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Covering space

Type of continuous map in topology

Intopology, a coveringorcovering projection is a map between topological spaces that, intuitively, locally acts like a projection of multiple copies of a space onto itself. In particular, coverings are special types of local homeomorphisms. If ${\displaystyle p:{\tilde {X}}\to X}$ is a covering, ${\displaystyle ({\tilde {X}},p)}$ is said to be a covering spaceorcoverof${\displaystyle X}$, and ${\displaystyle X}$ is said to be the base of the covering, or simply the base. By abuse of terminology, ${\displaystyle {\tilde {X}}}$ and ${\displaystyle p}$ may sometimes be called covering spaces as well. Since coverings are local homeomorphisms, a covering space is a special kind of étale space.

Covering spaces first arose in the context of complex analysis (specifically, the technique of analytic continuation), where they were introduced by Riemann as domains on which naturally multivalued complex functions become single-valued. These spaces are now called Riemann surfaces.^[1]: 10

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

Let ${\displaystyle X}$ be a topological space. A coveringof${\displaystyle X}$ is a continuous map

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

such that for every ${\displaystyle x\in X}$ there exists an open neighborhood ${\displaystyle U_{x}}$of${\displaystyle x}$ and a discrete space ${\displaystyle D_{x}}$ such that ${\displaystyle \pi ^{-1}(U_{x})=\displaystyle \bigsqcup _{d\in D_{x}}V_{d}}$ and ${\displaystyle \pi |_{V_{d}}:V_{d}\rightarrow U_{x}}$ is a homeomorphism for every ${\displaystyle d\in D_{x}}$. The open sets ${\displaystyle V_{d}}$ are called sheets, 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 the fiberof${\displaystyle x}$. If ${\displaystyle X}$ is connected, it can be shown that ${\displaystyle \pi }$issurjective, and the cardinalityof${\displaystyle D_{x}}$ is the same for all ${\displaystyle x\in X}$; this value is called the degree of the covering. If ${\displaystyle {\tilde {X}}}$ispath-connected, then the covering ${\displaystyle \pi :{\tilde {X}}\rightarrow X}$ is called a path-connected covering. This definition is equivalent to the statement that ${\displaystyle \pi }$ is a locally trivial Fiber bundle.

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

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

• 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 the unit 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 \bigsqcup _{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 \bigsqcup _{i=1}^{n}U} .

• A map which is a local homeomorphism but not a covering of the unit circle is ${\displaystyle p:\mathbb {R_{+}} \to S^{1}}$ 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 ${\displaystyle \pi :E\rightarrow X}$ 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 ${\displaystyle \pi |_{V}:V\rightarrow \pi ($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 and non-orientable manifold, then there is a covering ${\displaystyle \pi :{\tilde {X}}\rightarrow X}$ of degree ${\displaystyle 2}$, whereby ${\displaystyle {\tilde {X}}}$ is a connected and orientable manifold.^[2]: 234
• If${\displaystyle X}$ is a connected Lie group, then there is a covering ${\displaystyle \pi :{\tilde {X}}\rightarrow X}$ which is also a Lie group homomorphism and ${\displaystyle {\tilde {X}}:=\{\gamma :\gamma {\text{ is a path in X with }}\gamma (0)={\boldsymbol {1_{X}}}{\text{ modulo homotopy with fixed ends}}\}}$ is a Lie group.^[4]: 174
• If${\displaystyle X}$ is a graph, then it follows for a covering ${\displaystyle \pi :E\rightarrow X}$ that ${\displaystyle E}$ is also a graph.^[2]: 85
• If${\displaystyle X}$ is a connected manifold, then there is a covering ${\displaystyle \pi :{\tilde {X}}\rightarrow X}$, whereby ${\displaystyle {\tilde {X}}}$ is a connected and simply connected manifold.^[5]: 32
• If${\displaystyle X}$ is a connected Riemann surface, then there is a covering ${\displaystyle \pi :{\tilde {X}}\rightarrow X}$ 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 ${\displaystyle X'}$ be topological spaces and ${\displaystyle p:E\rightarrow X}$ and ${\displaystyle p':E'\rightarrow X'}$ be coverings, then ${\displaystyle p\times p':E\times E'\rightarrow X\times X'}$ with ${\displaystyle (p\times p')(e,e')=(p($e),p'(e'))} is a covering.^[6]: 339 However, coverings of ${\displaystyle X\times X'}$ are not all of this form in general.

Let ${\displaystyle X}$ be a topological space and ${\displaystyle p:E\rightarrow X}$ and ${\displaystyle p':E'\rightarrow X}$ be coverings. Both coverings are called equivalent, if there exists a homeomorphism ${\displaystyle h:E\rightarrow E'}$, such that the diagram

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

All coverings satisfy the lifting property, i.e.:

Let ${\displaystyle I}$ be the unit interval and ${\displaystyle p:E\rightarrow X}$ be a covering. Let ${\displaystyle F:Y\times I\rightarrow X}$ be a continuous map and ${\displaystyle {\tilde {F}}_{0}:Y\times \{0\}\rightarrow E}$ 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 ${\displaystyle {\tilde {F}}:Y\times I\rightarrow E}$ 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 a pathin${\displaystyle X}$ and for ${\displaystyle Y=I}$ it is a lift of a homotopy of paths in ${\displaystyle X}$.

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

Let ${\displaystyle X}$ be a path-connected space and ${\displaystyle p:E\rightarrow X}$ 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)\rightarrow 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 ${\displaystyle p:E\rightarrow X}$ a connected covering, then the induced group homomorphism

${\displaystyle p_{\#}:\pi _{1}($E)\rightarrow \pi _{1}(X)} with ${\displaystyle p_{\#}([\gamma ])=[p\circ \gamma ]}$,

isinjective and the subgroup ${\displaystyle p_{\#}(\pi _{1}($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 dimensional complex manifolds, and let ${\displaystyle f:X\rightarrow Y}$ be a continuous map. ${\displaystyle f}$isholomorphic in a point ${\displaystyle x\in X}$, if for any charts ${\displaystyle \phi _{x}:U_{1}\rightarrow V_{1}}$of${\displaystyle x}$ and ${\displaystyle \phi _{f($x)}:U_{2}\rightarrow V_{2}} of${\displaystyle f($x)} , with ${\displaystyle \phi _{x}(U_{1})\subset U_{2}}$, the map ${\displaystyle \phi _{f($x)}\circ f\circ \phi _{x}^{-1}:\mathbb {C} \rightarrow \mathbb {C} } isholomorphic.

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

The map ${\displaystyle F=\phi _{f($x)}\circ f\circ \phi _{x}^{-1}} is called the local expressionof${\displaystyle f}$in${\displaystyle x\in X}$.

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

Let ${\displaystyle f:X\rightarrow Y}$ 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 ${\displaystyle k_{x}\in \mathbb {N_{>0}} }$, 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 the ramification indexof${\displaystyle f}$in${\displaystyle x}$ and the point ${\displaystyle x\in X}$ is called a ramification pointif${\displaystyle k_{x}\geq 2}$. 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 a branch point.

Let ${\displaystyle f:X\rightarrow Y}$ be a non-constant, holomorphic map between compact Riemann surfaces. The degree ${\displaystyle \operatorname {deg} ($f)} of${\displaystyle f}$ is the cardinality of the fiber of an unramified point ${\displaystyle y=f($x)\in Y} , i.e. ${\displaystyle \operatorname {deg} ($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}=\operatorname {deg} (f)} ^[5]: 29

A continuous map ${\displaystyle f:X\rightarrow Y}$ is called a branched covering, if there exists a closed set with dense complement ${\displaystyle E\subset Y}$, such that ${\displaystyle f_{|X\smallsetminus f^{-1}($E)}:X\smallsetminus f^{-1}(E)\rightarrow Y\smallsetminus E} is a covering.

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

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

commutes.^[6]: 482

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

A universal covering does not always exist, but the following properties guarantee its existence:

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

${\displaystyle {\tilde {X}}}$ is defined as ${\displaystyle {\tilde {X}}:=\{\gamma :\gamma {\text{ is a path in }}X{\text{ with }}\gamma (0)=x_{0}\}/{\text{ homotopy with fixed ends}}}$ and ${\displaystyle p:{\tilde {X}}\rightarrow X}$by${\displaystyle p([\gamma ]):=\gamma ($1)} .^[2]: 64

The topologyon${\displaystyle {\tilde {X}}}$ is constructed as follows: Let ${\displaystyle \gamma :I\rightarrow X}$ 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 ${\displaystyle y\in U}$ the paths ${\displaystyle \sigma _{y}}$ inside ${\displaystyle U}$ from ${\displaystyle x}$to${\displaystyle y}$ are uniquely determined up to homotopy. Now consider ${\displaystyle {\tilde {U}}:=\{\gamma .\sigma _{y}:y\in U\}/{\text{ homotopy with fixed ends}}}$, then ${\displaystyle p_{|{\tilde {U}}}:{\tilde {U}}\rightarrow U}$ with ${\displaystyle p([\gamma .\sigma _{y}])=\gamma .\sigma _{y}($1)=y} is a bijection and ${\displaystyle {\tilde {U}}}$ can be equipped with the final topologyof${\displaystyle p_{|{\tilde {U}}}}$.

The fundamental group ${\displaystyle \pi _{1}(X,x_{0})=\Gamma }$ acts freely through ${\displaystyle ([\gamma ],[{\tilde {x}}])\mapsto [\gamma .{\tilde {x}}]}$on${\displaystyle {\tilde {X}}}$ and ${\displaystyle \psi :\Gamma \backslash {\tilde {X}}\rightarrow X}$ with ${\displaystyle \psi ([\Gamma {\tilde {x}}])={\tilde {x}}($1)} is a homeomorphism, i.e. ${\displaystyle \Gamma \backslash {\tilde {X}}\cong X}$.

• ${\displaystyle r:\mathbb {R} \to S^{1}}$ with ${\displaystyle r($t)=(\cos(2\pi t),\sin(2\pi t))} is the universal covering of the unit circle ${\displaystyle S^{1}}$.
• ${\displaystyle p:S^{n}\to \mathbb {R} P^{n}\cong \{+1,-1\}\backslash S^{n}}$ with ${\displaystyle p($x)=[x]} is the universal covering of the projective space ${\displaystyle \mathbb {R} P^{n}}$ for ${\displaystyle n>1}$.
• ${\displaystyle q:\mathrm {SU} ($n)\ltimes \mathbb {R} \to U(n)} with $${\displaystyle q(A,t)={\begin{bmatrix}\exp(2\pi it)&0\\0&I_{n-1}\end{bmatrix}}_{\vphantom {x}}A}$$ is the universal covering of the unitary group ${\displaystyle U($n)} .^{[7]: 5, Theorem 1}
• Since ${\displaystyle \mathrm {SU} ($2)\cong S^{3}} , it follows that the quotient map $${\displaystyle f:\mathrm {SU} ($$2)\rightarrow \mathrm {SU} (2)\backslash \mathbb {Z_{2}} \cong \mathrm {SO} (3)} is the universal covering of the ${\displaystyle \mathrm {SO} ($3)} .
• A topological space which has no universal covering is the Hawaiian earring: $${\displaystyle X=\bigcup _{n\in \mathbb {N} }\left\{(x_{1},x_{2})\in \mathbb {R} ^{2}:{\Bigl (}x_{1}-{\frac {1}{n}}{\Bigr )}^{2}+x_{2}^{2}={\frac {1}{n^{2}}}\right\}}$$ One can show that no neighborhood of the origin ${\displaystyle (0,0)}$ is simply connected.^{[6]: 487, Example 1}