# Homotopy Extension Property in Homotopy Theory for Topological Semigroups

Authors: Adem Kılıçman; Amin Saif
Journal: ISRN Geometry
 (2012)

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## Abstract

The purpose of this paper is to extend the concept of homotopy extension property in homotopy
theory for topological spaces to its analogical structure in homotopy theory for topological
semigroups. In this extension, we also give some results concerning on absolutely retract and its
properties.


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## Body

## 1. Introduction

The definition of homotopy theory for topological semigroups and most of the backgrounds for this paper have been worked out previously by Čerin in [1]. He introduced the concepts of S-homotopy relation, pathwise Sℵ-connectedness, S-homotopy domination, S-contractibility, and Sℵ-fibration.A topological semigroup(A,∘) is called an S-subspace of a topological semigroup (S,*) if A is a subspace of S, the map * takes the product A×A into A and *(x,y)=∘(x,y) for all x,y∈A. The pair (SI,p(*)) is a topological semigroup with the compact-open topology on a path space SI, where the multiplication p(*):SI×SI→SI on Pa(S) defined by(1.1)[p(*)(α,β)](t)=*(α(t),β(t)),forα,β∈SI,t∈I.P(S,*) will denote a topological semigroup (SI,p(*)). For s∈S, s̃ will denote the constant path into s in (S,*). The function f:(S1,*)→(S2,∘) is called anS-map of (S1,*) into (S2,∘) if f is a continuous map of space S1 into S2 such that(1.2)f(*(x,y))=∘(f(x),f(y)),forx,y∈S1.TwoS-maps f,g:(S,*)→(O,∘) are called S-homotopic (written f≃sg) if there is an S-map H:(S,*)→P(O,∘) (called S-homotopy) such that H(s)(0)=f(s) and H(s)(1)=g(s) for all s∈S, for more details see [1–3].Recall [1] that a surjective S-map f:(S,*)→(O,∘) is called Sℵ-fibration if it has an S-homotopy lifting property for any element in ℵ, where ℵ is the collection of all topological semigroup. Now we introduce the concept of an S-lifting function of an Sℵ-fibration as follows: let f:(S,*)→(O,∘) be an S-map. Then the function Lf:(Δf,*×p(∘))→P(S,*) is called an S-lifting function for f if it is satisfying the following:(1)
Lf is an S-map,(2)
Lf(s,α)(0)=s  for (s,α)∈Δf,(3)
f[Lf(s,α)]=α for (s,α)∈Δf,whereΔf={(s,α)∈S×Pa(O):f(s)=α(0)}. If Lf(s,f∘s̃)=s̃ for all s∈S, then the S-lifting function is called an Sℵ-regular lifting function. Also we say that the S-map f:(S,*)→(O,∘) is an Sℵ-regular fibration if it has Sℵ-regular lifting function.Note that we can easily prove thatS-map f is an Sℵ-regular fibration if and only if it has S-regular lifting function. Further, if we let f:(S,*)→(O,∘) be an Sℵ-fibration and (E,∘) be an S-subspace of (O,∘). Then S-map f|f-1(E):(f-1(E),*)→(E,∘) is an Sℵ-fibration, where f|E will be the Sℵ-fibration f|f-1(E). Thus, one by defining two Sℵ-fibrations f1:(S1,*1)→(O,∘) and f2:(S2,*2)→(O,∘), an S-map h:(S1,*1)→(S2,*2) and setting the relation f2∘h=f1, then we can also study two S-fiber maps h:(S1,*1)→(S2,*2) and g:(S1,*1)→(S2,*2) whether they are S-fiber homotopic.Theorem 1.1 (See [4]).
LetX,  Y, and  Z be topological spaces. Then the map  F:X×Y→Z always gives rise to a map  H:X→C(Y,Z) by defining  H(x)(y)=F(x,y) for all  x∈X,y∈Y. If  Y is locally compact and regular space, then the map  H:X→C(Y,Z) always gives rise to map  F:X×Y→Z by defining  F(x,y)=H(x)(y) for all  x∈X,y∈Y.By thenatural topological semigroup, we mean a topological semigroup (S,πi), where πi is continuous associative multiplication on S defined by π1(x,y)=x and π2(x,y)=y for all x,y∈S.


## 2.S-Homotopy Extension Property

In this section, we extend the notions of an absolute retract (AR), an absolute neighborhood retract (ANR) and homotopy extension property in homotopy theory for topological spaces into their analogical structure in homotopy theory for topological semigroups. Also we show the relations among them.Definition 2.1 (See [1]).
Let(O,*) be an S-subspace of topological semigroup (S,*). The S-retraction of (S,*) onto (O,*) is an S-map r:(S,*)→(O,*) such that r(s)=s for all s∈O. When there is an S-retraction of (S,*) onto (O,*), then we say that (O,*) is an S-retract of (S,*).Definition 2.2.
A topological semigroup(S,*) is called an Sℵ-absolute retract (Sℵ-AR) for normal topological semigroups (X,•) in the class ℵ if for every closed S-subspace (A,•) of (X,•), any S-map f:(A,•)→(S,*) has an extension S-map F:(X,•)→(S,*).Definition 2.3.
A topological semigroup(S,*) is called an Sℵ-absolute neighborhood retract (Sℵ-ANR) for normal topological semigroups (X,•) in the class ℵ if for every closed S-subspace (A,•) of (X,•), any S-map f:(A,•)→(S,*) can be extended to an open neighborhood S-subspace of (A,•).By using the above Definitions2.2 and 2.3, we can easily state the following theorem.Theorem 2.4.
If(S,*) is an  Sℵ-AR (resp.  Sℵ-ANR), then  S is an AR (resp. ANR).The converse of Theorem2.4 above needs not to be true for example: If · denotes the usual multiplication on the the unit closed interval I. J denotes the first half [0,1/2] of I. It is clear that the topological semigroup (J,·) is a closed S-subspace of a normal topological semigroup (I,·). Also it is clear that J is an ANR. Take A:=J and X:=I in Definition 2.3, let the identity S-map i:(J,·)→(J,·) has an extension S-map R:(M,·)→(J,·), where (M,·) is an open neighborhood S-subspace of (J,·) in (I,·). In the relative usual topology on I, we can get an element x∈M and x∉J, that is, 1/2<x≤1. Since x/2∈J and an S-map R would have to satisfy an impossible condition:(2.1)x2=r(x2)=R(x⋅12)=R(x)⋅R(12)=12⋅R(x)that impliesR(x)=x∉J. Hence (J,·) is not Sℵ-ANR. Similarly, J is an AR that but it is not Sℵ-AR.The converses of Theorem2.4 are true for the collection of natural topological semigroups 𝒩i as shown in the following theorem.Theorem 2.5.
A topological semigroup(S,πi) is an  S𝒩i-AR (resp. an  S𝒩i-ANR) for normal topological semigroups  (X,πi) if and only if the topological space  S is AR (resp. ANR).Proof.
It is clear that the functionf:(X,πi)→(S,πi) is an S-map if and only if the function f:X→S is a continuous. Hence a topological semigroup (S,πi) is an S𝒩i-AR (resp. an S𝒩i-ANR) for normal topological semigroups (X,πi) if and only if the topological space S is AR (resp. ANR).Theorem 2.6.
If(S,*)is anSℵ-AR,  thenP(S,*)is also anSℵ-AR.Proof.
Let(A,•) be a closed S-subspace of a normal topological semigroup (X,•) and let H:(A,•)→P(S,*) be an S-map. Since I is a locally compact and regular, then by Theorem 1.1, for t∈I, we can define an S-map ft:(A,•)→(S,*) by
(2.2)ft(a)=H(a)(t),fora∈A.
Since (S,*) is an Sℵ-AR, then, for t∈I, the S-map ft has an extension S-map Ft:(X,•)→(S,*). Hence H has an extension S-map F:(X,•)→P(S,*) defined by
(2.3)F(x)(t)=Ft(x),forx∈X,t∈I.
Hence, P(S,*) is an Sℵ-AR.Now in the following definitions, we give the concept of anS-homotopy extension property in homotopy theory for topological semigroups.Definition 2.7.
Let(S,*) be a topological semigroup and (A,*) be an S-subspace of (S,*). We will mean by (GA1,GS2,O)-maps for(A,*)in(S,*)with respect to a topological semigroup(O,∘) the two S-maps
(2.4)GA1:(A,*)⟶P(O,∘),GS2:(S,*)⟶(O,∘)
such that
(2.5)GA1(a)(0)=GS2(a)fora∈A.Definition 2.8.
Let(S,*) and (O,∘) be two topological semigroups. A closed S-subspace (A,*) of (S,*) is said to have S-homotopy extension property in(S,*)with respect to(O,∘) if any (GA1,GS2,O)-maps can be extended to an S-homotopy F:(S,*)→P(O,∘). That is, (2.6)F(s)(0)=GS2(s),F(a)(t)=GA1(a)(t),fora∈A,s∈S,t∈I.Theorem 2.9.
If a closedS-subspace (A,*) of (S,*) has S-homotopy extension property in (S,*) with respect to (O,∘), then A has homotopy extension property in a space S with respect to a space O.Proof.
LetG:(S×{0})∪(A×I)→O be any continuous map. Define (GA1,GS2,O)-maps for (A,π1) in (S,π1) with respect to a topological semigroup (O,π1) by GS2(s)=G(s,0) and GA1(a)(t)=G(a,t) for all t∈I, sx∈S, a∈A. Then by the hypothesis (GA1,GS2,O)-maps can be extended to an S-homotopy F:(S,π1)→P(O,π1). That is,
(2.7)F(s)(0)=GS2(s),F(a)(t)=GA1(a)(t),fora∈A,s∈S,t∈I.
Since I is a locally compact and regular space, then by Theorem 1.1 the function H:S×I→O which is defined by H(s,t)=F(s)(t) for all t∈I,s∈S is continuous. Hence H is an extension of G. That is, A has homotopy extension property in a space S with respect to a space O.In the following theorem, we clarify that any closedS-subspace (A,*) of a normal topological semigroup (S,*) has S-homotopy extension property in (S,*) with respect to any Sℵ-AR space (O,∘).Theorem 2.10.
Let(O,∘) be an Sℵ-AR and (A,*) be a closed S-subspace of a normal topological semigroup (S,*). Then (A,*) has S-homotopy extension property in (S,*) with respect to (O,∘).Proof.
Let that there is(GA1,GS2,O)-maps. Since I is a locally compact and regular, then by Theorem 1.1, for t∈I-{0}, define an S-map ft:(A,*)→(O,∘) by
(2.8)ft(a)=GA1(a)(t),fora∈A.
Since (O,∘) is an Sℵ-AR, then for t∈I-{0}, the S-map ft has an extension S-map Ft:(S,*)→(O,∘). While at the case t=0, we can take F0=GS2. Hence there is an S-map F:(S,*)→P(O,∘) defined by
(2.9)F(s)(t)={GS2(s),fors∈S,t=0,Ft(s),fors∈S,t∈I-{0}.
Also we observe that
(2.10)F(a)(t)=Ft(a)=ft(a)=GA1(a)(t),fora∈A.
That is, (A,*) has S-homotopy extension property in (S,*) with respect to (O,∘).


## 3.S-Homotopy Extension Theorems

The general extension theorem for maps of any closed subsets of spaceX into space Y discusses under which conditions on X and Y, for every closed A⊂X, each map f:A→Y will be extendable over X relative to Y. The concepts of an AR, an ANR, and homotopy extension property support this theorem in homotopy theory for topological spaces. Now in this section, we introduce the notions of homotopy extension theorem and fiber homotopy extension theorem in homotopy theory for topological semigroups via Sℵ-AR property.Lemma 3.1.
Let(S,*) be a topological semigroup. For each path α in (S,*) and r∈I, let αr be a path in (S,*) defined by αr(t)=α[h(r,t)] for all t∈I, where h:I×I→I be any map. Then the function H:P(S,*)→P(Pa(S),p(*)) defined by
(3.1)H(α)(r)=αrforr∈I,α∈Pa(S)
is an S-map.Proof.
By Theorem1.1 to prove that H is continuous, it is sufficient to prove that the function F:Pa(S)×I→Pa(S) defined by F(α,r)=αr for all r∈I, α∈Pa(S) is continuous. Let (β,ro)∈Pa(S)×I and W(K,U) be a neighborhood of βro in Pa(S). Then β(h({ro}×K))=βro(K)⊆U. By the continuity of β and U is an open set containing β(h({ro}×K)), there is open set G in I such that h({ro}×K)⊆G and β(G)⊆U. Also by the continuity of h, there are two open sets Iro and IK in I such that
(3.2){ro}×K⊆Iro×IK,h(Iro×IK)⊆G.
Since Iro is an open set in I containing ro, then there is a positive number ϵ>0 such that
(3.3)ro∈(ro-ϵ3,ro+ϵ3)⊂D=[ro-ϵ2,ro+ϵ2]⊂(ro-ϵ,ro+ϵ)⊆Iro.Since D and K are compact sets in I and h is a continuous, then h(D×K) is also a compact set in I. Now consider that W(h(D×K),U)×(ro-ϵ/3,ro+ϵ/3) a neighborhood of (β,ro) in Pa(S)×I. Hence for
(3.4)(α,r)∈W(h(D×K),U)×(ro-ϵ3,ro+ϵ3),F(α,r)(K)=αr(K)=α(h({r}×K))⊆α[h(D×K)]⊆U.
That is, F(α,r)∈W(K,U). Hence the function F is continuous.
Now forα,β∈P(S,*),
(3.5){H[p(*)(α,β)](r)}(t)=[p(*)(α,β)]r(t)=[p(*)(α,β)](h(r,t))=*[α(h(r,t)),β(h(r,t))]=*[αr(t),βr(t)]=[p(*)(αr,βr)](t)={p(p(*))[H(α),H(β)](r)}(t).
Hence H[p(*)(α,β)]=p(p(*))[H(α),H(β)], that is, H is an S-map.The following theorem clarifies the notion ofS-fiber homotopy extension theorem in homotopy theory for topological semigroups via Sℵ-AR propertyTheorem 3.2.
Letf:(S,*)→(O,∘) be an Sℵ-regular fibration with an Sℵ-AR (S,*).  Let (A,•) be a closed S-subspace of a normal topological semigroup (X,•).  If there is (GA1,GX2,S)-maps such that
(3.6)f[GA1(a)(t)]=f[GA1(a)(0)]fora∈A,t∈I,
then there is an S-map H:(X,•)→P(S,*) such that H is an extension of GA1, H0=GX2 and
(3.7)f[H(x)(t)]=f[H(x)(0)]forx∈X,t∈I.Proof.
Since(S,*) is an Sℵ-AR and (A,•) is a closed S-subspace of a normal topological semigroup (X,•), then, by Theorem 2.10, the S-map GA1 can be extended to S-map G:(X,•)→P(S,*) such that G0=GX2. Now for α∈P(S,*) and r∈I, we can define the path αr in P(S,*) by αr(t)=α[(1-t)r] for all t∈I. Then by Lemma 3.1, the function
(3.8)P(S,*)⟶P(Pa(S),p(*)):(α)(r)⟶αr
is an S-map. Hence, we can define the S-map H:(X,•)→P(S,*) by
(3.9)H(x)(t)=Lf[G(x)(t),f∘G(x)t](1)forx∈X,t∈I.
Firstly, we show that H0=GX2. By the Sℵ-regularity of Lf and since G is an extension for GA1, we observe that for x∈X,
(3.10)H(x)(0)=Lf(G(x)(0),f∘G(x)0)(1)=Lf(G(x)(0),f∘G(x)(0))(1)=Lf(G(x)(0),f∘G(x)(0)̃)(1)=[G(x)(0)̃](1)=G(x)(0)=GX2(x).
Secondly, we show that H is an extension of GA1. Since G is an extension for GA1 and by the hypothesis, we get that
(3.11)(f∘G(a)r)(t)=f[G(a)((1-t)r)]=f[GA1(a)((1-t)r)]=f[GA1(a)(0)]=f[GA1(a)(r)]=f[G(a)(r)]=[f∘G(a)(r)̃](t),
for all a∈A and r,t∈I. Hence, by the Sℵ-regularity of Lf, we get that
(3.12)H(a)(r)=G(a)(r)=GA1(a)(r)forr∈I,a∈A.
Hence H is an extension for GA1. Finally, we also observe that
(3.13)f[H(x)(t)]=f[Lf(G(x)(t),f∘G(x)t)(1)]=(f∘G(x)t)(1)=f[G(x)t(1)]=f[G(x)(0)]=f[GX2(x)]=f[H(x)(0)],
for all x∈X, t∈I.Now we can give the other rephrasing of above theorem in the following corollary.Corollary 3.3.
Letf:(S,*)→(O,∘) be an Sℵ-regular fibration with an Sℵ-AR (S,*) and (A,•) be a closed S-subspace of a normal topological semigroup (X,•). Let k1,k2:(A,•)→(S,*) be two S-maps and R:(A,•)→P(S,*) be an S-homotopy between them such that
(3.14)f[R(a)(t)]=f[R(a)(0)]fora∈A,t∈I.
If k1 has an extension S-map K1 to all of (X,•), then k2 has an extension S-map K2 to all of (X,•). Also there is an S-homotopy H:(X,•)→P(S,*) between K1 and K2 such that H is an extension of R and
(3.15)f[H(x)(t)]=f[H(x)(0)]forx∈X,t∈I.Proof.
Sincek1 has an extension S-map K1 to all of (X,•), that is,
(3.16)K1(a)=k1(a)=R(a)(0)fora∈A.
Hence there is (R,K1,S)-maps with the property
(3.17)f[R(a)(t)]=f[R(a)(0)]fora∈A,t∈I.
Then by Theorem 3.2, there is an S-map H:(X,•)→P(S,*) such that H is an extension of R, H0=K1 and
(3.18)f[H(x)(t)]=f[H(x)(0)]forx∈X,t∈I.
Also we can define the S-map K2:(X,•)→(S,*) by K2(x)=H(x)(1) for all x∈X. Then H is an S-homotopy between K1 and K2 and also
(3.19)K2(a)=H(a)(1)=k2(a)fora∈A.
That is, K2 is an extension of k2.The following theorem clarifies the notion ofS-fiber homotopy extension theorem via Sℵ-AR property.Theorem 3.4.
Letf1:(S1,*1)→(O,∘) and f2:(S2,*2)→(O,∘) be two Sℵ-fibrations with an Sℵ-AR (S2,*2), a normal (S1,*1) and (E,∘) be a closed S-subspace of (O,∘). Let k,k′:(f1-1(E),*1)→(f2-1(E),*2) be two S-fiber maps between two Sℵ-fibrations f1|E and f2|E such that K-≈fK′. If k is extendable to a full S-fiber map K:(S1,*1)→(S2,*2), then k′ is extendable to a full S-fiber map K′:(S1,*1)→(S2,*2) and K-≈fK′.Proof.
SinceK-≈fK′, then there is an S-homotopy
(3.20)R:(f1-1(E),*1)⟶P(f2-1(E),*2)⊆P(S2,*2),
such that R(s)(0)=k(s), R(s)(1)=k′(s) and
(3.21)f2[R(s)(t)]=f1(s)=f2[k(s)]=f2[R(s)(0)]fors∈f1-1(E),t∈I.
Since k is extendable to a full S-fiber map K, then there is (K,R,S2)-maps. Since (f1-1(E),*1) is a closed S-subspace of a normal (S1,*1), we can apply Theorem 3.2 on Sℵ-fibration f2, where A=f1-1(E) and X=(S1,*1). Hence there is an S-map H:(S1,*1)→P(S2,*2) such that H is an extension of R, H0=K and
(3.22)f2[H(s)(t)]=f2[H(s)(0)]fors∈S1,t∈I.
Now we can define K′ by K′=H1. Hence H is an S-homotopy between K and K′. From (3.22) and since K is an S-fiber map, we get that K′ is an S-fiber map and
(3.23)f2[H(s)(t)]=f2[H(s)(0)]=f2[K(s)]=f1(s)fors∈S1,t∈I.
HenceK-≈fK′.



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