A covering of a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
is a
continuous map
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in valu ...
with special properties.
Definition
![Covering space diagram](https://upload.wikimedia.org/wikipedia/commons/1/16/Covering_space_diagram.svg)
Let
be a topological space. A covering of
is a continuous map
:
such that there exists a
discrete space
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest to ...
and for every
an
open neighborhood
In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. It is closely related to the concepts of open set and interior. Intuitively speaking, a neighbourhood of a po ...
, such that
and
is a
homeomorphism
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
for every
.
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
Connected may refer to:
Film and television
* ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular''
* '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film
* ''Connected'' (2015 TV ...
.
For each
the discrete subset
is called the fiber of
. The degree of a covering is the
cardinality
In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
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
![Covering_map](https://upload.wikimedia.org/wikipedia/commons/2/24/Covering_map.svg)
* The map
with
is a covering of the
unit circle
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucl ...
. 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
In mathematics, more specifically topology, a local homeomorphism is a function between topological spaces that, intuitively, preserves local (though not necessarily global) structure.
If f : X \to Y is a local homeomorphism, X is said to be an ...
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
In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space i ...
, then there is a covering
of degree
, whereby
is a connected and orientable manifold.
* If
is a connected
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
, then there is a covering
which is also a
Lie group homomorphism
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the add ...
and
is a Lie group.
* If
is a
graph
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discre ...
, then it follows for a covering
that
is also a graph.
* If
is a connected
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
, then there is a covering
, whereby
is a connected and
simply connected manifold.
* If
is a connected
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...
, then there is a covering
which is also a holomorphic map and
is a connected and simply connected Riemann surface.
Factorisation
Let
and
be continuous maps, such that the diagram
![Commutativ coverings](https://upload.wikimedia.org/wikipedia/commons/e/ed/Commutativ_coverings.png)
commutes.
* If
and
are coverings, so is
.
* If
and
are coverings, so is
.
Product of coverings
Let
and
be topological spaces and
and
be coverings, then
with
is a covering.
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
![Kommutatives_Diagramm_Äquivalenz_von_Überlagerungen](https://upload.wikimedia.org/wikipedia/commons/6/64/Kommutatives_Diagramm_%C3%84quivalenz_von_%C3%9Cberlagerungen.png)
commutes. If such a homeomorphism exists, then one calls the covering spaces
and
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
.
Lifting property
An important property of the covering is, that it satisfies the
lifting property
In mathematics, in particular in category theory, the lifting property is a property of a pair of morphism (category theory), morphisms in a category (mathematics), category. It is used in homotopy theory within algebraic topology to define propert ...
, i.e.:
Let
be the
unit interval
In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analysis, ...
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.
.
If
is a path-connected space, then for
it follows that the map
is a lift of a
path
A path is a route for physical travel – see Trail.
Path or PATH may also refer to:
Physical paths of different types
* Bicycle path
* Bridle path, used by people on horseback
* Course (navigation), the intended path of a vehicle
* Desire p ...
in
and for
it is a lift of a
homotopy
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...
of paths in
.
Because of that property one can show, that the
fundamental group
In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...
of the unit circle is an
infinite cyclic group
In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative binar ...
, which is generated by the homotopy classes of the loop
with
.
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
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
.
If
is a path-connected space and
a connected covering, then the induced
group homomorphism
In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that
: h(u*v) = h(u) \cdot h(v)
wh ...
:
with