Fundamental Properties of Digital Simplicial Homology Groups

Transcription

1 Vol., No. 2, March 23, PP: 25-42, ISSN: (Online) Research article Fundamental Properties of Digital Simplicial Homology Groups Ozgur Ege*, Ismet Karaca *Department of Mathematics, Celal Bayar University, 454 Muradiye, Manisa, TURKEY Department of Mathematics, Ege University, Bornova, Izmir 35, TURKEY Abstract In this article we give characteristic properties of the simplicial homology groups of digital images which are based on the simplicial homology groups of topological spaces in algebraic topology. We then investigate Eilenberg- Steenrod axioms for the simplicial homology groups of digital images. We state universal coefficient theorem for digital images. We conclude that the Künneth formula for simplicial homology doesnt be hold in digital images. Moreover we show that the Hurewicz Theorem need not be hold in digital images. Copyright AJCTA, all rights reserved. Keywords: Digital image, Digital simplicial homology group, Künneth formula, Hurewicz theorem.. Introduction Homology theory is a branch of algebraic topology that attempts to distinguish between spaces by constructing algebraic invariants that reflect the properties of a space. The homology groups of a space characterize the number and type of holes in that space and therefore give a fundamental description of its structure. This type of information is used, for example, for determining similarities between proteins in molecular biology (see []). As a result, the study of computational homology is an active area of researches. 25

2 Vol., No. 2, March 23, PP: 25-42, ISSN: (Online) There are several [2] different ways of defining homology groups. The simplicial homology theory is a classical one. Simplicial homology theory [25] is defined by means of a covariant functor from the category of simplicial complexes to the category of chain complexes. The advantages of this theory are that some proofs appear too easy and are that it is usually easy to compute H( X ) for specific X space. The simplicial theory [9] is easy to calculate with, whereas in the singular theory it is uite hard to calculate the homology of many spaces. On the other hand, many of the theorems that have been proved for the singular theory such as the existence of induced homomorphisms, their homotopy invariance, and the connection between homology and homotopy groups can also be proved for the simplicial theory. Simplicial homology theory applies to the category of pairs (X, A) where X and A have triangulations K and L for which L is a subcomplex of K, while the singular homology theory applies to all pairs (X, A) where X is a topological space with subspace A. Simplicial homology is the easiest homology theory to adapt for implementation on a computer. This homology is feasible for application to real life situations, such as image analysis, medical imaging, and data analysis in general. Since computer graphics, digital imaging and visualization are depended on the describing of triangulated surfaces, simplices are significant building blocks. Eilenberg and Steenrod [] obtained a system of axioms which characterize homology theories with simplices and singular chains. Eilenberg and Steenrod showed that both simplicial and singular homology have these axioms. They also proved that, on triangulable spaces, every homology theory having these properties would give the same answer. Kaczynski et al. [6] proposed to compute homology groups with a seuence of reductions. The idea is to derive a new object with less cells while preserving homology at each step of the transformation. During the computations, to ensure invertible coefficients, Kaczynski et al. choose them in a field. Kaczynski, Mischaikow and Mrozek [7] show that homology is computable by presenting in detail an algorithm based on linear algebra over the integers. This is essential because it demonstrates that the homology group of a topological space made up of cubes is computable. However, for spaces made up of many cubes, the algorithm is of little immediate practical value. Therefore, they introduce combinatorial techniues for reducing the number of elements involved in the computation. Peltier, Alayrangues, Fuchs and Lachaud [22] focus on homology groups, which are known to be computable in finite dimensions, and which have a good topological characterization power at least in low dimensions. For instance Euler characteristic and Betti numbers are straightforwardly deduced from homology groups. They choose simplicial homology since it is widely used in geometric modeling and is straightforwardly applicable to digital objects. After that, they present approach for computing homology groups. They also show some experiments, both on simplicial and discrete objects. Arslan et al. [] introduce the simplicial homology groups of n-dimensional digital images from algebraic topology. They also compute simplicial homology groups of MSS 8. Boxer, Karaca and Oztel [8] expand knowledge of the simplicial homology groups of digital images. They study the simplicial homology groups of certain minimal simple closed surfaces, extend an earlier definition of the Euler characteristics of digital image, and compute the Euler characteristic of several digital surfaces. 26

3 Vol., No. 2, March 23, PP: 25-42, ISSN: (Online) Karaca and Ege [8] present the digital cubical homology groups of digital images. They study some fundamental properties of digital images. They show that the Mayer-Vietoris Theorem does not hold in digital images. Karaca and Ege [9] study some results related to the simplicial homology groups of 2D digital images. They show that if a bounded digital image is nonempty and -connected, then its homology groups at the first dimension are a trivial group. They also prove that the homology groups of the operands of a wedge of digital images need not be additive. This paper is organized as follows: In Section 2 we introduce the general notions of digital images with - adjacency relations, we give definitions and theorems that are related to digital homotopy, digital fundamental group. In Section 3 we present some fundamental properties and definitions with respect to digital simplicial homology groups. In Section 4 we give the Eilenberg-Steenrod axioms for digital images and we conclude that excision axiom doesnt exist in digital images. In Section 5 we express Universal Coefficient Theorem for digital images and we give an example about this theorem. In Section 6 we show that the Künneth formula for simplicial homology doesnt be hold in digital images. In Section 7 we show that the Hurewicz Theorem need not be hold in digital images. 2. Preliminaries Let be the set of integers. A (binary) digital image is a pair ( X, ), where n X for some positive integer n and represents certain adjacency relation for the members of X. Various adjacency relations are used in the study of digital images. Some of the well-known adjacencies are the following. Two points p and in 2 are 8-adjacent if they are distinct and differ by at most in each coordinate; points p and in 2 are 4-adjacent if they are 8- adjacent and differ in exactly one coordinate. Two points p and in 3 are 26-adjacent if they are distinct and differ by at most in each coordinate; they are 8-adjacent if they are 26-adjacent and differ in at most two coordinate; they are 6-adjacent if they are 8-adjacent and differ in exactly one coordinate. We generalize these adjacencies as follows. Let l, n be positive integers, l n and two distinct points p ( p, p2,, pn), (, 2,, n) in n, p and are k l -adjacent [5], if there at most l distinct coordinates j for which pj j and for all other coordinates j, p j j. A kl -adjacency relation represents the number of points n that are adjacent to a given point n p. From this point of view, the k -adjacency on may be denoted by the number 2 and k -adjacent points are called 2-adjacent. Similarly, k -adjacent points of 2 are called 4-adjacent; k2 -adjacent points of 2 are called 8-adjacent; and in 3, k, k 2 and k3 -adjacent points are called 6-adjacent, 8-adjacent and 26-adjacent, respectively. Let be an adjacency relation defined on n n. A -neighbor of p is a point of n n that is -adjacent to p. A digital image X is -connected [5] if and only if for every pair of different points x, y X, there is a set { x, x,, x r } of points of a digital image X such that x x, y xr and x i and xi are -neighbors where i,,, r. A -component of a digital image X is a maximal -connected subset of X. Let ab, with a<b. A digital interval [2] is a set of the form [ ab, ] { z a z b}. 27

4 Vol., No. 2, March 23, PP: 25-42, ISSN: (Online) n n Let ( X, ) and ( Y, ) be digital images. A function f : X Y is said to be (, ) - continuous [3] if for every -connected subset U of X, f(u) is a -connected subset of Y. Such a function is called digitally continuous. A function f : X Y is (, ) -continuous [3] if and only if for every - adjacent points { x, x } of X, either f ( x) f( x) or f ( x ) and f ( x ) are -adjacent in Y. By a digital -path [6] from x to y in a digital image X, we mean a (2, )-continuous function f :[, m] X such that f()=x and f(m)=y. If f()=f(m) then f is called digital -loop and the point f() 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 -curve of m 4 points in a digital image X is a seuence { f(), f(),, f( m )} of images of the -path f :[, m] X such that f(i) and f(j) are -adjacent if and only if j i mod m. n n Let ( X, ) and ( Y, ) be digital images. A function f : X Y is a (, ) -isomorphism [] if f is (, ) -continuous and bijective and further f : Y X is (, ) -continuous and it is denoted by X Y. n n Definition 2. [3]. Let X and Y be digital images with -adjacency and -adjacency respectively. Two (, ) -continuous functions f, g: X Y if there is a positive integer m and a function H : X [, m] Y such that for all x X, H( x,) f( x) and H( x, m) g( x) ; for all x X, the induced function H :[, m] Y (2, ) is -continuous; and for all t [, m], the induced function H : X Y (, ) is -continuous. x x are said to be digitally (, ) -homotopic in Y defined by H () t H( x,) t for all t [, m], t defined by H ( x) H( x, t) for all x X, t The function H is called a digital (, ) -homotopy between f and g. The notation f that functions f and g are digitally (, ) -homotopic in Y. The digital (, ) f :( X, ) ( Y, ). euivalence among digitally continuous functions, g is used to indicate -homotopy relation [3] is 28

5 Vol., No. 2, March 23, PP: 25-42, ISSN: (Online) Let f : X Y be a (, ) -continuous function and let g: Y X be a (, ) -continuous function such that f g, and g f,. Then we say X and Y have the same (, ) -homotopy type [3] and that X and Y are (, ) -homotopy euivalent. Definition 2.2 [2]. A digital image ( X, ) is said to be -contractible if its identity map is (, )-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 (, ) -continuous function f : X Y is -nullhomotopic if f is -homotopic to a constant function c in Y. For a digital image ( X, ) and its subset ( A, ), we call (X,A) a digital image pair with -adjacency. Moreover, if A is a singleton set { x }, then ( X, x ) is called a pointed digital image. Definition 2.3 [3]. Let (X,A) be a digital image pair with -adjacency. Let i: A X A is called a -retract of X if and only if there is a -continuous function r: X A such that ra ( ) a A. Then the function r is called a -retraction of X onto A. If f :[, m ] X and g:[, m2] X are digital -paths with f ( m ) g() ( f g):[, m m ] X [7] by 2 f(), t tm ( f g)( t) gt ( m), m t m +m 2. be the inclusion function. afor all, then define the product Let f and (, ) f be -loops in a pointed digital image X x {,,, }. We say sets of -paths f f2 f r and { F, F2,, F p } in X such that f is a trivial extension of f [3] if there are r p; f f f2 fr ; f F F F ; 2 p There are indices i i2 ir p such that Fi a trivial loop. j f j, j r ; and i {,, i 2, i r } implies F i is 29

6 Vol., No. 2, March 23, PP: 25-42, ISSN: (Online) If f, g:[, m] ( X, ) are -paths [3] such that f()=g() and f(m)=g(m), then we say that a homotopy H :[, m] [, M] X between f and g such that for all t [, M], H(, t) f() and Hmt (, ) fm ( ), holds the endpoints fixed. Two loops f, f with the same base point x X belong to the same loop class [ f ] X [3] if they have trivial extensions that can be joined by a homotopy that holds the endpoints fixed. Define ( X, x) to be the set of digital homotopy classes of -loops [ f ] X in X with base point x. ( X, x ) is a group under the product operation [3] defined by [ f ] [ g] [ f g]. X X X Lemma 2.4 [3]. Let (X, p) be a pointed digital image. Let p :[, m] X be the constant function p() t p. Then [ p ] is an identity element for ( X, p). Lemma 2.5 [3]. If f :[, m] X represents an element of ( X, p), then the function g:[, m] X defined by gt () f( mt) for t [, m] is an element of [ f ] in ( X, p). Theorem 2.6 [3]. Let X be a digital image with adjacency relation. If p and belong to the same -component of X, then ( X, p) and ( X, ) are isomorphic groups. A digital pointed image ( X, x ) is said to be simply -connected [4] if X is -connected and ( X, x) f :( X, x ) ( Y, y ) (, )-continuous map of pointed digital images, then [3] trivial group. If is a is a : (, ) (, ), defined by f ([ g]) [ f g] f X x Y y, is a group homomorphism. : (,,, ) Definition 2.7 [3]. Let c x x x n be a closed -curve in 2, where, 4,8. A point x of the complement c of a closed -curve c in 2 is said to be interior to c if it belongs to the bounded - connected component of c. The set of all interior points to c is denoted by Int( c ). Since a digital image is made up of discrete points, a closure of a digital image is again itself, that is, if A is a digital image, then its closure which is denoted by A is A A. 3. Some Properties of Digital Simplicial Homology Groups Before we give the digital version of Eilenberg-Steenrod axioms, we shall recall definitions, theorems and other important notions about [], [8] the digital simplicial homology groups. 3

7 Vol., No. 2, March 23, PP: 25-42, ISSN: (Online) Definition 3. [25]. Let S be a set of nonempty subset a digital image ( X, ). Then the members of S are called simplexes of ( X, ) if the following hold : a) If p and are distinct points of s S b) If s S and t s, then t S., then p and are -adjacent. An m-simplex is a simplex S such that S m. Let P be a digital m-simplex. If P is a nonempty proper subset of P, then Vert P to denote the vertex set of P, namely, the set of all P is called a face of P. We write digital -simplexes in P. A digital subcomplex A of a digital simplicial complex X with -adjacency is a digital Vert A Vert X. simplicial complex [24] contained in X with ( ) ( ) Let ( X, ) be a finite collection of digital m-simplices, m d for some non-negative integer d. If the following statements hold then ( X, ) is called [] a finite digital simplicial complex : () If P belongs to X, then every face of P also belongs to X. (2) If PQ, X, then P Q is either empty or a common face of P and Q. The dimension of a digital simplicial complex X is the largest integer m such that X has an m-simplex. C ( X) free abelian group [] with basis all digital (, ) -simplices in X. is a n Corollary 3.2 [8]. Let ( X, ) be a digital simplicial complex of dimension m. Then for all m C ( X) is a trivial group. n Let ( X, ) be a digital simplicial complex of dimension m. The homomorphism : C ( X) C ( X) defined (see []) by i p p pi p ( p, p,, p ) i ( ),,,,,, m, m, is called a boundary homomorphism, where p i means delete the point Proposition 3.3 []. For all m, we have. n Theorem 3.4 []. Let ( X, ) be a digital simplicial complex of dimension m. Then p i. 3

8 Vol., No. 2, March 23, PP: 25-42, ISSN: (Online) m m m m m C ( X): C ( X) C ( X) C ( X) is a chain complex. Definition 3.5 [5]. Let ( X, ) be a digital simplicial complex. () Z ( X) Ker is called the group of digital simplicial -cycles. (2) B ( X) Im is called the group of digital simplicial -boundaries. (3) H ( X) Z ( X)/ B ( X) is called the th digital simplicial homology group. Because we need to use relative simplicial homology for digital images, we give its definition. Let ( A, ) be a digital subcomplex of the digital simplicial complex ( X, ). The chain group C ( A) is a subgroup of the chain group C ( X). The uotient group C ( X)/ C ( A) is called the group of relative chains of X modulo A and is denoted by C ( X, A). The boundary operator : C( A) C ( A) is the restriction of the boundary operator on C ( X). A homomorphism C( X, A) C ( X, A) of the relative chain groups is induced by and is also denoted by. Definition 3.6 [2]. Let ( A, ) be a digital subcomplex of the digital simplicial complex ( X, ). Z ( X, A) Ker is called the group of digital relative simplicial -cycles. B( X, A) Im is called the group of digital relative simplicial -boundaries. H ( X, A) Z ( X)/ B ( X) is called the th digital relative simplicial homology group. Theorem 3.7 [8]. Let ( X, ) be a directed digital simplicial complex of dimension m. () H ( X) is a finitely generated abelian group for every. (2) H ( X) is a trivial group for all m. (3) H ( X) is a free abelian group, possible zero. m 32

9 Vol., No. 2, March 23, PP: 25-42, ISSN: (Online) Theorem 3.8 [8]. For each, H is a covariant functor from the category of digital simplicial complexes and simplicial maps to the category of abelian groups. Corollary 3.9 []. If f :( X, ) ( Y, 2) is a digitally (, 2) f H X H Y is a group isomorphism. 2 : ( ) ( ) -isomorphism, then Let :( X, ) ( Y, ) be a function between digital images. If for every digital (, m) -simplex P determined by the adjacency relation in X, ( P) is a (, n) -simplex in Y for some n m, then is called (see [8]) a digital simplicial map. Let :( X, ) ( Y, ) be a digital simplicial map. For, we define a homomorphism é : C ( X) C ( Y) by ( p,, p ) ( p ),, ( p ). é é é Theorem 3. [26]. Let f, g:( X, ) ( Y, 2) be the digitally (, 2) -continuous functions. Assume that 2 there are homomorphisms : C ( X) C ( Y) such that fé gé where 2 2 : C ( X) C ( X) and : C ( Y) C ( Y) are digital boundary operators of ( X, ) and 2 ( Y, 2), respectively. Then f g: H ( X) H ( Y) as group homomorphisms. Theorem 3. [26]. Let ( A, ) be a nonempty subset of digital image ( X, ) with -adjacency, and let i:( A, ) ( X, ) be the inclusion. Then i : H ( A) H ( X) is a monomorphism for every. 4. The Eilenberg-Steenrod Axioms We now list the digital version of Eilenberg-Steenrod axioms which is very significant in algebraic topology. Note that the first 5 axioms hold in digital images. Since these axioms are proved in a similar way in algebraic topology, we dont prove these axioms. Axiom (Identity axiom) [2]. Let X be a digital image with -adjacency. If i:( X, ) ( X, ) then i : H ( X) H ( X) is the identity. is the identity, Axiom 2 (Composition axiom) [2]. Let X, Y and Z be digital images with, and 2 -adjacency, respectively. h:( X, ) ( Y, ) k:( Y, ) ( Z, ) are digitally continuous functions, then ( k h) k h. If and 2 Axiom 3 (Commutativity axiom) [2]. Let (X,A) and (Y,B) be digital image pairs with and -adjacency, respectively. If f :( X, A) ( Y, B), then the following diagram commutes : 33

10 Vol., No. 2, March 23, PP: 25-42, ISSN: (Online) Axiom 4 (Exactness axiom) [2]. The seuence i p H ( A) H ( X) H ( X, A) H ( A) is exact, where i: A X and p : X ( X, A) are inclusion maps. Axiom 5 (Dimension axiom) []. If X is a one-point space with -adjacency, then H ( X) for and H ( X). Homotopy and excision axioms don t hold in digital images. We first give definition of these axioms in algebraic topology. Homotopy axiom [2]. Let X and Y be digital images with and 2 -adjacency, respectively. If h:( X, ) ( Y, 2) and k:( X, ) ( Y, 2) are homotopic, then h k. Excision axiom [2]. Given (X,A), let U be an open subset of X such that U IntA, then inclusion induces an isomorphism H ( X U, AU) H ( X, A). Proposition 4. Excision axiom for simplicial homology doesnt be hold in digital images. Proof. Let Figure ). X c c c c 2 { (,), (,), 2 (,), 3 (,)} be a digital image with 4-adjacency (see Figure. X 34

11 Vol., No. 2, March 23, PP: 25-42, ISSN: (Online) Since X is a digital simple closed 4-curve (see []), the digital simplicial homology groups of X are 4, =, H ( X),,. A{ c (,), c (,), c (,)} and U { c (,)} (see Figure 2). Let 2 Its clear that A X, U A and U IntA Figure 2. A and U 4 4, = H( A) H( U),. So we get following exact seuence :. The digital simplicial homology groups of A and U are H ( A) H ( X) H ( X, A) H ( A) For >, as H 4 ( A) we get H ( X) H ( X, A). For =, we have following exact seuence : 4 4 i j k l m n 4 4 H X A H X A (, ) (, ) 4 4 From this exact seuence, Im m Ker n H ( X, A), thus m is an epimorphism and H (, ) X A. As l is an inclusion map, from Theorem 3. we get that Ker l. Im k due to exact seuence. From first isomorphism theorem, H 4 ( X, A )/Ker k Im k and we find that H 4 ( X, A) Ker k since Im k. As 4 this seuence is exact, we have Ker k Im j and so we get Im j H ( X, A). Thus j is an epimorphism. On the other hand, from exactness of seuence, we have Im i Ker j. Since Im i, Ker j and thus j is a 4 monomorphism. Therefore j is an isomorphism and we find that H (, ) X A. Conseuently, we obtain 35

12 Vol., No. 2, March 23, PP: 25-42, ISSN: (Online) 4, =, H ( X, A),,. Now we compute H 4 ( X U, A U). X U { c (, ), c2 (,), c3 (,)} and AU { c (, ), c (,)} (see Figure 3). 2 Figure 3. X-U and A-U Its clear that AU X U. The digital simplicial homology groups of X-U and A-U are So we get following exact seuence : 4 4, = H( X U) H( AU),. H ( AU) H ( X U) H ( X U, AU) H ( AU) For >, as seuence : H 4 ( AU) we get H ( X U) H ( X U, A U). For =, we have following exact 4 4 From this exact seuence, H 4 X U A U 4 4 H X U AU H X U A U (, ) (, ) Im Ker H ( X U, A U) thus is an epimorphism and 4 (, ). As is an inclusion map, from Theorem 3. we get that Ker. Im because of exact seuence. From first isomorphism theorem, H ( X U, AU )/Ker Im 4 and we find that (, ) Ker since Im. As this seuence is exact, we have H 4 X U A U 36

13 Vol., No. 2, March 23, PP: 25-42, ISSN: (Online) Ker Im and so we get 4 H X A. As a result, we obtain (, ) 4, = H ( X U, AU),. 4 4 For =, since H( X U, A U) H( X, A), excision theorem for simplicial homology doesnt be hold in digital images. We can also say that the homotopy axiom for simplicial homology doesnt hold in digital images. For this purpose, we must give a counterexample to homotopy axiom for digital images. The simple closed curve X of Proposition H ( X) H ( x ). But from Proposition 4. we would have is 4-contractible (see []) so we have H X Therefore X x that is X and singleton set 4 ( ). (4,4) euivalent, but they don t have isomorphic homology groups. 5. Universal Coefficient Theorem for Digital Simplicial Homology We state universal coefficient theorem for homology of a digital simplicial complex. x are digitally (4,4) - homotopy Theorem 5. [2]. Let ( X, ) be a digital simplicial complex. For any abelian group G, there is exact seuence H ( X) G H ( X, G) Tor( H ( X), G), where H ( X) H ( X; ). This exact seuence is split; hence H ( X, G) H ( X) G Tor( H ( X), G). Example 5.2 Let MSC { c (,), c (, ), c (,), c (,)} be a digital image with adjacency (see Figure 4). 37

14 Vol., No. 2, March 23, PP: 25-42, ISSN: (Online) The digital simplicial homology groups of Figure 4. [4] MSC 8 are MSC 8 8, =, H ( MSC8),, (see [8]). We use universal coefficient theorem for homology to calculate simplicial homology groups of MSC 8 with 3 -coefficient. For =, we have 8 8 H ( MSC ; ). For =, we have H ( MSC ; ). For all 2, we get H ( MSC ; ) 8 3. As a result we find that 8 3, =, H ( MSC8; 3),,. 6. Künneth Formula for Digital Simplicial Homology We recall that Künneth formula for simplicial homology used in algebraic topology Theorem 6. [2]. Suppose X and Y are topological spaces. We then have the following relation between the homology groups of X, Y and the product space X Y. For any n and any module M over a principal ideal domain R, we have: ( ) ( ) H ( X Y; M) H ( X; M) H ( Y; M) Tor( H ( X; M), H ( Y; M)) n i j p ijn pn where Tor denotes the Tor functor. If M F is a field, then the Tor functor is always trivial and in this case Künneth formula can be stated as for any n. H ( X Y; F) H ( X; F) H ( Y; F ) n i j i jn We show that Künneth formula for simplicial homology doesnt be hold in digital images. Proposition 6.2 Künneth formula for simplicial homology doesnt be hold in digital images. Proof. Let X [,] and Y [,] [,] be digital images with 2-adjacency and 4-adjacency, respectively. Assume that M. Then simplicial homology groups of X are 38

15 Vol., No. 2, March 23, PP: 25-42, ISSN: (Online) 2, n= Hn ( X), n. Since Y is a digital simple closed 4-curve (see []), its digital simplicial homology groups are Since X Y is 4, n=, Hn ( Y), n,. MSS 6, its simplicial homology groups are, n= 6 5 Hn ( X Y), n=, n, (see [8]). We now use Künneth formula to compute H 6 n ( X Y) for n. Since M is a field, Hn( X Y; ) Hi ( X; ) H j( Y; ). For n=, we have i jn For n=, we have H ( X Y; ) H ( X; ) H ( Y; ) H ( X Y; ) ( H ( X; ) H ( Y; )) ( H ( X; ) H ( Y; )) ( ) ( ) For all n 2, we have H 6 ( X Y; ). Therefore we find that n 6, n=, Hn ( X Y; ), n,. However homology group of X Y for n= must be H ( X Y; ) Künneth formula for simplicial homology doesnt be hold in digital images., so we get a contradiction. As a result, The Hurewicz Theorem Need Not Be Hold In Digital Images Theorem 7. (Hurewicz Theorem) [24]. If X is path connected connected, then the Hurewicz map : ( X, x ) H ( X) is a surjection with ( X, x) ( X, x ), the commutator subgroup of ( X, x )/ ( X, x ) H ( X).. Hence 39

16 Vol., No. 2, March 23, PP: 25-42, ISSN: (Online) We now give an example which shows that the Hurewicz Theorem theorem neednt be hold in digital images. Example 7.2 Let MSS { c (,,), c (,,), c (, 2,), c (, 3,), c (, 2,), c (,,), c (,, ), c (,2,), c (,2,2), c (,,2)} be a digital image with 8-adjacency (see Figure 5). Figure 5. [4] MSS 8 Since MSS 8 is 8-contractible minimal simple closed 8-surface, H, = ( MSS ), =,, ( MSS, c ) {}. However 8 8 (see [8]). ( MSS, c )/ ( MSS, c ) H ( MSS ) So we can state the following. Proposition 7.3 Hurewicz Theorem does not be hold in digital images. 8. Conclusion The aim of this article is to determine differences between digital topology and algebraic topology. We state Eilenberg-Steenrod axioms for the simplicial homology groups of digital images. Especially we present that excision axiom which is the one of Eilenberg-Steenrod axioms that doesnt exist in digital images. Universal Coefficient theorem and Künneth formula for digital simplicial homology are presented with examples. Also we show that the Hurewicz Theorem need not be hold in digital images. Digital simplicial homology is needed to study digital cohomology groups and digital cohomology operations which are very important in algebraic topology and stable homotopy theory. We hope that this theory will help to the analysis and understanding of these topics. 4

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