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R E S E A R C H

Open Access

Lefschetz fixed point theorem for digital

images

Ozgur Ege

1*

and Ismet Karaca

2

*Correspondence:

ozgur.ege@cbu.edu.tr

1Department of Mathematics, Celal

Bayar University, Muradiye, Manisa 45140, Turkey

Full list of author information is available at the end of the article

Abstract

In this article we study the fixed point properties of digital images. Moreover, we prove the Lefschetz fixed point theorem for a digital image. We then give some examples about the fixed point property. We conclude that sphere-like digital images have the fixed point property.

MSC: 55N35; 68R10; 68U05; 68U10

Keywords: digital image; Lefschetz fixed point theorem; Euler characteristic

1 Introduction

Digital topology is important for computer vision, image processing and computer graph-ics. In this area many researchers, such as Rosenfeld, Kong, Kopperman, Kovalevsky, Boxer, Karaca, Han and others, have characterized the properties of digital images with tools from topology, especially algebraic topology.

The homology theory is the major subject of algebraic topology. It is used in the classi-fication problems of topological spaces. The simplicial homology is one of the homology theories. Homology groups are topological invariants which are related to the differentn -dimensional holes, connected components of a geometric object. For example, the torus contains one -dimensional hole, two -dimensional holes and one -dimensional hole.

The Lefschetz fixed point theorem is a formula that counts fixed points of a continuous mapping from a compact topological spaceXto itself by means of traces of the induced mappings on the homology groups ofX. In , Lefschetz introduced the Lefschetz num-ber of a map and proved that if the numnum-ber is nonzero, then the map has a fixed point. Since the Lefschetz number is defined in terms of the homology homomorphism induced by the map, it is a homotopy invariant. As a result, a nonzero Lefschetz number implies that each of the maps in the given homotopy class has at least one fixed point. The Lef-schetz fixed point theorem generalizes a collection of fixed point theorems for different topological spaces. We will work entirely with digital images that are digital simplicial complexes or retracts of digital simplicial complexes.

Arslanet al.[] introduced the simplicial homology groups ofn-dimensional digital im-ages from algebraic topology. They also computed simplicial homology groups ofMSS.

Boxeret al.[] expanded the knowledge of simplicial homology groups of digital images. They studied the simplicial homology groups of certain minimal simple closed surfaces, extended an earlier definition of the Euler characteristics of a digital image, and com-puted the Euler characteristic of several digital surfaces. Demir and Karaca [] comcom-puted

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the simplicial homology groups of some digital surfaces such asMSSMSS,MSSand MSSMSS.

Karaca and Ege [] studied some results related to the simplicial homology groups of Ddigital images. They showed that if a bounded digital imageX⊂Zis nonempty and κ-connected, then its homology groups at the first dimension are a trivial group. They also proved that the homology groups of the operands of a wedge of digital images need not be additive. Ege and Karaca [] gave characteristic properties of the simplicial homol-ogy groups of digital images and then investigated the Eilenberg-Steenrod axioms for the simplicial homology groups of digital images.

This paper is organized as follows. The second section provides the general notions of digital images withκ-adjacency relations, digital homotopy groups and digital homology groups. In Section  we present the Lefschetz fixed point theorem for digital images and provide some of its important applications. In the last section we make some conclusions about this topic.

2 Preliminaries

LetZbe the set of integers.A (binary) digital imageis a pair (X,κ), whereX⊂Znfor

some positive integernandκrepresents certain adjacency relation for the members ofX. A variety of adjacency relations are used in the study of digital images. We give one of them. Letl,nbe positive integers, ≤lnand two distinct pointsp= (p,p, . . . ,pn), q= (q,q, . . . ,qn) inZn,pandqarekl-adjacent[] if there are at mostldistinct

coordi-natesj, for which|pjqj|=  and for all other coordinatesj,pj=qj. The number of points q∈Znthat are adjacent to a given pointp∈Znis represented by akl-adjacency relation. From this viewpoint, thek-adjacency onZis denoted by the number  andk-adjacent

points are called -adjacent. In a similar way, we call -adjacentand -adjacentforkand k-adjacent points ofZ; and inZ, -adjacent, -adjacentand -adjacentfork,kand k-adjacent points, respectively.

Aκ-neighborofp∈Znis a point ofZnwhich isκ-adjacent top, whereκis an adjacency

relation defined onZn. A digital imageXZnisκ-connected[] if and only if for every

pair of different pointsx,yX, there is a set{x,x, . . . ,xr}of points of a digital imageX

such thatx=x,y=xrandxiandxi+areκ-neighbors wherei= , , . . . ,r– .A maximal

κ-connectedsubset of a digital imageXis calledaκ-componentofX. A set of the form

[a,b]Z=z∈Z|azb

is said to bea digital interval[], wherea,b∈Zwitha<b.

Let (X,κ)⊂Zn and (Y,κ)⊂Zn be digital images. A functionf :X−→Y is called

(κ,κ)-continuous[] if for everyκ-connected subsetU ofX,f(U) is aκ-connected

subset ofY.

In a digital imageX, if there is a (,κ)-continuous functionf : [,m]Z−→Xsuch that f() =xandf(m) =y, then we say that there existsa digitalκ-path[] fromxtoy. If

f() =f(m), thenfis calleddigitalκ-loopand the pointf() is the base point of the loopf. When a digital loopf is a constant function, it is said to bea trivial loop.A simple closed

κ-curveofm≥ points in a digital imageXis a sequence{f(),f(), . . . ,f(m– )}of images of theκ-pathf : [,m– ]Z−→Xsuch thatf(i) andf(j) areκ-adjacent if and only ifj=

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Definition . Let (X,κ)⊂Znand (Y,κ)⊂Znbe digital images. A functionf :X−→ Y isa(κ,κ)-isomorphism[] iff is (κ,κ)-continuous and bijective andf–:Y−→Xis

(κ,κ)-continuous and it is denoted byX≈(κ,κ)Y.

For a digital image (X,κ) and its subset (A,κ), we call (X,A) a digital image pair with κ-adjacency. IfAis a singleton set{x}, then (X,x) is calleda pointed digital image.

Definition . [] Let (X,κ)⊂Zn and (Y,κ)⊂Zn be digital images. Two (κ,κ

)-continuous functionsf,g:X−→Yare said to be digitally (κ,κ)-homotopic inY if there

is a positive integermand a functionH:X×[,m]Z−→Ysuch that • for allxX,H(x, ) =f(x)andH(x,m) =g(x);

• for allxX, the induced functionHx: [,m]Z−→Ydefined by

Hx(t) =H(x,t) for allt∈[,m]Z

is(,κ)-continuous; and

• for allt∈[,m]Z, the induced functionHt:X−→Ydefined by

Ht(x) =H(x,t) for allxX

is(κ,κ)-continuous.

The functionHis called a digital (κ,κ)-homotopy betweenf andg. If these functions

are digitally (κ,κ)-homotopic, this is denoted byf κ,κg. The digital (κ,κ)-homotopy

relation [] is equivalence among digitally continuous functionsf : (X,κ)−→(Y,κ).

Definition .[] (i) A digital image (X,κ) is said to beκ-contractible if its identity map is (κ,κ)-homotopic to a constant functionc¯for somecX, where the constant function ¯

c:X−→Xis defined byc¯(x) =cfor allxX.

(ii) Let (X,A) be a digital image pair withκ-adjacency. Leti:A−→Xbe the inclusion function.Ais called aκ-retract ofXif and only if there is aκ-continuous functionr:X−→ Asuch thatr(a) =afor allaA. Then the functionris called aκ-retraction ofXontoA. (iii) A digital homotopyH:X×[,m]Z−→Xis a deformation retract [] if the induced

mapH(–, ) is the identity map Xand the induced mapH(–,m) is retraction ofXonto H(X× {m})⊂X. The setH(X× {m}) is called a deformation retract ofX.

Definition .[] LetSbe a set of nonempty subsets of a digital image (X,κ). Then the members ofSare called simplexes of (X,κ) if the following hold:

(a) Ifpandqare distinct points ofsS, thenpandqareκ-adjacent. (b) IfsSand∅ =ts, thentS.

Anm-simplex is a simplexSsuch that|S|=m+ .

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Definition . Let (X,κ) be a finite collection of digitalm-simplexes, ≤mdfor some nonnegative integerd. If the following statements hold, then (X,κ) is called [] a finite digital simplicial complex:

() IfPbelongs toX, then every face ofPalso belongs toX.

() IfP,QX, thenPQis either empty or a common face ofPandQ.

The dimension of a digital simplicial complexXis the largest integermsuch thatXhas anm-simplex.

q(X) is a free abelian group [] with basis of all digital (κ,q)-simplexes

inX.

Corollary .[] Let(X,κ)⊂Znbe a digital simplicial complex of dimension m.Then,

for all q>m,

q(X)is a trivial group.

Let (X,κ)⊂Znbe a digital simplicial complex of dimensionm. The homomorphism

∂q:Cqκ(X)−→Cqκ–(X) defined (see []) by

∂q

p,p, . . . ,pq

=

⎧ ⎨ ⎩

q

i=(–)ip,p, . . . ,piˆ, . . . ,pq, qm,

, q>m

is called a boundary homomorphism, wherepˆimeans delete the pointpi. In [], it is shown

that for all ≤qm,

∂q–◦∂q= .

Definition .[] Let (X,κ) be a digital simplicial complex.

()

q(X) =Ker∂qis called the group of digital simplicialq-cycles. ()

q(X) =Im∂q+is called the group of digital simplicialq-boundaries. ()

q(X) =Zqκ(X)/Bκq(X)is called theqth digital simplicial homology group.

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

Example . LetX={(, ), (, ), (, ), (, )} ⊂Zbe a digital image with -adjacency.

In [], it is shown that digital homology groups ofXare:

Hq(X) =

⎧ ⎨ ⎩

Z, q= , , q= .

Example . LetX={(, ), (, ), (, )} ⊂Zbe a digital image with -adjacency. Havana et al.[] showed that digital homology groups of (X, ) are

Hq(X) =

⎧ ⎨ ⎩

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Figure 1 MSC4.

Figure 2 MSS6.

Example . MSC={(, ), (, –), (, ), (, ), (, –), (–, ), (–, ), (–, –)}is a digital

image with -adjacency inZ(see Figure ). The following result is given in [].

Hq(MSC) =

⎧ ⎨ ⎩

Z, q= , , , q= , .

Example . MSSis a digital image with -adjacency inZ(see Figure ). In [] Demir

and Karaca showed that the digital homology groups ofMSSare as follows:

Hq(MSS) =

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

Z, q= , Z, q= ,

, q= , .

Example . Let MSS={c= (, , ),c = (, , ),c= (, , ),c = (, , ),c= (,

, ),c= (, , ),c= (, , ),c= (, , )}be a digital image with -adjacency (see

Fig-ure ). In [], it is shown that

HqMSS=

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

Z, q= , Z, q= ,

, q= , .

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Figure 3 MSS6.

()

q(X)is a finitely generated abelian group for everyq≥. ()

q(X)is a trivial group for allq>m. ()

m(X)is a free abelian group,possible zero.

In [], it is proven that for eachq≥,

q is a covariant functor from the category of

digital simplicial complexes and simplicial maps to the category of abelian groups.

Definition .[] Letf: (X,κ)−→(Y,κ) be a function between two digital images. If

for every digital (κ,m)-simplexPdetermined byκinX,f(P) is a (κ,n)-simplex inYfor

somenm, thenf is called a digital simplicial map.

For a digital simplicial map f : (X,κ)−→(Y,κ) andq≥, two induced

homomor-phismsf#:Cκq(X)−→Cqκ(Y) andf∗:Hqκ(X)−→Hqκ(Y) are defined by (see [])

f#

p, . . . ,pq

=f#(p), . . . ,f#(pq)

,

fz+Bκq(X)

=f#(z) +q(Y),

wherez

q(X), respectively.

Definition .[] LetXandYbe digital images inZn, using the same adjacency notion,

denoted byκ, such thatXY={x}, wherexis the only point ofXadjacent to any point

ofYandxis the only point ofYadjacent to any point ofX. Then the wedge ofXandY,

denoted byXY, is the imageXY=XY, withκ-adjacency.

3 The Lefschetz fixed point theorem for digital images

A digital image (X,κ) is said to have the fixed point property if for any (κ,κ)-continuous functionf : (X,κ)−→(X,κ), there existsxXsuch thatf(x) =x. The fixed point property is a topological invariant,i.e., is preserved by any digital isomorphism. The fixed point property is also preserved by any retraction. The Lefschetz fixed point theorem deter-mines 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.

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numberλ(f) is defined as follows:

λ(f) =

n

i=

(–)itr(f∗),

wheref∗:Hiκ(X)−→H

κ

i(X).

Theorem . Let G be a free abelian group,and letG:GG be the identity homomor-phism.Thentr(G) =rank(G),wheretr(G)is the trace ofG.

Proof SinceGis a free abelian group, the rank ofGis ;i.e., we haverank(G) = . Moreover, the trace of Gis  because G:G−→Gis the identity map and has an identity ×-matrix

[]. So, we haverank(G) =tr(G).

Theorem . If(X,κ)is a finite digital simplicial complex,or the retract of some finite digital simplicial complex,and f : (X,κ)−→(X,κ)is a map withλ(f)= ,then f has a fixed point.

Proof Suppose thatfhas no fixed points. Then we must show thatλ(f) = . The approach would be to show that alltr(f∗:Hiκ(X)−→Hiκ(X)) are zero. For this purpose, we need the

following which is called the Lefschetz principle:

λ(f) =

n

i=

(–)itrf#:Ciκ(X)−→Ciκ(X)

.

To compute the digital homology groups of (X,κ), we use digital simplicial chainsCiκ(X). Sincef has no fixed points and (X,κ) is compact, f must move points at least a fixed positive distance, sayδ. Then we use simplexes which have diameter less thanδ/, and we approximatef by a digital simplicial mapgso that pointwisegis withinδ/ off. We may assume thatδis so small thatgis homotopic tof, so we may as well assume thatg=f. Becausef moves points at leastδand our digital simplexes are at mostδ/ in diameter, it is impossible forf to map any digital simplex to itself, so it is immediate that for allk,

trf#:Ciκ(X)−→Ciκ(X)

= .

Theorem .(One-dimensional Brouwer fixed point theorem) Every(, )-continuous function f : [, ]Z−→[, ]Zhas a fixed point.

Proof Letf : [, ]Z−→[, ]Z be (, )-continuous. Sincef() =  andf() = ,f has a

fixed point.

Theorem .(Two-dimensional Brouwer fixed point theorem) Let X={(, ), (, ), (, ), (, )} ⊂Z be a digital image with -adjacency. Every (, )-continuous function f :

(X, )−→(X, )has a fixed point.

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identity homomorphism. By Theorem .,

tr(f∗) =rank(Z) = .

For allq> , it follows from Example . thatf∗is a zero map. Thus we have

λ(f) =

n

i=

(–)itr(f∗)

=  –  +  –  +· · ·

= .

Since λ(f) =  = , Theorem . implies that f has a fixed point. As a result, two-dimensional Brouwer fixed point theorem holds for digital images.

We conclude now by showing that the fixed point property is a topological property;

i.e., two topological spaces which are topologically equivalent to each other either both have or both lack the fixed point property. Because one of the main goals of topology is to discover what properties are maintained when moving between topologically equivalent spaces, this is an incredibly important proposition to make.

Theorem . Let(X,κ)and(Y,κ)be digital images such that X≈(κ,κ)Y.If(X,κ)has the fixed point property,then(Y,κ)has the fixed point property.

Proof SinceX≈(κ,κ)Y, there exists a bijective functionf : (X,κ)−→(Y,κ) such thatf is

(κ,κ)-continuous and that its inversef–is (κ,κ)-continuous. Also,Xhas the fixed point

property;i.e., every (κ,κ)-continuous functiong: (X,κ)−→(X,κ) has a fixed point. Now, leth: (Y,κ)−→(Y,κ) be (κ,κ)-continuous. Thenhf : (X,κ)−→(Y,κ) and, conse-quently,f–hf : (X,κ)−→(X,κ) are certainly (κ,κ)-continuous and (κ,κ)-continuous,

respectively. However, sinceXhas the fixed point property,f–(h(f(x))) =xfor somexX.

It follows thatf(f–(h(f(x)))) =f(x), which implies thath(f(x)) =f(x), proving thathhas a

fixed point, namelyf(x).

Corollary . The fixed point property is a topological invariant for digital images.

Theorem . Let(X,κ)be any digital image,and let f : (X,κ)−→(X,κ)be any(κ,κ) -continuous map.If X isκ-contractible,then f has a fixed point.

Proof IfXisκ-contractible, then we havef κ,κX. So,Xand a single point image have

the same digital homology groups. Hence we have

Hiκ(X) =

⎧ ⎨ ⎩

Z, i= , , i= .

Letf : (X,κ)−→(X,κ). If we take the induced homomorphismsf∗:

q(X)−→Hqκ(X), for q= , we have the identity homomorphismf∗:Z−→Z. By Theorem .,

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Moreover, for allq> , it follows from Example . thatf∗is a zero map. So, the Lefschetz

number off is

λ(f) =

i=

tr(fi) =  –  +  –· · ·= .

By Theorem ., we have thatf has a fixed point.

Example . IfXis the digital image in Example ., then for any (, )-continuous map

f : (X, )−→(X, ), the Lefschetz numberλ(f) is given by the trace of the induced map on homology

f∗:H(X) =Z−→H(X) =Z

which is simply the identity function and thus has a nonzero trace. From Theorem ., every map on a digital image (X, ) has at least one fixed point.

Corollary . Any digital image(X,κ)with the same digital homology groups as a single point image always has a fixed point.

Theorem . Let(X,κ)be a digital image and(A,κ)be a digitalκ-retract of(X,κ).If

(X,κ)has the fixed point property,then(A,κ)has the fixed point property.

Proof Let (A,κ) be a digital κ-retract of (X,κ). Then there exists a digitalκ-retraction maprsuch thatri= (A,κ), wherei: (A,κ)−→(X,κ) is an inclusion map. Since (X,κ) has

the fixed point property, every (κ,κ)-continuous functionf : (X,κ)−→(X,κ) has a fixed point, that is, there exists a pointxXsuch thatf(x) =x. We have

(A,κ)−→i (X,κ)−→f (X,κ)−→r (A,κ).

For allx∈(A,κ),

rfi(x) =rf(x) =r(x) =x.

Hence there is a pointx∈(A,κ) such thath(x) =x, whereh=rfi. As a result, (A,κ)

has the fixed point property.

Proposition . Let A,B be digital images withκ-adjacency.The AB has the fixed point property if and only if both A and B have the fixed point property.

Proof Letube the common base point. For sufficient condition, supposef :AB−→ AB, and let us assume thatf(u)∈Aandf(u)=u. Letg:A−→Abep, the projection of

ABontoA. Theng(a) =afor somea∈A. Iff(a)∈A, then

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On the other hand, iff(a)∈B, then

a=g(a) =pf(a) =u

contrary to the assumption thatf(u)∈AB. So,f has the fixed point. Necessary condition

follows from Theorem ..

The boundaryBd(In+) of an (n+ )-cubeIn+ is homeomorphic ton-sphereSn. This

allows us to represent a digital sphere by using the boundary of a digital cube. We use n

to denote the origin ofZn. Boxer [] defines sphere-like digital image as follows:

Sn= [–, ]nZ+\ {n+} ⊂Zn+.

Converse of the Lefschetz fixed point theorem need not be true for digital images. Let us see this.

Example . S = {(, ), (, –), (, ), (, ), (, –), (–, ), (–, ), (–, –)} is a digital

-sphere with -adjacency inZ. It is the digital -closed curveMSC

. By Example .,

we have

Hq(S) =

⎧ ⎨ ⎩

Z, q= , , , q= , .

Letφ: (S, )−→(S, ) be a digital (, )-continuous map. This map is defined asφ(z) =z,

wherezS. The Lefschetz number ofφis

λ(φ) =

q

i=

trφi:Hi(S)−→Hi(S)

=  –  = ,

sinceφ,φ:Z−→Zare the identity functions. Thusλ(φ) = , butφhas the fixed point.

As a consequence, we have the following corollary.

Corollary . Converse of the Lefschetz fixed point theorem need not be true for digital images.

Now we compute the Lefschetz number ofS. We define and sketch it.

Example . S= [–, ]Z\{(, , )}is a digital -sphere with -adjacency inZ. It is the

digital imageMSSin Example .. Letf: (S, )−→(S, ) be a digital (, )-continuous

map. Using Theorem ., the Lefschetz number off is

λ(f) =

q

i=

trf∗:Hi(S)−→Hi(S)

=  –  +  = –,

sincef:Z−→Zis the identity function andf:Z−→Z. As a result,f has a fixed

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In [], Boxer et al.defined the Euler characteristic of digital images. Let (X,κ) be a digital image of dimensionm, and for eachq≥, letαqbe the number of digital (κ,q

)-simplexes inX. The Euler characteristic ofX, denoted byχ(X,κ), is defined by

χ(X,κ) =

m

q=

(–)qαq.

Moreover, they proved that if (X,κ) is a digital image of dimensionm, then

χ(X,κ) =

m

q=

(–)qrank

q(X).

Proposition . Let(X,κ)be a digital image.If a map f : (X,κ)−→(X,κ)is homotopic to the identity,thenλ(f) =χ(X,κ).

Proof We could equally have defined the Lefschetz number without homology as follows:

λ(f) =

n

i=

(–)itrf∗:Hiκ(X)−→Hiκ(X)=

n

i=

(–)itr(f#),

wheref#:Ciκ(X)−→Ciκ(X). Since homotopic maps induce the same map onn-chains, we

can consider #:i(X)−→Ciκ(X). The map #is theαq(X,κ)×αq(X,κ) identity matrix

and thus has the traceαq(X,κ). As a result,

λ(f) =

n

i=

(–)itr(f#) = n

i=

(–)iαi(X,κ) =χ(X,κ).

Example . LetMSSin Example . be denoted by (X, ), and letf : (X, )−→(X, ) be any (, )-continuous map. Forq> ,fq:H

q(X)−→Hq(X) is a zero map, sotr(fq) = .

Forq= ,tr(f∗) =  becausefis the identity function onZ. On the other hand, forq= ,

we gettr(f∗) =rank(Z) =  in that ×-matrix off

:Z−→Zis

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣                          ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ . We have

λ(f) =

i=

(–)itr(fi) =  –  +  = –= .

By Theorem .,f has a fixed point. The Euler characteristic ofMSSisχ(MSS, ) = –.

We define a digital version of the real projective planeRPvia a quotient map fromS

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Lef-Figure 4 Digital projective planeP2.

schetz number ofSin Example .. Let

S=

c= (–, –, ),c= (, –, ),c= (, –, ),c= (, , ),c= (, , ),

c= (–, , ),c= (–, , ),c= (, , ),c= (, , ),c= (, , ),

c= (, , ),c= (–, , ),c= (–, , ),c= (, , ),c= (, –, ),

c= (, –, ),c= (–, –, ),c= (–, –, ),c= (, –, ),

c= (, –, ),c= (, , ),c= (, , ),c= (–, , ),c= (–, , ),

c= (, , ),c= (, , )

.

If we take the quotient mapq:S−→S/x∼x, where –xis the antipodal point ofxS,

then we have the digital projective planeP(see Figure ).

Define a mapH:P×[, ]Z−→Pby the following.

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

H(c, ) =c, t= ,

H(c, ) =c,H(c, ) =c,H(c, ) =c,H(c, ) =c, t= ,

H(c, ) =c,H(c, ) =c,H(c, ) =c,H(c, ) =c, t= ,

H(c, ) =c,H(c, ) =c, t= ,

H(c, ) =H(c, ) =c, t= ,

where cP. It is clear that this map is a -deformation retract of P. So, Pis

-contractible image. As a result,Phas the same digital homology groups as a single

point image. By Theorem ., we have the following.

Corollary . Every(, )-continuous map f :P−→Phas a fixed point.

4 Conclusion

The main goal of this study is to determine fixed point properties for a digital image. More-over, we study the relations between the Euler characteristic and the Lefschetz number. At the end of this work, we give a digital version of the real projective plane and calculate its homology groups and the Lefschetz number. We expect that these properties will be useful for image processing.

Competing interests

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Authors’ contributions

All the authors read and approved the final manuscript.

Author details

1Department of Mathematics, Celal Bayar University, Muradiye, Manisa 45140, Turkey.2Departments of Mathematics, Ege

University, Bornova, Izmir 35100, Turkey.

Acknowledgements

The authors thank the editor and the referees for their valuable suggestions to improve the quality of this paper.

Received: 5 July 2013 Accepted: 30 September 2013 Published:07 Nov 2013 References

1. Arslan, H, Karaca, I, Oztel, A: Homology groups ofn-dimensional digital images XXI. Turkish National Mathematics Symposium, B1-13 (2008)

2. Boxer, L, Karaca, I, Oztel, A: Topological invariants in digital images. J. Math. Sci. Adv. Appl.11(2), 109-140 (2011) 3. Demir, E, Karaca, I: Simplicial Homology Groups of Certain Digital Surfaces. Preprint (2012)

4. Karaca, I, Ege, O: Some results on simplicial homology groups of 2D digital images. Int. J. Inf. Comput. Sci.1(8), 198-203 (2012)

5. Ege, O, Karaca, I: Fundamental properties of simplicial homology groups for digital images. Am. J. Comput. Technol. Appl.1(2), 25-42 (2013)

6. Boxer, L: Homotopy properties of sphere-like digital images. J. Math. Imaging Vis.24, 167-175 (2006) 7. Herman, GT: Oriented surfaces in digital spaces. CVGIP, Graph. Models Image Process.55, 381-396 (1993) 8. Boxer, L: Digitally continuous functions. Pattern Recognit. Lett.15, 833-839 (1994)

9. Boxer, L: A classical construction for the digital fundamental group. J. Math. Imaging Vis.10, 51-62 (1999) 10. Boxer, L: Digital products, wedges and covering spaces. J. Math. Imaging Vis.25, 169-171 (2006) 11. Boxer, L: Properties of digital homotopy. J. Math. Imaging Vis.22, 19-26 (2005)

12. Spanier, E: Algebraic Topology. McGraw-Hill, New York (1966)

10.1186/1687-1812-2013-253

Figure

Figure 1 MSC4.
Figure 3 MSS′6
Figure 4 Digital projective plane P2.

References

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