Intopology, acovering orcovering projection is amap betweentopological spaces that, intuitively,locally acts like aprojection of multiple copies of a space onto itself. In particular, coverings are special types oflocal homeomorphisms. If${\displaystyle p:\tilde{X}\to X}$ is a covering,${\displaystyle (\tilde{X},p)}$ is said to be acovering space orcover of${\displaystyle X}$, and${\displaystyle X}$ is said to be thebase of the covering, or simply thebase. Byabuse of terminology,${\displaystyle \tilde{X}}$ and${\displaystyle p}$ may sometimes be calledcovering spaces as well. Since coverings are local homeomorphisms, a covering space is a special kind ofétalé space.

Covering spaces first arose in the context ofcomplex analysis (specifically, the technique ofanalytic continuation), where they were introduced byRiemann as domains on which naturallymultivalued complex functions become single-valued. These spaces are now calledRiemann surfaces.^[1]: 10

Covering spaces are an important tool in several areas of mathematics. In moderngeometry, covering spaces (orbranched coverings, which have slightly weaker conditions) are used in the construction ofmanifolds,orbifolds, and themorphisms between them. Inalgebraic topology, covering spaces are closely related to thefundamental group: for one, since all coverings have thehomotopy lifting property, covering spaces are an important tool in the calculation ofhomotopy groups. A standard example in this vein is the calculation of thefundamental group of the circle by means of the covering of${\displaystyle S^1}$ by${\displaystyle \mathbb{R}}$ (seebelow).^[2]: 29 Under certain conditions, covering spaces also exhibit aGalois correspondence with the subgroups of the fundamental group.

Let${\displaystyle X}$ be a topological space. Acovering of${\displaystyle X}$ is a continuous map

${\displaystyle \pi :\tilde{X}\to X}$

such that for every${\displaystyle x\in X}$ there exists anopen neighborhood${\displaystyle U_x}$ of${\displaystyle x}$ and adiscrete space${\displaystyle D_x}$ such that${\displaystyle {\pi }^{-1}(U_x)={\displaystyle \underset{d\in D_x}{⨆}{V}_d}}$ and${\displaystyle \pi {|}_{V_d}:V_d\to U_x}$ is ahomeomorphism for every${\displaystyle d\in D_x}$.The open sets${\displaystyle V_d}$ are calledsheets, which are uniquely determined up to homeomorphism if${\displaystyle U_x}$ isconnected.^[2]: 56 For each${\displaystyle x\in X}$ the discrete set${\displaystyle {\pi }^{-1}(x)}$ is called thefiber of${\displaystyle x}$. If${\displaystyle X}$ is connected (and${\displaystyle \tilde{X}}$ is non-empty), it can be shown that${\displaystyle \pi }$ issurjective, and thecardinality of${\displaystyle D_x}$ is the same for all${\displaystyle x\in X}$; this value is called thedegree of the covering. If${\displaystyle \tilde{X}}$ ispath-connected, then the covering${\displaystyle \pi :\tilde{X}\to X}$ is called apath-connected covering. This definition is equivalent to the statement that${\displaystyle \pi }$ is a locally trivialFiber bundle.

Some authors also require that${\displaystyle \pi }$ be surjective in the case that${\displaystyle X}$ is not connected.^[3]

• For every topological space${\displaystyle X}$, theidentity map${\displaystyle \mathrm{id}:X\to X}$ is a covering. Likewise for any discrete space${\displaystyle D}$ the projection${\displaystyle \pi :X\times D\to X}$ taking${\displaystyle (x,i)\mapsto x}$ is a covering. Coverings of this type are calledtrivial coverings; if${\displaystyle D}$ has finitely many (say${\displaystyle k}$) elements, the covering is called thetrivial${\displaystyle k}$-sheeted covering of${\displaystyle X}$.

• The map${\displaystyle r:\mathbb{R}\to S^1}$ with${\displaystyle r(t)=(\cos (2\pi t),\sin (2\pi t))}$ is a covering of theunit circle${\displaystyle S^1}$. The base of the covering is${\displaystyle S^1}$ and the covering space is${\displaystyle \mathbb{R}}$. For any point${\displaystyle x=(x_1,x_2)\in S^1}$ such that${\displaystyle x_1>0}$, the set${\displaystyle U:=\{(x_1,x_2)\in S^1\mid x_1>0\}}$ is an open neighborhood of${\displaystyle x}$. The preimage of${\displaystyle U}$ under${\displaystyle r}$ is

  ${\displaystyle r^{-1}(U)={\displaystyle \underset{n\in \mathbb{Z}}{⨆}\left(n-\frac{1}{4},n+\frac{1}{4}\right)}}$

and the sheets of the covering are${\displaystyle V_n=(n-1/4,n+1/4)}$ for${\displaystyle n\in \mathbb{Z}.}$ The fiber of${\displaystyle x}$ is

${\displaystyle r^{-1}(x)=\{t\in \mathbb{R}\mid (\cos (2\pi t),\sin (2\pi t))=x\}.}$

• Another covering of the unit circle is the map${\displaystyle q:S^1\to S^1}$ with${\displaystyle q(z)=z^n}$ for some${\displaystyle n\in \mathbb{N}.}$ For an open neighborhood${\displaystyle U}$ of an${\displaystyle x\in S^1}$, one has:

${\displaystyle q^{-1}(U)={\displaystyle \underset{i=1}{\overset{n}{⨆}}U}}$.

• A map which is alocal homeomorphism but not a covering of the unit circle is${\displaystyle p:{\mathbb{R}}_{\mathbb{+}}\to S^1}$ with${\displaystyle p(t)=(\cos (2\pi t),\sin (2\pi t))}$. There is a sheet of an open neighborhood of${\displaystyle (1,0)}$, which is not mapped homeomorphically onto${\displaystyle U}$.

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

It follows that the covering space${\displaystyle E}$ and the base space${\displaystyle X}$ locally share the same properties.

• If${\displaystyle X}$ is a connected andnon-orientable manifold, then there is a covering${\displaystyle \pi :\tilde{X}\to X}$ of degree${\displaystyle 2}$, whereby${\displaystyle \tilde{X}}$ is a connected and orientable manifold.^[2]: 234
• If${\displaystyle X}$ is a connectedLie group, then there is a covering${\displaystyle \pi :\tilde{X}\to X}$ which is also aLie group homomorphism and${\displaystyle \tilde{X}:=\{\gamma :\gamma \text{ is a path in X with }\gamma (0)={\mathbf{1}}_{\mathit{X}}\text{ modulo homotopy with fixed ends}\}}$ is a Lie group.^[4]: 174
• If${\displaystyle X}$ is agraph, then it follows for a covering${\displaystyle \pi :E\to X}$ that${\displaystyle E}$ is also a graph.^[2]: 85
• If${\displaystyle X}$ is a connectedmanifold, then there is a covering${\displaystyle \pi :\tilde{X}\to X}$, whereby${\displaystyle \tilde{X}}$ is a connected andsimply connected manifold.^[5]: 32
• If${\displaystyle X}$ is a connectedRiemann surface, then there is a covering${\displaystyle \pi :\tilde{X}\to X}$ which is also a holomorphic map^[5]: 22 and${\displaystyle \tilde{X}}$ is a connected and simply connected Riemann surface.^[5]: 32

Let${\displaystyle X,Y}$ and${\displaystyle E}$ be path-connected, locally path-connected spaces, and${\displaystyle p,q}$ and${\displaystyle r}$ be continuous maps, such that the diagram

commutes.

• If${\displaystyle p}$ and${\displaystyle q}$ are coverings, so is${\displaystyle r}$.
• If${\displaystyle p}$ and${\displaystyle r}$ are coverings, so is${\displaystyle q}$.^[6]: 485

Let${\displaystyle X}$ and${\displaystyle X^{\prime }}$ be topological spaces and${\displaystyle p:E\to X}$ and${\displaystyle p^{\prime }:E^{\prime }\to X^{\prime }}$ be coverings, then${\displaystyle p\times p^{\prime }:E\times E^{\prime }\to X\times X^{\prime }}$ with${\displaystyle (p\times p^{\prime })(e,e^{\prime })=(p(e),p^{\prime }(e^{\prime }))}$ is a covering.^[6]: 339 However, coverings of${\displaystyle X\times X^{\prime }}$ are not all of this form in general.

Let${\displaystyle X}$ be a topological space and${\displaystyle p:E\to X}$ and${\displaystyle p^{\prime }:E^{\prime }\to X}$ be coverings. Both coverings are calledequivalent, if there exists a homeomorphism${\displaystyle h:E\to E^{\prime }}$, such that the diagram

commutes. If such a homeomorphism exists, then one calls the covering spaces${\displaystyle E}$ and${\displaystyle E^{\prime }}$isomorphic.

All coverings satisfy thelifting property, i.e.:

Let${\displaystyle I}$ be theunit interval and${\displaystyle p:E\to X}$ be a covering. Let${\displaystyle F:Y\times I\to X}$ be a continuous map and${\displaystyle {\tilde{F}}_0:Y\times \{0\}\to E}$ be a lift of${\displaystyle F{|}_{Y\times \{0\}}}$, i.e. a continuous map such that${\displaystyle p\circ {\tilde{F}}_0=F{|}_{Y\times \{0\}}}$. Then there is a uniquely determined, continuous map${\displaystyle \tilde{F}:Y\times I\to E}$ for which${\displaystyle \tilde{F}(y,0)={\tilde{F}}_0}$ and which is a lift of${\displaystyle F}$, i.e.${\displaystyle p\circ \tilde{F}=F}$.^[2]: 60

If${\displaystyle X}$ is a path-connected space, then for${\displaystyle Y=\{0\}}$ it follows that the map${\displaystyle \tilde{F}}$ is a lift of apath in${\displaystyle X}$ and for${\displaystyle Y=I}$ it is a lift of ahomotopy of paths in${\displaystyle X}$.

As a consequence, one can show that thefundamental group${\displaystyle {\pi }_1(S^1)}$ of the unit circle is aninfinite cyclic group, which is generated by the homotopy classes of the loop${\displaystyle \gamma :I\to S^1}$ with${\displaystyle \gamma (t)=(\cos (2\pi t),\sin (2\pi t))}$.^[2]: 29

Let${\displaystyle X}$ be a path-connected space and${\displaystyle p:E\to X}$ be a connected covering. Let${\displaystyle x,y\in X}$ be any two points, which are connected by a path${\displaystyle \gamma }$, i.e.${\displaystyle \gamma (0)=x}$ and${\displaystyle \gamma (1)=y}$. Let${\displaystyle \tilde{\gamma }}$ be the unique lift of${\displaystyle \gamma }$, then the map

${\displaystyle L_{\gamma }:p^{-1}(x)\to p^{-1}(y)}$ with${\displaystyle L_{\gamma }(\tilde{\gamma }(0))=\tilde{\gamma }(1)}$

isbijective.^[2]: 69

If${\displaystyle X}$ is a path-connected space and${\displaystyle p:E\to X}$ a connected covering, then the inducedgroup homomorphism

${\displaystyle p_{\mathrm{\#}}:{\pi }_1(E)\to {\pi }_1(X)}$ with${\displaystyle p_{\mathrm{\#}}([\gamma ])=[p\circ \gamma ]}$,

isinjective and thesubgroup${\displaystyle p_{\mathrm{\#}}({\pi }_1(E))}$ of${\displaystyle {\pi }_1(X)}$ consists of the homotopy classes of loops in${\displaystyle X}$, whose lifts are loops in${\displaystyle E}$.^[2]: 61

Let${\displaystyle X}$ and${\displaystyle Y}$ beRiemann surfaces, i.e. one dimensionalcomplex manifolds, and let${\displaystyle f:X\to Y}$ be a continuous map.${\displaystyle f}$ isholomorphic in a point${\displaystyle x\in X}$, if for anycharts${\displaystyle {\varphi }_x:U_1\to V_1}$ of${\displaystyle x}$ and${\displaystyle {\varphi }_{f(x)}:U_2\to V_2}$ of${\displaystyle f(x)}$, with${\displaystyle {\varphi }_x(U_1)\subset U_2}$, the map${\displaystyle {\varphi }_{f(x)}\circ f\circ {\varphi }_x^{-1}:\mathbb{C}\to \mathbb{C}}$ isholomorphic.

If${\displaystyle f}$ is holomorphic at all${\displaystyle x\in X}$, we say${\displaystyle f}$ isholomorphic.

The map${\displaystyle F={\varphi }_{f(x)}\circ f\circ {\varphi }_x^{-1}}$ is called thelocal expression of${\displaystyle f}$ in${\displaystyle x\in X}$.

If${\displaystyle f:X\to Y}$ is a non-constant, holomorphic map betweencompact Riemann surfaces, then${\displaystyle f}$ issurjective and anopen map,^[5]: 11 i.e. for every open set${\displaystyle U\subset X}$ theimage${\displaystyle f(U)\subset Y}$ is also open.

Let${\displaystyle f:X\to Y}$ be a non-constant, holomorphic map between compact Riemann surfaces. For every${\displaystyle x\in X}$ there exist charts for${\displaystyle x}$ and${\displaystyle f(x)}$ and there exists a uniquely determined${\displaystyle k_x\in {\mathbb{N}}_{\mathbb{>}\mathbb{0}}}$, such that the local expression${\displaystyle F}$ of${\displaystyle f}$ in${\displaystyle x}$ is of the form${\displaystyle z\mapsto z^{k_x}}$.^[5]: 10 The number${\displaystyle k_x}$ is called theramification index of${\displaystyle f}$ in${\displaystyle x}$ and the point${\displaystyle x\in X}$ is called aramification point if${\displaystyle k_x\ge 2}$. If${\displaystyle k_x=1}$ for an${\displaystyle x\in X}$, then${\displaystyle x}$ isunramified. The image point${\displaystyle y=f(x)\in Y}$ of a ramification point is called abranch point.

Let${\displaystyle f:X\to Y}$ be a non-constant, holomorphic map between compact Riemann surfaces. Thedegree${\displaystyle \mathrm{deg}(f)}$ of${\displaystyle f}$ is the cardinality of the fiber of an unramified point${\displaystyle y=f(x)\in Y}$, i.e.${\displaystyle \mathrm{deg}(f):=|f^{-1}(y)|}$.

This number is well-defined, since for every${\displaystyle y\in Y}$ the fiber${\displaystyle f^{-1}(y)}$ is discrete^[5]: 20 and for any two unramified points${\displaystyle y_1,y_2\in Y}$, it is:${\displaystyle |f^{-1}(y_1)|=|f^{-1}(y_2)|.}$

It can be calculated by:

${\displaystyle \sum _{x\in f^{-1}(y)}{k}_x=\mathrm{deg}(f)}$^[5]: 29

A continuous map${\displaystyle f:X\to Y}$ is called abranched covering, if there exists aclosed set withdense complement${\displaystyle E\subset Y}$, such that${\displaystyle f_{|X\setminus f^{-1}(E)}:X\setminus f^{-1}(E)\to Y\setminus E}$ is a covering.

• Let${\displaystyle n\in \mathbb{N}}$ and${\displaystyle n\ge 2}$, then${\displaystyle f:\mathbb{C}\to \mathbb{C}}$ with${\displaystyle f(z)=z^n}$ is a branched covering of degree${\displaystyle n}$, where by${\displaystyle z=0}$ is a branch point.
• Every non-constant, holomorphic map between compact Riemann surfaces${\displaystyle f:X\to Y}$ of degree${\displaystyle d}$ is a branched covering of degree${\displaystyle d}$.

Let${\displaystyle p:\tilde{X}\to X}$ be asimply connected covering. If${\displaystyle \beta :E\to X}$ is another simply connected covering, then there exists a uniquely determined homeomorphism${\displaystyle \alpha :\tilde{X}\to E}$, such that the diagram

commutes.^[6]: 482

This means that${\displaystyle p}$ is, up to equivalence, uniquely determined and because of thatuniversal property denoted as theuniversal covering of the space${\displaystyle X}$.

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

Let${\displaystyle X}$ be a connected,locally simply connected topological space. Then, there exists a universal covering${\displaystyle p:\tilde{X}\to X.}$

The set${\displaystyle \tilde{X}}$ is defined as${\displaystyle \tilde{X}=\{\gamma :\gamma \text{ is a path in }X\text{ with }\gamma (0)=x_0\}/\text{homotopy with fixed ends},}$ where${\displaystyle x_0\in X}$ is any chosen base point. The map${\displaystyle p:\tilde{X}\to X}$ is defined by${\displaystyle p([\gamma ])=\gamma (1).}$^[2]: 64

Thetopology on${\displaystyle \tilde{X}}$ is constructed as follows: Let${\displaystyle \gamma :I\to X}$ be a path with${\displaystyle \gamma (0)=x_0.}$ Let${\displaystyle U}$ be a simply connected neighborhood of the endpoint${\displaystyle x=\gamma (1).}$ Then, for every${\displaystyle y\in U,}$ there is apath${\displaystyle {\sigma }_y}$ inside${\displaystyle U}$ from${\displaystyle x}$ to${\displaystyle y}$ that is unique up tohomotopy. Now consider the set${\displaystyle \tilde{U}=\{\gamma {\sigma }_y:y\in U\}/\text{homotopy with fixed ends}.}$ The restriction${\displaystyle p{|}_{\tilde{U}}:\tilde{U}\to U}$ with${\displaystyle p([\gamma {\sigma }_y])=\gamma {\sigma }_y(1)=y}$ is a bijection and${\displaystyle \tilde{U}}$ can be equipped with thefinal topology of${\displaystyle p{|}_{\tilde{U}}.}$^{[further explanation needed]}

The fundamental group${\displaystyle {\pi }_1(X,x_0)=\mathrm{\Gamma }}$ actsfreely on${\displaystyle \tilde{X}}$ by${\displaystyle ([\gamma ],[\tilde{x}])\mapsto [\gamma \tilde{x}],}$ and the orbit space${\displaystyle \mathrm{\Gamma }\mathrm{\setminus }\tilde{X}}$ is homeomorphic to${\displaystyle X}$ through the map${\displaystyle [\mathrm{\Gamma }\tilde{x}]\mapsto \tilde{x}(1).}$

• ${\displaystyle r:\mathbb{R}\to S^1}$ with${\displaystyle r(t)=(\cos (2\pi t),\sin (2\pi t))}$ is the universal covering of the unit circle${\displaystyle S^1}$.
• ${\displaystyle p:S^n\to \mathbb{R}{P}^n\cong \{+1,-1\}\mathrm{\setminus }{S}^n}$ with${\displaystyle p(x)=[x]}$ is the universal covering of theprojective space${\displaystyle \mathbb{R}{P}^n}$ for${\displaystyle n>1}$.
• ${\displaystyle q:\mathrm{S}\mathrm{U}(n)⋉\mathbb{R}\to U(n)}$ with is the universal covering of theunitary group${\displaystyle U(n)}$.^{[7]: 5, Theorem 1}
• Since${\displaystyle \mathrm{S}\mathrm{U}(2)\cong S^3}$, it follows that thequotient map$${\displaystyle f:\mathrm{S}\mathrm{U}(2)\to \mathrm{S}\mathrm{U}(2)/{\mathbb{Z}}_{\mathbb{2}}\cong \mathrm{S}\mathrm{O}(3)}$$ is the universal covering of${\displaystyle \mathrm{S}\mathrm{O}(3)}$.
• A topological space which has no universal covering is theHawaiian earring:$${\displaystyle X=\bigcup _{n\in \mathbb{N}}\left\{(x_1,x_2)\in {\mathbb{R}}^2:(x_1-\frac{1}{n}{)}^2+x_2^2=\frac{1}{n^2}\right\}}$$ One can show that no neighborhood of the origin${\displaystyle (0,0)}$ is simply connected.^{[6]: 487, Example 1}

LetG be adiscrete groupacting on thetopological spaceX. This means that each elementg ofG is associated to a homeomorphism H_g ofX onto itself, in such a way that H_gh is always equal to H_g ∘ H_h for any two elementsg andh ofG. (Or in other words, a group action of the groupG on the spaceX is just a group homomorphism of the groupG into the group Homeo(X) of self-homeomorphisms ofX.) It is natural to ask under what conditions the projection fromX to theorbit spaceX/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 productX ×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 ofX andX/G is not so straightforward.

However the groupG does act on the fundamentalgroupoid ofX, and so the study is best handled by considering groups acting on groupoids, and the correspondingorbit groupoids. The theory for this is set down in Chapter 11 of the bookTopology and groupoids referred to below. The main result is that for discontinuous actions of a groupG on a Hausdorff spaceX which admits a universal cover, then the fundamental groupoid of the orbit spaceX/G is isomorphic to the orbit groupoid of the fundamental groupoid ofX, i.e. the quotient of that groupoid by the action of the groupG. This leads to explicit computations, for example of the fundamental group of the symmetric square of a space.

LetE andM besmooth manifolds with or withoutboundary. A covering${\displaystyle \pi :E\to M}$ is called asmooth covering if it is asmooth map and the sheets are mappeddiffeomorphically onto the corresponding open subset ofM. (This is in contrast to the definition of a covering, which merely requires that the sheets are mappedhomeomorphically onto the corresponding open subset.)

Let${\displaystyle p:E\to X}$ be a covering. Adeck transformation is a homeomorphism${\displaystyle d:E\to E}$, such that the diagram of continuous maps

commutes. Together with the composition of maps, the set of deck transformation forms agroup${\displaystyle \mathrm{Deck}(p)}$, which is the same as${\displaystyle \mathrm{Aut}(p)}$.

Now suppose${\displaystyle p:C\to X}$ is a covering map and${\displaystyle C}$ (and therefore also${\displaystyle X}$) is connected and locally path connected. The action of${\displaystyle \mathrm{Aut}(p)}$ on each fiber isfree. If this action istransitive on some fiber, then it is transitive on all fibers, and we call the coverregular (ornormal orGalois). Every such regular cover is aprincipal${\displaystyle G}$-bundle, where${\displaystyle G=\mathrm{Aut}(p)}$ is considered as a discrete topological group.

Every universal cover${\displaystyle p:D\to X}$ is regular, with deck transformation group being isomorphic to thefundamental group${\displaystyle {\pi }_1(X)}$.

• Let${\displaystyle q:S^1\to S^1}$ be the covering${\displaystyle q(z)=z^n}$ for some${\displaystyle n\in \mathbb{N}}$, then the map${\displaystyle d_k:S^1\to S^1:z\mapsto z{e}^{2\pi ik/n}}$ for${\displaystyle k\in \mathbb{Z}}$ is a deck transformation and${\displaystyle \mathrm{Deck}(q)\cong \mathbb{Z}/n\mathbb{Z}}$.
• Let${\displaystyle r:\mathbb{R}\to S^1}$ be the covering${\displaystyle r(t)=(\cos (2\pi t),\sin (2\pi t))}$, then the map${\displaystyle d_k:\mathbb{R}\to \mathbb{R}:t\mapsto t+k}$ for${\displaystyle k\in \mathbb{Z}}$ is a deck transformation and${\displaystyle \mathrm{Deck}(r)\cong \mathbb{Z}}$.
• As another important example, consider${\displaystyle \mathbb{C}}$ the complex plane and${\displaystyle {\mathbb{C}}^{\times }}$ the complex plane minus the origin. Then the map${\displaystyle p:{\mathbb{C}}^{\times }\to {\mathbb{C}}^{\times }}$ with${\displaystyle p(z)=z^n}$ is a regular cover. The deck transformations are multiplications with${\displaystyle n}$-throots of unity and the deck transformation group is therefore isomorphic to thecyclic group${\displaystyle \mathbb{Z}/n\mathbb{Z}}$. Likewise, the map${\displaystyle \exp :\mathbb{C}\to {\mathbb{C}}^{\times }}$ with${\displaystyle \exp (z)=e^z}$ is the universal cover.

Let${\displaystyle X}$ be a path-connected space and${\displaystyle p:E\to X}$ be a connected covering. Since a deck transformation${\displaystyle d:E\to E}$ isbijective, it permutes the elements of a fiber${\displaystyle p^{-1}(x)}$ with${\displaystyle x\in X}$ and is uniquely determined by where it sends a single point. In particular, only the identity map fixes a point in the fiber.^[2]: 70 Because of this property every deck transformation defines agroup action on${\displaystyle E}$, i.e. let${\displaystyle U\subset X}$ be an open neighborhood of a${\displaystyle x\in X}$ and${\displaystyle \tilde{U}\subset E}$ an open neighborhood of an${\displaystyle e\in p^{-1}(x)}$, then${\displaystyle \mathrm{Deck}(p)\times E\to E:(d,\tilde{U})\mapsto d(\tilde{U})}$ is agroup action.

A covering${\displaystyle p:E\to X}$ is called normal, if${\displaystyle \mathrm{Deck}(p)\mathrm{\setminus }E\cong X}$. This means, that for every${\displaystyle x\in X}$ and any two${\displaystyle e_0,e_1\in p^{-1}(x)}$ there exists a deck transformation${\displaystyle d:E\to E}$, such that${\displaystyle d(e_0)=e_1}$.

Let${\displaystyle X}$ be a path-connected space and${\displaystyle p:E\to X}$ be a connected covering. Let${\displaystyle H=p_{\mathrm{\#}}({\pi }_1(E))}$ be asubgroup of${\displaystyle {\pi }_1(X)}$, then${\displaystyle p}$ is a normal covering iff${\displaystyle H}$ is anormal subgroup of${\displaystyle {\pi }_1(X)}$.

If${\displaystyle p:E\to X}$ is a normal covering and${\displaystyle H=p_{\mathrm{\#}}({\pi }_1(E))}$, then${\displaystyle \mathrm{Deck}(p)\cong {\pi }_1(X)/H}$.

If${\displaystyle p:E\to X}$ is a path-connected covering and${\displaystyle H=p_{\mathrm{\#}}({\pi }_1(E))}$, then${\displaystyle \mathrm{Deck}(p)\cong N(H)/H}$, whereby${\displaystyle N(H)}$ is thenormaliser of${\displaystyle H}$.^[2]: 71

Let${\displaystyle E}$ be a topological space. A group${\displaystyle \mathrm{\Gamma }}$ actsdiscontinuously on${\displaystyle E}$, if every${\displaystyle e\in E}$ has an open neighborhood${\displaystyle V\subset E}$ with${\displaystyle V\ne \mathrm{\varnothing }}$, such that for every${\displaystyle d_1,d_2\in \mathrm{\Gamma }}$ with${\displaystyle d_{1}V\cap d_{2}V\ne \mathrm{\varnothing }}$ one has${\displaystyle d_1=d_2}$.

If a group${\displaystyle \mathrm{\Gamma }}$ acts discontinuously on a topological space${\displaystyle E}$, then thequotient map${\displaystyle q:E\to \mathrm{\Gamma }\mathrm{\setminus }E}$ with${\displaystyle q(e)=\mathrm{\Gamma }e}$ is a normal covering.^[2]: 72 Hereby${\displaystyle \mathrm{\Gamma }\mathrm{\setminus }E=\{\mathrm{\Gamma }e:e\in E\}}$ is thequotient space and${\displaystyle \mathrm{\Gamma }e=\{\gamma (e):\gamma \in \mathrm{\Gamma }\}}$ is theorbit of the group action.

• The covering${\displaystyle q:S^1\to S^1}$ with${\displaystyle q(z)=z^n}$ is a normal coverings for every${\displaystyle n\in \mathbb{N}}$.
• Every simply connected covering is a normal covering.

Let${\displaystyle \mathrm{\Gamma }}$ be a group, which acts discontinuously on a topological space${\displaystyle E}$ and let${\displaystyle q:E\to \mathrm{\Gamma }\mathrm{\setminus }E}$ be the normal covering.

• If${\displaystyle E}$ is path-connected, then${\displaystyle \mathrm{Deck}(q)\cong \mathrm{\Gamma }}$.^[2]: 72

• If${\displaystyle E}$ is simply connected, then${\displaystyle \mathrm{Deck}(q)\cong {\pi }_1(\mathrm{\Gamma }\mathrm{\setminus }E)}$.^[2]: 71

• Let${\displaystyle n\in \mathbb{N}}$. The antipodal map${\displaystyle g:S^n\to S^n}$ with${\displaystyle g(x)=-x}$ generates, together with the composition of maps, a group${\displaystyle D(g)\cong \mathbb{Z}\mathbb{/}\mathbb{2}\mathbb{Z}}$ and induces a group action${\displaystyle D(g)\times S^n\to S^n,(g,x)\mapsto g(x)}$, which acts discontinuously on${\displaystyle S^n}$. Because of${\displaystyle {\mathbb{Z}}_{\mathbb{2}}\mathrm{\setminus }{S}^n\cong \mathbb{R}{P}^n}$ it follows, that the quotient map${\displaystyle q:S^n\to {\mathbb{Z}}_{\mathbb{2}}\mathrm{\setminus }{S}^n\cong \mathbb{R}{P}^n}$ is a normal covering and for${\displaystyle n>1}$ a universal covering, hence${\displaystyle \mathrm{Deck}(q)\cong \mathbb{Z}\mathbb{/}\mathbb{2}\mathbb{Z}\cong {\pi }_1(\mathbb{R}{P}^n)}$ for${\displaystyle n>1}$.
• Let${\displaystyle \mathrm{S}\mathrm{O}(3)}$ be thespecial orthogonal group, then the map${\displaystyle f:\mathrm{S}\mathrm{U}(2)\to \mathrm{S}\mathrm{O}(3)\cong {\mathbb{Z}}_{\mathbb{2}}\mathrm{\setminus }\mathrm{S}\mathrm{U}(2)}$ is a normal covering and because of${\displaystyle \mathrm{S}\mathrm{U}(2)\cong S^3}$, it is the universal covering, hence${\displaystyle \mathrm{Deck}(f)\cong \mathbb{Z}\mathbb{/}\mathbb{2}\mathbb{Z}\cong {\pi }_1(\mathrm{S}\mathrm{O}(3))}$.
• With the group action${\displaystyle (z_1,z_2)\ast (x,y)=(z_1+(-1{)}^{z_2}x,z_2+y)}$ of${\displaystyle {\mathbb{Z}}^{\mathbb{2}}}$ on${\displaystyle {\mathbb{R}}^{\mathbb{2}}}$, whereby${\displaystyle ({\mathbb{Z}}^{\mathbb{2}},\ast )}$ is thesemidirect product${\displaystyle \mathbb{Z}⋊\mathbb{Z}}$, one gets the universal covering${\displaystyle f:{\mathbb{R}}^{\mathbb{2}}\to (\mathbb{Z}⋊\mathbb{Z})\mathrm{\setminus }{\mathbb{R}}^{\mathbb{2}}\cong K}$ of theklein bottle${\displaystyle K}$, hence${\displaystyle \mathrm{Deck}(f)\cong \mathbb{Z}⋊\mathbb{Z}\cong {\pi }_1(K)}$.
• Let${\displaystyle T=S^1\times S^1}$ be thetorus which is embedded in the${\displaystyle {\mathbb{C}}^{\mathbb{2}}}$. Then one gets a homeomorphism${\displaystyle \alpha :T\to T:(e^{ix},e^{iy})\mapsto (e^{i(x+\pi )},e^{-iy})}$, which induces a discontinuous group action${\displaystyle G_{\alpha }\times T\to T}$, whereby${\displaystyle G_{\alpha }\cong \mathbb{Z}\mathbb{/}\mathbb{2}\mathbb{Z}}$. It follows, that the map${\displaystyle f:T\to G_{\alpha }\mathrm{\setminus }T\cong K}$ is a normal covering of the klein bottle, hence${\displaystyle \mathrm{Deck}(f)\cong \mathbb{Z}\mathbb{/}\mathbb{2}\mathbb{Z}}$.
• Let${\displaystyle S^3}$ be embedded in the${\displaystyle {\mathbb{C}}^{\mathbb{2}}}$. Since the group action${\displaystyle S^3\times \mathbb{Z}\mathbb{/}\mathbb{p}\mathbb{Z}\to S^3:((z_1,z_2),[k])\mapsto (e^{2\pi ik/p}{z}_1,e^{2\pi ikq/p}{z}_2)}$ is discontinuously, whereby${\displaystyle p,q\in \mathbb{N}}$ arecoprime, the map${\displaystyle f:S^3\to {\mathbb{Z}}_{\mathbb{p}}\mathrm{\setminus }{S}^3=:L_{p,q}}$ is the universal covering of thelens space${\displaystyle L_{p,q}}$, hence${\displaystyle \mathrm{Deck}(f)\cong \mathbb{Z}\mathbb{/}\mathbb{p}\mathbb{Z}\cong {\pi }_1(L_{p,q})}$.

Let${\displaystyle X}$ be a connected andlocally simply connected space, then for everysubgroup${\displaystyle H\subseteq {\pi }_1(X)}$ there exists a path-connected covering${\displaystyle \alpha :X_H\to X}$ with${\displaystyle {\alpha }_{\mathrm{\#}}({\pi }_1(X_H))=H}$.^[2]: 66

Let${\displaystyle p_1:E\to X}$ and${\displaystyle p_2:E^{\prime }\to X}$ be two path-connected coverings, then they are equivalent iff the subgroups${\displaystyle H=p_{1\mathrm{\#}}({\pi }_1(E))}$ and${\displaystyle H^{\prime }=p_{2\mathrm{\#}}({\pi }_1(E^{\prime }))}$ areconjugate to each other.^[6]: 482

Let${\displaystyle X}$ be a connected and locally simply connected space, then, up to equivalence between coverings, there is a bijection:

For a sequence of subgroups${\displaystyle {\displaystyle \{\text{e}\}\subset H\subset G\subset {\pi }_1(X)}}$ one gets a sequence of coverings${\displaystyle \tilde{X}⟶X_H\cong H\mathrm{\setminus }\tilde{X}⟶X_G\cong G\mathrm{\setminus }\tilde{X}⟶X\cong {\pi }_1(X)\mathrm{\setminus }\tilde{X}}$. For a subgroup${\displaystyle H\subset {\pi }_1(X)}$ withindex${\displaystyle {\displaystyle [{\pi }_1(X):H]=d}}$, the covering${\displaystyle \alpha :X_H\to X}$ has degree${\displaystyle d}$.

Let${\displaystyle X}$ be a topological space. The objects of thecategory${\displaystyle \mathit{C}\mathit{o}\mathit{v}\mathbf{(}\mathit{X}\mathbf{)}}$ are the coverings${\displaystyle p:E\to X}$ of${\displaystyle X}$ and themorphisms between two coverings${\displaystyle p:E\to X}$ and${\displaystyle q:F\to X}$ are continuous maps${\displaystyle f:E\to F}$, such that the diagram

commutes.

Let${\displaystyle G}$ be atopological group. Thecategory${\displaystyle \mathit{G}\mathbf{-}\mathit{S}\mathit{e}\mathit{t}}$ is the category of sets which areG-sets. The morphisms areG-maps${\displaystyle \varphi :X\to Y}$ between G-sets. They satisfy the condition${\displaystyle \varphi (gx)=g\varphi (x)}$ for every${\displaystyle g\in G}$.

Let${\displaystyle X}$ be a connected and locally simply connected space,${\displaystyle x\in X}$ and${\displaystyle G={\pi }_1(X,x)}$ be the fundamental group of${\displaystyle X}$. Since${\displaystyle G}$ defines, by lifting of paths and evaluating at the endpoint of the lift, a group action on the fiber of a covering, thefunctor${\displaystyle F:\mathit{C}\mathit{o}\mathit{v}\mathbf{(}\mathit{X}\mathbf{)}⟶\mathit{G}\mathbf{-}\mathit{S}\mathit{e}\mathit{t}:p\mapsto p^{-1}(x)}$ is anequivalence of categories.^{[2]: 68–70}

An important practical application of covering spaces occurs incharts on SO(3), therotation group. This group occurs widely in engineering, due to 3-dimensional rotations being heavily used innavigation,nautical engineering, andaerospace engineering, among many other uses. Topologically, SO(3) is thereal projective spaceRP³, with fundamental groupZ/2, and only (non-trivial) covering space the hypersphereS³, which is the groupSpin(3), and represented by the unitquaternions. Thus quaternions are a preferred method for representing spatial rotations – seequaternions and spatial rotation.

However, it is often desirable to represent rotations by a set of three numbers, known asEuler angles (in numerous variants), both because this is conceptually simpler for someone familiar with planar rotation, and because one can build a combination of threegimbals to produce rotations in three dimensions. Topologically this corresponds to a map from the 3-torusT³ of three angles to the real projective spaceRP³ 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 asgimbal lock, and is demonstrated in the animation at the right – at some points (when the axes are coplanar) therank 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.

• Bethe lattice is the universal cover of aCayley graph
• Covering graph, a covering space for anundirected graph, and its special case thebipartite double cover
• Covering group
• Galois connection
• Quotient space (topology)

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