A covering of a topological space
X
{\displaystyle X}
is a continuous map
π
:
E
→
X
{\displaystyle \pi :E\rightarrow X}
with special properties.
Let
X
{\displaystyle X}
be a topological space. A covering of
X
{\displaystyle X}
is a continuous map
π
:
E
→
X
{\displaystyle \pi :E\rightarrow X}
s.t. there exists a discrete space
D
{\displaystyle D}
and for every
x
∈
X
{\displaystyle x\in X}
an open neighborhood
U
⊂
X
{\displaystyle U\subset X}
, s.t.
π
−
1
(
U
)
=
⨆
d
∈
D
V
d
{\displaystyle \pi ^{-1}(U)=\displaystyle \bigsqcup _{d\in D}V_{d}}
and
π
|
V
d
:
V
d
→
U
{\displaystyle \pi |_{V_{d}}:V_{d}\rightarrow U}
is a homeomorphism for every
d
∈
D
{\displaystyle d\in D}
.
Intuitively, a covering locally project a "stack of pancakes" above an open neighborhood
U
{\displaystyle U}
onto
U
{\displaystyle U}
Often, the notion of a covering is used for the covering space
E
{\displaystyle E}
as well as for the map
π
:
E
→
X
{\displaystyle \pi :E\rightarrow X}
. The open sets
V
d
{\displaystyle V_{d}}
are called sheets, which are uniquely determined if
U
{\displaystyle U}
is connected .[ 1]
p
.56
{\displaystyle \,^{p.56}}
For a
x
∈
X
{\displaystyle x\in X}
the discrete subset
π
−
1
(
x
)
{\displaystyle \pi ^{-1}(x)}
is called the fiber of
x
{\displaystyle x}
. The degree of a covering is the cardinality of the space
D
{\displaystyle D}
. If
E
{\displaystyle E}
is path-connected , then the covering
π
:
E
→
X
{\displaystyle \pi :E\rightarrow X}
is denoted as a path-connected covering .
For every topological space
X
{\displaystyle X}
there exists the covering
π
:
X
→
X
{\displaystyle \pi :X\rightarrow X}
with
π
(
x
)
=
x
{\displaystyle \pi (x)=x}
, which is denoted as the trivial covering of
X
.
{\displaystyle X.}
The space
Y
=
[
0
,
1
]
×
R
{\displaystyle Y=[0,1]\times \mathbb {R} }
is the covering space of
X
=
[
0
,
1
]
×
S
1
{\displaystyle X=[0,1]\times S^{1}}
. The disjoint open sets
S
i
{\displaystyle S_{i}}
are mapped homeomorphically onto
U
{\displaystyle U}
. The fiber of
x
{\displaystyle x}
consists of the points
y
i
{\displaystyle y_{i}}
.
The map
r
:
R
→
S
1
{\displaystyle r\colon \mathbb {R} \to S^{1}}
with
r
(
t
)
=
(
cos
(
2
π
t
)
,
sin
(
2
π
t
)
)
{\displaystyle r(t)=(\cos(2\pi t),\sin(2\pi t))}
is a covering of the unit circle
S
1
{\displaystyle S^{1}}
. For an open neighborhood
U
⊂
X
{\displaystyle U\subset X}
of an
x
∈
S
1
{\displaystyle x\in S^{1}}
, which has positiv
cos
(
2
π
t
)
{\displaystyle \cos(2\pi t)}
-value, one has:
r
−
1
(
U
)
=
⨆
n
∈
Z
(
n
−
1
4
,
n
+
1
4
)
{\displaystyle r^{-1}(U)=\displaystyle \bigsqcup _{n\in \mathbb {Z} }(n-{\frac {1}{4}},n+{\frac {1}{4}})}
.
Another covering of the unit circle is the map
q
:
S
1
→
S
1
{\displaystyle q\colon S^{1}\to S^{1}}
with
q
(
z
)
=
z
n
{\displaystyle q(z)=z^{n}}
for some
n
∈
N
{\displaystyle n\in \mathbb {N} }
. For an open neighborhood
U
⊂
X
{\displaystyle U\subset X}
of an
x
∈
S
1
{\displaystyle x\in S^{1}}
, one has:
q
−
1
(
U
)
=
⨆
i
=
1
n
U
{\displaystyle q^{-1}(U)=\displaystyle \bigsqcup _{i=1}^{n}U}
.
A map which is a local homeomorphism but not a covering of the unit circle is
p
:
R
+
→
S
1
{\displaystyle p\colon \mathbb {R_{+}} \to S^{1}}
with
p
(
t
)
=
(
cos
(
2
π
t
)
,
sin
(
2
π
t
)
)
{\displaystyle p(t)=(\cos(2\pi t),\sin(2\pi t))}
. There is a sheet of an open neighborhood of
(
1
,
0
)
{\displaystyle (1,0)}
, which is not mapped homeomorphically onto
U
{\displaystyle U}
.
Local homeomorphism [ edit ]
Since a covering
π
:
E
→
X
{\displaystyle \pi :E\rightarrow X}
maps each of the disjoint open sets of
π
−
1
(
U
)
{\displaystyle \pi ^{-1}(U)}
homeomorphically onto
U
{\displaystyle U}
it is a local homeomorphism, i.e.
π
{\displaystyle \pi }
is a continuous map and for every
e
∈
E
{\displaystyle e\in E}
there exists an open neighborhood
V
⊂
E
{\displaystyle V\subset E}
of
e
{\displaystyle e}
, s.t.
π
|
V
:
V
→
π
(
V
)
{\displaystyle \pi |_{V}:V\rightarrow \pi (V)}
is a homeomorphism.
It follows that the covering space
E
{\displaystyle E}
and the base space
X
{\displaystyle X}
locally share the same properties.
If
X
{\displaystyle X}
is a connected and non-orientable manifold , then there is a covering
π
:
X
~
→
X
{\displaystyle \pi :{\tilde {X}}\rightarrow X}
of degree
2
{\displaystyle 2}
, whereby
X
~
{\displaystyle {\tilde {X}}}
is a connected and orientable manifold.[ 1]
p
.234
{\displaystyle \,^{p.234}}
If
X
{\displaystyle X}
is a connected Lie group , then there is a covering
π
:
X
~
→
X
{\displaystyle \pi :{\tilde {X}}\rightarrow X}
which is also a Lie group homomorphism and
X
~
:=
{
γ
:
γ
is a path in X with
γ
(
0
)
=
1
X
modulo homotopy with fixed ends
}
{\displaystyle {\tilde {X}}:=\{\gamma :\gamma {\text{ is a path in X with }}\gamma (0)={\boldsymbol {1_{X}}}{\text{ modulo homotopy with fixed ends}}\}}
is a Lie group.[ 2]
p
.174
{\displaystyle ^{p.174}}
If
X
{\displaystyle X}
is a graph , then it follows for a covering
π
:
E
→
X
{\displaystyle \pi :E\rightarrow X}
that
E
{\displaystyle E}
is also a graph.[ 1]
p
.85
{\displaystyle ^{p.85}}
If
X
{\displaystyle X}
is a connected manifold , then there is a covering
π
:
X
~
→
X
{\displaystyle \pi :{\tilde {X}}\rightarrow X}
, whereby
X
~
{\displaystyle {\tilde {X}}}
is a connected and simply connected manifold.[ 3]
p
.32
{\displaystyle ^{p.32}}
If
X
{\displaystyle X}
is a connected Riemann surface , then there is a covering
π
:
X
~
→
X
{\displaystyle \pi :{\tilde {X}}\rightarrow X}
which is also a holomorphic map[ 3]
p
.22
{\displaystyle ^{p.22}}
and
X
~
{\displaystyle {\tilde {X}}}
is a connected and simply connected Riemann surface.[ 3]
p
.32
{\displaystyle ^{p.32}}
Let
p
,
q
{\displaystyle p,q}
and
r
{\displaystyle r}
be continuous maps, s.t. the diagram
commutes.
If
p
{\displaystyle p}
and
q
{\displaystyle q}
are coverings, so is
r
{\displaystyle r}
.[ 4]
p
.485
{\displaystyle ^{p.485}}
If
p
{\displaystyle p}
and
r
{\displaystyle r}
are coverings, so is
q
{\displaystyle q}
.[ 4]
p
.485
{\displaystyle ^{p.485}}
Product of coverings [ edit ]
Let
X
{\displaystyle X}
and
X
′
{\displaystyle X'}
be topological spaces and
p
:
E
→
X
{\displaystyle p:E\rightarrow X}
and
p
′
:
E
′
→
X
′
{\displaystyle p':E'\rightarrow X'}
be coverings, then
p
×
p
′
:
E
×
E
′
→
X
×
X
′
{\displaystyle p\times p':E\times E'\rightarrow X\times X'}
with
(
p
×
p
′
)
(
e
,
e
′
)
=
(
p
(
e
)
,
p
′
(
e
′
)
)
{\displaystyle (p\times p')(e,e')=(p(e),p'(e'))}
is a covering.[ 4]
p
.339
{\displaystyle ^{p.339}}
Equivalence of coverings [ edit ]
Let
X
{\displaystyle X}
be a topological space and
p
:
E
→
X
{\displaystyle p:E\rightarrow X}
and
p
′
:
E
′
→
X
{\displaystyle p':E'\rightarrow X}
be coverings. Both coverings are called equivalent , if there exists a homeomorphism
h
:
E
→
E
′
{\displaystyle h:E\rightarrow E'}
, s.t. the diagram
commutes. If such a homeomorphism exists, then one calls the covering spaces
E
{\displaystyle E}
and
E
′
{\displaystyle E'}
isomorphic .
An important property of the covering is, that it satisfies the lifting property , i.e.:
Let
I
{\displaystyle I}
be the unit interval and
p
:
E
→
X
{\displaystyle p:E\rightarrow X}
be a covering. Let
F
:
Y
×
I
→
X
{\displaystyle F:Y\times I\rightarrow X}
be a continuous map and
F
~
0
:
Y
×
{
0
}
→
E
{\displaystyle {\tilde {F}}_{0}:Y\times \{0\}\rightarrow E}
be a lift of
F
|
Y
×
{
0
}
{\displaystyle F|_{Y\times \{0\}}}
, i.e. a continuous map such that
p
∘
F
~
0
=
F
|
Y
×
{
0
}
{\displaystyle p\circ {\tilde {F}}_{0}=F|_{Y\times \{0\}}}
. Then there is a uniquely determined, continuous map
F
~
:
Y
×
I
→
E
{\displaystyle {\tilde {F}}:Y\times I\rightarrow E}
, which is a lift of
F
{\displaystyle F}
, i.e.
p
∘
F
~
=
F
{\displaystyle p\circ {\tilde {F}}=F}
.[ 1]
p
.60
{\displaystyle ^{p.60}}
If
X
{\displaystyle X}
is a path-connected space, then for
Y
=
{
0
}
{\displaystyle Y=\{0\}}
it follows that the map
F
~
{\displaystyle {\tilde {F}}}
is a lift of a path in
X
{\displaystyle X}
and for
Y
=
I
{\displaystyle Y=I}
it is a lift of a homotopy of paths in
X
{\displaystyle X}
.
Because of that property one can show, that the fundamental group
π
1
(
S
1
)
{\displaystyle \pi _{1}(S^{1})}
of the unit circle is an infinite cyclic group , which is generated by the homotopy classes of the loop
γ
:
I
→
S
1
{\displaystyle \gamma :I\rightarrow S^{1}}
with
γ
(
t
)
=
(
cos
(
2
π
t
)
,
sin
(
2
π
t
)
)
{\displaystyle \gamma (t)=(\cos(2\pi t),\sin(2\pi t))}
.[ 1]
p
.29
{\displaystyle ^{p.29}}
Let
X
{\displaystyle X}
be a path-connected space and
p
:
E
→
X
{\displaystyle p:E\rightarrow X}
be a connected covering. Let
x
,
y
∈
X
{\displaystyle x,y\in X}
be any two points, which are connected by a path
γ
{\displaystyle \gamma }
, i.e.
γ
(
0
)
=
x
{\displaystyle \gamma (0)=x}
and
γ
(
1
)
=
y
{\displaystyle \gamma (1)=y}
. Let
γ
~
{\displaystyle {\tilde {\gamma }}}
be the unique lift of
γ
{\displaystyle \gamma }
, then the map
L
γ
:
p
−
1
(
x
)
→
p
−
1
(
y
)
{\displaystyle L_{\gamma }:p^{-1}(x)\rightarrow p^{-1}(y)}
with
L
γ
(
γ
~
(
0
)
)
=
γ
~
(
1
)
{\displaystyle L_{\gamma }({\tilde {\gamma }}(0))={\tilde {\gamma }}(1)}
is bijective .[ 1]
p
.69
{\displaystyle ^{p.69}}
If
X
{\displaystyle X}
is a path-connected space and
p
:
E
→
X
{\displaystyle p:E\rightarrow X}
a connected covering, then the induced group homomorphism
p
#
:
π
1
(
E
)
→
π
1
(
X
)
{\displaystyle p_{\#}:\pi _{1}(E)\rightarrow \pi _{1}(X)}
with
p
#
(
[
γ
]
)
=
[
p
∘
γ
]
{\displaystyle p_{\#}([\gamma ])=[p\circ \gamma ]}
,
is injective and the subgroup
p
#
(
π
1
(
E
)
)
{\displaystyle p_{\#}(\pi _{1}(E))}
of
π
1
(
X
)
{\displaystyle \pi _{1}(X)}
consists of the homotopy classes of loops in
X
{\displaystyle X}
, whose lifts are loops in
E
{\displaystyle E}
.[ 1]
p
.61
{\displaystyle ^{p.61}}
Holomorphic maps between Riemann surfaces [ edit ]
Let
X
{\displaystyle X}
and
Y
{\displaystyle Y}
be Riemann surfaces , i.e. one dimensional complex manifolds , and let
f
:
X
→
Y
{\displaystyle f:X\rightarrow Y}
be a continuous map.
f
{\displaystyle f}
is holomorphic in a point
x
∈
X
{\displaystyle x\in X}
, if for any charts
ϕ
x
:
U
1
→
V
1
{\displaystyle \phi _{x}:U_{1}\rightarrow V_{1}}
of
x
{\displaystyle x}
and
ϕ
f
(
x
)
:
U
2
→
V
2
{\displaystyle \phi _{f(x)}:U_{2}\rightarrow V_{2}}
of
f
(
x
)
{\displaystyle f(x)}
, with
ϕ
x
(
U
1
)
⊂
U
2
{\displaystyle \phi _{x}(U_{1})\subset U_{2}}
, the map
ϕ
f
(
x
)
∘
f
∘
ϕ
x
−
1
:
C
→
C
{\displaystyle \phi _{f(x)}\circ f\circ \phi _{x}^{-1}:\mathbb {C} \rightarrow \mathbb {C} }
is holomorphic .
If
f
{\displaystyle f}
is for all
x
∈
X
{\displaystyle x\in X}
holomorphic, we say
f
{\displaystyle f}
is holomorphic.
The map
F
=
ϕ
f
(
x
)
∘
f
∘
ϕ
x
−
1
{\displaystyle F=\phi _{f(x)}\circ f\circ \phi _{x}^{-1}}
is called the local expression of
f
{\displaystyle f}
in
x
∈
X
{\displaystyle x\in X}
.
If
f
:
X
→
Y
{\displaystyle f:X\rightarrow Y}
is a non-constant, holomorphic map between compact Riemann surfaces , then
f
{\displaystyle f}
is surjective [ 3]
p
.11
{\displaystyle ^{p.11}}
and an open map [ 3]
p
.11
{\displaystyle ^{p.11}}
, i.e. for every open set
U
⊂
X
{\displaystyle U\subset X}
the image
f
(
U
)
⊂
Y
{\displaystyle f(U)\subset Y}
is also open.
Ramification point and branch point [ edit ]
Let
f
:
X
→
Y
{\displaystyle f:X\rightarrow Y}
be a non-constant, holomorphic map between compact Riemann surfaces. For every
x
∈
X
{\displaystyle x\in X}
there exist charts for
x
{\displaystyle x}
and
f
(
x
)
{\displaystyle f(x)}
and there exists a uniquely determined
k
x
∈
N
>
0
{\displaystyle k_{x}\in \mathbb {N_{>0}} }
, s.t. the local expression
F
{\displaystyle F}
of
f
{\displaystyle f}
in
x
{\displaystyle x}
is of the form
z
↦
z
k
x
{\displaystyle z\mapsto z^{k_{x}}}
.[ 3]
p
.10
{\displaystyle ^{p.10}}
The number
k
x
{\displaystyle k_{x}}
is called the ramification index of
f
{\displaystyle f}
in
x
{\displaystyle x}
and the point
x
∈
X
{\displaystyle x\in X}
is called a ramification point if
k
x
≥
2
{\displaystyle k_{x}\geq 2}
. If
k
x
=
1
{\displaystyle k_{x}=1}
for an
x
∈
X
{\displaystyle x\in X}
, then
x
{\displaystyle x}
is unramified . The image point
y
=
f
(
x
)
∈
Y
{\displaystyle y=f(x)\in Y}
of a ramification point is called a branch point.
Degree of a holomorphic map [ edit ]
Let
f
:
X
→
Y
{\displaystyle f:X\rightarrow Y}
be a non-constant, holomorphic map between compact Riemann surfaces. The degree
d
e
g
(
f
)
{\displaystyle deg(f)}
of
f
{\displaystyle f}
is the cardinality of the fiber of an unramified point
y
=
f
(
x
)
∈
Y
{\displaystyle y=f(x)\in Y}
, i.e.
d
e
g
(
f
)
:=
|
f
−
1
(
y
)
|
{\displaystyle deg(f):=|f^{-1}(y)|}
.
This number is well-defined, since for every
y
∈
Y
{\displaystyle y\in Y}
the fiber
f
−
1
(
y
)
{\displaystyle f^{-1}(y)}
is discrete[ 3]
p
.20
{\displaystyle ^{p.20}}
and for any two unramified points
y
1
,
y
2
∈
Y
{\displaystyle y_{1},y_{2}\in Y}
, it is:
|
f
−
1
(
y
1
)
|
=
|
f
−
1
(
y
2
)
|
.
{\displaystyle |f^{-1}(y_{1})|=|f^{-1}(y_{2})|.}
[ 3]
p
.29
{\displaystyle ^{p.29}}
It can be calculated by:
∑
x
∈
f
−
1
(
y
)
k
x
=
d
e
g
(
f
)
{\displaystyle \sum _{x\in f^{-1}(y)}k_{x}=deg(f)}
[ 3]
p
.29
{\displaystyle ^{p.29}}
A continuous map
f
:
X
→
Y
{\displaystyle f:X\rightarrow Y}
is called a branched covering , if there exists a closed set with dense complement
E
⊂
Y
{\displaystyle E\subset Y}
, s.t.
f
|
X
∖
f
−
1
(
E
)
:
X
∖
f
−
1
(
E
)
→
Y
∖
E
{\displaystyle f_{|X\smallsetminus f^{-1}(E)}:X\smallsetminus f^{-1}(E)\rightarrow Y\smallsetminus E}
is a covering.
Let
n
∈
N
{\displaystyle n\in \mathbb {N} }
and
n
≥
2
{\displaystyle n\geq 2}
, then
f
:
C
→
C
{\displaystyle f:\mathbb {C} \rightarrow \mathbb {C} }
with
f
(
z
)
=
z
n
{\displaystyle f(z)=z^{n}}
is branched covering of degree
n
{\displaystyle n}
, whereby
z
=
0
{\displaystyle z=0}
is a branch point.
Every non-constant, holomorphic map between compact Riemann surfaces
f
:
X
→
Y
{\displaystyle f:X\rightarrow Y}
of degree
d
{\displaystyle d}
is a branched covering of degree
d
{\displaystyle d}
.
Let
p
:
X
~
→
X
{\displaystyle p:{\tilde {X}}\rightarrow X}
be a simply connected covering and
β
:
E
→
X
{\displaystyle \beta :E\rightarrow X}
be a covering, then there exists a uniquely determined covering
α
:
X
~
→
E
{\displaystyle \alpha :{\tilde {X}}\rightarrow E}
, s.t. the diagram
commutes.[ 4]
p
.486
{\displaystyle ^{p.486}}
Let
p
:
X
~
→
X
{\displaystyle p:{\tilde {X}}\rightarrow X}
be a simply connected covering. If
β
:
E
→
X
{\displaystyle \beta :E\rightarrow X}
is another simply connected covering, then there exists a uniquely determined homeomorphism
α
:
X
~
→
E
{\displaystyle \alpha :{\tilde {X}}\rightarrow E}
, s.t. the diagram
commutes.[ 4]
p
.482
{\displaystyle ^{p.482}}
This means that
p
{\displaystyle p}
is, up to equivalence, uniquely determined and because of that universal property denoted as the universal covering of the space
X
{\displaystyle X}
.
A universal covering does not always exists, but the following properties guarantee the existence:
Let
X
{\displaystyle X}
be a connected, locally simply connected , then there exists a universal covering
p
:
X
~
→
X
{\displaystyle p:{\tilde {X}}\rightarrow X}
.
X
~
{\displaystyle {\tilde {X}}}
is defined as
X
~
:=
{
γ
:
γ
is a path in
X
with
γ
(
0
)
=
x
0
}
/
homotopy with fixed ends
{\displaystyle {\tilde {X}}:=\{\gamma :\gamma {\text{ is a path in }}X{\text{ with }}\gamma (0)=x_{0}\}/{\text{ homotopy with fixed ends}}}
and
p
:
X
~
→
X
{\displaystyle p:{\tilde {X}}\rightarrow X}
by
p
(
[
γ
]
)
:=
γ
(
1
)
{\displaystyle p([\gamma ]):=\gamma (1)}
.[ 1]
p
.64
{\displaystyle ^{p.64}}
The topology on
X
~
{\displaystyle {\tilde {X}}}
is constructed as follows: Let
γ
:
I
→
X
{\displaystyle \gamma :I\rightarrow X}
be a path with
γ
(
0
)
=
x
0
{\displaystyle \gamma (0)=x_{0}}
. Let
U
{\displaystyle U}
be a simply connected neighborhood of the endpoint
x
=
γ
(
1
)
{\displaystyle x=\gamma (1)}
, then for every
y
∈
U
{\displaystyle y\in U}
the paths
σ
y
{\displaystyle \sigma _{y}}
inside
U
{\displaystyle U}
from
x
{\displaystyle x}
to
y
{\displaystyle y}
are uniquely determined up to homotopy . Now consider
U
~
:=
{
γ
.
σ
y
:
y
∈
U
}
/
homotopy with fixed ends
{\displaystyle {\tilde {U}}:=\{\gamma .\sigma _{y}:y\in U\}/{\text{ homotopy with fixed ends}}}
, then
p
|
U
~
:
U
~
→
U
{\displaystyle p_{|{\tilde {U}}}:{\tilde {U}}\rightarrow U}
with
p
(
[
γ
.
σ
y
]
)
=
γ
.
σ
y
(
1
)
=
y
{\displaystyle p([\gamma .\sigma _{y}])=\gamma .\sigma _{y}(1)=y}
is a bijection and
U
~
{\displaystyle {\tilde {U}}}
can be equipped with the final topology of
p
|
U
~
{\displaystyle p_{|{\tilde {U}}}}
.
The fundamental group
π
1
(
X
,
x
0
)
=
Γ
{\displaystyle \pi _{1}(X,x_{0})=\Gamma }
acts freely through
(
[
γ
]
,
[
x
~
]
)
↦
[
γ
.
x
~
]
{\displaystyle ([\gamma ],[{\tilde {x}}])\mapsto [\gamma .{\tilde {x}}]}
on
X
~
{\displaystyle {\tilde {X}}}
and
ψ
:
Γ
∖
X
~
→
X
{\displaystyle \psi :\Gamma \backslash {\tilde {X}}\rightarrow X}
with
ψ
(
[
Γ
x
~
]
)
=
x
~
(
1
)
{\displaystyle \psi ([\Gamma {\tilde {x}}])={\tilde {x}}(1)}
is a homeomorphism, i.e.
Γ
∖
X
~
≅
X
{\displaystyle \Gamma \backslash {\tilde {X}}\cong X}
.
r
:
R
→
S
1
{\displaystyle r\colon \mathbb {R} \to S^{1}}
with
r
(
t
)
=
(
cos
(
2
π
t
)
,
sin
(
2
π
t
)
)
{\displaystyle r(t)=(\cos(2\pi t),\sin(2\pi t))}
is the universal covering of the unit circle
S
1
{\displaystyle S^{1}}
.
p
:
S
n
→
R
P
n
≅
{
+
1
,
−
1
}
∖
S
n
{\displaystyle p\colon S^{n}\to \mathbb {R} P^{n}\cong \{+1,-1\}\backslash S^{n}}
with
p
(
x
)
=
[
x
]
{\displaystyle p(x)=[x]}
is the universal covering of the projective space
R
P
n
{\displaystyle \mathbb {R} P^{n}}
for
n
>
1
{\displaystyle n>1}
.
q
:
S
U
(
n
)
⋉
R
→
U
(
n
)
{\displaystyle q\colon SU(n)\ltimes \mathbb {R} \to U(n)}
with
q
(
A
,
t
)
=
[
exp
(
2
π
i
t
)
0
0
I
n
−
1
]
A
{\displaystyle q(A,t)={\begin{bmatrix}\exp(2\pi it)&0\\0&I_{n-1}\end{bmatrix}}A}
is the universal covering of the unitary group
U
(
n
)
{\displaystyle U(n)}
.[ 5]
Since
S
U
(
2
)
≅
S
3
{\displaystyle SU(2)\cong S^{3}}
, it follows that the quotient map
f
:
S
U
(
2
)
→
Z
2
∖
S
U
(
2
)
≅
S
O
(
3
)
{\displaystyle f:SU(2)\rightarrow \mathbb {Z_{2}} \backslash SU(2)\cong SO(3)}
is the universal covering of the
S
O
(
3
)
{\displaystyle SO(3)}
.The Hawaiian earring. Only the ten largest circles are shown.
A topological space, which has no universal covering is the Hawaiian earring :
X
=
⋃
n
∈
N
{
(
x
1
,
x
2
)
∈
R
2
:
(
x
1
−
1
n
)
2
+
x
2
2
=
1
n
2
}
{\displaystyle X=\bigcup _{n\in \mathbb {N} }\left\{(x_{1},x_{2})\in \mathbb {R} ^{2}:{\Bigl (}x_{1}-{\frac {1}{n}}{\Bigr )}^{2}+x_{2}^{2}={\frac {1}{n^{2}}}\right\}}
One can show, that no neighborhood of the origin
(
0
,
0
)
{\displaystyle (0,0)}
is simply connected.[ 4]
p
.487
E
x
a
m
p
l
e
1
{\displaystyle ^{p.487\,Example\,1}}
Let
p
:
E
→
X
{\displaystyle p:E\rightarrow X}
be a covering. A deck transformation is a homeomorphism
d
:
E
→
E
{\displaystyle d:E\rightarrow E}
, s.t. the diagram of continuous maps
commutes. Together with the composition of maps, the set of deck transformation forms a group
D
e
c
k
(
p
)
{\displaystyle Deck(p)}
, which is the same as
A
u
t
(
p
)
{\displaystyle Aut(p)}
.
Let
q
:
S
1
→
S
1
{\displaystyle q\colon S^{1}\to S^{1}}
be the covering
q
(
z
)
=
z
n
{\displaystyle q(z)=z^{n}}
for some
n
∈
N
{\displaystyle n\in \mathbb {N} }
, then the map
d
:
S
1
→
S
1
:
z
↦
z
e
2
π
i
/
n
{\displaystyle d:S^{1}\rightarrow S^{1}:z\mapsto z\,e^{2\pi i/n}}
is a deck transformation and
D
e
c
k
(
q
)
≅
Z
/
n
Z
{\displaystyle Deck(q)\cong \mathbb {Z} /\mathbb {nZ} }
.
Let
r
:
R
→
S
1
{\displaystyle r\colon \mathbb {R} \to S^{1}}
be the covering
r
(
t
)
=
(
cos
(
2
π
t
)
,
sin
(
2
π
t
)
)
{\displaystyle r(t)=(\cos(2\pi t),\sin(2\pi t))}
, then the map
d
k
:
R
→
R
:
t
↦
t
+
k
{\displaystyle d_{k}:\mathbb {R} \rightarrow \mathbb {R} :t\mapsto t+k}
with
k
∈
Z
{\displaystyle k\in \mathbb {Z} }
is a deck transformation and
D
e
c
k
(
r
)
≅
Z
{\displaystyle Deck(r)\cong \mathbb {Z} }
.
Let
X
{\displaystyle X}
be a path-connected space and
p
:
E
→
X
{\displaystyle p:E\rightarrow X}
be a connected covering. Since a deck transformation
d
:
E
→
E
{\displaystyle d:E\rightarrow E}
is bijective , it permutes the elements of a fiber
p
−
1
(
x
)
{\displaystyle p^{-1}(x)}
with
x
∈
X
{\displaystyle x\in X}
and is uniquely determined by where it sends a single point. In particular, only the identity map fixes a point in the fiber.[ 1]
p
.70
{\displaystyle ^{p.70}}
Because of this property every deck transformation defines a group action on
E
{\displaystyle E}
, i.e. let
U
⊂
X
{\displaystyle U\subset X}
be an open neighborhood of a
x
∈
X
{\displaystyle x\in X}
and
U
~
⊂
E
{\displaystyle {\tilde {U}}\subset E}
an open neighborhood of an
e
∈
p
−
1
(
x
)
{\displaystyle e\in p^{-1}(x)}
, then
D
e
c
k
(
p
)
×
E
→
E
:
(
d
,
U
~
)
↦
d
(
U
~
)
{\displaystyle Deck(p)\times E\rightarrow E:(d,{\tilde {U}})\mapsto d({\tilde {U}})}
is a group action .
A covering
p
:
E
→
X
{\displaystyle p:E\rightarrow X}
is called normal, if
D
e
c
k
(
p
)
∖
X
≅
E
{\displaystyle Deck(p)\backslash X\cong E}
. This means, that for every
x
∈
X
{\displaystyle x\in X}
and any two
e
0
,
e
1
∈
p
−
1
(
x
)
{\displaystyle e_{0},e_{1}\in p^{-1}(x)}
there exists a deck transformation
d
:
E
→
E
{\displaystyle d:E\rightarrow E}
, s.t.
d
(
e
0
)
=
e
1
{\displaystyle d(e_{0})=e_{1}}
.
Let
X
{\displaystyle X}
be a path-connected space and
p
:
E
→
X
{\displaystyle p:E\rightarrow X}
be a connected covering. Let
H
=
p
#
(
π
1
(
E
)
)
{\displaystyle H=p_{\#}(\pi _{1}(E))}
be a subgroup of
π
1
(
X
)
{\displaystyle \pi _{1}(X)}
, then
p
{\displaystyle p}
is a normal covering iff
H
{\displaystyle H}
is a normal subgroup of
π
1
(
X
)
{\displaystyle \pi _{1}(X)}
.[ 1]
p
.71
{\displaystyle ^{p.71}}
If
p
:
E
→
X
{\displaystyle p:E\rightarrow X}
is a normal covering and
H
=
p
#
(
π
1
(
E
)
)
{\displaystyle H=p_{\#}(\pi _{1}(E))}
, then
D
e
c
k
(
p
)
≅
π
1
(
X
)
/
H
{\displaystyle Deck(p)\cong \pi _{1}(X)/H}
.[ 1]
p
.71
{\displaystyle ^{p.71}}
If
p
:
E
→
X
{\displaystyle p:E\rightarrow X}
is a path-connected covering and
H
=
p
#
(
π
1
(
E
)
)
{\displaystyle H=p_{\#}(\pi _{1}(E))}
, then
D
e
c
k
(
p
)
≅
N
(
H
)
/
H
{\displaystyle Deck(p)\cong N(H)/H}
, whereby
N
(
H
)
{\displaystyle N(H)}
is the normaliser of
H
{\displaystyle H}
.[ 1]
p
.71
{\displaystyle ^{p.71}}
Let
E
{\displaystyle E}
be a topological space. A group
Γ
{\displaystyle \Gamma }
acts discontinuously on
E
{\displaystyle E}
, if every
e
∈
E
{\displaystyle e\in E}
has an open neighborhood
V
⊂
E
{\displaystyle V\subset E}
with
V
≠
∅
{\displaystyle V\neq \emptyset }
, such that for every
γ
∈
Γ
{\displaystyle \gamma \in \Gamma }
with
γ
V
∩
V
≠
∅
{\displaystyle \gamma V\cap V\neq \emptyset }
one has
d
1
=
d
2
{\displaystyle d_{1}=d_{2}}
.
If a group
Γ
{\displaystyle \Gamma }
acts discontinuously on a topological space
E
{\displaystyle E}
, then the quotient map
q
:
E
→
Γ
∖
E
{\displaystyle q:E\rightarrow \Gamma \backslash E}
with
q
(
e
)
=
Γ
e
{\displaystyle q(e)=\Gamma e}
is a normal covering.[ 1]
p
.72
{\displaystyle ^{p.72}}
Hereby
Γ
∖
E
=
{
Γ
e
:
e
∈
E
}
{\displaystyle \Gamma \backslash E=\{\Gamma e:e\in E\}}
is the quotient space and
Γ
e
=
{
γ
(
e
)
:
γ
∈
Γ
}
{\displaystyle \Gamma e=\{\gamma (e):\gamma \in \Gamma \}}
is the orbit of the group action.
The covering
q
:
S
1
→
S
1
{\displaystyle q\colon S^{1}\to S^{1}}
with
q
(
z
)
=
z
n
{\displaystyle q(z)=z^{n}}
is a normal coverings for every
n
∈
N
{\displaystyle n\in \mathbb {N} }
.
Every simply connected covering is a normal covering.
Let
Γ
{\displaystyle \Gamma }
be a group, which acts discontinuously on a topological space
E
{\displaystyle E}
and let
q
:
E
→
Γ
∖
E
{\displaystyle q:E\rightarrow \Gamma \backslash E}
be the normal covering.
If
E
{\displaystyle E}
is path-connected, then
D
e
c
k
(
q
)
≅
Γ
{\displaystyle Deck(q)\cong \Gamma }
.[ 1]
p
.72
{\displaystyle ^{p.72}}
If
E
{\displaystyle E}
is simply connected, then
D
e
c
k
(
q
)
≅
π
1
(
X
)
{\displaystyle Deck(q)\cong \pi _{1}(X)}
.[ 1]
p
.71
{\displaystyle ^{p.71}}
Let
n
∈
N
{\displaystyle n\in \mathbb {N} }
. The antipodal map
g
:
S
n
→
S
n
{\displaystyle g:S^{n}\rightarrow S^{n}}
with
g
(
x
)
=
−
x
{\displaystyle g(x)=-x}
generates, together with the composition of maps, a group
D
(
g
)
≅
Z
/
2
Z
{\displaystyle D(g)\cong \mathbb {Z/2Z} }
and induces a group action
D
(
g
)
×
S
n
→
S
n
,
(
g
,
x
)
↦
g
(
x
)
{\displaystyle D(g)\times S^{n}\rightarrow S^{n},(g,x)\mapsto g(x)}
, which acts discontinuously on
S
n
{\displaystyle S^{n}}
. Because of
Z
2
∖
S
n
≅
R
P
n
{\displaystyle \mathbb {Z_{2}} \backslash S^{n}\cong \mathbb {R} P^{n}}
it follows, that the quotient map
q
:
S
n
→
Z
2
∖
S
n
≅
R
P
n
{\displaystyle q\colon S^{n}\rightarrow \mathbb {Z_{2}} \backslash S^{n}\cong \mathbb {R} P^{n}}
is a normal covering and for
n
>
1
{\displaystyle n>1}
a universal covering, hence
D
e
c
k
(
q
)
≅
Z
/
2
Z
≅
π
1
(
R
P
n
)
{\displaystyle Deck(q)\cong \mathbb {Z/2Z} \cong \pi _{1}({\mathbb {R} P^{n}})}
for
n
>
1
{\displaystyle n>1}
.
Let
S
O
(
3
)
{\displaystyle SO(3)}
be the special orthogonal group , then the map
f
:
S
U
(
2
)
→
S
O
(
3
)
≅
Z
2
∖
S
U
(
2
)
{\displaystyle f:SU(2)\rightarrow SO(3)\cong \mathbb {Z_{2}} \backslash SU(2)}
is a normal covering and because of
S
U
(
2
)
≅
S
3
{\displaystyle SU(2)\cong S^{3}}
, it is the universal covering, hence
D
e
c
k
(
f
)
≅
Z
/
2
Z
≅
π
1
(
S
O
(
3
)
)
{\displaystyle Deck(f)\cong \mathbb {Z/2Z} \cong \pi _{1}(SO(3))}
.
With the group action
(
z
1
,
z
2
)
∗
(
x
,
y
)
=
(
z
1
+
(
−
1
)
z
2
x
,
z
2
+
y
)
{\displaystyle (z_{1},z_{2})*(x,y)=(z_{1}+(-1)^{z_{2}}x,z_{2}+y)}
of
Z
2
{\displaystyle \mathbb {Z^{2}} }
on
R
2
{\displaystyle \mathbb {R^{2}} }
, whereby
(
Z
2
,
∗
)
{\displaystyle (\mathbb {Z^{2}} ,*)}
is the semidirect product
Z
⋊
Z
{\displaystyle \mathbb {Z} \rtimes \mathbb {Z} }
, one gets the universal covering
f
:
R
2
→
(
Z
⋊
Z
)
∖
R
2
≅
K
{\displaystyle f:\mathbb {R^{2}} \rightarrow (\mathbb {Z} \rtimes \mathbb {Z} )\backslash \mathbb {R^{2}} \cong K}
of the klein bottle
K
{\displaystyle K}
, hence
D
e
c
k
(
f
)
≅
Z
⋊
Z
≅
π
1
(
K
)
{\displaystyle Deck(f)\cong \mathbb {Z} \rtimes \mathbb {Z} \cong \pi _{1}(K)}
.
Let
T
=
S
1
×
S
1
{\displaystyle T=S^{1}\times S^{1}}
be the torus which is embedded in the
C
2
{\displaystyle \mathbb {C^{2}} }
. Then one gets a homeomorphism
α
:
T
→
T
:
(
e
i
x
,
e
i
y
)
↦
(
e
i
(
x
+
π
)
,
e
−
i
y
)
{\displaystyle \alpha :T\rightarrow T:(e^{ix},e^{iy})\mapsto (e^{i(x+\pi )},e^{-iy})}
, which induces a discontinuous group action
G
α
×
T
→
T
{\displaystyle G_{\alpha }\times T\rightarrow T}
, whereby
G
α
≅
Z
/
2
Z
{\displaystyle G_{\alpha }\cong \mathbb {Z/2Z} }
. It follows, that the map
f
:
T
→
G
α
∖
T
≅
K
{\displaystyle f:T\rightarrow G_{\alpha }\backslash T\cong K}
is a normal covering of the klein bottle, hence
D
e
c
k
(
f
)
≅
Z
/
2
Z
{\displaystyle Deck(f)\cong \mathbb {Z/2Z} }
.
Let
S
3
{\displaystyle S^{3}}
be embedded in the
C
2
{\displaystyle \mathbb {C^{2}} }
. Since the group action
S
3
×
Z
/
p
Z
→
S
3
:
(
(
z
1
,
z
2
)
,
[
k
]
)
↦
(
e
2
π
i
k
/
p
z
1
,
e
2
π
i
k
q
/
p
z
2
)
{\displaystyle S^{3}\times \mathbb {Z/pZ} \rightarrow S^{3}:((z_{1},z_{2}),[k])\mapsto (e^{2\pi ik/p}z_{1},e^{2\pi ikq/p}z_{2})}
is discontinuously, whereby
p
,
q
∈
N
{\displaystyle p,q\in \mathbb {N} }
are coprime , the map
f
:
S
3
→
Z
p
∖
S
3
=:
L
p
,
q
{\displaystyle f:S^{3}\rightarrow \mathbb {Z_{p}} \backslash S^{3}=:L_{p,q}}
is the universal covering of the lens space
L
p
,
q
{\displaystyle L_{p,q}}
, hence
D
e
c
k
(
f
)
≅
Z
/
p
Z
≅
π
1
(
L
p
,
q
)
{\displaystyle Deck(f)\cong \mathbb {Z/pZ} \cong \pi _{1}(L_{p,q})}
.
Galois correspondence [ edit ]
Let
X
{\displaystyle X}
be a connected and locally simply connected space, then for every subgroup
H
⊆
π
1
(
X
)
{\displaystyle H\subseteq \pi _{1}(X)}
there exists a path-connected covering
α
:
X
H
→
X
{\displaystyle \alpha :X_{H}\rightarrow X}
with
α
#
(
π
1
(
X
H
)
)
=
H
{\displaystyle \alpha _{\#}(\pi _{1}(X_{H}))=H}
.[ 1]
p
.66
{\displaystyle ^{p.66}}
Let
p
1
:
E
→
X
{\displaystyle p_{1}:E\rightarrow X}
and
p
2
:
E
′
→
X
{\displaystyle p_{2}:E'\rightarrow X}
be two path-connected coverings, then they are equivalent iff the subgroups
H
=
p
1
#
(
π
1
(
E
)
)
{\displaystyle H=p_{1\#}(\pi _{1}(E))}
and
H
′
=
p
2
#
(
π
1
(
E
′
)
)
{\displaystyle H'=p_{2\#}(\pi _{1}(E'))}
are conjugate to each other.[ 4]
p
.482
{\displaystyle ^{p.482}}
Let
X
{\displaystyle X}
be a connected and locally simply connected space, then, up to equivalence between coverings, there is a bijection:
{
Subgroup of
π
1
(
X
)
}
⟷
{
path-connected covering
p
:
E
→
X
}
H
⟶
α
:
X
H
→
X
p
#
(
π
1
(
E
)
)
⟵
p
{
normal subgroup of
π
1
(
X
)
}
⟷
{
normal covering
p
:
E
→
X
}
H
⟶
α
:
X
H
→
X
p
#
(
π
1
(
E
)
)
⟵
p
{\displaystyle {\begin{matrix}\qquad \displaystyle \{{\text{Subgroup of }}\pi _{1}(X)\}&\longleftrightarrow &\displaystyle \{{\text{path-connected covering }}p:E\rightarrow X\}\\H&\longrightarrow &\alpha :X_{H}\rightarrow X\\p_{\#}(\pi _{1}(E))&\longleftarrow &p\\\displaystyle \{{\text{normal subgroup of }}\pi _{1}(X)\}&\longleftrightarrow &\displaystyle \{{\text{normal covering }}p:E\rightarrow X\}\\H&\longrightarrow &\alpha :X_{H}\rightarrow X\\p_{\#}(\pi _{1}(E))&\longleftarrow &p\end{matrix}}}
For a sequence of subgroups
{
e
}
⊂
H
⊂
G
⊂
π
1
(
X
)
{\displaystyle \displaystyle \{{\text{e}}\}\subset H\subset G\subset \pi _{1}(X)}
one gets a sequence of coverings
X
~
⟶
X
H
≅
H
∖
X
~
⟶
X
G
≅
G
∖
X
~
⟶
X
≅
π
1
(
X
)
∖
X
~
{\displaystyle {\tilde {X}}\longrightarrow X_{H}\cong H\backslash {\tilde {X}}\longrightarrow X_{G}\cong G\backslash {\tilde {X}}\longrightarrow X\cong \pi _{1}(X)\backslash {\tilde {X}}}
. For a subgroup
H
⊂
π
1
(
X
)
{\displaystyle H\subset \pi _{1}(X)}
with index
[
π
1
(
X
)
:
H
]
=
d
{\displaystyle \displaystyle [\pi _{1}(X):H]=d}
, the covering
α
:
X
H
→
X
{\displaystyle \alpha :X_{H}\rightarrow X}
has degree
d
{\displaystyle d}
.
Category of coverings [ edit ]
Let
X
{\displaystyle X}
be a topological space. The objects of the category
C
o
v
(
X
)
{\displaystyle {\boldsymbol {Cov(X)}}}
are the coverings
p
:
E
→
X
{\displaystyle p:E\rightarrow X}
of
X
{\displaystyle X}
and the morphisms between two coverings
p
:
E
→
X
{\displaystyle p:E\rightarrow X}
and
q
:
F
→
X
{\displaystyle q:F\rightarrow X}
are continuous maps
f
:
E
→
F
{\displaystyle f:E\rightarrow F}
, s.t. the diagram
commutes.
Let
G
{\displaystyle G}
be a topological group . The category
G
−
S
e
t
{\displaystyle {\boldsymbol {G-Set}}}
is the category of sets which are G-sets . The morphisms are G-maps
ϕ
:
X
→
Y
{\displaystyle \phi :X\rightarrow Y}
between G-sets. They satisfy the condition
ϕ
(
g
x
)
=
g
ϕ
(
x
)
{\displaystyle \phi (gx)=g\,\phi (x)}
for every
g
∈
G
{\displaystyle g\in G}
.
Let
X
{\displaystyle X}
be a connected and locally simply connected space,
x
∈
X
{\displaystyle x\in X}
and
G
=
π
1
(
X
,
x
)
{\displaystyle G=\pi _{1}(X,x)}
be the fundamental group of
X
{\displaystyle X}
. Since
G
{\displaystyle G}
defines, by lifting of paths and evaluating at the endpoint of the lift, a group action on the fiber of a covering, the functor
F
:
C
o
v
(
X
)
⟶
G
−
S
e
t
:
p
↦
p
−
1
(
x
)
{\displaystyle F:{\boldsymbol {Cov(X)}}\longrightarrow {\boldsymbol {G-Set}}:p\mapsto p^{-1}(x)}
is an equivalence of categories .[ 1]
p
.68
−
70
{\displaystyle ^{p.68-70}}
Allen Hatcher: Algebraic Topology . Cambridge Univ. Press, Cambridge, ISBN 0-521-79160-X
Otto Forster: Lectures on Riemann surfaces . Springer Berlin, München 1991, ISBN 978-3-540-90617-9
James Munkres: Topology . Upper Saddle River, NJ: Prentice Hall, Inc., ©2000, ISBN 978-0-13-468951-7
Wolfgang Kühnel: Matrizen und Lie-Gruppen . Springer Fachmedien Wiesbaden GmbH, Stuttgart, ISBN 978-3-8348-9905-7
Maximiliano Aguilar and Miguel Socolovsky: The Universal Covering Group of U(n) and Projective Representations . Hrsg.: International Journal of Theoretical Physics. Dezember 1999
^ a b c d e f g h i j k l m n o p q Hatcher, Allen (2001). Algebraic Topology . Cambridge: Cambridge Univ. Press. ISBN 0-521-79160-X .
^ Kühnel, Wolfgang. Matrizen und Lie-Gruppen . Stuttgart: Springer Fachmedien Wiesbaden GmbH. ISBN 978-3-8348-9905-7 .
^ a b c d e f g h i Forster, Otto (1991). Lectures on Riemann surfaces . München: Springer Berlin. ISBN 978-3-540-90617-9 .
^ a b c d e f g Munkres, James (2000). Topology . Upper Saddle River, NJ: Prentice Hall, Inc. ISBN 978-0-13-468951-7 .
^ Aguilar, Maximiliano; Socolovsky, Miguel (December 1999). "The Universal Covering Group of U(n) and Projective Representations". International Journal of Theoretical Physics : 5 Theorem 1.