equation (1) is valid because the identity element 1 of G is represented by the constant map b 0. To prove (2), lift a κ 1 -loop representing α to a κ 0 -path f from e 0 to e 0 α and a κ 1 -loop representing β to a κ 0 -path g from e 0 α to (e 0 α) β. Then f g is a lift of a path representing α β, starting at e 0 and ending at e 0 (αβ). Therefore, (e 0 α) β = e 0 (αβ). Lemma 3.1. [10] Let p : (E, e 0 ) (B, b 0 ) be a (κ 0, κ 1 )-covering map that is a radius 2 local isomorphism. Let E be κ 0 -connected. Let e F. Then the group G acts transitively on the set F, i.e, given e, e 0 F, there exists α G such that e = e 0 α.

6 LAURENCE BOXER AND ISMET KARACA Lemma 3.2. Let p : (E, e 0 ) (B, b 0 ) be a (κ 0, κ 1 )-covering map that is a radius 2 local isomorphism. Let G e0 = {α G e 0 α = e 0 } (called the stabilizer of a point e 0 F ). Then G e0 = p (π κ 0 1 (E, e 0)). Proof. Note that α G e0 (α = [f] for some κ 1 -path f that lifts to a κ 0 -loop based at e 0 ) α Im (p ). Theorem 3.3. Let p : (E, e 0 ) (B, b 0 ) be a (κ 0, κ 1 )-covering map that is a radius 2 local isomorphism. Let E be κ 0 -connected. Then there is a bijection between the set π κ 1 1 (B, b 0)/p (π κ 0 1 (E, e 0)) of the right cosets, and the set p 1 (b 0 ). Proof. Define the map ϕ : G/G e0 F taking the right coset G e0 α to e 0 (G e0 α) = e 0 α. By Lemma 3.1, the map ϕ is a surjection. Let ϕ(g e0 α) = ϕ(g e0 β) where α, β G. Then e 0 α = e 0 β e 0 α β 1 = e 0 α β 1 G e0 G e0 α = G e0 β. Therefore ϕ is bijective. Corollary 3.4. If p 1 (b 0 ) is a group, then the bijection of Theorem 3.3 is a group isomorphism. Proof. This follows from Theorem 3.3 and property (2) of the dot operator. In [10], statement (4.3), given without proof, claims that if p 1 (b 0 ) is a group, then p 1 (b 0 ) N(p (π κ 0 1 (E, e 0)))/p (π κ 0 1 (E, e 0)), where N(p (π κ 0 1 (E, e 0))) is the normalizer of p (π κ 0 1 (E, e 0)) in π κ 1 In light of the Corollary 3.4, Han s statement must be false unless π κ 1 1 (B, b 0)/p (π κ 0 1 (E, e 0)) N(p (π κ 0 1 (E, e 0)))/p (π κ 0 1 (E, e 0)), which is true, for example, when p (π κ 0 1 (E, e 0)) is a normal subgroup of π κ 1 See also Corollary 4.9, Theorem 4.13, and Corollary 4.14, below. Definition 3.5. Let p : (E, e 0 ) (B, b 0 ) be a (κ 0, κ 1 )-covering map that is a radius 2 local isomorphism. If p 1 (b 0 ) has n elements, then the number n is called the number of sheets of the digital covering space. The order of the fundamental group π κ 1 1 (B, b 0) is the number of elements in π κ 1 The index of the subgroup p (π κ 0 1 (E, e 0)) in π κ 1 1 (B, b 0) is the number of distinct cosets of 1 (E, e 0))]. the subgroup p (π κ 0 1 (E, e 0)) in π κ 1 It is denoted by [π κ 1 1 (B, b 0) : p (π κ 0 Theorem 3.6. The number of sheets of a digital covering map p : (E, e 0 ) (B, b 0 ) is the index of p (π κ 0 1 (E, e 0)) in π κ 1 1 (B, b 0)). Proof. This follows from Theorem 3.3. Corollary 3.7. Let p : (E, e 0 ) (B, b 0 ) be a (κ 0, κ 1 )-covering map. If E is a simply κ 0 -connected, then the number of sheets equals to the order of π κ 1 1 (B, b 0)).

SOME PROPERTIES OF DIGITAL COVERING SPACES 7 Lemma 3.8. [7] A function h α : π κ 1 (B, b 1 ) π κ 1 (B, b 0 ) defined by h α ([β]) = [α 1 β α] is an isomorphism of groups where β is a κ-loop at b 1, α : [0, m] Z B is a κ-path with α(0) = b 1 and α(m) = b 0, and α 1 is the path that reverses α. Theorem 3.9. Let p : (E, e 0 ) (B, b 0 ) be a (κ 0, κ 1 )-covering map where E is κ 0 - connected. For b 0, b 1 B, let F 0 = p 1 (b 0 ) and F 1 = p 1 (b 1 ). Then F 0 = F 1. Proof. Let e 0 F 0 and e 1 F 1. Let λ be a κ 0 -path in E from e 0 to e 1, and λ = p λ denote the corresponding κ 1 -path in B from b 0 to b 1. By Lemma 3.8, h λ : π κ 0 1 (E, e 0) π κ 0 1 (E, e 1), defined by h λ([ f]) = [ λ 1 f λ], and h λ : π κ 1 1 (B, b 0) π κ 1 1 (B, b 1), defined by h λ ([f]) = [λ 1 f λ], are isomorphisms. By Corollary 2.6, p is an injection, so it follows that h λ induces a bijection between cosets: [π κ 1 1 (B, b 0) : p (π κ 0 1 (E, e 0))] = [π κ 1 1 (B, b 1) : p (π κ 0 1 (E, e 1))]. Theorem 3.6 now gives the result. Theorem 3.10. [7] Let p : (E, e 0 ) (B, b 0 ) be a (κ 0, κ 1 )-covering map. If e 0, e 1 F, then p (π κ 0 1 (E, e 0)) and p (π κ 0 1 (E, e 1)) are conjugate subgroups of π κ 1 1 (B, b 0)). 4. A deck transformation or automorphism of the digital covering In this section, we investigate maps between digital coverings. Definition 4.1. (1) Let q : D B and p : E B be digital covering maps. A homomorphism [16,11] of digital covering spaces is a digitally continuous function f : D E such that pf(d) = q(d) for every d in D. (2) Let p : (E, e 0 ) (B, b 0 ) be a (κ 0, κ 1 )-covering map that is a radius 2 local isomorphism. A deck transformation or automorphism [16,7] of the digital covering is a (κ 0, κ 0 )-continuous map h : E E that is a homomorphism. In [11], Han implicitly calls a self-isomorphism an automorphism. In Theorem 4.5 below, we prove that what we call an automorphism is an isomorphism. Theorem 4.2. [7] If p : (E, e 0 ) (B, b 0 ) is a pointed (κ 0, κ 1 )-covering map, then as e ranges over the points of p 1 (b 0 ), p (π κ 0 1 (E, e)) ranges over all conjugates of p (π κ 0 1 (E, e 0)) in π κ 1 Proposition 4.3. [7] Suppose that q : (D, d 0 ) (B, b 0 ) is a pointed (κ 0, κ 2 )-covering map and p : (E, e 0 ) (B, b 0 ) is a pointed (κ 1, κ 2 )-covering map where D is a κ 0 -connected digital image and E is a κ 1 -connected digital image. If f, g : D E are homomorphisms of digital (κ 0, κ 1 )-covering spaces such that f(d 0 ) = g(d 0 ), then f and g are identical on (D, d 0 ). Theorem 4.4. [7] Suppose that q : D B is a digital (κ 0, κ 2 )-covering map and p : E B is a digital (κ 1, κ 2 )-covering map where D is κ 0 -connected. Let d 0 q 1 (b 0 ) and e 0 p 1 (b 0 ). Consider the following statements. (1) There is an isomorphism f : D E with f(d 0 ) = e 0. (2) q (π κ 0 1 (D, d 0)) and p (π κ 1 1 (E, e 0)) are conjugate in π κ 2 Then (1) implies (2). Further, if p and q are radius 2 local isomorphisms, then (2) implies (1).

8 LAURENCE BOXER AND ISMET KARACA Theorem 4.5. Let p : (E, e) (B, b) be a (κ 1, κ 2 )-covering map that is a radius 2 local isomorphism. Let h : E E be an automorphism. Then h is an isomorphism. Proof. We have p(e) = ph(e). By Theorem 4.2, p (π κ 1 1 (E, e)) and p (π κ 1 1 (E, h(e))) are conjugate in π κ 2 1 (B, b). By Theorem 4.4, there is an isomorphism f : E E with f(e) = h(e). By Proposition 4.3, h and f must be the same function. It follows from the previous result that the set of all automorphisms of a digital covering space (E, p) is a group under composition of functions. We denote this group by A(E, p). Proposition 4.6. Let p : (E, e 0 ) (B, b 0 ) be a (κ 0, κ 1 )-covering map that is a radius 2 local isomorphism. If h A(E, p), α π κ 1 1 (B, b 0), and e p 1 (b 0 ) then (he) α = h(e α). Proof. Let f be a κ 1 -loop at b 0 representing α and let g : [0, m f ] Z E be a lifting of f with starting point e. From the definition we have g(m f ) = e α. The path h g is a lift of f and starts at h(e). Thus it ends at (he) α. On the other hand it ends at h(e α). Lemma 4.7. G e0 α = α 1 G e0 α where α G. Proof. Notice Thus we have G e0 α = {β (e 0 α) β = (e 0 α)} = {β e 0 α β α 1 = e 0 } = {β α β α 1 G e0 }. G e0 α = α 1 G e0 α. Theorem 4.8. Let p : (E, e 0 ) (B, b 0 ) be a (κ 0, κ 1 )-covering map that is a radius 2 local isomorphism and let e p 1 (b 0 ). Then the following statements are equivalent: (1) There is an automorphism h A(E, p) such that h(e 0 ) = e. (2) There is an α N(p π κ 0 1 (E, e 0)) such that e = e 0 α where N(p π κ 0 1 (E, e 0)) is the normalizer of the subgroup p π κ 0 1 (E, e 0) of the group π κ 1 (3) p π κ 0 1 (E, e 0) = p π κ 0(E, e). 1 Proof. (1) (3) By Theorem 4.5, h is an isomorphism, so h 1 exists. Since h is a lifting of p, Theorem 2.7 yields that p (π κ 0 1 (E, e 0)) p (π κ 0 1 (E, e)). Similarly, h 1 is a lifting of p, so p (π κ 0 1 (E, e)) p (π κ 0 1 (E, e 0)). Statement (3) follows. (3) (1) By Theorem 2.7, there is a map h covering the identity such that h(e 0 ) = e. Similarly, there is a map h covering the identity such that h (e) = e 0. Then h h and h h both cover the identity, and each has a point in common with the identity map, so by Proposition 4.3, h h = 1 = h h. (2) (3) If e = e 0 α for some α N(G e0 ), then G e = G e0 α = ( by Lemma 4.7) α 1 G e0 α = G e0, so (3) follows from Lemma 3.2. (3) (2) By Lemma 3.2, G e0 = G e. By Lemma 3.1, e = e 0 α for some α G. Then G e0 = G e = G e0 α = (by Lemma 4.7) α 1 G e0 α, so α N(G e0 ). We immediately have the following results.

SOME PROPERTIES OF DIGITAL COVERING SPACES 9 Corollary 4.9. Let p : (E, e 0 ) (B, b 0 ) be a (κ 0, κ 1 )-covering map that is a radius 2 local isomorphism. The subgroup p π κ 0 1 (E, e 0) is normal in π κ 1 1 (B, b 0) if and only if the group A(E, p) acts transitively on p 1 (b 0 ). Proof. Suppose that G e0 is normal in G. Let e F. By Lemma 3.1, there exist α G such that e = e 0 α. Then α N(G e0 ). By the equivalence of (1) and (2) of Theorem 4.8, there is an automorphism h A(E, p) such that h(e 0 ) = e, as desired. Conversely, assume that A(E, p) acts transitively on F : if e, e 0 F, then there exists h A(E, p) with h(e 0 ) = e. Now h (G e0 ) = G e. Since p = ph, it follows that p = p h, hence p (π κ 0 1 (E, e 0)) = p h ((π κ 0 1 (E, e 0))) = p (π κ 0 1 (E, e)). Since e is an arbitrary member of F, it follows from Theorem 4.2 that every conjugate of G e0 in G is equal to G e0. Thus G e0 is a normal subgroup of G. Example 4.10. We give an example of a covering for which A(E, p) does not act transitively on p 1 (b 0 ). Let B = {b i } 12 i=0 (Z2, c 2 ), where b 0 = (0, 0), b 1 = (1, 1), b 2 = (1, 2), b 3 = (1, 3), b 4 = (0, 4), b 5 = ( 1, 3), b 6 = ( 1, 2), b 7 = ( 1, 1), b 8 = (1, 1), b 9 = (1, 2), b 10 = (0, 3), b 11 = ( 1, 2), b 12 = ( 1, 1). Note we have B = B 6 B 8, where B 8 = {b i } 7 i=0 and B 6 = {b 0 } {b i } 12 i=8 are both simple closed c 2-curves. See Figure 1. For j Z, let h j : Z 2 Z 2 be the translation defined by Let E (Z 2, c 2 ) be the set E =( j Z\{0} j= ({6j} Z)) ( ([Z\{0}] {8j 1}). h j (x, y) = (x, y + j). j= h 8j (B 8 [{i} 1 i= 5 { 1}] [{i}5 i=1 { 1}])) See Figure 2. Let e 0 = b 0. Let p : (E, e 0 ) (B, b 0 ) be as follows: p wraps each vertical digital line {6j} Z in E around B 8 such that p(6j, y) = b 0 if and only if y 7 (mod 8); p maps each h 8j (B 8 ) isomorphically to B 8 via h 1 8j ; { b13+i if 5 i 1; p(h 8j (i, 1)) = b 7+i if 1 i 5; { b0 if z 0, z = 0 (mod 6); p(z, 8j 1) = b 7+(z (mod 6)) if z 0 (mod 6). Then p is easily seen to be a (c 2, c 2 )-covering map. Note {(0, 0), (6, 1)} p 1 (b 0 ), but no automorphism h of B satisfies h(0, 0) = (6, 1). The following appears as Theorem 6.2 of [11]. However, Han s attempt to prove this assertion merely appeals to Theorem 5.3 (our Theorem 2.7); some readers will find Han s argument unclear.

10 LAURENCE BOXER AND ISMET KARACA Theorem 4.11. [11] Let e, e 0 E, where E is κ 0 -connected. Let p : (E, e 0 ) (B, b 0 ) and q : (E, e 0 ) (B, b 0 ) be (κ 0, κ 1 ) coverings that are radius 2 local isomorphisms. Then there is an automorphism f : (E, e) (E, e 0 ) such that q f = p if and only if p (π κ 0 1 (E, e)) = q (π κ 0 1 (E, e 0)). Proof. Suppose there is an automorphism f : (E, e) (E, e 0 ) such that q f = p. By Theorem 2.7, p (π κ 0 1 (E, e)) q (π κ 0 1 (E, e 0)). By Theorem 4.5, f is an isomorphism, so the automorphism f 1 exists, and q = p f 1. By Theorem 2.7, q (π κ 0 1 (E, e 0)) p (π κ 0 1 (E, e)), and it follows that p (π κ 0 1 (E, e)) = q (π κ 0 1 (E, e 0)). Suppose p (π κ 0 1 (E, e)) = q (π κ 0 1 (E, e 0)). By Theorem 2.7, there are liftings f : (E, e) (E, e 0 ) of p with respect to q, and g : (E, e 0 ) (E, e) of q with respect to p. Then gf(e) = e and fg(e 0 ) = e 0. By Proposition 4.3, we have gf = 1 E = fg. It follows that f is the desired automorphism. Definition 4.12. Let p : (E, e 0 ) (B, b 0 ) be a (κ 0, κ 1 )-covering map that is a radius 2 local isomorphism. The map p is said to be regular if the group A(E, p) acts transitively on the set p 1 (b 0 ). If p is a radius 2 local isomorphism, regularity is equivalent (by Corollary 4.9) to the normality of p π κ 0 1 (E, e 0) in π κ 1 Definition 4.13. Let p : (E, e 0 ) (B, b 0 ) be a (κ 0, κ 1 )-covering map that is a radius 2 local isomorphism. Define a function Θ : N(G e0 ) A(E, p) by Θ(α) = h α where h α is the unique deck transformation such that h α (e 0 ) = e 0 α. By the equivalence (1) (2) of Theorem 4.8, Θ is well defined. Theorem 4.14. Let p : (E, e 0 ) (B, b 0 ) be a (κ 0, κ 1 )-covering map that is a radius 2 local isomorphism. The map Θ : N(G e0 ) A(E, p) is an epimorphism with kernel G e0. Consequently, we have A(E, p) N(p π κ 0 1 (E, e 0))/p π κ 0 1 (E, e 0). Proof. First compute h β h α (e 0 ) = h β (e 0 α) = (by Proposition 4.6) (h β (e 0 )) α = (e 0 β) α = e 0 (βα) = h βα (e 0 ). Thus Θ is a homomorphism by Proposition 4.3. Next note that if h A(E, p) then, by Theorem 4.8, there is an α N(G e0 ) such that h(e 0 ) = e 0 α = h α (e 0 ). By Proposition 4.3, h = h α, which shows that Θ is onto. Finally we compute the kernel of Θ: h α = 1 e 0 α = e 0 α G e0. By Lemma 3.2, the assertion follows.

SOME PROPERTIES OF DIGITAL COVERING SPACES 11 Corollary 4.15. Let p : (E, e 0 ) (B, b 0 ) be a (κ 0, κ 1 )-covering map that is a radius 2 local isomorphism. (1) If the map p is regular then A(E, p) π κ 1 1 (B, b 0)/p π κ 0 1 (E, e 0). (2) If the digital image E is simply κ 0 -connected then A(E, p) π κ 0 Proof. (1) This follows from Theorem 4.14 and Corollary 4.9. (2) Since E is simply κ 0 -connected, π κ 0 1 (E, e 0) is a trivial group. Using Corollary 4.9, we see that p is regular. By (1), it follows that A(E, p) π κ 0 References 1. L. Boxer, Digitally continuous functions, Pattern Recognition Letters 15 (1994), 833 839. 2., A classical construction for the digital fundamental group, Journal of Mathematical Imaging and Vision 10 (1999), 51 62. 3., Properties of digital homotopy, Journal of Mathematical Imaging and Vision 22 (2005), 19 26. 4., Homotopy properties of sphere-like digital images, Journal of Mathematical Imaging and Vision 24 (2006), 167 175. 5., Digital products, wedges, and covering spaces, Journal of Mathematical Imaging and Vision 25 (2006), 159 171. 6., Fundamental groups of unbounded digital images, Journal of Mathematical Imaging and Vision 27 (2007), 121 127. 7. L. Boxer and I. Karaca, The classification of digital covering spaces, preprint (2007). 8. S.E. Han, Non-product property of the digital fundamental group, Information Sciences 171 (2005), 73 91. 9., Digital coverings and their applications, Journal of Applied Mathematics and Computing 18 (2005), 487 495. 10., The k-fundamental group of a closed k-surface, Information Sciences 177 (2007), 3731 3748. 11., Equivalent (k 0, k 1 )-covering and generalized digital lifting, Information Sciences (2007), doi: 10.1016/j.ins.2007.02.004. 12. G.T. Herman, Oriented surfaces in digital spaces, CVGIP: Graphical Models and Image Processing 55 (1993), 381 396. 13. E. Khalimsky, Motion, deformation, and homotopy in finite spaces, Proceedings IEEE International Conference on Systems, Man, and Cybernetics (1987), 227 234. 14. T.Y. Kong, A digital fundamental group, Computers and Graphics 13 (1989), 159 166. 15. T.Y. Kong, A.W. Roscoe, and A. Rosenfeld, Concepts of digital topology, Topology and its Applications 46 (1992), 219 262. 16. W.S. Massey, Algebraic Topology, Springer-Verlag, New York, 1977. 17. A. Rosenfeld, Digital topology, American Mathematical Monthly 86 (1979), 76 87. 18., Continuous functions on digital pictures, Pattern Recognition Letters 4 (1986), 177 184. Department of Computer and Information Sciences, Niagara University, NY 14109, USA; and Department of Computer Science and Engineering, State University of New York at Buffalo E-mail address: boxer@niagara.edu Department of Mathematics, Ege University, Bornova, Izmir 35100 TURKEY E-mail address: ismet.karaca@ege.edu.tr