A covering of a topological space
is a continuous map
with special properties.
Definition
Let
be a topological space. A covering of
is a continuous map
![{\displaystyle \pi :E\rightarrow X}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f562f74bd72a4b43a0891fcce8ccc71ba695fd1)
such that there exists a discrete space
and for every
an open neighborhood
, such that
and
is a homeomorphism for every
.
Intuitively, a covering locally projects a "stack of pancakes" above an
open neighborhood ![{\displaystyle U}](https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025)
onto
![{\displaystyle U}](https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025)
Often, the notion of a covering is used for the covering space
as well as for the map
. The open sets
are called sheets, which are uniquely determined up to a homeomorphism if
is connected.[1]: 56 For a
the discrete subset
is called the fiber of
. The degree of a covering is the cardinality of the space
. If
is path-connected, then the covering
is denoted as a path-connected covering.
Examples
- For every topological space
there exists the covering
with
, which is denoted as the trivial covering of ![{\displaystyle X.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ba76c5a460c4a0bb1639a193bc1830f0a773e03)
The space
![{\displaystyle Y=[0,1]\times \mathbb {R} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b566709fb62d0c99d75506726ea4f22fc3f98078)
is the covering space of
![{\displaystyle X=[0,1]\times S^{1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ecfebcfe87fcb8d16f87d9f6dfecca71bf16af1)
. The disjoint open sets
![{\displaystyle S_{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de6e810a93f67802ecb603ee0e3324005c6e583e)
are mapped homeomorphically onto
![{\displaystyle U}](https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025)
. The fiber of
![{\displaystyle x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4)
consists of the points
![{\displaystyle y_{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/67d30d30b6c2dbe4d6f150d699de040937ecc95f)
.
- 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
![{\displaystyle r^{-1}(U)=\displaystyle \bigsqcup _{n\in \mathbb {Z} }\left(n-{\frac {1}{4}},n+{\frac {1}{4}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2a8da94a813d6ed8cb634a0f3013852fd27d20e)
- and the sheets of the covering are
for
The fiber of
is
![{\displaystyle r^{-1}(x)=\{t\in \mathbb {R} \mid (\cos(2\pi t),\sin(2\pi t))=x\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f81a923e9d92c80de5b6ab0c128739e43254bbc3)
- 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
Local homeomorphism
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.[1]: 234
- If
is a connected Lie group, then there is a covering
which is also a Lie group homomorphism and
is a Lie group.[2]: 174
- If
is a graph, then it follows for a covering
that
is also a graph.[1]: 85
- If
is a connected manifold, then there is a covering
, whereby
is a connected and simply connected manifold.[3]: 32
- If
is a connected Riemann surface, then there is a covering
which is also a holomorphic map[3]: 22 and
is a connected and simply connected Riemann surface.[3]: 32
Factorisation
Let
and
be continuous maps, such that the diagram
commutes.
- If
and
are coverings, so is
.
- If
and
are coverings, so is
.[4]: 485
Product of coverings
Let
and
be topological spaces and
and
be coverings, then
with
is a covering.[4]: 339
Equivalence of coverings
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
An important property of the covering is, that it satisfies 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
, which is a lift of
, i.e.
.[1]: 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
.
Because of that property 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
.[1]: 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 ![{\displaystyle L_{\gamma }({\tilde {\gamma }}(0))={\tilde {\gamma }}(1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/91068ffce17d1bb1975ff761cbe16a04f471b466)
is bijective.[1]: 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
.[1]: 61
Branched covering
Definitions
Holomorphic maps between Riemann surfaces
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,[3]: 11 i.e. for every open set
the image
is also open.
Ramification point and branch point
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
.[3]: 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
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[3]: 20 and for any two unramified points
, it is:
It can be calculated by:
[3]: 29
Branched covering
Definition
A continuous map
is called a branched covering, if there exists a closed set with dense complement
, such that
is a covering.
Examples
- Let
and
, then
with
is branched covering of degree
, whereby
is a branch point.
- Every non-constant, holomorphic map between compact Riemann surfaces
of degree
is a branched covering of degree
.
Universal covering
Definition
Let
be a simply connected covering. If
is another simply connected covering, then there exists a uniquely determined homeomorphism
, such that the diagram
commutes.[4]: 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
A universal covering does not always exist, but the following properties guarantee its existence:
Let
be a connected, locally simply connected, then there exists a universal covering
.
is defined as
and
by
.[1]: 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
![{\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\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/85a55ffa77d8593ae838c36da05bbbab8b0ae306)
- One can show that no neighborhood of the origin
is simply connected.[4]: 487, Example 1
Deck transformation
Definition
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
.
Examples
- 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
.
Properties
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.[1]: 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
Definition
A covering
is called normal, if
. This means, that for every
and any two
there exists a deck transformation
, such that
.
Properties
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
.[1]: 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.[1]: 72 Hereby
is the quotient space and
is the orbit of the group action.
Examples
- The covering
with
is a normal coverings for every
.
- Every simply connected covering is a normal covering.
Calculation
Let
be a group, which acts discontinuously on a topological space
and let
be the normal covering.
- If
is path-connected, then
.[1]: 72
- If
is simply connected, then
.[1]: 71
Examples
- 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
Let
be a connected and locally simply connected space, then for every subgroup
there exists a path-connected covering
with
.[1]: 66
Let
and
be two path-connected coverings, then they are equivalent iff the subgroups
and
are conjugate to each other.[4]: 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
Definitions
Category of coverings
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
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
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.[1]: 68–70
Applications
Gimbal lock occurs because any map
T3 → RP3 is not a covering map. In particular, the relevant map carries any element of
T3, that is, an ordered triple (a,b,c) of angles (real numbers mod 2
π), to the composition of the three coordinate axis rotations R
x(a)∘R
y(b)∘R
z(c) by those angles, respectively. Each of these rotations, and their composition, is an element of the rotation group
SO(3), which is topologically
RP3. This animation shows a set of three gimbals mounted together to allow
three degrees of freedom. When all three gimbals are lined up (in the same plane), the system can only move in two dimensions from this configuration, not three, and is in
gimbal lock. In this case it can pitch or yaw, but not roll (rotate in the plane that the axes all lie in).
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 RP3, with fundamental group Z/2, and only (non-trivial) covering space the hypersphere S3, 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 T3 of three angles to the real projective space RP3 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
Literature
- 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
References