Editing Covering space

{{Short description|Type of continuous map in topology}}

[[File:Covering space diagram.svg|upright=1|thumb|Intuitively, a covering locally projects a "stack of pancakes" above an [[open neighborhood]] <math>U</math> onto <math>U</math>]]

In [[topology]], a '''covering''' or '''covering projection''' is a [[continuous function|map]] between [[topological space]]s that, intuitively, [[Local property|locally]] acts like a [[Projection (mathematics)|projection]] of multiple copies of a space onto itself. In particular, coverings are special types of [[local homeomorphism]]s. If <math> p : \tilde X \to X </math> is a covering, <math>(\tilde X, p)</math> is said to be a '''covering space''' or '''cover''' of <math>X</math>, and <math>X</math> is said to be the '''base of the covering''', or simply the '''base'''. By [[abuse of terminology]], <math>\tilde X</math> and <math>p</math> may sometimes be called '''covering spaces''' as well. Since coverings are local homeomorphisms, a covering space is a special kind of [[étalé space]].

Covering spaces first arose in the context of [[complex analysis]] (specifically, the technique of [[analytic continuation]]), where they were introduced by [[Bernhard Riemann|Riemann]] as domains on which naturally [[multivalued function|multivalued]] complex functions become single-valued. These spaces are now called [[Riemann surface]]s.<ref name="Gillian">{{Cite book|translator=Bruce Gillian|last=Forster|first=Otto|title=Lectures on Riemann Surfaces|publisher=Springer|year=1981|series=GTM|number=81|chapter=Chapter 1: Covering Spaces|location=New York|ISBN=9781461259633}}</ref>{{rp|p=10}}

Covering spaces are an important tool in several areas of mathematics. In modern [[geometry]], covering spaces (or [[branched covering]]s, which have slightly weaker conditions) are used in the construction of [[manifold]]s, [[orbifold]]s, and the [[morphism]]s 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 [[Circle|<math>S^1</math>]] by [[Real number|<math>\mathbb{R}</math>]] (see [[Covering_space#Lifting_property|below]]).<ref name="Hatcher">{{Cite book|last=Hatcher|first=Allen|title=Algebraic Topology|publisher=Cambridge Univ. Press|year=2001|isbn=0-521-79160-X|location=Cambridge}}</ref>{{rp|p=29}} Under certain conditions, covering spaces also exhibit a [[Galois_connection#Algebraic_topology:_covering_spaces|Galois correspondence]] with the subgroups of the fundamental group. 

== Definition ==

Let <math>X</math> be a topological space. A '''covering''' of <math>X</math> is a continuous map
: <math>\pi : \tilde X \rightarrow X</math>
such that for every <math>x \in X</math> there exists an [[Neighbourhood (mathematics)|open neighborhood]] <math>U_x</math> of <math>x</math> and a [[discrete space]] <math>D_x</math> such that <math>\pi^{-1}(U_x)= \displaystyle \bigsqcup_{d \in D_x} V_d </math> and <math>\pi|_{V_d}:V_d \rightarrow U_x </math> is a [[homeomorphism]] for every <math>d \in D_x </math>.
The open sets <math>V_{d}</math> are called '''sheets''', which are uniquely determined up to homeomorphism if <math>U_x</math> is [[Connected space|connected]].{{r|Hatcher|p=56}} For each <math>x \in X</math> the discrete set <math>\pi^{-1}(x)</math> is called the '''fiber''' of <math>x</math>. If <math>X</math> is connected (and <math>\tilde X</math> is non-empty), it can be shown that <math>\pi</math> is [[surjective]], and the [[cardinality]] of <math>D_x</math> is the same for all <math>x \in X</math>; this value is called the '''degree''' of the covering. If <math>\tilde X</math> is [[Path connected|path-connected]], then the covering <math> \pi : \tilde X \rightarrow X</math> is called a '''path-connected covering'''. This definition is equivalent to the statement that <math>\pi</math> is a locally trivial [[Fiber bundle]].

Some authors also require that <math>\pi</math> be surjective in the case that <math>X</math> is not connected.<ref>Rowland, Todd. "Covering Map." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/CoveringMap.html</ref>

== Examples ==
* For every topological space <math>X</math>, the [[Identity function|identity map]] <math>\operatorname{id}:X \rightarrow X</math> is a covering. Likewise for any discrete space <math>D</math> the projection <math>\pi:X \times D \rightarrow X</math> taking <math>(x, i) \mapsto x</math> is a covering. Coverings of this type are called '''trivial coverings'''; if <math>D</math> has finitely many (say <math>k</math>) elements, the covering is called the '''trivial ''<math>k</math>-sheeted'' covering''' of <math>X</math>. 
{{Dark mode invert|[[File:Covering_map.svg|thumb|The space <math>Y=[0,1] \times \mathbb{R}</math> is a covering space of <math>X=[0,1] \times S^1</math>. The disjoint open sets <math>S_i</math> are mapped homeomorphically onto <math>U</math>. The fiber of <math>x</math> consists of the points <math>y_i</math>.]]}}
* The map <math>r : \mathbb{R} \to S^1</math> with <math>r(t)=(\cos(2 \pi t), \sin(2 \pi t))</math> is a covering of the [[unit circle]] <math>S^1</math>. The base of the covering is <math>S^1</math> and the covering space is <math>\mathbb{R}</math>. For any point <math>x = (x_1, x_2) \in S^1</math> such that <math>x_1 > 0</math>, the set <math>U := \{(x_1, x_2) \in S^1 \mid x_1 > 0 \}</math> is an open neighborhood of <math>x</math>. The preimage of <math>U</math> under <math>r</math> is 
*: <math>r^{-1}(U)=\displaystyle\bigsqcup_{n \in \mathbb{Z}} \left( n - \frac 1 4, n + \frac 1 4\right)</math> 
: and the sheets of the covering are <math>V_n = (n - 1/4, n+1/4)</math> for <math>n \in \mathbb{Z}.</math> The fiber of <math>x</math> is
:: <math>r^{-1}(x) = \{t \in \mathbb{R} \mid (\cos(2 \pi t), \sin(2 \pi t)) = x\}.</math> 
* Another covering of the unit circle is the map <math>q : S^1 \to S^1</math> with <math>q(z)=z^{n}</math> for some positive <math>n \in \mathbb{N}.</math> For an open neighborhood <math>U</math> of an <math>x \in S^1</math>, one has: 
:: <math>q^{-1}(U)=\displaystyle\bigsqcup_{i=1}^{n} U</math>.
* A map which is a [[local homeomorphism]] but not a covering of the unit circle is <math>p : \mathbb{R_{+}} \to S^1</math> with <math>p(t)=(\cos(2 \pi t), \sin(2 \pi t))</math>. There is a sheet of an open neighborhood of <math>(1,0)</math>, which is not mapped homeomorphically onto <math>U</math>.

== Properties ==
=== Local homeomorphism ===
Since a covering <math>\pi:E \rightarrow X</math> maps each of the disjoint open sets of <math>\pi^{-1}(U)</math> homeomorphically onto <math>U</math> it is a local homeomorphism, i.e. <math>\pi</math> is a continuous map and for every <math>e \in E</math> there exists an open neighborhood <math>V \subset E</math> of <math>e</math>, such that <math>\pi|_V : V \rightarrow \pi(V)</math> is a homeomorphism.

It follows that the covering space <math>E</math> and the base space <math>X</math> locally share the same properties.

* If <math>X</math> is a connected and [[non-orientable manifold]], then there is a covering <math>\pi:\tilde X \rightarrow X</math> of degree <math>2</math>, whereby <math>\tilde X</math> is a connected and orientable manifold.{{r|Hatcher|p=234}}
* If <math>X</math> is a connected [[Lie group]], then there is a covering <math>\pi:\tilde X \rightarrow X</math> which is also a [[Lie group homomorphism]] and <math>\tilde X := \{\gamma:\gamma \text{ is a path in X with  }\gamma(0)= \boldsymbol{1_X} \text{ modulo homotopy with fixed ends}\}</math> is a Lie group.<ref>{{Cite book|last=Kühnel |first=Wolfgang |title=Matrizen und Lie-Gruppen|date=6 December 2010 |publisher=Springer Fachmedien Wiesbaden GmbH|location=Stuttgart|isbn=978-3-8348-9905-7}}</ref>{{rp|p=174}}
* If <math>X</math> is a [[Graph theory#Graph|graph]], then it follows for a covering <math>\pi:E \rightarrow X</math> that <math>E</math> is also a graph.{{r|Hatcher|p=85}}
* If <math>X</math> is a connected [[manifold]], then there is a covering <math>\pi:\tilde X \rightarrow X</math>, whereby <math>\tilde X</math> is a connected and [[Simply connected space|simply connected]] manifold.{{r|Forster|p=32}}
* If <math>X</math> is a connected [[Riemann surface]], then there is a covering <math>\pi:\tilde X \rightarrow X</math> which is also a holomorphic map{{r|Forster|p=22}} and <math>\tilde X</math> is a connected and simply connected Riemann surface.{{r|Forster|p=32}}

=== Factorisation ===
Let <math> X, Y</math> and <math>E</math> be path-connected, locally path-connected spaces, and <math>p,q</math> and <math>r</math> be continuous maps, such that the diagram 
[[File:Commutativ coverings.png|center|frameless]]
commutes.

* If <math>p</math> and <math>q</math> are coverings, so is <math>r</math>.
* If <math>p</math> and <math>r</math> are coverings, so is <math>q</math>.{{r|Munkres|p=485}}

=== Product of coverings ===
Let <math>X</math> and <math>X'</math> be topological spaces and <math>p:E \rightarrow X</math> and <math>p':E' \rightarrow X'</math> be coverings, then <math>p \times p':E \times E' \rightarrow X \times X'</math> with <math>(p \times p')(e, e') = (p(e), p'(e'))</math> is a covering.<ref name="Munkres">{{Cite book|last=Munkres|first=James|title=Topology|publisher=Upper Saddle River, NJ: Prentice Hall, Inc.|year=2000|isbn=978-0-13-468951-7}}</ref>{{rp|p=339}} However, coverings of <math>X\times X'</math> are not all of this form in general.

=== Equivalence of coverings ===
Let <math>X</math> be a topological space and <math>p:E \rightarrow X</math> and <math>p':E' \rightarrow X</math> be coverings. Both coverings are called '''equivalent''', if there exists a homeomorphism <math>h:E \rightarrow E'</math>, such that the diagram
[[File:Kommutatives_Diagramm_Äquivalenz_von_Überlagerungen.png|center|frameless]]
commutes. If such a homeomorphism exists, then one calls the covering spaces <math>E</math> and <math>E'</math> [[Isomorphism|isomorphic]].

=== Lifting property ===
All coverings satisfy the [[lifting property]], i.e.:

Let <math>I</math> be the [[unit interval]] and <math>p:E \rightarrow X</math> be a covering. Let <math>F:Y \times I \rightarrow X</math> be a continuous map and <math>\tilde F_0:Y \times \{0\} \rightarrow E</math> be a lift of <math>F|_{Y \times \{0\}}</math>, i.e. a continuous map such that <math>p \circ \tilde F_0 = F|_{Y \times \{0\}}</math>. Then there is a uniquely determined, continuous map <math>\tilde F:Y \times I \rightarrow E</math> for which <math>\tilde F(y,0) = \tilde F_0</math> and which is a lift of <math>F</math>, i.e. <math>p \circ \tilde F = F</math>.{{r|Hatcher|p=60}}

If <math>X</math> is a path-connected space, then for <math>Y=\{0\}</math> it follows that the map <math>\tilde F</math> is a lift of a [[Path (topology)|path]] in <math>X</math> and for <math>Y=I</math> it is a lift of a [[homotopy]] of paths in <math>X</math>.

As a consequence, one can show that the [[fundamental group]] <math>\pi_{1}(S^1)</math> of the unit circle is an [[Cyclic group|infinite cyclic group]], which is generated by the homotopy classes of the loop <math>\gamma: I \rightarrow S^1</math> with <math>\gamma (t) = (\cos(2 \pi t), \sin(2 \pi t))</math>.{{r|Hatcher|p=29}}

Let <math>X</math> be a path-connected space and <math>p:E \rightarrow X</math> be a connected covering. Let <math>x,y \in X</math> be any two points, which are connected by a path <math>\gamma</math>, i.e. <math>\gamma(0)= x</math> and <math>\gamma(1)= y</math>. Let <math>\tilde \gamma</math> be the unique lift of <math>\gamma</math>, then the map 
: <math>L_{\gamma}:p^{-1}(x) \rightarrow p^{-1}(y)</math> with <math>L_{\gamma}(\tilde \gamma (0))=\tilde \gamma (1)</math>

is [[Bijection|bijective]].{{r|Hatcher|p=69}}

If <math>X</math> is a path-connected space and <math>p: E \rightarrow X</math> a connected covering, then the induced [[group homomorphism]] 
: <math> p_{\#}: \pi_{1}(E) \rightarrow \pi_{1}(X)</math> with <math> p_{\#}([\gamma])=[p \circ \gamma]</math>,
is [[Injective function|injective]] and the [[subgroup]] <math>p_{\#}(\pi_1(E))</math> of <math>\pi_1(X)</math> consists of the homotopy classes of loops in <math>X</math>, whose lifts are loops in <math>E</math>.{{r|Hatcher|p=61}}

== Branched covering ==
=== Definitions ===
==== Holomorphic maps between Riemann surfaces ====
Let <math>X</math> and <math>Y</math> be [[Riemann surface|Riemann surfaces]], i.e. one dimensional [[Complex manifold|complex manifolds]], and let <math>f: X \rightarrow Y</math> be a continuous map. <math>f</math> is '''holomorphic in a point''' <math>x \in X</math>, if for any [[Chart (mathematics)|charts]] <math>\phi _x:U_1 \rightarrow V_1</math> of <math>x</math> and <math>\phi_{f(x)}:U_2 \rightarrow V_2</math> of <math>f(x)</math>, with <math>\phi_x(U_1) \subset U_2</math>, the map <math>\phi _{f(x)} \circ f \circ \phi^{-1} _x: \mathbb{C} \rightarrow \mathbb{C}</math> is [[Holomorphic function|holomorphic]].

If <math>f</math> is holomorphic at all <math>x \in X</math>, we say <math>f</math> is '''holomorphic.'''

The map <math>F =\phi _{f(x)} \circ f \circ \phi^{-1} _x</math> is called the '''local expression''' of <math>f</math> in <math>x \in X</math>.

If <math>f: X \rightarrow Y</math> is a non-constant, holomorphic map between [[Compact riemann surface|compact Riemann surfaces]], then <math>f</math> is [[Surjective function|surjective]] and an [[open map]],<ref name="Forster">{{Cite book|last=Forster|first=Otto|title=Lectures on Riemann surfaces|publisher=Springer Berlin|year=1991|isbn=978-3-540-90617-9|location=München}}</ref>{{rp|p=11}} i.e. for every open set <math>U \subset X</math> the [[Image (mathematics)|image]] <math>f(U) \subset Y</math> is also open.

==== Ramification point and branch point ====
Let <math>f: X \rightarrow Y</math> be a non-constant, holomorphic map between compact Riemann surfaces. For every <math>x \in X</math> there exist charts for <math>x</math> and <math>f(x)</math> and there exists a uniquely determined <math>k_x \in \mathbb{N_{>0}}</math>, such that the local expression <math>F</math> of <math>f</math> in <math>x</math> is of the form <math>z \mapsto z^{k_{x}}</math>.{{r|Forster|p=10}} The number <math>k_x</math> is called the '''ramification index''' of <math>f</math> in <math>x</math> and the point <math>x \in X</math> is called a '''ramification point''' if <math>k_x \geq 2</math>. If <math>k_x =1</math> for an <math>x \in X</math>, then <math>x</math> is '''unramified'''. The image point <math>y=f(x) \in Y</math> of a ramification point is called a '''branch point.'''

==== Degree of a holomorphic map ====
Let <math>f: X \rightarrow Y</math> be a non-constant, holomorphic map between compact Riemann surfaces. The '''degree <math>\operatorname{deg}(f)</math>''' of <math>f</math> is the cardinality of the fiber of an unramified point <math>y=f(x) \in Y</math>, i.e. <math>\operatorname{deg}(f):=|f^{-1}(y)|</math>.

This number is well-defined, since for every <math>y \in Y</math> the fiber <math>f^{-1}(y)</math> is discrete{{r|Forster|p=20}} and for any two unramified points <math>y_1,y_2 \in Y</math>, it is: <math>|f^{-1}(y_1)|=|f^{-1}(y_2)|.</math>

It can be calculated by: 
: <math>\sum_{x \in f^{-1}(y)} k_x = \operatorname{deg}(f)</math> {{r|Forster|p=29}}

=== Branched covering ===
==== Definition ====
A continuous map <math>f: X \rightarrow Y</math> is called a '''branched covering''', if there exists a [[closed set]] with [[Dense set|dense]] complement <math>E \subset Y</math>, such that <math>f_{|X \smallsetminus f^{-1}(E)}:X \smallsetminus f^{-1}(E) \rightarrow Y \smallsetminus E</math> is a covering.

==== Examples ====
* Let <math>n \in \mathbb{N}</math> and <math>n \geq 2</math>, then <math>f:\mathbb{C} \rightarrow \mathbb{C}</math> with <math>f(z)=z^n</math> is a branched covering of degree <math>n</math>, where by <math>z=0</math> is a branch point.
* Every non-constant, holomorphic map between compact Riemann surfaces <math>f: X \rightarrow Y</math> of degree <math>d</math> is a branched covering of degree <math>d</math>.

== Universal covering ==

=== Definition ===
Let <math>p: \tilde X \rightarrow X</math> be a [[Simply connected space|simply connected]] covering. If <math>\beta : E \rightarrow X</math> is another simply connected covering, then there exists a uniquely determined homeomorphism <math>\alpha : \tilde X \rightarrow E</math>, such that the diagram
[[File:Universelle_Überlagerung_2.0.png|center|frameless]]
commutes.{{r|Munkres|p=482}}

This means that <math>p</math> is, up to equivalence, uniquely determined and because of that [[universal property]] denoted as the '''universal covering''' of the space <math>X</math>.

=== Existence ===
A universal covering does not always exist. The following theorem guarantees its existence for a certain class of base spaces.

Let <math>X</math> be a connected, [[Locally simply connected space|locally simply connected]] topological space. Then, there exists a universal covering <math>p:\tilde X \rightarrow X.</math>

The set <math>\tilde X</math> is defined as <math>\tilde X = \{\gamma:\gamma \text{ is a path in }X \text{ with }\gamma(0) = x_0 \}/\text{homotopy with fixed ends},</math> where <math>x_0 \in X</math> is any chosen base point. The map <math>p:\tilde X \rightarrow X</math> is defined by <math>p([\gamma])=\gamma(1).</math>{{r|Hatcher|p=64}}

The [[topology]] on <math>\tilde X</math> is constructed as follows: Let <math>\gamma:I \rightarrow X</math> be a path with <math>\gamma(0)=x_0.</math> Let <math>U</math> be a simply connected neighborhood of the endpoint <math>x=\gamma(1).</math> Then, for every <math>y \in U,</math> there is a [[path (topology)|path]] <math>\sigma_y</math> inside <math>U</math> from <math>x</math> to <math>y</math> that is unique up to [[homotopy]]. Now consider the set <math>\tilde U=\{\gamma\sigma_y:y \in U \}/\text{homotopy with fixed ends}.</math> The restriction <math>p|_{\tilde U}: \tilde U \rightarrow U</math> with <math>p([\gamma\sigma_y])=\gamma\sigma_y(1)=y</math> is a bijection and <math>\tilde U</math> can be equipped with the [[final topology]] of <math>p|_{\tilde U}.</math>{{explain|date=December 2024|reason=How do these topologies on the tilde-U combine into one on tilde-X?}}

The fundamental group <math>\pi_{1}(X,x_0) = \Gamma</math> acts [[Free group action|freely]] on <math>\tilde X</math> by <math>([\gamma],[\tilde x]) \mapsto [\gamma\tilde x],</math> and the orbit space <math>\Gamma \backslash \tilde X</math> is homeomorphic to <math>X</math> through the map <math>[\Gamma \tilde x]\mapsto\tilde x(1).</math>

=== Examples ===
[[File:Hawaiian_Earrings.svg|right|thumb|250x250px|The Hawaiian earring. Only the ten largest circles are shown.]]

* <math>r : \mathbb{R} \to S^1</math> with <math>r(t)=(\cos(2 \pi t), \sin(2 \pi t))</math> is the universal covering of the unit circle <math>S^1</math>.
* <math>p : S^n \to \mathbb{R}P^n \cong \{+1,-1\}\backslash S^n</math> with <math>p(x)=[x]</math> is the universal covering of the [[projective space]] <math>\mathbb{R}P^n</math> for <math>n>1</math>.
* <math>q : \mathrm{SU}(n) \ltimes \mathbb{R} \to U(n)</math> with <math display=block>q(A,t)= \begin{bmatrix}
 \exp(2 \pi i t) & 0\\
 0 & I_{n-1}
\end{bmatrix}_\vphantom{x} A </math> is the universal covering of the [[unitary group]] <math>U(n)</math>.<ref>{{Cite journal
|last1=Aguilar |first1=Marcelo Alberto
|last2=Socolovsky |first2=Miguel
|date=23 November 1999
|title=The Universal Covering Group of U(n) and Projective Representations
|journal=[[International Journal of Theoretical Physics]]
|publisher=Springer US
|publication-date=April 2000
|volume=39
|issue=4
|pages=997–1013
|arxiv=math-ph/9911028
|doi=10.1023/A:1003694206391
|bibcode=1999math.ph..11028A|s2cid=18686364
}}</ref>{{rp|p=5|at=Theorem 1}}
* Since <math>\mathrm{SU}(2) \cong S^3</math>, it follows that the [[quotient map (topology)|quotient map]] <math display=block>f : \mathrm{SU}(2) \rightarrow \mathrm{SU}(2) /  \mathbb{Z_2} \cong \mathrm{SO}(3)</math> is the universal covering of <math>\mathrm{SO}(3)</math>.
* A topological space which has no universal covering is the [[Hawaiian earring]]: <math display=block>
X = \bigcup_{n\in \N}\left\{(x_1,x_2)\in\R^{2} : \Bigl(x_1-\frac{1}{n}\Bigr)^2+x_2^2=\frac{1}{n^2}\right\}
</math> One can show that no neighborhood of the origin <math>(0,0)</math> is simply connected.{{r|Munkres|p=487|at=Example 1}}

== G-coverings ==
Let ''G'' be a [[discrete group]] [[Group action (mathematics)|acting]] on the [[topological space]] ''X''. This means that each element ''g'' of ''G'' is associated to a homeomorphism H<sub>''g''</sub> of ''X'' onto itself, in such a way that H<sub>''g'' ''h''</sub> is always equal to H<sub>''g''</sub> <math>\circ</math> H<sub>''h''</sub> 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 {{nowrap|''X'' × ''X''}} by the twist action where the non-identity element acts by {{nowrap|(''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.

== Smooth coverings ==
Let {{math|''E''}} and {{math|''M''}} be [[smooth manifolds]] with or without [[Boundary of a manifold|boundary]]. A covering <math>\pi : E \to M</math> is called a '''smooth covering''' if it is a [[smooth map]] and the sheets are mapped ''diffeomorphically'' onto the corresponding open subset of {{math|''M''}}. (This is in contrast to the definition of a covering, which merely requires that the sheets are mapped ''homeomorphically'' onto the corresponding open subset.)

== Deck transformation ==
=== Definition ===
Let <math>p:E \rightarrow X</math> be a covering. A '''deck transformation''' is a homeomorphism <math>d:E \rightarrow E</math>, such that the diagram of continuous maps
[[File:Diagramm_Decktrafo.png|center|frameless]]
commutes. Together with the composition of maps, the set of deck transformation forms a [[Group (mathematics)|group]] <math>\operatorname{Deck}(p)</math>, which is the same as <math>\operatorname{Aut}(p)</math>.

Now suppose <math>p:C \to X</math> is a covering map and <math>C</math> (and therefore also <math>X</math>) is connected and locally path connected. The action of <math>\operatorname{Aut}(p)</math> on each fiber is [[Group action (mathematics)#Notable properties of actions|free]]. If this action is [[Group action (mathematics)#Remarkable properties of actions|transitive]] on some fiber, then it is transitive on all fibers, and we call the cover '''regular''' (or '''normal''' or '''Galois'''). Every such regular cover is a [[principal bundle|principal {{nowrap|<math>G</math>-bundle}}]], where <math>G = \operatorname{Aut}(p)</math> is considered as a discrete topological group.

Every universal cover <math>p:D \to X </math> is regular, with deck transformation group being isomorphic to the [[fundamental group]] {{nowrap|<math>\pi_1(X)</math>.}}

=== Examples ===
* Let <math>q : S^1 \to S^1</math> be the covering <math>q(z)=z^{n}</math> for some <math>n \in \mathbb{N} </math>, then the map <math>d_k:S^1 \rightarrow S^1 : z \mapsto z \, e^{2\pi ik/n} </math> for <math>k \in \mathbb{Z}</math> is a deck transformation and <math>\operatorname{Deck}(q)\cong \mathbb{Z}/ n\mathbb{Z}</math>.
* Let <math>r : \mathbb{R} \to S^1</math> be the covering <math>r(t)=(\cos(2 \pi t), \sin(2 \pi t))</math>, then the map <math>d_k:\mathbb{R} \rightarrow \mathbb{R} : t \mapsto t + k</math> for <math>k \in \mathbb{Z}</math> is a deck transformation and <math>\operatorname{Deck}(r)\cong \mathbb{Z}</math>.
* As another important example, consider <math>\Complex</math> the complex plane and <math>\Complex^{\times}</math> the complex plane minus the origin.  Then the map <math>p: \Complex^{\times} \to \Complex^{\times}</math> with <math> p(z) = z^{n} </math> is a regular cover. The deck transformations are multiplications with <math>n</math>-th [[root of unity|roots of unity]] and the deck transformation group is therefore isomorphic to the [[cyclic group]] <math>\Z/n\Z</math>. Likewise, the map <math>\exp : \Complex \to \Complex^{\times}</math> with <math>\exp(z) = e^{z}</math> is the universal cover.

=== Properties ===
Let <math>X</math> be a path-connected space and <math>p:E \rightarrow X</math> be a connected covering. Since a deck transformation <math>d:E \rightarrow E</math> is [[Bijection|bijective]], it permutes the elements of a fiber <math>p^{-1}(x)</math> with <math>x \in X</math> and is uniquely determined by where it sends a single point. In particular, only the identity map fixes a point in the fiber.{{r|Hatcher|p=70}} Because of this property every deck transformation defines a [[group action]] on <math>E</math>, i.e. let <math>U \subset X</math> be an open neighborhood of a <math>x \in X</math> and <math>\tilde U \subset E</math> an open neighborhood of an <math>e \in p^{-1}(x)</math>, then <math>\operatorname{Deck}(p) \times E \rightarrow E: (d,\tilde U)\mapsto d(\tilde U)</math> is a [[group action]].

=== Normal coverings ===
==== Definition ====
A covering <math>p:E \rightarrow X</math> is called normal, if <math>\operatorname{Deck}(p) \backslash E \cong X</math>. This means, that for every <math>x \in X</math> and any two <math>e_0,e_1 \in p^{-1}(x)</math> there exists a deck transformation <math>d:E \rightarrow E</math>, such that <math>d(e_0)=e_1</math>.

==== Properties ====
Let <math>X</math> be a path-connected space and <math>p:E \rightarrow X</math> be a connected covering. Let <math>H=p_{\#}(\pi_1(E))</math> be a [[subgroup]] of <math>\pi_1(X)</math>, then <math>p</math> is a normal covering iff <math>H</math> is a [[normal subgroup]] of <math>\pi_1(X)</math>.

If <math>p:E \rightarrow X</math> is a normal covering and <math>H=p_{\#}(\pi_1(E))</math>, then <math>\operatorname{Deck}(p) \cong \pi_1(X)/H</math>.

If <math>p:E \rightarrow X</math> is a path-connected covering and <math>H=p_{\#}(\pi_1(E))</math>, then <math>\operatorname{Deck}(p) \cong N(H)/H</math>, whereby <math>N(H)</math> is the [[normaliser]] of <math>H</math>.{{r|Hatcher|p=71}}

Let <math>E</math> be a topological space. A group <math>\Gamma</math> acts ''discontinuously'' on <math>E</math>, if every <math>e \in E</math> has an open neighborhood <math>V \subset E</math>  with <math>V \neq \empty</math>, such that for every <math>d_1, d_2 \in \Gamma
</math> with <math>d_1 V \cap d_2 V \neq \empty
</math> one has <math>d_1 = d_2</math>.

If a group <math>\Gamma</math> acts discontinuously on a topological space <math>E</math>, then the [[quotient map (topology)|quotient map]] <math>q: E \rightarrow \Gamma \backslash E </math> with <math>q(e)=\Gamma e</math> is a normal covering.{{r|Hatcher|p=72}} Hereby <math>\Gamma \backslash E = \{\Gamma e: e \in E\}</math> is the [[Quotient space (topology)|quotient space]] and <math>\Gamma e = \{\gamma(e):\gamma \in \Gamma\}</math> is the [[Orbit (group theory)|orbit]] of the group action.

==== Examples ====
* The covering <math>q : S^1 \to S^1 </math> with <math>q(z)=z^{n}</math> is a normal coverings for every <math>n \in \mathbb{N}</math>.
* Every simply connected covering is a normal covering.

=== Calculation ===
Let <math>\Gamma</math> be a group, which acts discontinuously on a topological space <math>E</math> and let <math>q: E \rightarrow \Gamma \backslash E </math> be the normal covering. 

* If <math>E</math> is path-connected, then <math>\operatorname{Deck}(q) \cong \Gamma</math>.{{r|Hatcher|p=72}}

* If <math>E</math> is simply connected, then <math>\operatorname{Deck}(q)\cong \pi_1(\Gamma \backslash E)</math>.{{r|Hatcher|p=71}}

==== Examples ====
* Let <math>n \in \mathbb{N}</math>. The antipodal map <math>g:S^n \rightarrow S^n</math> with <math>g(x)=-x</math> generates, together with the composition of maps, a group <math>D(g) \cong \mathbb{Z/2Z}</math> and induces a group action <math>D(g) \times S^n \rightarrow S^n, (g,x)\mapsto g(x)</math>, which acts discontinuously on <math>S^n</math>. Because of <math>\mathbb{Z_2} \backslash S^n \cong \mathbb{R}P^n</math> it follows, that the quotient map <math>q : S^n \rightarrow \mathbb{Z_2}\backslash S^n \cong \mathbb{R}P^n</math> is a normal covering and for <math>n > 1</math> a universal covering, hence <math>\operatorname{Deck}(q)\cong \mathbb{Z/2Z}\cong \pi_1({\mathbb{R}P^n})</math> for <math>n > 1</math>.
* Let <math>\mathrm{SO}(3)</math> be the [[special orthogonal group]], then the map <math>f : \mathrm{SU}(2) \rightarrow \mathrm{SO}(3) \cong \mathbb{Z_2} \backslash \mathrm{SU}(2)</math> is a normal covering and because of <math>\mathrm{SU}(2) \cong S^3</math>, it is the universal covering, hence <math>\operatorname{Deck}(f) \cong \mathbb{Z/2Z} \cong \pi_1(\mathrm{SO}(3))</math>.
* With the group action <math>(z_1,z_2)*(x,y)=(z_1+(-1)^{z_2}x,z_2+y)</math> of <math>\mathbb{Z^2}</math> on <math>\mathbb{R^2}</math>, whereby <math>(\mathbb{Z^2},*)</math> is the [[semidirect product]] <math>\mathbb{Z} \rtimes \mathbb{Z} </math>, one gets the universal covering <math>f: \mathbb{R^2} \rightarrow (\mathbb{Z} \rtimes \mathbb{Z}) \backslash \mathbb{R^2} \cong K </math> of the [[klein bottle]] <math>K</math>, hence  <math>\operatorname{Deck}(f) \cong \mathbb{Z} \rtimes \mathbb{Z} \cong \pi_1(K)</math>.
* Let <math>T = S^1 \times S^1</math> be the [[Torus#Topology|torus]] which is embedded in the <math>\mathbb{C^2}</math>. Then one gets a homeomorphism <math>\alpha: T \rightarrow T: (e^{ix},e^{iy}) \mapsto (e^{i(x+\pi)},e^{-iy})</math>, which induces a discontinuous group action <math>G_{\alpha} \times T \rightarrow T</math>, whereby <math>G_{\alpha} \cong \mathbb{Z/2Z}</math>. It follows, that the map <math>f: T \rightarrow G_{\alpha} \backslash T \cong K</math> is a normal covering of the klein bottle, hence <math>\operatorname{Deck}(f) \cong \mathbb{Z/2Z}</math>.
* Let <math>S^3</math> be embedded in the <math>\mathbb{C^2}</math>. Since the group action <math>S^3 \times \mathbb{Z/pZ} \rightarrow S^3: ((z_1,z_2),[k]) \mapsto (e^{2 \pi i k/p}z_1,e^{2 \pi i k q/p}z_2)</math> is discontinuously, whereby <math>p,q \in \mathbb{N}</math> are [[Coprime integers|coprime]], the map <math>f:S^3 \rightarrow \mathbb{Z_p} \backslash S^3 =: L_{p,q}</math> is the universal covering of the [[lens space]] <math>L_{p,q}</math>, hence <math>\operatorname{Deck}(f) \cong \mathbb{Z/pZ} \cong \pi_1(L_{p,q})</math>.

== Galois correspondence ==
Let <math>X</math> be a connected and [[Locally simply connected space|locally simply connected]] space, then for every [[subgroup]] <math>H\subseteq \pi_1(X)</math> there exists a path-connected covering <math>\alpha:X_H \rightarrow X</math> with <math>\alpha_{\#}(\pi_1(X_H))=H</math>.{{r|Hatcher|p=66}}

Let  <math>p_1:E \rightarrow X</math> and <math>p_2: E' \rightarrow X</math> be two path-connected coverings, then they are equivalent iff the subgroups <math>H = p_{1\#}(\pi_1(E))</math> and <math>H'=p_{2\#}(\pi_1(E'))</math> are [[Conjugacy class#Definition|conjugate]] to each other.{{r|Munkres|p=482}}

Let <math>X</math> be a connected and locally simply connected space, then, up to equivalence between coverings, there is a bijection:

<math>
\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\}
\end{matrix}
</math>

For a sequence of subgroups <math>
\displaystyle \{\text{e}\} \subset H \subset G \subset \pi_1(X)
</math> one gets a sequence of coverings <math>
\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
</math>. For a subgroup <math>
H \subset \pi_1(X)
</math> with [[Index of a group|index]] <math>
\displaystyle[\pi_1(X):H] = d
</math>, the covering <math>
\alpha:X_H \rightarrow X
</math> has degree <math>d</math>.

== Classification ==
=== Definitions ===
==== Category of coverings ====
Let <math>X</math> be a topological space. The objects of the [[Category theory|category]] '''<math>\boldsymbol{Cov(X)}</math>''' are the coverings <math>p:E \rightarrow X</math> of <math>X</math> and the [[Morphism (category theory)|morphisms]] between two coverings <math>p:E \rightarrow X</math> and <math>q:F\rightarrow X</math> are continuous maps <math>f:E \rightarrow F</math>, such that the diagram 
[[File:Kommutierendes_Diagramm_Cov.png|center|frameless]]
commutes.

==== G-Set ====
Let <math>G</math> be a [[topological group]]. The [[Category theory|category]] <math>\boldsymbol{G-Set}</math> is the category of sets which are [[G-set|G-sets]]. The morphisms are [[Group action#Morphisms and isomorphisms between G-sets|G-maps]] <math>\phi:X \rightarrow Y</math> between G-sets. They satisfy the condition <math>\phi(gx)=g \, \phi(x)</math> for every <math>g \in G</math>.

=== Equivalence ===
Let <math>X</math> be a connected and locally simply connected space, <math>x \in X</math> and <math>G = \pi_1(X,x)</math> be the fundamental group of <math>X</math>. Since <math>G</math> defines, by lifting of paths and evaluating at the endpoint of the lift, a group action on the fiber of a covering, the [[functor]] <math>F:\boldsymbol{Cov(X)} \longrightarrow \boldsymbol{G-Set}: p \mapsto p^{-1}(x)</math> is an [[equivalence of categories]].{{r|Hatcher|pp=68-70}}

== Applications ==
[[Image:Rotating gimbal-xyz.gif|thumb|300px|[[Gimbal lock]] occurs because any map {{nowrap|''T''<sup>3</sup> → '''RP'''<sup>3</sup>}} is not a covering map. In particular, the relevant map carries any element of ''T''<sup>3</sup>, that is, an ordered triple (a,b,c) of angles (real numbers mod&nbsp;2{{pi}}), to the composition of the three coordinate axis rotations R<sub>x</sub>(a)<math>\circ</math>R<sub>y</sub>(b)<math>\circ</math>R<sub>z</sub>(c) by those angles, respectively. Each of these rotations, and their composition, is an element of the rotation group SO(3), which is topologically '''RP'''<sup>3</sup>.

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 SO(3)|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'''<sup>3</sup>, with fundamental group '''Z'''/2, and only (non-trivial) covering space the hypersphere ''S''<sup>3</sup>, which is the group [[spin 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 [[gimbal]]s to produce rotations in three dimensions. Topologically this corresponds to a map from the 3-torus ''T''<sup>3</sup> of three angles to the real projective space '''RP'''<sup>3</sup> 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 (differential topology)|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 ==
* [[Bethe lattice]] is the universal cover of a [[Cayley graph]]
* [[Covering graph]], a covering space for an [[undirected graph]], and its special case the [[bipartite double cover]]
* [[Covering group]]
* [[Galois connection]]
* [[Quotient space (topology)]]

== Literature ==
* {{cite book | last=Hatcher | first=Allen | title=Algebraic topology | publisher=Cambridge University Press | publication-place=Cambridge | date=2002 | isbn=0-521-79160-X | oclc=45420394}}
* {{cite book | last=Forster | first=Otto | title=Lectures on Riemann surfaces | publication-place=New York | date=1981 | isbn=0-387-90617-7 | oclc=7596520}}
* {{cite book | last=Munkres | first=James R. | title=Topology | publication-place=New York, NY | date=2018 | isbn=978-0-13-468951-7 | oclc=964502066}}
* {{cite book | first = Wolfgang | last = Kühnel| title=Matrizen und Lie-Gruppen Eine geometrische Einführung | publisher = Vieweg+Teubner Verlag |  publication-place=Wiesbaden | date=2011 | isbn=978-3-8348-9905-7 | oclc=706962685 | language=de | doi=10.1007/978-3-8348-9905-7}}

== References ==
<references />

[[Category:Algebraic topology]]
[[Category:Homotopy theory]]
[[Category:Fiber bundles]]
[[Category:Topological graph theory]]