Group of the Covering Space
A. Tekcan, M. Bayraktar and O. Bizim
Dedicated to the Memory of Grigorios TSAGAS (1935-2003), President of Balkan Society of Geometers (1997-2003)
Abstract
In this paper some properties of covering space and the automorphism group of the covering space are obtained.
Mathematics Subject Classification: 55E17
Key words: covering space, automorphism group of covering space, universal cover- ing, regular covering, fundamental group
1 Introduction
LetXe be a connected space,X be a space and letp:Xe −→X be a continious map.
If for everyx∈X has an path connected open neighbourhoodU such thatp−1(U) is open inXe and each component ofp−1(U) is mapped topologically ontoU bypthen pis called acovering map.In this case the pair (X, p) is called ae covering spaceofX.
Let (X, p) be a covering space ofe X,xe0 ∈X,ande p(ex0) =x0.Then, for any path αin X with initial pointx0,there exists a unique pathβin Xe with initial pointxe0
such thatpβ=α.
Let (X, p) be a covering space ofe X andx∈X.For any pointex∈p−1(x) and any α∈π(X, x) we defineexα∈p−1(x) as follows: From above there exists a unique path classαe inXe such thatp∗(eα) =αand the initial point ofαeis the pointx.e Definexαe to be the terminal point of the path classα.e Then it is easily verify that
(exα)β = ex(αβ), e
xe = ex.
Thus π(X, x) be a group of right operators on the set p−1(x).Moreover the group π(X, x) acts transitively on the set p−1(x).To show this let ex0,xe1 ∈ p−1(x).Since Xe is arcwise connected, there exists a path class αe in Xe with initial point ex0 and terminal pointex1.Letp∗(eα) =α. Then,αis an equivalence class of closed paths, and obviously ex0α = ex1.Thus, the set p−1(x) is a homogeneous rightπ(X, x)− space.
Balkan Journal of Geometry and Its Applications, Vol.8, No.1, 2003, pp. 101-108.∗
c
°Balkan Society of Geometers, Geometry Balkan Press 2003.
From the definition, we see that for any point xe ∈ p−1(x), the isotropy subgroup corresponding to this point is precisely the subgroup p∗π(X,e x) ofe π(X, x). Hence p−1(x) is isomorphic to the space of cosets, π(X, x)/ p∗π(X,e ex), and the number of sheets of the covering is equal to the index of the subgroupp∗π(X,e x) ine π(X, x)
If X is simply connected then the fundamental group π(X, x) is trival and the index of p∗π(X,e ex) in π(X, x) is 1. So (X, p) is an one-sheeted covering ofe X and thereforepis a homeomorphism. Similarly if Xe is simply connected then π(X,e ex) is trival and the index ofp∗π(X,e ex) in π(X, x) is equal to the order of π(X, x).
A covering transformationof a covering space (X, p) ofe X is a homeomorphism h:Xe −→Xe such thatph=p.The set of all covering transformations of (X, p) forme a group denoted byA(X, p).e
Let (X, p) be a covering space ofe X and p(ex1) = p(ex2) = x for ex1,xe2 ∈ Xe and x∈X. Let consider the homomorphisms
p∗ : π(X,e xe1)→π(X, x), p∗ : π(X,e xe2)→π(X, x)
Let {γi: i∈I}be a path class in Xe with inital point xe1 and terminal point xe2. Define
u∗:π(X,e xe1)→π(X,e ex2)
to beu∗(α) =γ−1αγ forγ∈ {γi: i∈I}.Then we have the commutative diagram in Figure 1.
Figura 1.
Here v∗(β) = (p∗γ)−1β(p∗γ). Since p∗γ is a closed it is a path in π(X, x). So the images of the fundamental groupsπ(X,e ex1) andπ(X,e ex2) under p∗are conjugate subgroups ofπ(X, x).
Lemma 1.1 Let (X, p)e be a path connected covering space of a locally pathwise connected space X and p(ex1) = p(ex2) = x for xe1,xe2 ∈ Xe and x ∈ X. Then p∗π(X,e xe1)and p∗π(X,e xe2) are conjugate subgroups of π(X, x) iff there exists a ϕ∈A(X, p)e such that ϕ(ex1) =xe2.[4]
Lemma 1.2 Let (Xe1, p1) and (Xe2, p2) be two covering space of a locally pathwise connected spaceX. Then these two covering are isomorphic iff there exists a homeo- morphismh:Xe1→Xe2 such that p2h=p1.[4]
Lemma 1.3 If two space are homeomorphic, then their fundamental groups are iso- morphic, i.e. if h : (X, x) → (Y, y) is a homeomorphism, then h∗ : π(X, x) → π(Y, y)is an isomorphism.[3]
Lemma 1.4 If(Xe1, p1)and(Xe2, p2) are two simply connected covering space of a lo- cally pathwise connected spaceX,then there exists a homeomorphismh: (Xe1, p1)−→
(Xe2, p2)such thatp2h=p1.[2]
Lemma 1.5 Let (Xe1, p1) and (Xe2, p2) be two covering space of a locally pathwise connected spaceX. Then there exists a morphismϕfrom(Xe1, p1)into(Xe2, p2)such thatϕ(ex1) =ex2 iff
p1∗π(Xe1,xe1) = p2∗π(Xe2,ex2).[4]
Lemma 1.6 Let (X, p)e be path connected covering space of a locally pathwise con- nected spaceX. Thenpis a homeomorphism iff p∗π(X,e ex) =π(X, x).[6]
From Lemma 1.1 we have if (Xe1, p1) and (Xe2, p2) are two covering spaces of a locally pathwise connected space X, then these two coverings are isomorphic iff p1∗π(Xe1,ex1) and p2∗π(Xe2,ex2) are conjugate subgroups of π(X, x).
LetX be a connected space. The category of connected spaces of X has objects which are covering projectionsp:Xe →X, whereXe is connected, and morphisms
p1:Xe1→X , p2:Xe2→X and f :Xe1→Xe2
such thatp2f =p1.
Definition. LetX be a connected space andXe be a locally path connected space.
A universal covering space ofX is an object p:Xe →X of the category of connected covering spaces of X such that for any object p1 : Xe1 → X there is a morphism f :Xe →Xe1such that p1f =p.[6]
Lemma 1.7 If (X, p)e is an universal covering space of a locally pathwise connected spaceX,then the automorphism groupA(X, p)e is isomorphic to the fundamental group π(X, x), and the number of the sheets of the covering is equal to the order of the fundamental group π(X, x).[4]
Let Xe be be a pathwise connected space and let (X, p) be a covering space of ae locally pathwise connected space X. Then (X, p) is ae regular covering space ofX iff p∗π(X,e x) is a normal subgroup ofe π(X, x).
Lemma 1.8 Let p:Xe →X be a covering map such thatp(ex1) =p(ex2)forxe1,xe2∈ X.e Thenpis regular iff p∗π(X,e ex1) = p∗π(X,e ex2).[6]
Lemma 1.9 Let Xe be be a pathwise connected space and let(X, p)e is a regular cov- ering space of a locally pathwise connected space X. Then X homeomorphic to the quatient spaceX / A(e X, p).e [6]
Let G be a group of homeomorphisms of X. If for every x ∈ X, there exists a neighbourhoodV ofxsuch thatgV ∩V =θ, for allg∈Gdifferent from the unity of G, then we sayGacts discontiniouslyonX.
Lemma 1.10 LetGbe a discontinious proper group of homeomorphisms of a locally pathwise connected spaceX and let q: X →X/Gbe a naturel projection defined by q(x) = [x]. Then(X, q)e is a regular covering space ofX/Gand A(X, q)e is isomorphic toG.[6]
2 On the covering space and its automorphism group
In this paper we obtain some properties of covering spaces and their automorphism and fundamental groups.
Theorem 2.1 Let (Xe1, p1)and(Xe2, p2) be two universal covering space of a locally pathwise connected spaceX. Then
1. these two covering spaces are homeomorphic, and therefore the fundamental groupsπ(Xe1,ex1)andπ(Xe2,ex2)are isomorphic.
2. A(Xe1, p1)andA(Xe2, p2)are isomorphic.
Proof. 1. Let (Xe1, p1) and (Xe2, p2) be two universal covering space of X. Then from Lemma 1.4. there exists a homeomorphism h: (Xe1, p1)−→(Xe2, p2) such that p2h=p1. Therefore these two covering are isomorphic. Therefore form Lemma 1.3.
the fundamental groupsπ(Xe1,ex1) andπ(Xe2,xe2) are isomorphic.
2. Let p1(ex1) = p2(ex2) = x for xe1 ∈ Xe1 and ex2 ∈ Xe2.Since (Xe1, p1) and (Xe2, p2) are universal covering space of X, from Lemma 1.8. the automorphism group A(Xe1, p1) is isomorphic to the fundamental groupπ(X, x), and the automor- phism group A(Xe2, p2) is isomorphic to the fundamental group π(X, x). Therefore A(Xe1, p1) andA(Xe2, p2) are isomorphic.
Theorem 2.2 Let (X, p)e be an universal covering space of a locally pathwise con- nected space X and p(ex1) = p(ex2) = x for xe1,xe2 ∈ X.e Then there exists an auto- morphism ϕ∈A(X, p)e such thatϕ(ex1) =ex2,i.e.A(X, p)e acts transitively on the set p−1(x).
Proof. Since (X, p) is an universal covering space ofe X ,there is a pathγin Xe with initial pointxe1and terminal pointxe2.Usingγ define
u∗:π(X,e xe1)→π(X,e ex2)
to be u∗(α) = γ−1αγ. Then u∗ is an isomorphism and thus p∗π(X,e xe1) and p∗π(X,e xe2) are conjugate subgroups ofπ(X, x).Therefore from Lemma 1.1.ϕ(ex1) = e
x2.
Theorem 2.3 Let (X, p)e be a covering space of a locally pathwise connected space X .Then there exists a ϕ ∈ A(X, p)e such that ϕ(ex1) = ex2 iff (X, p)e is a regular covering space ofX .
Proof.Let assume that there exists aϕ∈A(X, p) such thate ϕ(ex1) =ex2.Then from Lemma 1.5 p∗π(X,e xe1) = p∗π(X,e ex2).Thus from Lemma 1.9 (X, p) is a regulare covering space ofX .
Conversely let (X, p) is a regular covering space ofe X . Then from Lemma 1.9p∗π(X,e xe1) =p∗π(X,e ex2), and from Lemma 1.5 there exists a ϕ∈ A(X, p) suche thatϕ(ex1) =ex2.
Theorem 2.4 Universal covering is regular.
Proof.Let (X, p) be an universal covering space of a locally pathwise connected spacee X ,and let xe1 and ex2 be two points of Xe such thatp(ex1) =p(ex2) =x.Since (X, p)e is an universal covering space ofX, there exists a path γin Xe with initial point xe1
and terminal pointxe2.Define u∗ :π(X,e xe1)→π(X,e ex2) to beu∗(α) =γ−1αγ. Then u∗ is an isomorphism and thusp∗π(X,e xe1) and p∗π(X,e ex2) are conjugate subgroups of π(X, x). Thus from Lemma 1.1. there exists a ϕ ∈ A(X, p) such thate ϕ(ex1) = e
x2.Therefore from above Theorem (X, p) is regular covering space ofe X .
Let (X, p) is a regular covering space of a locally pathwise connected spacee X . Then we know from Lemma 1.10 that X is homeomorphic to the quatient space X/A(e X, p),e i.e. there exists a homeomorphismr:X/A(e X, p)e →X.
Theorem 2.5 If(X, p)e is an universal covering space of a locally pathwise connected spaceX,then the order of the automorphism group A
³X/A(e X, p), re
´
of the quatient spaceX/A(e X, p)e is equal to the number of the sheets of the covering (X, p)e of X.
Proof. Since universal covering space is regular (X, p) is a regular covering spacee of X . So there exists a homeomorphism r : X/A(e X, p)e → X.On the other hand since (X, p) is an universal covering space ofe X , (X, q) is an universal covering spacee of X/A(e X, p),e and the number of the sheets of the covering is equal to the order of the group π(X, x). r is an universal covering map since p and q are universal covering. So from Lemma 1.8.A
³X/A(e X, p), re
´
is isomorphic to the fundamental groupπ(X, x).Thus the number of the sheets of the covering (X, p) ofe X is equal the the order of the automorphism groupA
³X/A(e X, p), re
´ .
Theorem 2.6 If(X, p)e is an universal covering space of a locally pathwise connected spaceX andA(X, p)e is the discontinious proper group of homeomorphisms ofX, thene the fundamental groupsπ(X, x)andπ
³X/A(e X, p),e [ex]
´
are isomorphic under the na- turel projection
q: Xe →X/A(e X, p)e defined byq(ex) = [ex].
Proof. Since A(X, p) be the discontinious proper group of homeomorphisms ofe X,e q is a regular map and the automorphism group A(X, q) is isomorphic toe A(X, p).e Since (X, p) is an universal covering space, (e X, q) is an universal covering space ofe X/A(e X, p).e Therefore the automorphism groupA(X, p) is isomorphic to the funda-e mental group π(X, x) and A(X, q) is isomorphic toe π
³X/A(e X, p),e [ex]
´
. Hence the fundamental groups π(X, x) and π( ˜X/A( ˜X, p),[˜x]) are isomorphic since A( ˜X, q) is isomorphic toA( ˜X, p).
Let (X, p) be a universal covering space of a locally pathwise connected spacee X .Then from above Theorem we have following corollaries.
Corollary 2.7 The number of the sheets of the covering is equal to the order of the fundamental groupπ
³X/A(e X, p),e [ex]
´ . Corollary 2.8 A(X, p)e is isomorphic toπ
³X/A(e X, p),e [ex]
´ .
Theorem 2.9 Let (Xe1, p1)and(Xe2, p2)be two universal covering space of a simply connected space X and let G1 and G2 be discontinious proper group of homeomor- phisms ofXe1 andXe2,respectively. Then the diagram in Figure 2 is commutative.
Proof. Since X is simply connected,p1 and p2 are homeomorphisms and therefore p1∗ and p2∗ are isomorphisms. On the other hand, since (Xe1, p1) and (Xe2, p2) are universal covering space ofX, there exists a homeomorphismh:Xe1→Xe2 such that p2h = p1.So h∗ : π(Xe1,xe1) → π(Xe2,xe2) is an isomorphism for xe1 ∈ Xe1 and xe2 ∈ Xe2. Since these covering space are universalA(Xe1, p1) andA(Xe2, p2) are isomorphic to the fundamental group π(X, x),i.e. there exist the isomorphisms ϕ1∗ and ϕ2∗. From Theorem 2.6. there exist the isomorphisms r1∗ and r1∗. Hence the diagram is commutative.
Figure 2.
Theorem 2.10 If (X, p)e is an universal covering space of a locally pathwise con- nected spaceX,then the automorphism groupA
³X/A(e X, p), re
´
of the quatient space X/A(e X, p)e is isomorphic to the automorphism groupA(X, p).e
Proof.Since the universal covering is regular this covering is regular. Therefore there exists a homeomorphism r : X/A(e X, p)e → X. On the other hand since this cov- ering is universal from Lemma 1.8. the automorphism group A(X, p) is isomorphice to the fundamental group π(X, x). Moreover from Theorem 2.5. the automorphism groupA
³X/A(e X, p), re
´
is isomorphic to theπ(X, x).ThereforeA
³X/A(e X, p), re
´ is isomorphic to the automorphism groupA(X, p).e
From above Theorem
Corollary 2.11 If(Xe1, p1)and(Xe2, p2)are two universal covering spaces of a locally pathwise connected spaceX then Xe1/A(Xe1, p1)andXe2/A(Xe2, p2) are isomorphic.
Proof.Let (Xe1, p1) and (Xe2, p2) be two universal covering spaces of X.Then there exists a homeomorphismh:Xe1→Xe2 such thatp2h=p1.On the other hand since these coverings are regular there exist homeomorphismsr1 :Xe1/A(Xe1, p1)→X and r2:Xe2/A(Xe2, p2)→X.Hence Xe1/A(Xe1, p1) andXe2/A(Xe2, p2) are isomorphic.
Theorem 2.12 Let(X, p)e be a covering space of a locally pathwise connected spaceX.Then the number of the elements in the orbit [ex] of the point ex∈ p−1(x) is equal to the number of the sheets of the covering(X, p)e ofX.
Proof.We know that the number of the elements in the orbit [ex] of the point ex∈ p−1(x) is equal to the index of the isotropy subgroup which corresponding to exin π(X, x).Moreover isotropy subgroup which correspondingxeis the subgroupp∗π(X,e x)e ofπ(X, x).Since the number of the sheets of the covering is equal to the index of the subgroupp∗π(X,e ex) in π(X, x), the number of the sheets of the covering is equal to the elements in the orbit [ex] of the pointex∈p−1(x).
Theorem 2.13 If(X, p)e is an universal covering space of a locally pathwise connected spaceX, then the order of the fundamental groupπ³
X/A(e X, p),e [ex]´
of the quatient spaceX/A(e X, p)e is equal to the number of sheets of the covering(X, p)e of X.
Proof.Since universal covering is regular this covering is regular, and therefore there exists a homeomorphism r : X/A(e X, p)e → X. Thus r∗ is an isomorphism, i.e. the fundamental groupsπ³
X/A(e X, p),e [ex]´
andπ(X, x) are isomorphic. Since this cov- ering is universalA(X, p) is isomorphic to the fundamental groupe π(X, x), and the number of the sheets of the covering is equal to the order of the fundamental group π(X, x). Therefore the number of the sheets of the covering is equal to the order of the fundamental group ³
X/A(e X, p),e [ex]´
since π³
X/A(e X, p),e [ex]´
and π(X, x) are isomorphic.
Definition.Let (X, p) be a covering space of a locally pathwise connected spacee X . If the automorphism groupA(X, p) acts transitively on the sete p−1(x),for everyx∈ X,then (X, p) is callede Galois.
Theorem 2.14 Regular covering is Galois.
Proof. Let (X, p) be a regular covering space ofe X . Then from Theorem 2.3. there exist aϕ∈A(X, p) such thate ϕ(ex1) =xe2for xe1,xe2∈X,e i.e.A(X, p) acts transitivelye on the setp−1(x) forx∈X.Thus from definition (X, p) is Galois covering ofe X .
We know from Theorem 2.4. that universal covering is regular. Thus from above Theorem
Corollary 2.15 Universal covering is Galois.
Theorem 2.16 If (X, p)e is a 2− sheeted covering space of a locally pathwise con- nected space X, then this covering is Galois.
Proof.Let (X, p) be 2−e sheeted covering space ofX. Since the number of the sheets of the covering is equal to the index of the subgroupp∗π(X,e x) ine π(X, x), p∗π(X,e ex) is a normal subgroup of π(X, x).Therefore (X, p) is a regular covering space ofe X and thus (X, p) is a Galois covering ofe X.
References
[1] J.B. Fraleigh, A First Course in Abstract Algebra, Addision-Wesley Publishing Company, 1988.
[2] M.J. Greenberg and J.R. Harper,Algebraic Topology. A First Course, Addision- Wesley Publishing Company, Inc., 1980.
[3] C. Kosniovski, A First Course in Algebraic Topology, Cambridge University Press, 1980.
[4] W.S. Massey, Algebraic Topology an Introduction, Harcourt, Brace&World, Inc., 1967.
[5] J.R. Munkres,Topology. A First Course, Prencite-Hall.Inc. New Jersey, 1967.
[6] E.H., Spanier, Algebraic Topology, Tata McGraw-Hill Publishing Company Ltd, New Delhi, 1978.
University of Uludag, Faculty of Science, Dept. of Mathematics, G¨or¨ukle 16059, Bursa -TURKEY,
email: [email protected], [email protected], [email protected]
