On the Covering Space and the Automorphism
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 covering, regular covering, fundamental group
1
Introduction
e be a connected space, X be a space and let p : X
e −→ X be a continious map.
Let X
If for every x ∈ X has an path connected open neighbourhood U such that p−1 (U ) is
e and each component of p−1 (U ) is mapped topologically onto U by p then
open in X
e p) is called a covering space of X.
p is called a covering map. In this case the pair (X,
e p) be a covering space of X, x
e
Let (X,
e0 ∈ X,and
p(e
x0 ) = x0 . Then, for any path
e with initial point x
α in X with initial point x0 , there exists a unique path β in X
e0
such that pβ = α.
e p) be a covering space of X and x ∈ X. For any point x
Let (X,
e ∈ p−1 (x) and any
−1
α ∈ π(X, x) we define x
eα ∈ p (x) as follows: From above there exists a unique path
e
class α
e in X such that p∗ (e
α) = α and the initial point of α
e is the point x
e. Define x
eα
to be the terminal point of the path class α
e. Then it is easily verify that
(e
xα)β
x
ee
= x
e(αβ),
= x
e.
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 x
e0 , x
e1 ∈ p−1 (x). Since
e is arcwise connected, there exists a path class α
e with initial point x
X
e in X
e0 and
terminal point x
e1 . Let p∗ (e
α) = α. Then, α is an equivalence class of closed paths, and
obviously x
e0 α = x
e1 . Thus, the set p−1 (x) is a homogeneous right π(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.
°
102
A. Tekcan, M. Bayraktar and O. Bizim
From the definition, we see that for any point x
e ∈ p−1 (x), the isotropy subgroup
e x
corresponding to this point is precisely the subgroup p∗ π(X,
e) of π(X, x). Hence
−1
e
p (x) is isomorphic to the space of cosets, π(X, x)/ p∗ π(X, x
e), and the number of
e x
sheets of the covering is equal to the index of the subgroup p∗ π(X,
e) in π(X, x)
If X is simply connected then the fundamental group π(X, x) is trival and the
e x
e p) is an one-sheeted covering of X and
index of p∗ π(X,
e) in π(X, x) is 1. So (X,
e is simply connected then π(X,
e x
therefore p is a homeomorphism. Similarly if X
e) is
e x
trival and the index of p∗ π(X,
e) in π(X, x) is equal to the order of π(X, x).
e p) of X is a homeomorphism
A covering transformation of a covering space (X,
e
e
e p) form
h : X −→ X such that ph = p. The set of all covering transformations of (X,
e p).
a group denoted by A(X,
e
e and
Let (X, p) be a covering space of X and p(e
x1 ) = p(e
x2 ) = x for x
e1 , x
e2 ∈ X
x ∈ X. Let consider the homomorphisms
p∗
:
p∗
:
e x
π(X,
e1 ) → π(X, x),
e x
π(X,
e2 ) → π(X, x)
e with inital point x
Let {γi : i ∈ I} be a path class in X
e1 and terminal point x
e2 .
Define
e x
e x
u∗ : π(X,
e1 ) → π(X,
e2 )
to be u∗ (α) = γ −1 αγ for γ ∈ {γi : i ∈ I}.Then we have the commutative diagram in
Figure 1.
Figura 1.
Here v∗ (β) = (p∗ γ)−1 β(p∗ γ). Since p∗ γ is a closed it is a path in π(X, x). So
e x
e x
the images of the fundamental groups π(X,
e1 ) and π(X,
e2 ) under p∗ are conjugate
subgroups of π(X, x).
e p) be a path connected covering space of a locally pathwise
Lemma 1.1 Let (X,
e and x ∈ X. Then
connected space X and p(e
x1 ) = p(e
x2 ) = x for x
e1 , x
e2 ∈ X
e
e
p∗ π(X, x
e1 ) and p∗ π(X, x
e2 ) are conjugate subgroups of π(X, x) iff there exists a
e p) such that ϕ(e
ϕ ∈ A(X,
x1 ) = x
e2 .[4]
On the Covering Space and the Automorphism Group of the Covering Space
103
e1 , p1 ) and (X
e2 , p2 ) be two covering space of a locally pathwise
Lemma 1.2 Let (X
connected space X. Then these two covering are isomorphic iff there exists a homeoe1 → X
e2 such that p2 h = p1 .[4]
morphism h : X
Lemma 1.3 If two space are homeomorphic, then their fundamental groups are isomorphic, i.e. if h : (X, x) → (Y, y) is a homeomorphism, then h∗ : π(X, x) →
π(Y, y) is an isomorphism.[3]
e1 , p1 ) and (X
e2 , p2 ) are two simply connected covering space of a loLemma 1.4 If (X
e1 , p1 ) −→
cally pathwise connected space X, then there exists a homeomorphism h : (X
e
(X2 , p2 ) such that p2 h = p1 .[2]
e1 , p1 ) and (X
e2 , p2 ) be two covering space of a locally pathwise
Lemma 1.5 Let (X
e1 , p1 ) into (X
e2 , p2 ) such
connected space X. Then there exists a morphism ϕ from (X
that ϕ(e
x1 ) = x
e2 iff
e1 , x
e2 , x
p1∗ π(X
e1 ) = p2∗ π(X
e2 ). [4]
e p) be path connected covering space of a locally pathwise conLemma 1.6 Let (X,
e x
nected space X. Then p is a homeomorphism iff p∗ π(X,
e) = π(X, x). [6]
e1 , p1 ) and (X
e2 , p2 ) are two covering spaces of
From Lemma 1.1 we have if (X
a locally pathwise connected space X , then these two coverings are isomorphic iff
e1 , x
e2 , x
p1∗ π(X
e1 ) and p2∗ π(X
e2 ) are conjugate subgroups of π(X, x).
Let X be a connected space. The category of connected spaces of X has objects
e → X , where X
e is connected, and morphisms
which are covering projections p : X
e 1 → X , p2 : X
e2 → X and f : X
e1 → X
e2
p1 : X
such that p2 f = p1 .
e be a locally path connected space.
Definition. Let X be a connected space and X
e → X of the category of connected
A universal covering space of X is an object p : X
e1 → X there is a morphism
covering spaces of X such that for any object p1 : X
e
e
f : X → X1 such that p1 f = p. [6]
e p) is an universal covering space of a locally pathwise connected
Lemma 1.7 If (X,
e p) is isomorphic to the fundamental group
space X,then the automorphism group A(X,
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]
e be be a pathwise connected space and let (X,
e p) be a covering space of a
Let X
e p) is a regular covering space of X iff
locally pathwise connected space X . Then (X,
e
p∗ π(X, x
e) is a normal subgroup of π(X, x).
e → X be a covering map such that p(e
Lemma 1.8 Let p : X
x1 ) = p(e
x2 ) for x
e1 , x
e2 ∈
e
e x
e x
X. Then p is regular iff p∗ π(X,
e1 ) = p∗ π(X,
e2 ).[6]
104
A. Tekcan, M. Bayraktar and O. Bizim
e be be a pathwise connected space and let (X,
e p) is a regular covLemma 1.9 Let X
ering space of a locally pathwise connected space X . Then X homeomorphic to the
e / A(X,
e p). [6]
quatient space X
Let G be a group of homeomorphisms of X. If for every x ∈ X, there exists a
neighbourhood V of x such that gV ∩ V = θ , for all g ∈ G different from the unity of
G, then we say G acts discontiniously on X.
Lemma 1.10 Let G be a discontinious proper group of homeomorphisms of a locally
pathwise connected space X and let q : X → X/G be a naturel projection defined by
e q) is a regular covering space of X/G and A(X,
e q) is isomorphic
q(x) = [x]. Then (X,
to G. [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.
e1 , p1 ) and (X
e2 , p2 ) be two universal covering space of a locally
Theorem 2.1 Let (X
pathwise connected space X . Then
1. these two covering spaces are homeomorphic, and therefore the fundamental
e1 , x
e2 , x
groups π(X
e1 ) and π(X
e2 ) are isomorphic.
e1 , p1 ) and A(X
e2 , p2 ) are isomorphic.
2. A(X
e1 , p1 ) and (X
e2 , p2 ) be two universal covering space of X . Then
Proof. 1. Let (X
e1 , p1 ) −→ (X
e2 , p2 ) such that
from Lemma 1.4. there exists a homeomorphism h : (X
p2 h = p1 . Therefore these two covering are isomorphic. Therefore form Lemma 1.3.
e1 , x
e2 , x
the fundamental groups π(X
e1 ) and π(X
e2 ) are isomorphic.
e1 and x
e2 . Since (X
e1 , p1 ) and
2. Let p1 (e
x1 ) = p2 (e
x2 ) = x for x
e1 ∈ X
e2 ∈ X
e
(X2 , p2 ) are universal covering space of X, from Lemma 1.8. the automorphism
e1 , p1 ) is isomorphic to the fundamental group π(X, x), and the automorgroup A(X
e2 , p2 ) is isomorphic to the fundamental group π(X, x). Therefore
phism group A(X
e
e2 , p2 ) are isomorphic.
A(X1 , p1 ) and A(X
e p) be an universal covering space of a locally pathwise conTheorem 2.2 Let (X,
e Then there exists an autonected space X and p(e
x1 ) = p(e
x2 ) = x for x
e1 , x
e2 ∈ X.
e p) such that ϕ(e
e p) acts transitively on the set
morphism ϕ ∈ A(X,
x1 ) = x
e2 , i.e. A(X,
p−1 (x).
e p) is an universal covering space of X , there is a path γ in X
e with
Proof. Since (X,
initial point x
e1 and terminal point x
e2 . Using γ define
e x
e x
u∗ : π(X,
e1 ) → π(X,
e2 )
e x
to be u∗ (α) = γ −1 αγ. Then u∗ is an isomorphism and thus p∗ π(X,
e1 ) and
e
p∗ π(X, x
e2 ) are conjugate subgroups of π(X, x). Therefore from Lemma 1.1. ϕ(e
x1 ) =
x
e2 .
On the Covering Space and the Automorphism Group of the Covering Space
105
e p) be a covering space of a locally pathwise connected space
Theorem 2.3 Let (X,
e p) such that ϕ(e
e p) is a regular
X . Then there exists a ϕ ∈ A(X,
x1 ) = x
e2 iff (X,
covering space of X .
e p) such that ϕ(e
Proof. Let assume that there exists a ϕ ∈ A(X,
x1 ) = x
e2 . Then from
e
e
e
Lemma 1.5 p∗ π(X, x
e1 ) = p∗ π(X, x
e2 ). Thus from Lemma 1.9 (X, p) is a regular
covering space of X .
e p) is a regular covering space of X . Then from Lemma
Conversely let (X,
e
e x
e p) such
1.9 p∗ π(X, x
e1 ) = p∗ π(X,
e2 ), and from Lemma 1.5 there exists a ϕ ∈ A(X,
that ϕ(e
x1 ) = x
e2 .
Theorem 2.4 Universal covering is regular.
e p) be an universal covering space of a locally pathwise connected space
Proof. Let (X,
e such that p(e
e p)
X , and let x
e1 and x
e2 be two points of X
x1 ) = p(e
x2 ) = x. Since (X,
e with initial point x
is an universal covering space of X , there exists a path γ in X
e1
e x
e x
and terminal point x
e2 . Define u∗ : π(X,
e1 ) → π(X,
e2 ) to be u∗ (α) = γ −1 αγ. Then
e x
e x
u∗ is an isomorphism and thus p∗ π(X,
e1 ) and p∗ π(X,
e2 ) are conjugate subgroups
e p) such that ϕ(e
of π(X, x). Thus from Lemma 1.1. there exists a ϕ ∈ A(X,
x1 ) =
e
x
e2 . Therefore from above Theorem (X, p) is regular covering space of X .
e p) is a regular covering space of a locally pathwise connected space X .
Let (X,
Then we know from Lemma 1.10 that X is homeomorphic to the quatient space
e
e p), i.e. there exists a homeomorphism r : X/A(
e
e p) → X.
X/A(
X,
X,
e p) is an universal covering space of a locally pathwise connected
Theorem 2.5 If (X,
³
´
e
e p), r of the quatient
space X, then the order of the automorphism group A X/A(
X,
e
e p) is equal to the number of the sheets of the covering (X,
e p) of X.
space X/A(
X,
e p) is a regular covering space
Proof. Since universal covering space is regular (X,
e
e p) → X. On the other hand
of X . So there exists a homeomorphism r : X/A(X,
e
e
since (X, p) is an universal covering space of X , (X, q) is an universal covering space
e
e p), and the number of the sheets of the covering is equal to the order
of X/A(
X,
of the group π(X, x). r is an universal
covering
³
´ map since p and q are universal
e
e
covering. So from Lemma 1.8. A X/A(X, p), r is isomorphic to the fundamental
e p) of X is equal the
group π(X, x). Thus the number of the sheets
of the covering
(X,
³
´
e
e
the order of the automorphism group A X/A(X, p), r .
e p) is an universal covering space of a locally pathwise connected
Theorem 2.6 If (X,
e
e
space X and A(X, p) is the discontinious
³ proper group of
´ homeomorphisms of X, then
e
e p), [e
the fundamental groups π(X, x) and π X/A(
X,
x] are isomorphic under the naturel projection
defined by q(e
x) = [e
x].
e → X/A(
e
e p)
q: X
X,
106
A. Tekcan, M. Bayraktar and O. Bizim
e p) be the discontinious proper group of homeomorphisms of X,
e
Proof. Since A(X,
e
e
q is a regular map and the automorphism group A(X, q) is isomorphic to A(X, p).
e p) is an universal covering space, (X,
e q) is an universal covering space of
Since (X,
e
e p). Therefore the automorphism group A(X,
e p) is isomorphic to the fundaX/A(
X,
³
´
e
e
e p), [e
mental group π(X, x) and A(X, q) is isomorphic to π X/A(
X,
x] . Hence the
fundamental groups π(X, x) and π(X̃/A(X̃, p), [x̃]) are isomorphic since A(X̃, q) is
isomorphic to A(X̃, p).
e p) be a universal covering space of a locally pathwise connected space
Let (X,
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
´
³
e
e p), [e
X,
x] .
fundamental group π X/A(
³
´
e p) is isomorphic to π X/A(
e
e p), [e
Corollary 2.8 A(X,
X,
x] .
e1 , p1 ) and (X
e2 , p2 ) be two universal covering space of a simply
Theorem 2.9 Let (X
connected space X and let G1 and G2 be discontinious proper group of homeomore1 and X
e2 , respectively. Then the diagram in Figure 2 is commutative.
phisms of X
Proof. Since X is simply connected, p1 and p2 are homeomorphisms and therefore
e1 , p1 ) and (X
e2 , p2 ) are
p1∗ and p2∗ are isomorphisms. On the other hand, since (X
e
e
universal covering space of X, there exists a homeomorphism h : X1 → X2 such that
e1 , x
e2 , x
e1 and x
p2 h = p1 . So h∗ : π(X
e1 ) → π(X
e2 ) is an isomorphism for x
e1 ∈ X
e2 ∈
e
e
e
X2 . Since these covering space are universal A(X1 , p1 ) and A(X2 , 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.
e p) is an universal covering space of a locally pathwise conTheorem 2.10 If (X,
³
´
e
e p), r of the quatient space
nected space X, then the automorphism group A X/A(
X,
e
e p) is isomorphic to the automorphism group A(X,
e p).
X/A(
X,
On the Covering Space and the Automorphism Group of the Covering Space
107
Proof. Since the universal covering is regular this covering is regular. Therefore there
e
e p) → X. On the other hand since this covexists a homeomorphism r : X/A(
X,
e p) is isomorphic
ering is universal from Lemma 1.8. the automorphism group A(X,
to the fundamental
group
³
´ π(X, x). Moreover from Theorem 2.5. the
³ automorphism
´
e
e p), r is isomorphic to the π(X, x). Therefore A X/A(
e
e p), r is
group A X/A(
X,
X,
e p).
isomorphic to the automorphism group A(X,
From above Theorem
e1 , p1 ) and (X
e2 , p2 ) are two universal covering spaces of a locally
Corollary 2.11 If (X
e1 /A(X
e1 , p1 ) and X
e2 /A(X
e2 , p2 ) are isomorphic.
pathwise connected space X then X
e1 , p1 ) and (X
e2 , p2 ) be two universal covering spaces of X. Then there
Proof. Let (X
e1 → X
e2 such that p2 h = p1 . On the other hand since
exists a homeomorphism h : X
e1 /A(X
e1 , p1 ) → X and
these coverings are regular there exist homeomorphisms r1 : X
e
e
e
e
e
e
r2 : X2 /A(X2 , p2 ) → X. Hence X1 /A(X1 , p1 ) and X2 /A(X2 , p2 ) are isomorphic.
e p) be a covering space of a locally pathwise connected space X. Then
Theorem 2.12 Let (X,
the number of the elements in the orbit [e
x] of the point x
e ∈ p−1 (x) is equal to the
e p) of X.
number of the sheets of the covering (X,
Proof. We know that the number of the elements in the orbit [e
x] of the point x
e∈
p−1 (x) is equal to the index of the isotropy subgroup which corresponding to x
e in
e x
π(X, x). Moreover isotropy subgroup which corresponding x
e is the subgroup p∗ π(X,
e)
of π(X, x). Since the number of the sheets of the covering is equal to the index of the
e x
subgroup p∗ π(X,
e) in π(X, x), the number of the sheets of the covering is equal to
the elements in the orbit [e
x] of the point x
e ∈ p−1 (x).
e p) is an universal covering space of a locally pathwise connected
Theorem 2.13 If (X,
³
´
e
e p), [e
space X, then the order of the fundamental group π X/A(
X,
x] of the quatient
e
e p) is equal to the number of sheets of the covering (X,
e p) of X.
space X/A(
X,
Proof. Since universal covering is regular this covering is regular, and therefore there
e
e p) → X. Thus r∗ is an isomorphism, i.e. the
exists a homeomorphism
³ r : X/A(X, ´
e
e p), [e
X,
x] and π(X, x) are isomorphic. Since this covfundamental groups π X/A(
e p) is isomorphic to the fundamental group π(X, x), and the
ering is universal A(X,
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
³
´
³ covering is equal
´ to the order of
e
e p), [e
e
e p), [e
the fundamental group X/A(
X,
x] since π X/A(
X,
x] and π(X, x) are
isomorphic.
e p) be a covering space of a locally pathwise connected space X .
Definition. Let (X,
e p) acts transitively on the set p−1 (x) , for every x ∈
If the automorphism group A(X,
e
X, then (X, p) is called Galois.
Theorem 2.14 Regular covering is Galois.
108
A. Tekcan, M. Bayraktar and O. Bizim
e p) be a regular covering space of X . Then from Theorem 2.3. there
Proof. Let (X,
e p) such that ϕ(e
e i.e. A(X,
e p) acts transitively
exist a ϕ ∈ A(X,
x1 ) = x
e2 for x
e1 , x
e2 ∈ X,
−1
e
on the set p (x) for x ∈ X. Thus from definition (X, p) is Galois covering of X .
We know from Theorem 2.4. that universal covering is regular. Thus from above
Theorem
Corollary 2.15 Universal covering is Galois.
e p) is a 2− sheeted covering space of a locally pathwise conTheorem 2.16 If (X,
nected space X, then this covering is Galois.
e p) be 2− sheeted covering space of X. Since the number of the sheets
Proof. Let (X,
e x
e x
of the covering is equal to the index of the subgroup p∗ π(X,
e) in π(X, x), p∗ π(X,
e)
e
is a normal subgroup of π(X, x). Therefore (X, p) is a regular covering space of X
e p) is a Galois covering of X.
and thus (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, AddisionWesley 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,
Görükle 16059, Bursa -TURKEY,
email: fahmet@uludag.edu.tr, bmustafa@uludag.edu.tr,
obizim@uludag.edu.tr