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Applications of the Lefschetz number to digital images.

1 Introduction

Digital topology with algebraic properties is a growing area in computer vision, image processing and fixed point theory. Many researchers, such as Rosenfeld, Kong, Kopperman, Kovalevsky, Boxer, Karaca, Han and others, have studied the properties of digital images using topology and algebraic topology.

The Lefschetz fixed point theorem counts fixed points of a continuous mapping from a compact topological space X to itself via traces of the induced mappings on the homology groups of X. The Lefschetz number has been used in order to treat fixed point theory. Since the Lefschetz number is a homotopy invariant, it can be used to classify digital images. The main advantage is that this number can be easily computed.

Arslan et al. [1] introduce the simplicial homology groups of n-dimensional digital images. Boxer et al. [7] expanded knowledge of the simplicial homology groups of digital images. They study the simplicial homology groups of certain minimal simple closed surfaces, extended an earlier definition of the Euler characteristics of a digital image, and computed the Euler characteristic of several digital surfaces.

Karaca and Ege [14] study some results related to the simplicial homology groups of 2D digital images. They show that if a bounded digital image X [subset] Z is nonempty and [kappa]-connected, then its homology groups at the first dimension are trivial. They also prove that the homology groups of the operands of a wedge of digital images are not necessarily additive. Ege and Karaca [11] give characteristic properties of the simplicial homology groups of digital images and investigate the Eilenberg-Steenrod axioms for the simplicial homology groups of digital images. Ege and Karaca [10] construct Lefschetz fixed point theory for digital images and get some nice results. Ege et al. [12] study relative homology groups of digital images and compute relative homology groups of some digital images.

This paper is organized as follows. The second section provides the general notions of digital images with [kappa]-adjacency relations, digital homotopy and homology groups. In Section 3, we provide some important applications about the Lefschetz number, relative and reduced versions of the Lefschetz number, and we present some examples. In the final Section we draw some conclusions.

2 Preliminaries

Let Z be the set of integers. A digital image is a pair (X, [kappa]), where X [subset] [Z.sup.n] for some positive integer n and [kappa] represents a certain adjacency relation for the elements of X. There are various adjacency relations in the study of digital images but we give only one of them. Let l, n be positive integers, 1 [less than or equal to] l [less than or equal to] n and two distinct points p = ([p.sub.1], [p.sub.2], ..., [p.sub.n]), q = ([q.sub.1], [q.sub.2], ..., [q.sub.n]) in [Z.sup.n], p and q are [k.sub.l]-adjacent [5], if there are at most l distinct coordinates j for which [absolute value of [p.sub.j] - [q.sub.j]] = 1 and for all other coordinates j, [p.sub.j] = [q.sub.j]. The number of points q [member of] [Z.sup.n] that are adjacent to a given point p [member of] [Z.sup.n] is represented by a [k.sub.l]-adjacency relation. From this viewpoint, the [k.sub.1]-adjacency on Z is denoted by the number 2 and [k.sub.1]-adjacent points are called 2-adjacent. In a similar way, we call 4-adjacent and 8-adjacent for [k.sub.1] and [k.sub.2]-adjacent points of [Z.sup.2]; and in [Z.sup.3], 6-adjacent, 18-adjacent and 26-adjacent for [k.sub.1], [k.sub.2] and [k.sub.3]-adjacent points, respectively.

A [kappa]-neighbor of p [member of] [Z.sup.n] is a point of [Z.sup.n] which is [kappa]-adjacent to p, where [kappa] is an adjacency relation defined on [Z.sup.n]. A digital image X [subset] [Z.sup.n] is [kappa]-connected [13] if and only if for every pair of different points x, y [member of] X, there is a set {[x.sub.0], [x.sub.1], ..., [x.sub.r]} of points of a digital image X such that x = [x.sub.0], y = [x.sub.r] and [x.sub.i] and [x.sub.i+1] are [kappa]-neighbors where i = 0, 1, ..., r - 1. A set of the form

[[a, b].sub.Z] = {z [member of] Z|a [less than or equal to] z [less than or equal to] b},

is said to be a digital interval [2], where a, b [member of] Z with a < b.

Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be digital images. A function f : X [right arrow] Y is called ([[kappa].sub.0], [[kappa].sub.1])-continuous [3] if for every [[kapa].sub.0]-connected subset U of X, f (U) is a [[kappa].sub.1]-connected subset of Y.

In a digital image (X, [kappa]), if there is a (2, [kappa])-continuous function f : [[0, m].sub.Z] [right arrow] X such that f(0) = x and f(m) = y, then we say that there exists a digital [kappa]-path [6] from x to y. If f (0) = f (m) then f is called digital [kappa]-loop and the point f (0) is the base point of the loop f. When a digital loop f is a constant function, it is said to be a trivial loop. A simple closed [kappa]-curve of m [greater than or equal to] 4 points in a digital image X is a sequence {f (0), f (1), ..., f (m - 1)} of images of the [kappa]-path f : [[0, m - 1].sub.Z] [right arrow] X such that f (i) and f (j) are K-adjacent if and only if j = i [+ or -] mod m.

Definition 2.1. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be digital images. A function f : X [right arrow] Y is a ([[kappa].sup.0], [[kappa].sub.1])-isomorphism [1] if f is ([k.sub.0], [[kappa].sub.1])-continuous and bijective and [f.sup.-1] : Y [right arrow] X is ([[kappa].sub.1], [[kappa].sub.0])-continuous and it is denoted by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For a digital image (X, [kappa]) and its subset (A, [kappa]), we call (X, A) a digital image pair with [kappa]-adjacency. If A is a singleton set {[x.sub.0]}, then (X, [x.sub.0]) is called a pointed digital image.

Definition 2.2. [3]. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be digital images. Two ([[kappa].sub.0], [[kappa].sub.1])-continuous functions f, g : X [right arrow] Y are said to be digitally ([[kappa].sub.0], [[kappa].sub.1]) homotopic in Y if there is a positive integer m and a function H : X x [[0, m].sub.Z] [right arrow] Y such that

* for all x [member of] X, H(x, 0) = f (x) and H(x, m) = g(x);

* for all x [member of] X, the induced function [H.sub.x] : [[0, m].sub.Z] [right arrow] Y defined by

[H.sub.x] (t) = H(x, t) for all t [member of] [[0, m].sub.Z], is (2, [[kappa].sub.1])-continuous; and

* for all t [member of] [[0, m].sub.Z], the induced function [H.sub.t] : X [right arrow] Y defined by

[H.sub.t](x) = H(x, t) for all x [member of] X,

is ([[kappa].sub.0], [[kappa].sub.1])-continuous.

The function H is called a digital ([[kappa].sub.0], [[kappa].sub.1])-homotopy between f and g. If these functions are digitally ([[kappa].sub.0], [[kappa].sub.1])-homotopic, this is denoted [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] g. The digital ([[kappa].sub.0], [[kappa].sub.1])-homotopy relation [3] is equivalence among digitally continuous functions f : (X, [[kappa].sub.0]) [right arrow] (Y, [[kappa].sub.1]).

Definition 2.3. [3]. Let f: X [right arrow] Y and g: Y [right arrow] X be ([[kappa].sub.0],[[kappa].sub.1]) and ([[kappa].sub.1],[[kappa].sub.0]) continuous functions respectively such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We say that X and Y have the same ([[kappa].sub.0], [[kappa].sub.1])-homotopy type and that X and Y are ([[kappa].sub.0], [[kappa].sub.1])-homotopy equivalent.

Definition 2.4. [3]. (i) Let (X, [kappa]) and (Y, [kappa]') be digital images. A digitally continuous function f : X [right arrow] Y is digitally nullhomotopic if f is digitally homotopic in Y to a constant function. A digital image (X, [kappa]) is [kappa]-contractible if its identity map is digitally nullhomotopic.

(ii) Let (X, A) be a digital image pair with [kappa]-adjacency. A is called a [kappa]-retract of X if and only if there is a [kappa]-continuous function r : X [right arrow] A such that r(a) = a for all a [member of] A. Then the function r is called a [kappa]-retraction of X onto A.

Definition 2.5. [15]. Let S be a set of nonempty subsets of a digital image (X, [kappa]). Then the members of S are called simplexes of (X, [kappa]) if the following hold :

a) If p and q are distinct points of s [member of] S, then p and q are [kappa]-adjacent.

b) If s [member of] S and [empty set] [not equal to] t [subset] s, then t [member of] S.

An m-simplex is a simplex S such that [absolute value of S] = m + 1.

Let P be a digital m-simplex. If P' is a nonempty proper subset of P, then P' is called a face of P. We write Vert(P) to denote the vertex set of P, namely, the set of all digital 0-simplexes in P. A digital subcomplex A of a digital simplicial complex X with [kappa]-adjacency is a digital simplicial complex [15] contained in X with Vert(A) [subset] Vert(X).

Definition 2.6. Let (X, [kappa]) be a finite collection of digital m-simplexes, 0 [less than or equal to] m [less than or equal to] d for some nonnegative integer d. If the following statements hold, then (X, [kappa]) is called [1] a finite digital simplicial complex:

(1) If P belongs to X, then every face of P also belongs to X.

(2) If P, Q [member of] X, then P [intersection] 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.sup.[kappa].sub.q](X) is a free abelian group [1] with basis of all digital ([kappa], q)-simplexes in X.

Corollary 2.7. [1]. Let (X, [kappa]) [subset] [Z.sup.n] be a digital simplicial complex of dimension m. Then, for all q > m, [C.sup.[kappa].sub.q](X) is a trivial group.

Let (X, kappa]) [subset] [Z.sup.n] be a digital simplicial complex of dimension m. The homomorphism [[partial derivative].sub.q] : [C.sup.[kappa]](X) [right arrow] [C.sup.[kappa].sub.q-1](X) defined (see [1]) by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is called a boundary homomorphism, where [[??].sub.i] means delete the point [p.sub.i]. In [1], it is shown that for all 1 [less than or equal to] q [less than or equal to] m,

[[partial derivative].sub.q-1] [omicron] [[partial derivative].sub.q] = 0.

Definition 2.8. [7]. Let (X, [kappa]) be a digital simplicial complex.

(1) [Z.sup.[kappa].sub.q](X) = Ker [[partial derivative].sub.q] is called the group of digital simplicial q-cycles.

(2) [B.sup.[kappa].sub.q](X) = Im [[partial derivative].sub.q+1] is called the group of digital simplicial q- boundaries.

(3) [H.sup.[kappa].sub.q](X) = [Z.sup.[kappa].sub.q](X) /[B.sup.[kappa].sub.q](X) is called the qth digital simplicial homology group.

We recall some important examples about digital homology groups of certain digital images.

Example 2.9. Let X = {(0, 0), (1, 0), (0, 1), (1, 1)} [subset] [Z.sup.2] be a digital image with 4-adjacency (see Figure 1). In [1], it is shown that digital homology groups of X are:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Example 2.10. Let [MSS'.sub.6] = {[c.sub.0] = (0, 0, 0), [c.sub.1] = (1, 0, 0), [c.sub.2] = (1, 1, 0), [c.sub.3] = (0, 1, 0), [c.sub.4] = (0, 0, 1), [c.sub.5] = (1, 0, 1), [c.sub.6] = (1, 1, 1), [c.sub.7] = (0, 1, 1)} be a digital image with 6-adjacency (see Figure 2). Boxer et al. [7] show that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Example 2.11. Let [MSS'.sub.18] = {c.sub.0] = (1, 1, 0), [c.sub.1] = (0, 2, 0), [c.sub.2] = (-1, 1, 0), [c.sub.3] = (0, 0, 0), [c.sub.4] = (0, 1, -1), [c.sub.5] = (0,1,1)} be a digital image with 18-adjacency (see Figure 3). Boxer et al. [7] get the following

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Theorem 2.12. [7]. Let (X, [kappa]) be a directed digital simplicial complex of dimension m.

(1) [H.sup.[kappa].sub.q](X) is a finitely generated abelian group for every q [greater than or equal to] 0.

(2) [H.sup.[kappa].sub.q](X) is a trivial group for all q > m.

(3) [H.sup.[kappa].sub.q](X) is a free abelian group, possible zero.

Boxer et al. [7] conclude that for each q [greater than or equal to] 0, [H.sup.[kappa].sub.q] is a covariant functor from the category of digital simplicial complexes and simplicial maps to the category of abelian groups.

Definition 2.13. [7]. Let f : (X, [[kappa].sub.0]) [right arrow] (Y, [[kappa].sub.1]) be a function between two digital images. If for every digital ([[kappa].sub.0], m)-simplex P determined by [[kappa].sub.0] in X, f(P) is a ([[kappa].sub.1], n)-simplex in Y for some n [less than or equal to] m, then f is called a digital simplicial map.

For a digital simplicial map f : (X, [[kappa].sub.0]) [right arrow] (Y, [[kappa].sub.1]) and q [greater than or equal to] 0, two induced homomorphisms [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are defined by (see [7])

[f.sub.#](< [p.sub.0], ..., [p.sub.q] >) = < [f.sub.#] [(.sub.p0]), ..., [f.sub.#] ([p.sub.q]) >,

[f.sub.*] (z + [B.sup.[kappa].sub.q](X))= [f.sub.#](z) + [B.sup.[kappa].sub.q](Y),

where z [member of] [Z.sup.[kappa].sub.q](X), respectively.

Definition 2.14. [12]. Let (A, [kappa]) be a digital subcomplex of the digital simplicial complex (X, [kappa]). Then the chain group [C.sup.[kappa].sub.q](A) is a subgroup of the chain group [C.sup.[kappa].sub.q](X). The quotient group

[C.sup.[kappa].sub.q] (X, A) = [C.sup.[kappa].sub.q](X) / [C.sup.[kappa].sub.q](A)

is called the group of relative chains of X modulo A. The boundary operator

[[partial derivative].sub.q]: [C.sup.[kappa].sub.q](A) [right arrow] [C.sup.[kappa].sub.q-1] (A)

is the restriction of the boundary operator on [C.sup.[kappa].sub.q](X). It induces a homomorphism

[C.sup.[kappa].sub.q](X, A) [right arrow] [C.sup.[kappa].sub.q-1] (X, A)

of the relative chain groups and this is also denoted by [[partial derivative].sub.q].

Definition 2.15. [12]. Let (A, [kappa]) be a digital subcomplex of the digital simplicial complex (X, [kappa]).

* [Z.sup.[kappa].sub.q] (X, A) = Ker [[partial derivative].sub.q] is called the group of digital relative simplicial q- cycles.

* [B.sup.[kappa].sub.q] (X, A) = Im [[partial derivative].sub.q+1] is called the group of digital relative simplicial q-boundaries.

* [H.sup.[kappa].sub.q] (X, A) = [Z.sup.[kappa].sub.q](X, A)/[B.sup.[kappa].sub.q] (X, A) is called the qth digital relative simplicial homology group.

Definition 2.16. [12]. Let (X, [kappa]) be a digital simplicial complex.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is called the zero dimensional reduced digital homology group of (X, [kappa]), where

[member of] : [C.sup.[kappa].sub.0](X) [right arrow] Z

is defined by [member of]([upsilon]) = 1 and [[partial derivative].sub.1] : [C.sup.[kappa].sub.1] (X) [right arrow] [C.sup.[kappa].sub.0](X) is the boundary homomorphism. If [[??].sup.[kappa].sub.p](X) = [H.sup.[kappa].sub.p](X) for each p > 1, then

{[[??].sup.[kappa].sub.i] (X), i = 0,1, ...}

are called the reduced digital homology groups of (X, [kappa]).

Theorem 2.17. [12]. For a digital simplicial complex (X, [kappa]), there are the following formulas which are related to reduced homology groups:

[H.sup.[kappa].sub.0](X) [congruent to] [[??].sup.[kappa].sub.0](X) [direct sum] Z,

[H.sup.[kappa].sub.p](X) [congruent to] [[??].sup.[kappa].sub.p](X), p [greater than or equal to] 1.

3 Applications of the Lefschetz fixed point theorem to digital images

Let (X, [kappa]) be a digital image and f : (X, [kappa]) [right arrow] (X, [kappa]) be any ([kappa], [kappa])- continuous function. We say the digital image (X, [kappa]) has the fixed point property [10] if for every ([kappa], [kappa])-continuous map f : X [right arrow] X there exists x [member of] X such that f (x) = x. The fixed point property is preserved by any digital isomorphism, i.e., it is a topological invariant. The Lefschetz fixed point theorem determines when there exist fixed points of a map on a finite digital simplicial complex using a characteristic of the map known as the Lefschetz number. We can give a definition of trace for a digital map as in Algebraic Topology.

Definition 3.1. [9]. Given some digital map f : X [right arrow] X where (X, [kappa]) is a digital image, the trace of this map is defined by considering the trace of matrix representation of f, that is, choosing a basis for X and describing f as a matrix to this basis, and taking the trace of this square matrix.

Theorem 3.2. [9]. Let (X, [kappa]) and (Y, [kappa']) be digital images. If f : X [right arrow] Y and g : Y [right arrow] X are digital maps, then tr(g o f) = tr(f o g).

Proof. The proof is the same as in Algebraic Topology.

Definition 3.3. [10]. For a map f : (X,[kappa]) [right arrow] (X, [kappa]), where (X, [kappa]) is a digital image whose digital homology groups are finitely generated and vanish above some dimension, the Lefschetz number [lambda](f) is defined as follows:

[lambda](f) [n.summation over (i=0)][(-1).sup.i] tr([f.sub.*]),

where [f.sub.*] : [H.sup.[kappa].sub.i](X) [right arrow] [H.sup.[kappa].sub.i] (X) and tr denotes the trace.

Theorem 3.4. [10]. If (X, [kappa]) is a finite digital simplicial complex, or the retract of some finite digital simplicial complex, and f : (X, [kappa]) [right arrow] (X, [kappa]) is a map with [lambda](f) [not equal to] 0, then f has a fixed point.

Theorem 3.5. Let (X, [kappa]) be a digital image. If f : (X, [kappa]) [right arrow] (X, [kappa]) has [lambda](f) [not equal to] 0, then any map homotopic to f has a fixed point.

Proof. Assume that g : (X, [kappa]) [right arrow] (X, [kappa]) is ([kappa], [kappa])-homotopic to f. Then by homotopy axiom, we have

[f.sub.*] = [g.sub.*] : [H.sup.[kappa].sub.*](X) [right arrow] [H.sup.[kappa].sub.0](X)

and hence

[lambda](f) = [n.summation over (q=0)][(-1).sup.q]tr([f.sub.*]) = [n.summation over (q=0)][(-1).sup.q]tr([g.sub.*]) = [lambda](g).

So g has a fixed point.

Boxer et al. [6] define the Euler characteristic of digital images. Let (X, [kappa]) be a digital image of dimension m, and for each q [greater than or equal to] 0, let [[alpha].sub.q] be the number of digital ([kappa], q)-simplexes in X. The Euler characteristic of X is defined by

[chi](X [kappa]) = [n.summation over (q=0)][(-1).sup.q][[alpha].sub.q].

They also prove that if (X, [kappa]) is a digital image of dimension m, then

[chi square](X [kappa]) = [m.summation over (q=0)] [(-1).sup.q] rank [H.sup.[kappa].sub.q] (x).

Proposition 3.6. [10]. Let (X, [kappa]) be a digital image. If a map f : (X, [kappa]) [right arrow] (X, [kappa]) is homotopic to the identity, then [lambda](f) = [chi](X, [kappa]).

Theorem 3.7. Let (X, [kappa]) be a digital image. If [chi](X) [not equal to] 0, then any map homotopic to identity has a fixed point.

Proof. From Proposition 3.6, if [chi](X) = 0, we have [lambda]([1.sub.x]) [not equal to] 0. Let g be ([kappa], [kappa])-homotopic to [1.sub.X]. By Theorem 3.5, we have [lambda](g) [not equal to] 0 and hence g has a fixed point.

Theorem 3.8. Let (X, [[kappa].sub.1]) and (Y, [[kappa].sub.2]) be two digital images. If f : X [right arrow] Y and g : Y [right arrow] X are digital maps, then

[lambda](g [omicron] f) = [lambda](f [omicron] g).

Proof. By Theorem 3.2, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Theorem 3.9. Let (X, [[kappa].sub.1]) and (Y, [k.sub.2]) be two digital images. If f : X [right arrow] X is a map and h : X [right arrow] Y is a digital homotopy equivalence with a homotopy inverse k : Y [right arrow] X, then

[lambda](f) = [lambda](h [omicron] f [omicron] [kappa]).

Proof. From the digital homotopy equivalence of h and k, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Using Theorem 3.8, we conclude that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

As a result, we have [lambda](f) = [lambda](h [omicron] f [omicron] k).

The boundary Bd([I.sup.n+1]) of an (n + 1)-cube [I.sup.n+1] is homeomorphic to n-sphere [S.sup.n]. This allows us to represent a digital sphere by using the boundary of a digital cube. We use 0n to denote the origin of [Z.sup.n]. Boxer [6] defines sphere-like digital image as follows:

[S.sub.n] = [[-1,1].sup.n+1.sub.Z] \ {[0.sub.n+1]} [subset] [Z.sup.n+1].

For instance,

[s.sub.1] = {[c.sub.0] = (1,0), [c.sub.1] = (1,1), [c.sub.2] = (0,1), [c.sub.3] = (-1,1), [c.sub.4] = (-1,0), [c.sub.5] = (-1,-1), [c.sub.6] = (0,-1), [c.sub.7] =(1,-1)}

is digital 1-sphere with 4-adjacency in [Z.sup.2] (see Figure 4).

The following result is given in [1].

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

[S.sub.2], called [MSS.sub.6], is a digital image with 6-adjacency in [Z.sup.3] where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

[c.sub.24] = (0,1,2), [c.sub.25] = (1,1,2)}

(see Figure 5). Demir and Karaca [8] show that the digital homology groups of [S.sub.2] are as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Definition 3.10. Let f : ([S.sub.n], [[kappa].sub.n]) [right arrow] ([S.sub.n], [[kappa].sub.n]) be a ([[kappa].sub.n], [[kappa].sub.n])-continuous map where ([S.sub.n], [[kappa].sub.n]) is digital n-sphere, n [member of] {1,2} and [[kappa].sub.1] = 4, [[kappa].sub.2] = 6. Then f induces homomorphisms

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Definition of degree of f can be given similarly in algebraic topology (see [15]). We see that [f.sub.*] must be of the form

[f.sub.*] ([x]) = m[x],

for some fixed m [member of] Z, where [x] is a generator of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] This m is the called the degree of f.

Theorem 3.11. Let [S.sub.n] be a digital n-sphere where n [member of] {1,2}. If f : [S.sub.n] [right arrow] [S.sub.n] is a map of degree m [not equal to] 1, then f has fixed point.

Proof. Let f be a map of degree m [not equal to] 1.

[f.sub.*] : [H.sup.[kappa].sub.*] ([S.sub.n]) [right arrow] [H.sup.[kappa].sub.*] ([S.sub.n])

The trace of [f.sub.*] must be m. Since the trace of [f.sub.0]: [H.sup.[kappa].sub.0]([S.sub.n]) [right arrow] [H.sup.[kappa].sub.n]([S.sub.n]) is 1, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since m [not equal to] 1, we get [lambda](f) [not equal to] 0. By Theorem 3.4, f has a fixed point theorem.

Proposition 3.12. Let (X, [kappa]) be digital image and f : (X, [kappa]) [right arrow] (X, [kappa]) be a map. If

[f.sub.*] : [H.sup.[kappa].sub.*](X) [right arrow] [H.sup.[kappa].sub.*](X)

z [??] [f.sub.*] (z) = kz

where k is an integer, then [lambda](f) = k[chi](X).

Proof. It is easy to see that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

As a result, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Definition 3.13. If x = ([x.sub.1], ..., [x.sub.n+1]) [member of] [S.sub.n], its antipode is - x = (-[x.sub.1], ... , -[x.sub.n+1]).

The antipodal map [alpha] = [[alpha].sup.n] : [S.sub.n] [right arrow] [S.sub.n] is defined by [alpha](x) = -x.

Theorem 3.14. Let [[alpha].sub.i]: ([S.sub.i], [[kappa].sub.i]) [right arrow] ([S.sub.i], [[kappa].sub.i]) be the antipodal map between two digital i-spheres where i = {1,2} and [[kappa].sub.i] = {4,6}. Then a has degree [(- 1).sup.i+1].

Proof. By definition, we have deg ([[alpha].sub.i]) = d, where [([[alpha].sub.i]).sub.*] : [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is multiplication by d. Thus we get

tr[([[alpha].sub.i]).sub.*] = d = deg([[alpha].sub.i]),

so that

[lambda]([[alpha].sub.i]) = 1 + [(-1).sup.i]d.

but from Theorem 3.4, we have [lambda]([[alpha].sub.i]) = 0 because the antipodal map has no fixed points. As a result, d = [(-1).sup.i+1].

Theorem 3.15. Let h : ([S.sub.1], 4) [right arrow] ([S.sub.1], 4) be a digital (4,4)-continuous map. If h is digitally nullhomotopic, then h has a fixed point.

Proof. Since h : ([S.sub.1], 4) [right arrow] ([S.sub.1], 4) is digitally nullhomotopic, h [[congruent to].sub.(4,4)] c where c is a constant map on ([S.sub.1], 4). By Theorem 3.5, we have [h.sub.*] = [c.sub.*]. Since the Lefschetz number of c is

[lambda](c) = [[infinity].summation over (i=0)] [(-1).sup.i]tr([c.sub.*]) = 1,

we conclude that h has a fixed point.

Corollary 3.16. If h : ([S.sub.1], 4) [right arrow] ([S.sub.1], 4) is defined by h([c.sub.i]) = [c.sub.i+1] where [c.sub.i] [member of] [S.sub.1], then h is not nullhomotopic.

Proof. It's clear that h has not a fixed point. So from Theorem 3.15, we conclude that h is not nullhomotopic.

Corollary 3.17. There is at least one digital (4,4}-continuous map

h: ([S.sub.1], 4) [right arrow] ([S.sub.1], 4)

[c.sub.i] [??] h([c.sub.i]) [c.sub.i+4 (mod 8)]

for all [c.sub.i] [member of] [S.sub.1] which maps some point x [member of] [S.sub.1] to its antipode -x.

Lemma 3.18. If A [subset] [S.sub.1] is digital homeomorphic to [I.sup.k] for k [member of] {0,1} where I = [[0,1].sub.Z], then [H.sup.4.sub.q]([S.sub.1] - A) = 0 for all q [greater than or equal to] 0.

Proof. Since [S.sub.1]--A is digital 4-contractible image, digital homology groups of [S.sub.1]--A and one-point digital image are digital isomorphic. Thus we have

[H.sup.4.sub.q] ([S.sub.1] - A) = 0

for all q [greater than or equal to] 0.

Ege et al. in [12] give definition and properties of relative homology groups in digital images. We now would like to discuss the relative Lefschetz number for digital images.

Definition 3.19. Let f : (X, A) [right arrow] (X, A) be a digital map, where (A, [kappa]) is a digital subcomplex of (X, [kappa]). The map f induces homomorphisms

[f.sub.*] : [H.sup.[kappa].sub.q] (X, A) [right arrow] [H.sup.[kappa].sub.q](X, A).

The relative Lefschetz number [lambda](f; X, A) is defined by

[lambda](f; X, A) = [n.summation over (q=0)][(-1).sup.q] tr([f.sub.*]),

where [f.sub.*]:[H.sup.[kappa].sub.q] (X, A) [right arrow] [H.sup.[kappa].sub.q] (X, A).

Example 3.20. We compute the relative Lefschetz number of ([MSS.sub.6], A) where

A = {[c.sup.0], [c.sub.1], [c.sub.2], [c.sub.3]} [subset] [MSS'.sub.6]

has 6-adjacency. Ege et al. [12] show that for all q [greater than or equal to] 0, [H.sup.4.sub.q]([MSS'.sub.6], A) = 0. Let f : ([MSS'.sub.6], A) [right arrow] ([MSS'.sub.6], A) be a digital map. The relative Lefschetz number of f is given by

[lambda](f; [MSS'.sub.6], A) = [n.summation over (q=0)] [(-1).sup.q] tr([f.sub.*]) = 0,

where [f.sub.*] : [H.sup.6.sub.q] ([MSS'.sub.6], A) [right arrow] ([MSS'.sub.6], A).

We now would like to deal with a property on the relative Lefschetz number, which is satisfied in algebraic topology but is not satisfied in digital images. The property is given as follows. Let (X, [kappa]) be a digital complex, A and B be digital subcomplexes of X. Then

[lambda](f) = [lambda]([f.sub.A]) + [lambda] ([f.sub.B]) - [lambda]([f.sub.A[intersection]B]),

where f : X [right arrow] X, [f.sub.A] : A [right arrow] A, [f.sub.B] : B [right arrow] B and [f.sub.A [intersection] B] : A [intersection] B [right arrow] A [intersection] B. The above property is not satisfied in digital images. Let's show it by an example.

Example 3.21. Let X = [MSS.sub.6], A = {[c.sub.0], [c.sub.1], [c.sub.2], [c.sub.3]}, B = {[c.sub.4], [c.sub.5], [c.sub.6], [c.sub.7]} and A [intersection] B = [empty set]. Moreover, we take maps

f : X [right arrow] X, [f.sub.A] : A [right arrow] A, [f.sub.B] : B [right arrow] B and [f.sub.A[intersection]B] : A [intersection] B [right arrow] A [intersection] B.

Since A and B are digital simple closed 4-curves [1],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Therefore, we have the following results.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

From the following inequality

-4 [not equal to] 0 + 0 - 0,

we conclude that the equality

[lambda](f) = [lambda]([f.sub.A]) + [lambda]([f.sub.B]) - [lambda]([f.sub.A[intersection]B])

does not yield for digital images.

Theorem 3.5 can be given for relative digital images as follows.

Theorem 3.22. Let (X, [kappa]) be a digital complex and A [subset] X be a digital subcomplex. If f : (X, A, [kappa]) [right arrow] (X, A, [kappa]) has A(f; X, A) [not equal to] 0, then any map homotopic to f has a fixed point.

Proof. Let g : (X, A, [kappa]) [right arrow] (X, A, [kappa]) be ([kappa], [kappa])-homotopic to f. From homotopy axiom, we have

[f.sub.*] = [g.sub.*] : [H.sup.[kappa].sub.*](X, A)[right arrow][H.sup.[kappa].sub.*] (X, A).

We conclude that

[lambda](f; X, A) = [n.summation over (q=0)] [(-1).sup.q]tr([f.sub.*]) = [n.summation over (q=0)] [(-1).sup.q]tr([g.sub.*]) = [lambda](g; X, A).

As a result, g has a fixed point.

Definition 3.23. Let (X, [kappa]) be a digital image and f : (X, [kappa]) [right arrow] (X, [kappa]) be a digital map. The reduced Lefschetz number [??](f) is defined by

[??](f) = [lambda](f) - 1,

where [lambda](f) is the Lefschetz number of f.

Example 3.24. Let's compute the reduced Lefschetz number of

f : [MSS'.sub.8] [right arrow] [MSS'.sub.18],

where [MSS.sub.18] and its digital homology groups are given in Example 2.11. From Theorem 2.17, we know that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. So the reduced homology groups of [MSS'.sub.18] are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The Lefschetz number of f is

[lambda](f) = [n.summation over (q=0)] [(-1).sup.q] tr([f.sub.q]) = 1 - 0 + 1 - 0 + ... = 2,

where [f.sub.q] : [H.sup.18.sub.q]([MSS'.sub.18]) [right arrow] [H.sup.18.sub.q]([MSS'.sub.18]). As a result, the reduced Lefschetz number of f is

[??] (f)= [lambda](f) - 1 = 2 - 1 = 1.

4 Conclusion

The essential aim of this study is to determine some important applications of fixed point theory for a digital image. We study the relation between the Euler characteristic and the Lefschetz number. We deal with some characterizations about the Lefschetz number. We give also calculations about theorems and properties. Relative and reduced Lefschetz number is defined for digital images. We expect that properties in the paper will be useful for fixed point theory.

References

[1] H. Arslan, I. Karaca and A. Oztel, "Homology groups of n-dimensional digital images", XXI. Turkish National Mathematics Symposium, B1-13, 2008.

[2] L. Boxer, "Digitally continuous functions", Pattern Recognition Letters 15, 833-839, 1994.

[3] L. Boxer, "A classical construction for the digital fundamental group", Journal of Mathematical Imaging and Vision 10, 51-62, 1999.

[4] L. Boxer, "Properties of digital homotopy", Journal of Mathematical Imaging and Vision 22, 19-26, 2005.

[5] L. Boxer, "Homotopy properties of sphere-like digital images", Journal of Mathematical Imaging and Vision 24, 167-175, 2006.

[6] L. Boxer, Digital products, wedges and covering spaces, Journal of Mathematical Imaging and Vision 25, 169-171, 2006.

[7] L. Boxer, I. Karaca and A. Oztel, "Topological invariants in digital images", Journal of Mathematical Sciences: Advances and Applications 11(2), 109-140, 2011.

[8] E. Demir and I. Karaca, Simplicial Homology Groups of Certain Digital Surfaces, Hacettepe Journal of Mathematics and Statistics, In press, 2014.

[9] A. Granas and J. Dugundji, "Fixed Point Theory", Springer, 2003.

[10] O. Ege and I. Karaca, "The Lefschetz Fixed Point Theorem for Digital Images", Fixed Point Theory and Applications, doi:10.1186/10.1186/16871812-2013-253,2013.

[11] O. Ege and I. Karaca, "Fundamental properties of simplicial homology groups for digital images", American Journal of Computer Technology and Application 1 No.2, 25-42, 2013.

[12] O. Ege, I. Karaca and M.E. Ege, "Relative Homology Groups of Digital Images", Applied Mathematics and Information Science, 8(5), 2337-2345, 2014.

[13] G.T. Herman, "Oriented surfaces in digital spaces", CVGIP: Graphical Models and Image Processing 55,381-396, 1993.

[14] I. Karaca and O. Ege, Some results on simplicial homology groups of 2D digital images, International Journal of Information and Computer Science 1 no.8, 198-203, 2012.

[15] E. Spanier, Algebraic Topology, McGraw-Hill, New York, 1966.

Ozgur Ege *, Ismet Karaca

Department of Mathematics, Celal Bayar University, Muradiye, Manisa, 45140, Turkey

email:ozgur.ege@cbu.edu.tr

url: http://www.bayar.edu.tr/fef/matematik/ozgecmis/oege.pdf

Tel.:+90 236 2013228

Departments of Mathematics, Ege University, Bornova, Izmir, 35100, Turkey

email:ismet.karaca@ege.edu.tr url:fen.ege.edu.tr\~ismetkaraca

* Corresponding Author.

Received by the editors in February 2014.

Communicated by Y. Felix.

2010 Mathematics Subject Classification : 55N35,55M20, 68R10, 68U05,68U10.
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Author:Ege, Ozgur; Karaca, Ismet
Publication:Bulletin of the Belgian Mathematical Society - Simon Stevin
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Date:Dec 1, 2014
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