Laurence Boxer and Ismet KARACA

Transcription

1 THE CLASSIFICATION OF DIGITAL COVERING SPACES Laurence Boxer and Ismet KARACA Abstract. In this paper we classify digital covering spaces using the conjugacy class corresponding to a digital covering space. 1. Introduction The main purpose of Digital Topology is the study of topological properties of discrete objects which are obtained digitizing continuous objects. Digital Topology plays a very important role in computer vision, image processing, and computer graphics. The digital fundamental group of a discrete object was introduced in Digital Topology by Kong [13]. Boxer [2] shows how classical methods of Algebraic Topology may be used to construct the digital fundamental group. Knowledge of the digital fundamental group is a very important tool for Image Analysis. A general algorithm to decide whether two discrete objects have isomorphic fundamental groups would be a very powerful tool for Image Analysis. The paper [17] of Stout makes a contribution to this problem. The digital covering space is an important tool for computing digital fundamental groups of digital images. In [7] Han has introduced a digital covering space and given some properties of it. 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. We use the fundamental group of a digital image as a tool for studying digital covering spaces. In this paper we show the digital covering spaces are classified by the conjugacy class corresponding to a digital covering space. 2. Preliminaries Let Z be the set of integers. Then Z n is the set of lattice points in the n-dimensional Euclidean space. A (binary) digital image is a subset of Z n with an adjacency relation. A variety of adjacency relations are used in the study of digital images. Some of the betterknown adjacencies are the following Mathematics Subject Classification. Primary 55N35, 68R10, 68U05, 68U10. Key words and phrases. digital image, digital topology, homotopy, covering space. 1 Typeset by AMS-TEX

2 2 LAURENCE BOXER AND ISMET KARACA For a positive integer l with 1 l n and distinct two 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 i q i 1, p j = q j. It is common to denote an adjacency relation κ by the number of κ-adjacent points of a given point. For example, 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. More general adjacency relations are studied in [10]. A κ-neighbor of a lattice point p is κ-adjacent to p. Let κ be an adjacency relation defined on Z n. A digital image X Z n is κ-connected [10] 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], [16]) 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], [16]. 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. 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) then the κ-path is said to be closed, and the function f is called a κ-loop. Let f : [0, m 1] Z X be a (2, κ)-continuous function such that f(i) and f(j) are κ-adjacent if and only if j = i ± 1 mod m. Then the set f([0, m 1] Z ) is a simple closed κ-curve. 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 (κ 0, κ 1 )-homeomorphism [1] if f is (κ 0, κ 1 )-continuous and bijective and further f 1 : Y X is (κ 1, κ 0 )-continuous. In [5], Boxer suggests using the terminology isomorphic digital images rather than homeomorphic images. Throughout this paper we will prefer isomorphic to homeomorphic. 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),

3 THE CLASSIFICATION 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 (κ 0, κ 1 )-continuous function f : X Y is a (κ 0, κ 1 )-homotopy equivalence ([3], [6]) if there exists a (κ 1, κ 0 )-continuous function g : Y X such that g f is (κ 0, κ 0 )-homotopic to the identity function 1 X and f g is (κ 1, κ 1 )-homotopic to the identity function 1 Y. The function g is said to be a (κ 1, κ 0 )-homotopy inverse for the function f. Two digital images that are (κ 0, κ 1 )-homotopy equivalent are said to have the same (κ 0, κ 1 )-homotopy type. 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. We say that a (κ 0, κ 1 )-continuous function f : X Y is (κ 0, κ 1 )-nullhomotopic [1] if f is (κ 0, κ 1 )-homotopic to a constant function c in Y. A pointed digital image is a pair (X, x 0 ) where (X, κ) is a digital image and x 0 X. Let = A X. Assume κ-adjacency for A and X. A is called a κ-retract of X [2] if and only if there is a κ-continuous function r : X A such that r(a) = a for all a A. The function r is called a κ-retraction of X onto A. Let i : A X be the inclusion function. We say that A is a κ-deformation retract of X [3] if there exists a κ-homotopy H : X [0, m] Z X between the identity map 1 X and i r, for some κ-retraction r of X onto A. 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 [11] by { f(t) if t [0, m1 ] Z ; (f g)(t) = g(t m 1 ) if t [m 1, 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 (a constant function).

4 4 LAURENCE BOXER AND ISMET KARACA 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. If we would use the weaker assumption that the homotopy is loop-preserving instead of the assumption that the homotopy keeps the endpoints fixed, then all digital fundamental groups would be abelian. This contradicts Theorem 6.1 of [5] (which corrects the proof of the same assertion in [7] and gives an example of a non-abelian digital fundamental group). Therefore we use homotopies that keep the endpoints fixed. Define π κ 1 (X, x 0 ) to be the set of digital homotopy classes of κ-loops [f] X in X with base point x 0. Theorem 2.2. [2] π κ 1 (X, x 0 ) is a group under the product operation defined by [f] X [g] X = [f g] X. Proposition 2.3. [3] Let x 0 X. If a pointed digital image (X, x 0 ) is κ-contractible, then π κ 1 (X, x 0 ) is a trivial group. Let (E, κ) be a digital image and let ε N. The κ-neighborhood [7] 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 [7] was simplified in [5] as follows. Proposition 2.4. [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) = ; 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 2.5. [8] 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. Given a digital simple closed κ-curve S = {s i } m 1 i=0 such that s i and s j are κ-adjacent if and only if either j i + 1 mod m or j i 1 mod m, the function p : Z S defined by p(z) = s z mod m is a (2, κ)-covering [7]. Every covering is a radius 1 local isomorphism, by (3) of Proposition 2.4 [5]. However, if S Z 2 is an 8-curve of 4 points, then the map p is not a radius 2 local isomorphism [5]. A digital version of the notion of lifting from algebraic topology often yields efficient calculation of digital fundamental groups. 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 [7]).

5 THE CLASSIFICATION OF DIGITAL COVERING SPACES 5 Theorem 2.6. [7] 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 2.7. [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. 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 [7] 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 2.7, we immediately have the following result; Corollary 2.8. [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. In [7] Han attempts to obtain the digital fundamental groups of non-contractible digital simple closed curves. However, there are major holes in the proof of his claim. Boxer [5] repairs these errors. Let s state Han s claim: Theorem 2.9. [7] π 8 1(MSC 8 ) is isomorphic to 6Z as a group where MSC 8 Z 2 is a simple closed 8-curve of 6 points. 3. The Classification of Digital Covering Spaces The results of this section are based on analogs of Euclidean algebraic topology, as found in section 5.6 of [14]. The following theorem 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. Theorem 3.1. [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). An example of the use of Theorem 3.1 is the following.

6 6 LAURENCE BOXER AND ISMET KARACA Corollary 3.2. 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 that is a radius 2 local isomorphism. Suppose (X, x 0 ) is a pointed digital image such that π κ 2 1 (X, x 0) is a trivial group. Then every (κ 2, κ 1 )-continuous pointed map φ : (X, x 0 ) (B, b 0 ) has a lifting φ : (X, x 0 ) (E, e 0 ) with respect to φ. Proof. Since π κ 2 1 (X, x 0) is a trivial group, statement (2) of Theorem 3.1 is realized for any map φ satisfying the hypotheses. The conclusion follows from statement (1) of Theorem 3.1. An obvious question that must be considered is how the choice of the base point b 0 in a digital image B affects the digital fundamental group. Clearly, π1 κ (B, b 0 ) can carry information only on the κ-component of B containing b 0. Let b 1 be another point of the κ-component of B. Select a κ-path α : [0, m] Z B with α(0) = b 1 and α(m) = b 0. We use the notation [2] α 1 for the path that reverses α: α 1 (t) = α(m t) for t [0, m] Z. Given a κ-loop β at b 1, we produce a κ-loop at b 0 by taking composition α 1 β α. Define a function h α : π κ 1 (B, b 1 ) π κ 1 (B, b 0 ) by h α ([β]) = [α 1 β α]. Then we immediately have the following results. Lemma 3.3. (1) h α is an isomorphism of groups. (2) If α is another κ-path from b 0 to b 1 which is digitally homotopic to α via a digital homotopy fixing the end points, then h α = h α. (3) (h α ) 1 = h α 1. Proof. This is elementary and is left to the reader. Note that if we take b 1 = b 0, we can write h α ([β]) = [α 1 ][β][α] and h α becomes an inner automorphism of π κ 1 (B, b 0 ). Of course, if π κ 1 (B, x 0 ) is abelian, then any such isomorphism is the identity. Now suppose p : E B is a digital (κ 0, κ 1 )-covering map. Theorem 3.1 was concerned with the image of the homomorphism p : π κ 0 1 (E, e 0) π κ 1 1 (B, b 0). Is this subject to change with the choice of e 0 in p 1 (b 0 )? In general the answer is yes, but the variation can be characterized. Theorem 3.4. 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 1 (B, b 0). Proof. Let e p 1 (b 0 ) and let α be a κ 0 -path in E from e to e 0. Then α = p α is a κ 1 -loop at b 0, and we have p (π κ 0 1 (E, e)) = p (h α (π κ 0 1 (E, e 0))) = h α (p (π κ 0 1 (E, e 0))).

7 THE CLASSIFICATION OF DIGITAL COVERING SPACES 7 Thus p (π κ 0 1 (E, e)) is a conjugate of p (π κ 0 1 (E, e 0)). Let [φ] π κ 1 1 (B, b 0) and consider the conjugate [φ] 1 p (π κ 0 1 (E, e 0))[φ]. By Theorem 2.6, there is a lifting φ of φ: φ is a κ0 -path in E from some e p 1 (b 0 ) to e 0. Then p (π κ 0 1 (E, e)) = p (h φ(π κ 0 1 (E, e 0))) = h φ (p (π κ 0 1 (E, e 0))) = [φ] 1 (p (π κ 0 1 (E, e 0)))[φ]. Therefore each conjugate of p (π κ 0 1 (E, e 0)) in π κ 1 1 (B, b 0) is an image for some choice of e. This result allows us to state the Theorem 3.1 in a more general form: Corollary 3.5. 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. Let e p 1 (b 0 ). Consider the following statements. (1) There exists a lifting φ : (X, x 0 ) (E, e) of φ with respect to p. (2) φ (π κ 2 1 (X, x 0)) is contained in some conjugate of 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). Note that two choices for e in p 1 (b 0 ) may yield the same conjugate in π κ 1 1 (B, b 0). In fact, all e will yield the same conjugate in the case that the subgroup p (π κ 0 1 (E, e 0)) is normal in π κ 1 1 (B, b 0). In particular, this is the case if π κ 1 1 (B, b 0) is abelian. When does there exist a mapping of digital covering spaces that preserves the digital covering relationship? The answer will rely on the Theorem 3.1, but first we consider the more general setting. Definition 3.6. Let q : D B and p : E B be digital covering maps. A homomorphism [14,9] of digital covering spaces is a digitally continuous function f : D E such that pf(d) = q(d) for every d in D. Note that the composition of two homomorphisms is again a homomorphism, and that, if E is a digital covering space of B, then the identity map E E is a homomorphism. Proposition 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 B is a κ 2 -connected digital image. If a homomorphism f : D E is onto, then f itself is a digital (κ 0, κ 1 )- covering map. Proof. Let e E. Consider the neighborhood N κ2 (p(e), 1) of p(e). By Proposition 2.4, N κ1 (e, 1) is (κ 1, κ 2 )-isomorphic to N κ2 (p(e), 1). Then f 1 (p 1 (N κ2 (p(e), 1)) = q 1 (N κ2 (p(e), 1)) and each κ 1 -component of this set is isomorphic to N κ2 (p(e), 1) via q. Since f is onto, at least one of these components must contain a point of f 1 (e). By composing isomorphisms we see that each κ 0 -component of f 1 (N κ1 (e, 1)) is isomorphic to N κ2 (e, 1). Thus, f is a digital covering map.

8 8 LAURENCE BOXER AND ISMET KARACA Theorem 3.8. 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, E is a κ 1 -connected digital image, and B is a κ 2 -connected digital image. Then the homomorphism f : D E is a digital (κ 0, κ 1 )-covering map. Proof. From Proposition 3.7, it is enough to show that f is onto. For a given point e E, pick a point d q 1 (p(e)). Since E is κ 1 -connected, there is a κ 1 -path φ : [0, m] Z E from f(d) to e. Projecting this path down into B via p gives a κ 2 -path based at q(d) = p(e). By Theorem 2.6, there is a unique lift, with respect to q, of this κ 2 -path to D with initial point d; call this κ 0 -path ξ. f ξ and φ are κ 1 -paths in E with initial point f(d), projecting via p into the same path in B. By uniqueness of lifts f ξ and φ must be same path. Thus e = φ(m) = f(ξ(m)) is in the image of f and f is onto. By Proposition 3.7, f is a digital covering map. Note that this result places significant limitations on the continuous functions from D to E that can be homomorphisms of digital covering spaces. The following propositions provide further restrictions, as well as specific conditions for the existence of such a function. Proposition 3.9. 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 ) Proof. Note, by Theorem 3.8, f and g are both liftings of q to E. Suppose there exists d D such that f(d) g(d). Since D is connected, there is a path r : [0, m] Z D such that r(0) = d 0 and r(m) = d. Since f(r(0)) = g(r(0)), we may assume without loss of generality that m > 0 is the smallest t [0, m] Z such that f(r(t)) g(r(t)). By continuity of f and g, it follows that f(d) and g(d) are distinct members of N 0 = N κ1 (f(r(m 1)), 1). Since f and g are both homomorphisms for the pair of coverings (p, q), we have q(f(d)) = p(d) = q(g(d)). But this is impossible, since q N0 identical on (D, d 0 ). is one-to-one. This contradiction shows that f and g are In the special case that D = E, an isomorphism of the digital covering space is called an automorphism. The following is an immediate consequence of Proposition 3.9: Corollary Let q : (E, e 0 ) (B, b 0 ) be a digital (κ 1, κ 2 )-covering map. If an automorphism f : E E for the pair (q, q) is not the identity, then f is fixed-point free. Note that, in general, the assertion analogous to Corollary 3.10 for a pair (p, q) of distinct covering maps, is false. For example, consider p : Z Z to be the identity map, and f, q : Z Z to be the maps defined by f(z) = q(z) = z. It is easily seen that p and q are (2, 2)-covering maps and f is a homomorphism for (p, q), but 0 is a fixed point of f.

9 THE CLASSIFICATION OF DIGITAL COVERING SPACES 9 Proposition 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 a κ 0 -connected digital image. If p is a radius 2 local isomorphism and q (π κ 0 1 (D, d 0)) is contained in a conjugate of p (π κ 1 1 (E, e 0)), then there exists a homomorphism f : D E for the pair (p, q). Proof. This follows directly from Corollary 3.5. Proposition 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 a κ 0 -connected digital image. Consider the following statements. (1) There is a homomorphism f : D E with f(d 0 ) = e 0. (2) q (π κ 0 1 (D, d 0)) p (π κ 1 1 (E, e 0)). Then (1) implies (2). Further, if p is a radius 2 local isomorphism, then (2) implies (1). Proof. This follows from Theorem 3.1. We can summarize these results in the following theorem: Theorem 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. 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 1 (B, b 0). Then (1) implies (2). Further, if p and q are radius 2 local isomorphisms, then (2) implies (1). This theorem motivates the title of our paper. It classifies covering spaces with radius 2 local isomorphic covering maps, in that two such covering spaces are isomorphic if and only if the images of their fundamental groups under the covering maps are conjugate subgroups. Proof. If q (π κ 0 1 (D, d 0)) and p (π κ 1 1 (E, e 0)) are conjugate in π κ 2 1 (B, b 0) and both p and q are radius 2 local isomorphisms, then using Theorem 3.4, we can assume, without loss of generality (by changing the base point in D if necessary) that the images of q and p are equal. Applying Proposition 3.12, there exist homomorphisms f : (D, d 0 ) (E, e 0 ) and h : (E, e 0 ) (D, d 0 ). Therefore, q = p f and p = q h, hence p f h = q h = p and q h f = p f = q. Thus, f h is an automorphism for the pair (p, p), and h f is an automorphism for the pair (q, q). Now, by Corollary 3.10, each composition must be the respective identity since h f(d 0 ) = d 0 and f h(e 0 ) = e 0. Therefore, f is an isomorphism. Conversely an isomorphism f : (D, d 0 ) (E, e 0 ) implies the images of p and q are equal for one choice of base points. By Theorem 3.4, varying the base points within q 1 (b 0 ) and p 1 (b 0 ) will produce conjugate subgroups as images. Corollary Suppose that p : (E, e 0 ) (B, b 0 ) is a (κ 0, κ 1 )-covering map of connected pointed digital images. Suppose that (B, b 0 ) is κ 1 -simply connected. If p is a radius 2 local isomorphism, then p is an isomorphism. Proof. Since (B, b 0 ) is κ 1 -simply connected, we must have {0} = π κ 1 1 (B, b 0) = (1 B ) (π κ 1 1 (B, b 0)) = p (π κ 0 1 (E, e 0)). From Theorem 3.13, there is an isomorphism f : E B for the pair (p, 1 B ). Therefore, p = 1 B f = f is an isomorphism.

10 10 LAURENCE BOXER AND ISMET KARACA 4. Acknowledgement We thank the anonymous reviewers for their helpful suggestions. References 1. L. Boxer, Digitally continuous functions, Pattern Recognition Letters 15 (1994), , A classical construction for the digital fundamental group, Journal of Mathematical Imaging and Vision 10 (1999), , Properties of digital homotopy, Journal of Mathematical Imaging and Vision 22 (2005), , Homotopy properties of sphere-like digital images, Journal of Mathematical Imaging and Vision 24 (2006), , Digital products, wedges, and covering spaces, Journal of Mathematical Imaging and Vision 25 (2006), S.E. Han, On the classification of the digital images up to digital homotopy equivalence, Journal of Computer and Communications Research 10 (2000), , Non-product property of the digital fundamental group, Information Sciences 171 (2005), , Digital coverings and their applications, Journal of Applied Mathematics and Computing 18 (2005), , Equivalent (k 0, k 1 )-covering and generalized digital lifting, Information Sciences (2007), doi: /j.ins G.T. Herman, Oriented surfaces in digital spaces, CVGIP: Graphical Models and Image Processing 55 (1993), E. Khalimsky, Motion, deformation, and homotopy in finite spaces, Proceedings IEEE International Conference on Systems, Man, and Cybernetics (1987), T.Y. Kong, A digital fundamental group, Computers and Graphics 13 (1989), T.Y. Kong, A.W. Roscoe, and A. Rosenfeld, Concepts of digital topology, Topology and its Applications 46 (1992), W.S. Massey, Algebraic Topology, Springer-Verlag, New York, A. Rosenfeld, Digital topology, American Mathematical Monthly 86 (1979), , Continuous functions on digital pictures, Pattern Recognition Letters 4 (1986), Q.F. Stout, Topological matching, Proceedings 15 th Annual Symposium on Theory of Computing (1983), 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 address: boxer@niagara.edu Department of Mathematics, Ege University, Bornova, Izmir TURKEY address: ismet.karaca@ege.edu.tr