Type of continuous map in topology
In
topology
, a
covering
or
covering projection
is a
map
between
topological spaces
that, intuitively,
locally
acts like a
projection
of multiple copies of a space onto itself. In particular, coverings are special types of
local homeomorphisms
. If
is a covering,
is said to be a
covering space
or
cover
of
, and
is said to be the
base of the covering
, or simply the
base
. By
abuse of terminology
,
and
may sometimes be called
covering spaces
as well. Since coverings are local homeomorphisms, a covering space is a special kind of
etale space
.
Covering spaces first arose in the context of
complex analysis
(specifically, the technique of
analytic continuation
), where they were introduced by
Riemann
as domains on which naturally
multivalued
complex functions become single-valued. These spaces are now called
Riemann surfaces
.
[1]
: 10
Covering spaces are an important tool in several areas of mathematics. In modern
geometry
, covering spaces (or
branched coverings
, which have slightly weaker conditions) are used in the construction of
manifolds
,
orbifolds
, and the
morphisms
between them. In
algebraic topology
, covering spaces are closely related to the
fundamental group
: for one, since all coverings have the
homotopy lifting property
, covering spaces are an important tool in the calculation of
homotopy groups
. A standard example in this vein is the calculation of the
fundamental group
of the circle by means of the covering of
by
(see
below
).
[2]
: 29
Under certain conditions, covering spaces also exhibit a
Galois correspondence
with the subgroups of the fundamental group.
Definition
[
edit
]
Let
be a topological space. A
covering
of
is a continuous map
such that for every
there exists an
open neighborhood
of
and a
discrete space
such that
and
is a
homeomorphism
for every
.
The open sets
are called
sheets
, which are uniquely determined up to homeomorphism if
is
connected
.
[2]
: 56
For each
the discrete set
is called the
fiber
of
. If
is connected, it can be shown that
is
surjective
, and the
cardinality
of
is the same for all
; this value is called the
degree
of the covering. If
is
path-connected
, then the covering
is called a
path-connected covering
. This definition is equivalent to the statement that
is a locally trivial
Fiber bundle
.
Some authors also require that
be surjective in the case that
is not connected.
[3]
Examples
[
edit
]
- For every topological space
, the
identity map
is a covering. Likewise for any discrete space
the projection
taking
is a covering. Coverings of this type are called
trivial coverings
; if
has finitely many (say
) elements, the covering is called the
trivial
-sheeted
covering
of
.
- The map
with
is a covering of the
unit circle
. The base of the covering is
and the covering space is
. For any point
such that
, the set
is an open neighborhood of
. The preimage of
under
is
- and the sheets of the covering are
for
The fiber of
is
- Another covering of the unit circle is the map
with
for some
For an open neighborhood
of an
, one has:
- .
- A map which is a
local homeomorphism
but not a covering of the unit circle is
with
. There is a sheet of an open neighborhood of
, which is not mapped homeomorphically onto
.
Properties
[
edit
]
Local homeomorphism
[
edit
]
Since a covering
maps each of the disjoint open sets of
homeomorphically onto
it is a local homeomorphism, i.e.
is a continuous map and for every
there exists an open neighborhood
of
, such that
is a homeomorphism.
It follows that the covering space
and the base space
locally share the same properties.
- If
is a connected and
non-orientable manifold
, then there is a covering
of degree
, whereby
is a connected and orientable manifold.
[2]
: 234
- If
is a connected
Lie group
, then there is a covering
which is also a
Lie group homomorphism
and
is a Lie group.
[4]
: 174
- If
is a
graph
, then it follows for a covering
that
is also a graph.
[2]
: 85
- If
is a connected
manifold
, then there is a covering
, whereby
is a connected and
simply connected
manifold.
[5]
: 32
- If
is a connected
Riemann surface
, then there is a covering
which is also a holomorphic map
[5]
: 22
and
is a connected and simply connected Riemann surface.
[5]
: 32
Factorisation
[
edit
]
Let
and
be path-connected, locally path-connected spaces, and
and
be continuous maps, such that the diagram
commutes.
- If
and
are coverings, so is
.
- If
and
are coverings, so is
.
[6]
: 485
Product of coverings
[
edit
]
Let
and
be topological spaces and
and
be coverings, then
with
is a covering.
[6]
: 339
However covering of
are not all of this form in general.
Equivalence of coverings
[
edit
]
Let
be a topological space and
and
be coverings. Both coverings are called
equivalent
, if there exists a homeomorphism
, such that the diagram
commutes. If such a homeomorphism exists, then one calls the covering spaces
and
isomorphic
.
Lifting property
[
edit
]
All coverings satisfy the
lifting property
, i.e.:
Let
be the
unit interval
and
be a covering. Let
be a continuous map and
be a lift of
, i.e. a continuous map such that
. Then there is a uniquely determined, continuous map
for which
and which is a lift of
, i.e.
.
[2]
: 60
If
is a path-connected space, then for
it follows that the map
is a lift of a
path
in
and for
it is a lift of a
homotopy
of paths in
.
As a consequence, one can show that the
fundamental group
of the unit circle is an
infinite cyclic group
, which is generated by the homotopy classes of the loop
with
.
[2]
: 29
Let
be a path-connected space and
be a connected covering. Let
be any two points, which are connected by a path
, i.e.
and
. Let
be the unique lift of
, then the map
- with
is
bijective
.
[2]
: 69
If
is a path-connected space and
a connected covering, then the induced
group homomorphism
- with
,
is
injective
and the
subgroup
of
consists of the homotopy classes of loops in
, whose lifts are loops in
.
[2]
: 61
Branched covering
[
edit
]
Definitions
[
edit
]
Holomorphic maps between Riemann surfaces
[
edit
]
Let
and
be
Riemann surfaces
, i.e. one dimensional
complex manifolds
, and let
be a continuous map.
is
holomorphic in a point
, if for any
charts
of
and
of
, with
, the map
is
holomorphic
.
If
is holomorphic at all
, we say
is
holomorphic.
The map
is called the
local expression
of
in
.
If
is a non-constant, holomorphic map between
compact Riemann surfaces
, then
is
surjective
and an
open map
,
[5]
: 11
i.e. for every open set
the
image
is also open.
Ramification point and branch point
[
edit
]
Let
be a non-constant, holomorphic map between compact Riemann surfaces. For every
there exist charts for
and
and there exists a uniquely determined
, such that the local expression
of
in
is of the form
.
[5]
: 10
The number
is called the
ramification index
of
in
and the point
is called a
ramification point
if
. If
for an
, then
is
unramified
. The image point
of a ramification point is called a
branch point.
Degree of a holomorphic map
[
edit
]
Let
be a non-constant, holomorphic map between compact Riemann surfaces. The
degree
of
is the cardinality of the fiber of an unramified point
, i.e.
.
This number is well-defined, since for every
the fiber
is discrete
[5]
: 20
and for any two unramified points
, it is:
It can be calculated by:
- [5]
: 29
Branched covering
[
edit
]
Definition
[
edit
]
A continuous map
is called a
branched covering
, if there exists a
closed set
with
dense
complement
, such that
is a covering.
Examples
[
edit
]
- Let
and
, then
with
is branched covering of degree
, where by
is a branch point.
- Every non-constant, holomorphic map between compact Riemann surfaces
of degree
is a branched covering of degree
.
Universal covering
[
edit
]
Definition
[
edit
]
Let
be a
simply connected
covering. If
is another simply connected covering, then there exists a uniquely determined homeomorphism
, such that the diagram
commutes.
[6]
: 482
This means that
is, up to equivalence, uniquely determined and because of that
universal property
denoted as the
universal covering
of the space
.
Existence
[
edit
]
A universal covering does not always exist, but the following properties guarantee its existence:
Let
be a connected,
locally simply connected
topological space; then, there exists a universal covering
.
is defined as
and
by
.
[2]
: 64
The
topology
on
is constructed as follows: Let
be a path with
. Let
be a simply connected neighborhood of the endpoint
, then for every
the
paths
inside
from
to
are uniquely determined up to
homotopy
. Now consider
, then
with
is a bijection and
can be equipped with the
final topology
of
.
The fundamental group
acts
freely
through
on
and
with
is a homeomorphism, i.e.
.
Examples
[
edit
]
- with
is the universal covering of the unit circle
.
- with
is the universal covering of the
projective space
for
.
- with
is the universal covering of the
unitary group
.
[7]
: 5, Theorem 1
- Since
, it follows that the
quotient map
is the universal covering of the
.
- A topological space which has no universal covering is the
Hawaiian earring
:
One can show that no neighborhood of the origin
is simply connected.
[6]
: 487, Example 1
G-coverings
[
edit
]
Let
G
be a
discrete group
acting
on the
topological space
X
. This means that each element
g
of
G
is associated to a homeomorphism H
g
of
X
onto itself, in such a way that H
g
h
is always equal to H
g
? H
h
for any two elements
g
and
h
of
G
. (Or in other words, a group action of the group
G
on the space
X
is just a group homomorphism of the group
G
into the group Homeo(
X
) of self-homeomorphisms of
X
.) It is natural to ask under what conditions the projection from
X
to the
orbit space
X
/
G
is a covering map. This is not always true since the action may have fixed points. An example for this is the cyclic group of order 2 acting on a product
X
×
X
by the twist action where the non-identity element acts by
(
x
,
y
) ? (
y
,
x
)
. Thus the study of the relation between the fundamental groups of
X
and
X
/
G
is not so straightforward.
However the group
G
does act on the fundamental
groupoid
of
X
, and so the study is best handled by considering groups acting on groupoids, and the corresponding
orbit groupoids
. The theory for this is set down in Chapter 11 of the book
Topology and groupoids
referred to below. The main result is that for discontinuous actions of a group
G
on a Hausdorff space
X
which admits a universal cover, then the fundamental groupoid of the orbit space
X
/
G
is isomorphic to the orbit groupoid of the fundamental groupoid of
X
, i.e. the quotient of that groupoid by the action of the group
G
. This leads to explicit computations, for example of the fundamental group of the symmetric square of a space.
Deck transformation
[
edit
]
Definition
[
edit
]
Let
be a covering. A
deck transformation
is a homeomorphism
, such that the diagram of continuous maps
commutes. Together with the composition of maps, the set of deck transformation forms a
group
, which is the same as
.
Now suppose
is a covering map and
(and therefore also
) is connected and locally path connected. The action of
on each fiber is
transitive
. If this action is
free
on some fiber, then it is free on all fibers, and we call the cover
regular
(or
normal
or
Galois
). Every such regular cover is a
principal
-bundle
, where
is considered as a discrete topological group.
Every universal cover
is regular, with deck transformation group being isomorphic to the
fundamental group
.
Examples
[
edit
]
- Let
be the covering
for some
, then the map
is a deck transformation and
.
- Let
be the covering
, then the map
with
is a deck transformation and
.
- As another important example, consider
the complex plane and
the complex plane minus the origin. Then the map
with
is a regular cover. The deck transformations are multiplications with
-th
roots of unity
and the deck transformation group is therefore isomorphic to the
cyclic group
. Likewise, the map
with
is the universal cover.
Properties
[
edit
]
Let
be a path-connected space and
be a connected covering. Since a deck transformation
is
bijective
, it permutes the elements of a fiber
with
and is uniquely determined by where it sends a single point. In particular, only the identity map fixes a point in the fiber.
[2]
: 70
Because of this property every deck transformation defines a
group action
on
, i.e. let
be an open neighborhood of a
and
an open neighborhood of an
, then
is a
group action
.
Normal coverings
[
edit
]
Definition
[
edit
]
A covering
is called normal, if
. This means, that for every
and any two
there exists a deck transformation
, such that
.
Properties
[
edit
]
Let
be a path-connected space and
be a connected covering. Let
be a
subgroup
of
, then
is a normal covering iff
is a
normal subgroup
of
.
If
is a normal covering and
, then
.
If
is a path-connected covering and
, then
, whereby
is the
normaliser
of
.
[2]
: 71
Let
be a topological space. A group
acts
discontinuously
on
, if every
has an open neighborhood
with
, such that for every
with
one has
.
If a group
acts discontinuously on a topological space
, then the
quotient map
with
is a normal covering.
[2]
: 72
Hereby
is the
quotient space
and
is the
orbit
of the group action.
Examples
[
edit
]
- The covering
with
is a normal coverings for every
.
- Every simply connected covering is a normal covering.
Calculation
[
edit
]
Let
be a group, which acts discontinuously on a topological space
and let
be the normal covering.
- If
is path-connected, then
.
[2]
: 72
- If
is simply connected, then
.
[2]
: 71
Examples
[
edit
]
- Let
. The antipodal map
with
generates, together with the composition of maps, a group
and induces a group action
, which acts discontinuously on
. Because of
it follows, that the quotient map
is a normal covering and for
a universal covering, hence
for
.
- Let
be the
special orthogonal group
, then the map
is a normal covering and because of
, it is the universal covering, hence
.
- With the group action
of
on
, whereby
is the
semidirect product
, one gets the universal covering
of the
klein bottle
, hence
.
- Let
be the
torus
which is embedded in the
. Then one gets a homeomorphism
, which induces a discontinuous group action
, whereby
. It follows, that the map
is a normal covering of the klein bottle, hence
.
- Let
be embedded in the
. Since the group action
is discontinuously, whereby
are
coprime
, the map
is the universal covering of the
lens space
, hence
.
Galois correspondence
[
edit
]
Let
be a connected and
locally simply connected
space, then for every
subgroup
there exists a path-connected covering
with
.
[2]
: 66
Let
and
be two path-connected coverings, then they are equivalent iff the subgroups
and
are
conjugate
to each other.
[6]
: 482
Let
be a connected and locally simply connected space, then, up to equivalence between coverings, there is a bijection:
For a sequence of subgroups
one gets a sequence of coverings
. For a subgroup
with
index
, the covering
has degree
.
Classification
[
edit
]
Definitions
[
edit
]
Category of coverings
[
edit
]
Let
be a topological space. The objects of the
category
are the coverings
of
and the
morphisms
between two coverings
and
are continuous maps
, such that the diagram
commutes.
G-Set
[
edit
]
Let
be a
topological group
. The
category
is the category of sets which are
G-sets
. The morphisms are
G-maps
between G-sets. They satisfy the condition
for every
.
Equivalence
[
edit
]
Let
be a connected and locally simply connected space,
and
be the fundamental group of
. Since
defines, by lifting of paths and evaluating at the endpoint of the lift, a group action on the fiber of a covering, the
functor
is an
equivalence of categories
.
[2]
: 68?70
Applications
[
edit
]
An important practical application of covering spaces occurs in
charts on SO(3)
, the
rotation group
. This group occurs widely in engineering, due to 3-dimensional rotations being heavily used in
navigation
,
nautical engineering
, and
aerospace engineering
, among many other uses. Topologically, SO(3) is the
real projective space
RP
3
, with fundamental group
Z
/2, and only (non-trivial) covering space the hypersphere
S
3
, which is the group
Spin(3)
, and represented by the unit
quaternions
. Thus quaternions are a preferred method for representing spatial rotations ? see
quaternions and spatial rotation
.
However, it is often desirable to represent rotations by a set of three numbers, known as
Euler angles
(in numerous variants), both because this is conceptually simpler for someone familiar with planar rotation, and because one can build a combination of three
gimbals
to produce rotations in three dimensions. Topologically this corresponds to a map from the 3-torus
T
3
of three angles to the real projective space
RP
3
of rotations, and the resulting map has imperfections due to this map being unable to be a covering map. Specifically, the failure of the map to be a local homeomorphism at certain points is referred to as
gimbal lock
, and is demonstrated in the animation at the right ? at some points (when the axes are coplanar) the
rank
of the map is 2, rather than 3, meaning that only 2 dimensions of rotations can be realized from that point by changing the angles. This causes problems in applications, and is formalized by the notion of a covering space.
See also
[
edit
]
Literature
[
edit
]
References
[
edit
]
- ^
Forster, Otto (1981). "Chapter 1: Covering Spaces".
Lectures on Riemann Surfaces
. GTM. Translated by Bruce Gillian. New York: Springer.
ISBN
9781461259633
.
- ^
a
b
c
d
e
f
g
h
i
j
k
l
m
n
o
p
Hatcher, Allen (2001).
Algebraic Topology
. Cambridge: Cambridge Univ. Press.
ISBN
0-521-79160-X
.
- ^
Rowland, Todd. "Covering Map." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.
https://mathworld.wolfram.com/CoveringMap.html
- ^
Kuhnel, Wolfgang (6 December 2010).
Matrizen und Lie-Gruppen
. Stuttgart: Springer Fachmedien Wiesbaden GmbH.
ISBN
978-3-8348-9905-7
.
- ^
a
b
c
d
e
f
g
Forster, Otto (1991).
Lectures on Riemann surfaces
. Munchen: Springer Berlin.
ISBN
978-3-540-90617-9
.
- ^
a
b
c
d
e
Munkres, James (2000).
Topology
. Upper Saddle River, NJ: Prentice Hall, Inc.
ISBN
978-0-13-468951-7
.
- ^
Aguilar, Marcelo Alberto; Socolovsky, Miguel (23 November 1999). "The Universal Covering Group of U(n) and Projective Representations".
International Journal of Theoretical Physics
.
39
(4). Springer US (published April 2000): 997?1013.
arXiv
:
math-ph/9911028
.
Bibcode
:
1999math.ph..11028A
.
doi
:
10.1023/A:1003694206391
.
S2CID
18686364
.