On Generalized Covering Groups of Topological Groups
aa r X i v : . [ m a t h . A T ] A ug On Generalized Covering Groups of Topological Groups
Hamid Torabi ∗ , Mehdi Abdullahi Rashid, Majid Kowkabi Department of Pure Mathematics, Ferdowsi University of Mashhad,P.O.Box 1159-91775, Mashhad, Iran.
Abstract
It is well-known that a homomorphism p : e G → G between topological groups is acovering homomorphism if and only if p is an open epimorphism with discrete kernel.In this paper we generalize this fact, in precisely, we show that for a connected locallypath connected topological group G , a continuous map p : e G → G is a generalizedcovering if and only if e G is a topological group and p is an open epimorphism withprodiscrete (i.e, product of discrete groups) kernel. To do this we first show that if G is a topological group and H is any generalized covering subgroup of π ( G, e ), then H is as intersection of all covering subgroups, which contain H . Finally, we show thatevery generalized covering of a connected locally path connected topological groupis a fibration. Keywords:
Topological group, Fundamental group, Generalized Covering map.
1. Introduction
Chevalley [8] introduced a covering group theory for connected, locally path con-nected, and semi locally simply connected topological groups. Rotman [12, Theorem10.42] proved that for every covering space ( e X, p ) of a connected, locally path con-nected, and semi locally simply connected topological group G , e X is a topologicalgroup and p is a homomorphism. Recently, Torabi [15] developed this theory for con-nected locally path connected topological groups and gave a classification for coveringgroups of them. He showed that the natural structure, which has been called thepath space, and its relative endpoint projection map for a connected locally path ∗ Corresponding author
Email addresses: [email protected] (Hamid Torabi), [email protected] (Mehdi Abdullahi Rashid), [email protected] (Majid Kowkabi)
Preprint submitted to August 28, 2018 onnected topological group are a topological group and a group homomorphism,respectively. Recall that for an arbitrary subgroup H of the fundamental group π ( X, x ), the path space is the set of all paths starting at x , which is denoted by P ( X, x ), with an equivalence relation ∼ H as follows: α ∼ H α if and only if α (1) = α (1) and [ α ∗ α − ] ∈ H . The equivalence classof α is denoted by h α i H . One can consider the quotient space e X H = P ( X, x ) / ∼ H and the endpoint projection map p H : ( e X H , e H ) → ( X, x ) defined by h α i H α (1),where e H is the class of the constant path at x .If α ∈ P ( X, x ) and U is an open neighbourhood of α (1), then a continuationof α in U is a path β = α ∗ γ , where γ is a path in U with γ (0) = α (1). Put N ( h α i H , U ) = {h β i H ∈ e X H | β is a continuation of α in U } . It is well known thatthe subsets N ( h α i H , U ) form a basis for a topology on e X H for which the function p H : ( e X H , e H ) → ( X, x ) is continuous (see [13, Page 82]). Brodskiy et al. [7] calledthis topology on e X H the whisker topology.Moreover, they were interested in studying the spaces whose local properties canextend to the entire space and introduced the notions of SLT and strong
SLT spaces.
Definition 1.1. [7, Definition 4.18] A topological space X is called strong small looptransfer space (strong SLT space for short) at x if for each point x ∈ X and forevery neighbourhood U of x , there is a neighbourhood V of x such that for everypath α : I → X from x to x and every loop β in V based at x there is a loop γ in U based at x which is homotopic to α ∗ β ∗ α − relative ˙ I . The space X is calledstrong SLT space, if X is strong SLT space at x for every x ∈ X . On the other hand, Torabi showed that every topological group is a strong
SLT at the identity element [15, Theorem 2.4]. In Section 2, we introduce the generalizedcovering group of a topological group and give some examples to clarify the differencebetween covering and generalized covering groups (Examples 2.7 and 2.8). Moreover,we show that for a connected locally path connected strong
SLT at x space, everygeneralized covering subgroup H of the fundamental group can be written as theintersection of all covering subgroups, which contains H and vice versa.In Section 3, we attempt to provide a method for classifying generalized coveringgroups of a topological group by studying the topology of the kernel of the relativegeneralized covering homomorphism. Of cores, we extend the well-known resultabout covering groups (Remark 2.2) for generalized covering groups and show thatif G is a connected locally path connected topological group, ( e G, p ) is a generalizedcovering group of G if and only if p is an open epimorphism with prodiscrete (i.e,product of discrete groups) kernel (Corollary 3.10). In this regard, we first show2hat the central fibre of a generalized covering map of an arbitrary topological spaceis totally path disconnected ( Proposition 3.2). Counterexample 3.4 show that itis not a sufficient condition for generalized covering subgroups, even in the caseof topological groups. After that we present our desirable definition of prodiscretesubgroups (Definition 3.5) and show that for a connected locally path connectedtopological group G and a generalized covering subgroup H ≤ π ( G, e ), the kernel ofthe endpoint projection homomorphism is a prodiscrete subgroup ( Theorem 3.6).Berestovskii et al. in [4] provided a new definition by extending the concept ofa cover of a topological group G such as the pair ( e G, p ), where e G is a topologicalgroup and the homomorphism p : e G → G is an open epimorphism with prodiscretekernel. Note that the meaning of prodiscrete kernel in the sense of them was as theinverse limit of discrete groups. In this paper we use Definition 3.5 for a prodiscreteconcept and show that if H is a prodiscrete normal subgroup of topological group G ,then the pair ( G, ϕ H ) is a generalized covering group of GH (Theorem 3.7). Using thistheorem, we conclude the main result in Corollary 3.10. Finally, we show that in thecase of topological groups the concepts of rigid covering fibrations (which was firstlyintroduced by Biss [5]) and generalized covering groups are coincide and concludethat every generalized covering group of a topological group is also a fibration.
2. Generalized Coverings of Topological Groups
It is well-known that a continuous map p : ( e X, ˜ x ) → ( X, x ) has the uniquelifting property , if for every connected, locally path connected space ( Y, y ) and everycontinuous map f : ( Y, y ) → ( X, x ) with f ∗ π ( Y, y ) ⊆ p ∗ π ( e X, ˜ x ) for ˜ x ∈ p − ( x ),there exists a unique continuous map ˜ f : ( Y, y ) → ( e X, ˜ x ) with p ◦ ˜ f = f . If e X is aconnected, locally path connected space and p : e X → X has unique lifting property,then p and e X are called a generalized covering map and a generalized covering spacefor X , respectively. Definition 2.1.
Let G be a topological group. By a generalized covering group of G ,a pair ( e G, p ) is composed of a topological group e G and a homomorphism p : ( e G, ˜ e ) → ( G, e ) such that ( e G, p ) is a generalized covering space of G . It is easy to check that for an arbitrary pointed topological space (
X, x ) if p : ( e X, ˜ x ) → ( X, x ) is a generalized covering space, then the induced map p ∗ : π ( e X, ˜ x ) → π ( X, x ) is one to one. Therefore, the image H = p ∗ π ( e X, ˜ x ) is asubgroup of π ( X, x ), which is called generalized covering subgroup .3 emark 2.2. There is a well-known result about covering groups of topological groupswhich we will extend it for generalized covering groups. A homomorphism p : e G → G between topological groups is a covering homomorphism if and only if p is an openepimorphism with discrete kernel. As mentioned in [2] and [9], the endpoint projection map p H : e X H → X issurjective and open if G is path connected and locally path connected, respectively.On the other hand, for an arbitrary pointed topological space ( X, x ), Brazas [6,Lemma 5.10] showed the relationship between the image of a generalized coveringmap and the space e X H as follows: Lemma 2.3.
Suppose that ˆ p : ( ˆ X, ˆ x ) → ( X, x ) has the unique lifting property with ˆ p ∗ ( π ( ˆ X, ˆ x )) = H . Then there is a homeomorphism h : ( ˆ X, ˆ x ) → ( e X H , e H ) such that p H ◦ h = ˆ p . Remark 2.4.
It has been indicated in [15, proof of Theorem 3.6] that for everysubgroup H ≤ π ( G, e G ) of the fundamental group of a topological group G , onecan construct a multiplication on e G H , which makes e G H a topological group and thecontinuous map p H : e G H → G a homomorphism. The following proposition can be obtained by using the above remark and lemma.
Proposition 2.5. If p : ( e G, ˜ e ) → ( G, e ) is a continuous map with unique lifting prop-erty, G and e G are tow topological groups, and e G is connected locally path connected,then ( e G, p ) is a generalized covering group.Proof. If p : ( e G, ˜ e ) → ( G, e ) has unique lifting property, then by the above lemma,there is a homeomorphism h : ( e G, ˜ e ) → ( e G H , e e G H ) such that p H ◦ h = p . As itwas discussed in [15, proof of Theorem 3.6], e G H and so e G are topological groups.Moreover, Since the map p H is a group homomorphism, then for g , g ∈ e G , p ( g g ) = p H ◦ h ( g g ) = p H ( h ( g g )) = p H ( h ( g ) h ( g )) = p H ( h ( g )) p H ( h ( g )) = p H ◦ h ( g ) p H ◦ h ( g ) = p ( g ) p ( g ). Therefore, p is a homomorphism. Corollary 2.6.
Let G be a topological group, and let p : ( e G, ˜ e ) → ( G, e ) . Then ( e G, p ) is a generalized covering group if and only if it is a generalized covering space, orequivalently, p is a surjection with unique lifting property.Proof. It is the immediate result of the definition.Clearly, in the case of semilocally simply connected spaces the category of cov-ering and generalized covering spaces are equivalent. Out of semilocally simply con-nected spaces there may be a generalized covering, which is not a covering. It seemsinteresting to find some examples in the case of topological groups.4 xample 2.7.
Let Y be the Tychonoff product of the infinite number of S ’s. Clearly, Y is a topological group since the product of topological groups is also a topologicalgroup. Moreover, every open neighbourhood of any y ∈ Y contains many S ’s exceptfinite number and so Y is not a semilocally simply connected space at y ∈ Y . Since Y is locally path connected, hence Y does not have the classical universal coveringspace. Then the universal path space p : Q i ∈ I R → Y is not a covering group of Y .Although, it is a generalized covering group. Since p i : R → S is a covering map forevery i ∈ I , then it is also a generalized covering map. Therefore, from [6, Lemma2.31] the product of p i ’s, p , is also a generalized covering map. In the above example, p ∗ π ( Q i ∈ I R , ) is a trivial subgroup of π ( Y, ). It seemsinteresting to introduce a nontrivial generalized covering subgroup of π ( Y, ), whichis not a covering subgroup. Example 2.8.
From the above example, let q : Q i ∈ I R × S → Y be as q | Q i ∈ I R = p and identity on S . Clearly, p ∗ π ( Q i ∈ I R × S , ( , s )) = Z is a subgroup of thefundamental group of Y and q has unique path lifting property by the similar way ofthe above example. Thus q is a generalized covering homomorphism. We show that q is not a local homeomorphism, and so it is not a covering homomorphism. Let ( x , s ) ∈ Q i ∈ I R × S , and let U be an open subset containing ( x , s ) . By the definitionof Tychonoff product topology, U and q ( U ) contain many R ’s and S ’s except finitenumber, respectively. Then for some i ∈ I the restriction q | U i : R → S is notinjective. This fact implies that q is not a local homeomorphism. It was shown in [3, Theorem 1.4.5] that if G is a left topological group withcontinuous inverse operation and β e a local base of the space G at the identityelement e , then, for every subset A of Gcl ( A ) = A = \ { AU | U ∈ β e } . Recall from [7, 11] that for any path connected space X , any subgroup H ≤ π ( X, x ) and the endpoint projection map p H : e X H → X , the fibres p − H ( x ) and p − H ( y ) are homeomorphic, for every x, y ∈ X , if and only if X is an SLT space.Note that X is called a small loop transfer space ( SLT space for short) if for every x, y ∈ X , for every path α : I → X from x to y and for every neighbourhood U of x there is a neighbourhood V of y such that for every loop β in V based at y there isa loop γ in U based at x , which is homotopic to α ∗ β ∗ α − relative ˙ I . Clearly, everystrong SLT space is also an
SLT space.For an arbitrary pointed topological sapce (
X, x ), Abdullahi et al. in [2, Propo-sition 3.2 and Lemma 3.1] showed that the fundamental group equipped with the5hisker topology, π wh ( X, x ), is a left topological group and the collection β = { i ∗ π ( U, x ) | U is an open subset of X contaning x } forms a local basis at theidentity element in π wh ( X, x ). Now the following proposition is obtained using theseresults. Proposition 2.9. If ( X, x ) is SLT at x , then every closed subset A of π wh ( X, x ) can be written as A = T U { Ai ∗ π ( U, x ) | U is an open subset of X contaning x } .Proof. It is easily concluded from [11, Corollary 2.4] that π wh ( X, x ) is a topologicalgroup, since X is SLT at x . Then the inverse operation is continuous. Now theresult comes from [2, Proposition 3.2 and Lemma 3.1] and [3, Theorem 1.4.5] .If ( X, x ) is a strong SLT at x space, then it is easily concluded from thedefinition that for any open neighbourhood U of x in X , there is an open covering U of X such that π ( U , x ) ≤ i ∗ π ( U, x ). On the other hand, by [2, Corollary 3.10] if H ≤ π ( X, x ) is a generalized covering subgroup, then H is closed under the whiskertopology on the fundamental group. The following theorem states a nice result ofthis fact. Theorem 2.10.
Let ( X, x ) be a connected locally path connected space which isstrong SLT at x , and let p : e X → X be a map. The pair ( e X, p ) is a generalizedcovering of X with p ∗ π ( e X, ˜ x ) = H ≤ π ( X, x ) if and only if H is as intersectionof some covering subgroups in π ( X, x ) .Proof. It is easily concluded from [2, Corollary 2.13] that the intersection of anycollection of covering subgroups of the fundamental group is a generalized coveringsubgroup. Conversely, Since X is strong SLT at x , then for any open neighbourhood U of x in X , there is an open covering U of X such that π ( U , x ) ≤ i ∗ π ( U, x ). Itmeans that open subgroups of the whisker topology are covering subgroups. Thus forany open neighbourhood U of X at x and any subgroup H of π ( X, x ), the subgroup Hi ∗ π ( U, x ) is a covering subgroup of π ( X, x ) by [13, Theorem 2.5.13]. Finally, if H is a generalized covering subgroup, then it is closed under the whisker topology on thefundamental group (see [2, Corollary 3.10]) and so H = H = T U { Hi ∗ π ( U, x ) } . Corollary 2.11.
Let ( X, x ) be a connected locally path connected space which isstrong SLT at x . A subgroup H ≤ π ( X, x ) is a generalized covering subgroup ifand only if H is an intersection of all covering subgroups which contain H .Proof. Assume that Γ H is the collection of all covering subgroups of π ( X, x ) con-taining H . Clearly, T K ∈ Γ H K is a generalized covering subgroup. Conversely, by [1,Theorem 3.1] and [2, Lemma 3.1] for every K ∈ Γ H , there is an open neighbourhood6 K of X at x such that i ∗ π ( U K , x ) ≤ K . Thus Hi ∗ π ( U K , x ) ≤ K . As mentionedabove for every open neighbourhood U of X at x , there is an open covering U of X such that π ( U , x ) ≤ i ∗ π ( U, x ) ≤ Hi ∗ π ( U, x ), which shows that Hi ∗ π ( U, x ) is acovering subgroup. Therefore, Hi ∗ π ( U, x ) ∈ Γ H , for every open neighbourhood U of X at x . It implies that T K ∈ Γ H K = T K ∈ Γ H Hi ∗ π ( U K , x ) = H = H .
3. Kernel of Generalized Coverings of Topological Group If p : e X → X is a generalized covering map of a path connected SLT space X such that p ∗ π ( e X, ˜ x ) = H ≤ π ( X, x ), since e X and e X H are homeomorphic, byLemma 2.3, then for every x, y ∈ X the fibres p − ( x ) and p − ( y ) are homeomorphic. Corollary 3.1. If ( e G, p ) is a generalized covering group of a path connected topo-logical group G , then the fibres of all elements of G are homeomorphic subspaces of e G .Proof. By Theorem 2.11 in [15] every path connected topological group G is strong SLT and so
SLT space. Therefore, the result comes from the above assertion.
Proposition 3.2.
Let ( X, x ) be an arbitrary topological space. If ( e X, p ) is a gen-eralized covering space, then the fibre p − ( x ) ⊆ e X is totally path disconnected.Proof. Let α : I → p − ( x ) be a path with α (0) = ˜ x , and let C ˜ x be the constantpath in ˜ x . Clearly, p ◦ α = p ◦ C ˜ x = C x and α (0) = C ˜ x (0). Since p has uniquepath lifting property, hence α = C ˜ x . Therefore, p − ( x ) has no nonconstant path;that is, it is totally path disconnected space. Corollary 3.3. If ( e G, p ) is a generalized covering group of connected locally pathconnected topological group G , then p is an open epimorphism with totally path dis-connected kernel.Proof. It is easy to check that p is epic and open, since G is connected and locallypath connected, respectively. The result comes from combination of Propositions 2.5and 3.2.It seems interesting whether the converse statement of the above corollary can becorrect. There are some counterexamples even with extra conditions in very specialcase. For instance, where p : G → G/H is an open epimorphism with totally pathdisconnected kernel and H is a normal subgroup of a topological group G . In the nextsection we show that the necessary and sufficient condition to make p : G → G/H ageneralized covering is the prodiscrete H .7 xample 3.4. Let H = S ∩ ( Q × Q ) be the subgroup of Abelian topological group S (hence H E S ), and let p : S → S H be the natural canonical map. Theorem4.14 from [14] showed that p is an onto, continuous, and open map. It is easy toshow that the kernel of p is equal to H . Moreover, it is clear that every path in H is constant and thus it is a totally path disconnected subspace of S . Note thatsince H is a dense subgroup of S , the quotient space S H and its relative fundamentalgroup, π ( S H , H ) , are trivial space and group, respectively. Therefore, if ( S , p ) is ageneralized covering group of S H , then it is also a covering group, because the imageof induced map p ∗ π ( S , is equal to π ( S H , H ) . But it is impossible, since H is notdiscrete. The concept of prodiscrete space has been defined in different ways in varioussources. In this paper, we define the concept of prodiscrete subgroup of a topologicalgroup as follows based on the need that we felt.
Definition 3.5.
Let G be a topological group. We call H ≤ G a prodiscrete subgroupif there are some discrete groups H i , i ∈ I , and an isomorphism homeomorphism ψ : Q i ∈ I H i → H such that ψ ( Q i ∈ I,i = j H i ) E G for every j ∈ I . Recall from [6, Section 2.3] that the pull-back construction helped to show thatthe intersection of any collection of generalized covering subgroup is also a gener-alized covering subgroup. Although, there is another simple proof in [2, Corollary2.13]; using pull-backs of generalized coverings will be useful to study the fibres ofgeneralized covering maps: Let p : e X → X be a generalized covering of locallypath connected space X , and let f : Y → X be a map. The topological pull-back ♦ = { (˜ x, y ) ∈ e X × Y | p (˜ x ) = f ( y ) } is a subspace of the direct product e X × Y . Nowfor the base points y ∈ Y and x = f ( y ) ∈ X , pick ˜ x ∈ p − ( x ), and let f e X bethe path component of ♦ containing (˜ x , y ). The projection f p : f e X → Y with f p (˜ x, y ) = y is called the pull-back of e X by f . Brazas showed that for a generalizedcovering p : e X → X and a map f : Y → X , the pull-back f p : f e X → Y is also ageneralized covering [6, Lemma 2.34]. Theorem 3.6. If ( G, e ) is a connected locally path connected topological group and H is a generalized covering subgroup of π ( G, e ) , then the kernel of the endpointprojection homomorphism p H : ( e G H , ˜ e H ) → ( G, e ) is a prodiscrete subgroup.Proof. Let { K j | j ∈ J } be the collection of all covering subgroups of π ( G, e ), whichcontain H . By Corollary 2.11, H = T j ∈ J K j . For every j ∈ J , put p j : ( e G j , ˜ e j ) → ( G, e ) as the relative covering homomorphism of K j ; that is, K j = ( p j ) ∗ π ( e G j , ˜ e j ).8ake the direct product e G = Q j ∈ J e G j and p : ( e G, ˜ e ) → ( Q j ∈ J G, e ) is the producthomomorphism defined by p | e G j = p j , where e = ( e, e, e, . . . ). By Lemma 2.31 from[6], p is a generalized covering homomorphism. Now consider the diagonal map∆ : ( G, e ) → ( Q j ∈ J G, e ), and the pull-back of e G by ∆ is denoted by ∆ p : ∆ e G → ( G, e ) where ∆ e G = { (˜ t, g ) ∈ e G × G | p (˜ t ) = ∆( g ) } . By Lemma 2.34 of [6],(∆ e G, ∆ p ) is also a generalized covering group of G . Let x = (( t j ) , g ) be thebase point of ∆ e G . At first, we show that the image of (∆ p ) ∗ in π ( G, e ) is H . This implies that ( e G H , p H ) is a generalized covering group by Lemma 2.3, since(∆ e G, ∆ p ) is a generalized covering group, and therefore (∆ e G, ∆ p ) and ( e G H , p H )are equivalent generalized covering groups. After that by calculating the kernel of∆ p , we find that the kernel of p H is prodiscrete. To do this, we claim that for a loop α ∈ Ω( G, e ) its unique lift β ∈ P (∆ e G, x ) (such that ∆ p ◦ β = α ) is itself a loop ifand only if [ α ] ∈ H . Let q : ∆ e G → e G be the projection; so that p ◦ q = ∆ ◦ ∆ p . Bythe definition it is clear that β = ( γ, α ) for a path γ ∈ P ( e G, ( t j )). Let γ j ∈ P ( e G j , ˜ e j )be the j th component of γ . Now taking the j th component of the equation p ◦ γ = p ◦ q ◦ β = ∆ ◦ ∆ p ◦ β = ∆ ◦ α, which concludes that p j ◦ γ j = α for every j ∈ J . Therefore β is a loop ⇔ γ is a loop, ⇔ γ j is a loop f or every j ∈ J, ⇔ [ α ] ∈ ( p j ) ∗ ( π ( e G j , ˜ e j ) = K j f or every j ∈ J, ⇔ [ α ] ∈ H. Clearly, ker (∆ p ) = { (˜ t, g ) | ∆ p (˜ t, g ) = e } . This support that g = e and p (˜ t ) = ∆( e ) = e = ( e, e, . . . ). Hence p j ((˜ t ) j ) = e for every j ∈ J . It implies that˜ t ∈ ker ( p ) = Q j ∈ J ker ( p j ). Then ker (∆ p ) = { (˜ t, e ) | ˜ t ∈ Q j ∈ J ker ( p j ) } . Now sincefor every j ∈ J , p j : e G j → G is a covering homomorphism; thus its kernel is discrete.Let h : ∆ e G → e G H be the homeomorphism obtained from Lemma 2.3 (see Diagram1). It is easy to check that h is an isomorphism homeomorphism. Therefore ker ( p H )is a prodiscrete subgroup of e G H , since ker (∆ p ) is prodiscrete. Theorem 3.7. If G is a topological group and H is a prodiscrete normal subgroup of G such that GH is a connected locally path connected space, then the natural canonicalhomomorphism ϕ : G → GH is a generalized covering homomorphism. (cid:8) (∆ e G, x ) q / / ∆ p (cid:15) (cid:15) h z z ✈ ✈ ✈ ✈ ✈ ✈ ✈ ( e G = Q j ∈ J e G j , e ) p (cid:15) (cid:15) ( e G H , e H ) p H / / ( G, e ) ∆ / / (cid:16) Q j ∈ J G, ( e, e, . . . ) (cid:17) Figure 1: Diagram of e G H . Proof.
It is clear that GH is a topological group and by Theorem 4.14 from [14] ϕ isan open epimorphism. Let ψ : Q j ∈ J H j → H be the isomorphism homeomorphismof Definition 3.5. For every j ∈ J put K j = ψ ( Q i ∈ J,i = j H i ). Let p j : GK j → GH be thenatural canonical homomorphism. It is clear from the following diagram that p j iscontinuous and open, because ϕ and ϕ ′ are continuous and open. G ϕ ′ (cid:127) (cid:127) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) ϕ (cid:30) (cid:30) ❂❂❂❂❂❂❂❂ GK j p j / / GH Moreover, for every j ∈ J , p j is also an epimorphism with the kernel HK j . Since ψ ( H j ) is discrete, then by Remark 2.2 the pair ( GK j , p j ) is a covering group of GH forevery j ∈ J .Take the direct product e G = Q j ∈ J GK j and p : ( e G, ˜ e ) → ( Q j ∈ J GH , e ) is the producthomomorphism; that is, for every j ∈ J , p | GKj = p j and e = ( H, H, . . . ). It impliesfrom Lemma 2.31 of [6] that p is a generalized covering homomorphism. Now considerthe diagonal map ∆ : ( GH , H ) → ( Q j ∈ J GH , e ), and the pull-back of e G by ∆ is denotedby ∆ p : ∆ e G → ( GH , H ) where ∆ e G = { (( g j K j ) j ∈ J , gH ) ∈ e G × GH | p (( g j K j ) j ∈ J ) =∆( gH ) } . By Lemma 2.34 of [6], (∆ e G, ∆ p ) is also a generalized covering groupof GH . To complete the proof, it is enough to show that (∆ e G, ∆ p ), and ( G, e )are equivalent generalized covering groups of GH . Let x = (( K j ) j ∈ J , H ) be thebase point of ∆ e G , and define homomorphism θ : ( G, e ) → (∆ e G, x ) with θ ( g ) =(( gK j ) j ∈ J , gH ). Since for every g ∈ G , ∆ p ◦ θ ( g ) = ∆ p (( gK j ) j ∈ J , gH ) = gH ,10ence the right triangle of Diagram 2 is commutative. e G × GH (cid:9) (cid:8) (cid:9) υ / / ∆ e G q / / ∆ p (cid:15) (cid:15) = = θ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ e G = Q j ∈ J GK j p (cid:15) (cid:15) G ϕ / / σ O O GH ∆ / / Q j ∈ J GH
Figure 2: Commutative diagram
We show that θ is an isomorphism homeomorphism. Clearly, ker ( θ ) = { g ∈ G | (( gK j ) j ∈ J , gH ) = x = (( K j ) j ∈ J , H ) } = \ j ∈ J K j . Since T j ∈ J K j is the trivial subgroup, thus θ is one to one. To show that θ is onto, let(( g j K j ) j ∈ J , gH ) be an arbitrary element of ∆ e G , and let π j : H j → Q j ∈ J H j be theinclusion map. By the definition of ∆ e G , for every j ∈ J , there exists h j ∈ H j suchthat g j = gψ ( π j ( h j )). If h = ( h j ) j ∈ J ∈ Q j ∈ J H j and s ∈ J , then h = π s ( h s ) . ( l j ) j ∈ J where l j = h j if j = s and l s = 1. Hence, ψ ( h ) = ψ ( π s ( h s )( l j ) j ∈ J ) = ψ ( π s ( h s )) ψ (( l j ) j ∈ J ) . Then ψ ( h ) K s = ψ ( π s ( h s )) K s . Therefore, θ ( gψ ( h )) = (( gψ ( h ) K j ) j ∈ J , ghH ) = (( gψ ( π j ( h j )) K j ) j ∈ J , gH ) = (( g j K j ) j ∈ J , gH ) . It is clear from Diagram 2 that θ = υ ◦ σ , where σ : G → e G × GH is the productof natural canonical maps and υ : e G × GH → ∆ e G is the quotient map. Hence θ is continuous because σ and υ are continuous. The continuity of θ − follows fromopenness of ϕ and continuity of ∆ p . Corollary 3.8.
Let G be a topological group, and let H E G be such that the quotientgroup GH is connected locally path connected. The natural canonical homomorphism ϕ : G → GH is a generalized covering homomorphism if and only if H is a prodiscretesubgroup of G .Proof. It is clear that GH is a topological group. If ϕ is a generalized covering ho-momorphism, then by Theorem 3.6 the kernel of ϕ , H , is prodiscrete. The conversestatement is obtained from Theorem 3.7.11 orollary 3.9. Let G be connected locally path connected, and let p : e G → G be ahomomorphism on topological groups. The pair ( e G, p ) is a generalized covering groupof G if and only if p is an open epimorphism and the kernel of p is a prodiscretesubgroup of e G .Proof. Let p : e G → G be a generalized covering homomorphism, and let H = p ∗ π ( e G, ˜ e ). By Lemma 2.3, there is a homeomorphism h : e G → e G H such that p H ◦ h = p . Since p H is open onto, then also p is. Moreover, Theorem 3.6 implies that thekernel of p H and so p is prodiscrete. Conversely, consider ϕ : e G → e Gker ( p ) as the naturalcanonical homomorphism. Since ϕ is onto, then the homomorphism θ : e Gker ( p ) → G is an isomorphism homeomorphism, where e Gker ( p ) is equipped with the quotienttopology. Since ker ( p ) is a prodiscrete subgroup of e G and e Gker ( p ) is connected locallypath connected (derived from G is connected locally path connected), it impliesfrom Corollary 3.8 that ( e G, ϕ ) is a generalized covering group of e Gker ( p ) . Therefore,( e G, p = ϕ ◦ θ ) is a generalized covering group of G .By Remark 2.4 and Corollary 3.9 one can easily conclude the following corollarywhich was promised in Remark 2.2. Corollary 3.10.
Let G be a connected locally path connected topological group, andlet p : e G → G be a continuous map. The pair ( e G, p ) is a generalized covering groupof G if and only if e G is a topological group and p is an open epimorphism withprodiscrete kernel. Example 3.11.
It is easy to show that the kernel of generalized covering maps p : Q i ∈ I R → Y and q : Q i ∈ I R × S → Y in Examples 2.7 and 2.8, respectively, both are Q i ∈ I Z which is the product of discrete groups. For another example, let r n : S → S with r n ( z ) = z n be the well-known covering homomorphism. As mentioned before, theproduct homomorphism r = Q n ∈ N r n : Q n ∈ N S → Q n ∈ N S is a generalized coveringhomomorphism and ker ( r ) = Q n ∈ N A n where A n = { z ∈ S | z n = 1 } , which isalso a product of discrete groups. Moreover, the kernel of p in Example 3.4 is notprodiscrete. In [5], Biss investigated on a kind of fibrations which is called rigid coveringfibration with properties similar to covering spaces. Nasri et al. [10] simplified thedefinition of rigid covering fibration such as a fibration p : E → X with uniquepath lifting (unique lifting with respect to paths) property and concluded from [13,Theorem 2.2.5] that a fibration p : E → X is a rigid covering fibration if and only if12ach fibre of p is totally path disconnected. On the other hand it was shown in [13,Theorem 2.4.5] that a rigid covering fibration has the unique lifting property and soit is a generalized covering spaces. Although, the converse statement may not hold,in general, we show that it is right in the case of connected locally path connectedtopological groups. Proposition 3.12.
For a connected locally path connected topological group G , ( e G, p ) is a generalized covering group if and only if p : e G → G is rigid covering fibration.Proof. Let p : e G → G be a generalized covering homomorphism, and let H = p ∗ π ( e G, ˜ e ). By Lemma 2.3, ( e G, p ) and ( e G H , p H ) are equivalent generalized coveringgroups of G , and so the kernel of p H is a prodiscrete subgroup, since the kernelof p is prodiscrete. On the other hand, it was shown in [2, Section 3] that thekernel of p H and the left coset space π wh ( G,e ) H are homeomorphic, which implies that π wh ( G,e ) H is a prodiscrete space. Then π wh ( G,e ) H is totally path disconnected; that is, π wh ( G,e ) H has no nonconstant paths. As mentioned above, π qtop ( G,e ) H = π wh ( G,e ) H hasno nonconstant paths. Now use [5, Theorem 4.3] which guarantees the existenceof a rigid covering fibration q : E → G with p ∗ π ( E ) = H . Since every rigidcovering fibration has unique lifting property (see [13, Theorem 2.4.5]), there is ahomeomorphism g between E and e G H (Lemma 2.3). Therefore, it is done as shownin Diagram 3. e G (cid:8) (cid:9) p ! ! ❈❈❈❈❈❈❈❈❈ h / / e G Hp H (cid:15) (cid:15) E q } } ④④④④④④④④④ g o o ( G, e ) Figure 3: Diagram
The converse statement obtains from [13, Theorem 2.4.5].The following corollary is the immediate result of the definition of rigid coveringfibrations and Proposition 3.12.
Corollary 3.13. If G is a connected locally path connected topological group and ( e G, p ) is a generalized covering group of G , then p : e G → G is a fibration. eferenceReferences [1] M. Abdullahi Rashid, N. Jamali, B. Mashayekhy, S.Z. Pashaei and H. TorabiOn subgroup topologies on fundamental groups, arXiv:1807.00982v1. [2] M. Abdullahi Rashid, B. Mashayekhy, H. Torabi and S.Z. Pashaei, On sub-groups of topologized fundamental groups and generalized coverings, Bull.Iranian Math. Soc. no. 7, (2017), 2349–2370.[3] A. Arhangelskii and M. Tkachenko, Topological Groups and Related Structures,Atlantis Studies in Mathematics, 1. Atlantis Press, Paris,
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