Covering space#Lifting property

Let $X$ be a topological space. A covering of $X$ is a continuous map

: $\backslash pi\; :\; \backslash tilde\; X\; \backslash rightarrow\; X$

such that for every $x\; \backslash in\; X$ there exists an open neighborhood $U\_x$ of $x$ and a discrete space $D\_x$ such that $\backslash pi^\{-1\}(U\_x)=\; \backslash displaystyle\; \backslash bigsqcup\_\{d\; \backslash in\; D\_x\}\; V\_d$ and $\backslash pi|\_\{V\_d\}:V\_d\; \backslash rightarrow\; U\_x$ is a homeomorphism for every $d\; \backslash in\; D\_x$.

The open sets $V\_\{d\}$ are called sheets, which are uniquely determined up to homeomorphism if $U\_x$ is connected.{{r|Hatcher|p=56}} For each $x\; \backslash in\; X$ the discrete set $\backslash pi^\{-1\}(x)$ is called the fiber of $x$. If $X$ is connected (and $\backslash tilde\; X$ is non-empty), it can be shown that $\backslash pi$ is surjective, and the cardinality of $D\_x$ is the same for all $x\; \backslash in\; X$; this value is called the degree of the covering. If $\backslash tilde\; X$ is path-connected, then the covering $\backslash pi\; :\; \backslash tilde\; X\; \backslash rightarrow\; X$ is called a path-connected covering. This definition is equivalent to the statement that $\backslash pi$ is a locally trivial Fiber bundle.

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

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

{{Dark mode invert|File:Covering_map.svg}}

• The map $r\; :\; \backslash mathbb\{R\}\; \backslash to\; S^1$ with $r(t)=(\backslash cos(2\; \backslash pi\; t),\; \backslash sin(2\; \backslash pi\; t))$ is a covering of the unit circle $S^1$. The base of the covering is $S^1$ and the covering space is $\backslash mathbb\{R\}$. For any point $x\; =\; (x\_1,\; x\_2)\; \backslash in\; S^1$ such that $x\_1\; >\; 0$, the set $U\; :=\; \backslash \{(x\_1,\; x\_2)\; \backslash in\; S^1\; \backslash mid\; x\_1\; >\; 0\; \backslash \}$ is an open neighborhood of $x$. The preimage of $U$ under $r$ is
• : $r^\{-1\}(U)=\backslash displaystyle\backslash bigsqcup\_\{n\; \backslash in\; \backslash mathbb\{Z\}\}\; \backslash left(\; n\; -\; \backslash frac\; 1\; 4,\; n\; +\; \backslash frac\; 1\; 4\backslash right)$

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

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

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

:: $q^\{-1\}(U)=\backslash displaystyle\backslash bigsqcup\_\{i=1\}^\{n\}\; U$.

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

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

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

• If $X$ is a connected and non-orientable manifold, then there is a covering $\backslash pi:\backslash tilde\; X\; \backslash rightarrow\; X$ of degree $2$, whereby $\backslash tilde\; X$ is a connected and orientable manifold.{{r|Hatcher|p=234}}
• If $X$ is a connected Lie group, then there is a covering $\backslash pi:\backslash tilde\; X\; \backslash rightarrow\; X$ which is also a Lie group homomorphism and $\backslash tilde\; X\; :=\; \backslash \{\backslash gamma:\backslash gamma\; \backslash text\{\; is\; a\; path\; in\; X\; with\; \}\backslash gamma(0)=\; \backslash boldsymbol\{1\_X\}\; \backslash text\{\; modulo\; homotopy\; with\; fixed\; ends\}\backslash \}$ is a Lie group.{{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}}{{rp|p=174}}
• If $X$ is a graph, then it follows for a covering $\backslash pi:E\; \backslash rightarrow\; X$ that $E$ is also a graph.{{r|Hatcher|p=85}}
• If $X$ is a connected manifold, then there is a covering $\backslash pi:\backslash tilde\; X\; \backslash rightarrow\; X$, whereby $\backslash tilde\; X$ is a connected and simply connected manifold.{{r|Forster|p=32}}
• If $X$ is a connected Riemann surface, then there is a covering $\backslash pi:\backslash tilde\; X\; \backslash rightarrow\; X$ which is also a holomorphic map{{r|Forster|p=22}} and $\backslash tilde\; X$ is a connected and simply connected Riemann surface.{{r|Forster|p=32}}

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

File:Commutativ coverings.png

commutes.

• If $p$ and $q$ are coverings, so is $r$.
• If $p$ and $r$ are coverings, so is $q$.{{r|Munkres|p=485}}

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

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

File:Kommutatives_Diagramm_Äquivalenz_von_Überlagerungen.png

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

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

Let $I$ be the unit interval and $p:E\; \backslash rightarrow\; X$ be a covering. Let $F:Y\; \backslash times\; I\; \backslash rightarrow\; X$ be a continuous map and $\backslash tilde\; F\_0:Y\; \backslash times\; \backslash \{0\backslash \}\; \backslash rightarrow\; E$ be a lift of $F|\_\{Y\; \backslash times\; \backslash \{0\backslash \}\}$, i.e. a continuous map such that $p\; \backslash circ\; \backslash tilde\; F\_0\; =\; F|\_\{Y\; \backslash times\; \backslash \{0\backslash \}\}$. Then there is a uniquely determined, continuous map $\backslash tilde\; F:Y\; \backslash times\; I\; \backslash rightarrow\; E$ for which $\backslash tilde\; F(y,0)\; =\; \backslash tilde\; F\_0$ and which is a lift of $F$, i.e. $p\; \backslash circ\; \backslash tilde\; F\; =\; F$.{{r|Hatcher|p=60}}

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

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

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

: $L\_\{\backslash gamma\}:p^\{-1\}(x)\; \backslash rightarrow\; p^\{-1\}(y)$ with $L\_\{\backslash gamma\}(\backslash tilde\; \backslash gamma\; (0))=\backslash tilde\; \backslash gamma\; (1)$

is bijective.{{r|Hatcher|p=69}}

If $X$ is a path-connected space and $p:\; E\; \backslash rightarrow\; X$ a connected covering, then the induced group homomorphism

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

is injective and the subgroup $p\_\{\backslash \#\}(\backslash pi\_1(E))$ of $\backslash pi\_1(X)$ consists of the homotopy classes of loops in $X$, whose lifts are loops in $E$.{{r|Hatcher|p=61}}

Let $X$ and $Y$ be Riemann surfaces, i.e. one dimensional complex manifolds, and let $f:\; X\; \backslash rightarrow\; Y$ be a continuous map. $f$ is holomorphic in a point $x\; \backslash in\; X$, if for any charts $\backslash phi\; \_x:U\_1\; \backslash rightarrow\; V\_1$ of $x$ and $\backslash phi\_\{f(x)\}:U\_2\; \backslash rightarrow\; V\_2$ of $f(x)$, with $\backslash phi\_x(U\_1)\; \backslash subset\; U\_2$, the map $\backslash phi\; \_\{f(x)\}\; \backslash circ\; f\; \backslash circ\; \backslash phi^\{-1\}\; \_x:\; \backslash mathbb\{C\}\; \backslash rightarrow\; \backslash mathbb\{C\}$ is holomorphic.

If $f$ is holomorphic at all $x\; \backslash in\; X$, we say $f$ is holomorphic.

The map $F\; =\backslash phi\; \_\{f(x)\}\; \backslash circ\; f\; \backslash circ\; \backslash phi^\{-1\}\; \_x$ is called the local expression of $f$ in $x\; \backslash in\; X$.

If $f:\; X\; \backslash rightarrow\; Y$ is a non-constant, holomorphic map between compact Riemann surfaces, then $f$ is surjective and an open map,{{Cite book|last=Forster|first=Otto|title=Lectures on Riemann surfaces|publisher=Springer Berlin|year=1991|isbn=978-3-540-90617-9|location=München}}{{rp|p=11}} i.e. for every open set $U\; \backslash subset\; X$ the image $f(U)\; \backslash subset\; Y$ is also open.

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

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

This number is well-defined, since for every $y\; \backslash in\; Y$ the fiber $f^\{-1\}(y)$ is discrete{{r|Forster|p=20}} and for any two unramified points $y\_1,y\_2\; \backslash in\; Y$, it is: $|f^\{-1\}(y\_1)|=|f^\{-1\}(y\_2)|.$

It can be calculated by:

: $\backslash sum\_\{x\; \backslash in\; f^\{-1\}(y)\}\; k\_x\; =\; \backslash operatorname\{deg\}(f)$ {{r|Forster|p=29}}

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

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

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

File:Universelle_Überlagerung_2.0.png

commutes.{{r|Munkres|p=482}}

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

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

Let $X$ be a connected, locally simply connected topological space. Then, there exists a universal covering $p:\backslash tilde\; X\; \backslash rightarrow\; X.$

The set $\backslash tilde\; X$ is defined as $\backslash tilde\; X\; =\; \backslash \{\backslash gamma:\backslash gamma\; \backslash text\{\; is\; a\; path\; in\; \}X\; \backslash text\{\; with\; \}\backslash gamma(0)\; =\; x\_0\; \backslash \}/\backslash text\{homotopy\; with\; fixed\; ends\},$ where $x\_0\; \backslash in\; X$ is any chosen base point. The map $p:\backslash tilde\; X\; \backslash rightarrow\; X$ is defined by $p([\backslash gamma])=\backslash gamma(1).${{r|Hatcher|p=64}}

The topology on $\backslash tilde\; X$ is constructed as follows: Let $\backslash gamma:I\; \backslash rightarrow\; X$ be a path with $\backslash gamma(0)=x\_0.$ Let $U$ be a simply connected neighborhood of the endpoint $x=\backslash gamma(1).$ Then, for every $y\; \backslash in\; U,$ there is a path $\backslash sigma\_y$ inside $U$ from $x$ to $y$ that is unique up to homotopy. Now consider the set $\backslash tilde\; U=\backslash \{\backslash gamma\backslash sigma\_y:y\; \backslash in\; U\; \backslash \}/\backslash text\{homotopy\; with\; fixed\; ends\}.$ The restriction $p|\_\{\backslash tilde\; U\}:\; \backslash tilde\; U\; \backslash rightarrow\; U$ with $p([\backslash gamma\backslash sigma\_y])=\backslash gamma\backslash sigma\_y(1)=y$ is a bijection and $\backslash tilde\; U$ can be equipped with the final topology of $p|\_\{\backslash tilde\; U\}.${{explain|date=December 2024|reason=How do these topologies on the tilde-U combine into one on tilde-X?}}

The fundamental group $\backslash pi\_\{1\}(X,x\_0)\; =\; \backslash Gamma$ acts freely on $\backslash tilde\; X$ by $([\backslash gamma],[\backslash tilde\; x])\; \backslash mapsto\; [\backslash gamma\backslash tilde\; x],$ and the orbit space $\backslash Gamma\; \backslash backslash\; \backslash tilde\; X$ is homeomorphic to $X$ through the map $[\backslash Gamma\; \backslash tilde\; x]\backslash mapsto\backslash tilde\; x(1).$

File:Hawaiian_Earrings.svg

• $r\; :\; \backslash mathbb\{R\}\; \backslash to\; S^1$ with $r(t)=(\backslash cos(2\; \backslash pi\; t),\; \backslash sin(2\; \backslash pi\; t))$ is the universal covering of the unit circle $S^1$.
• $p\; :\; S^n\; \backslash to\; \backslash mathbb\{R\}P^n\; \backslash cong\; \backslash \{+1,-1\backslash \}\backslash backslash\; S^n$ with $p(x)=[x]$ is the universal covering of the projective space $\backslash mathbb\{R\}P^n$ for $n>1$.
• $q\; :\; \backslash mathrm\{SU\}(n)\; \backslash ltimes\; \backslash mathbb\{R\}\; \backslash to\; U(n)$ with $$q(A,t)=\; \backslash begin\{bmatrix\}$$

\exp(2 \pi i t) & 0\\

0 & I_{n-1}

\end{bmatrix}_\vphantom{x} A is the universal covering of the unitary group $U(n)$.{{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

}}{{rp|p=5|at=Theorem 1}}

• Since $\backslash mathrm\{SU\}(2)\; \backslash cong\; S^3$, it follows that the quotient map $$f\; :\; \backslash mathrm\{SU\}(2)\; \backslash rightarrow\; \backslash mathrm\{SU\}(2)\; /\; \backslash mathbb\{Z\_2\}\; \backslash cong\; \backslash mathrm\{SO\}(3)$$ is the universal covering of $\backslash mathrm\{SO\}(3)$.
• A topological space which has no universal covering is the Hawaiian earring: $$$$

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\}

One can show that no neighborhood of the origin $(0,0)$ is simply connected.{{r|Munkres|p=487|at=Example 1}}

Let G be a discrete group acting on the topological space X. This means that each element g of G is associated to a homeomorphism H_g of X onto itself, in such a way that H_{g h} is always equal to H_g ∘ H_h 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.

Let {{math|E}} and {{math|M}} be smooth manifolds with or without boundary. A covering $\backslash pi\; :\; E\; \backslash to\; M$ 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.)

Let $p:E\; \backslash rightarrow\; X$ be a covering. A deck transformation is a homeomorphism $d:E\; \backslash rightarrow\; E$, such that the diagram of continuous maps

File:Diagramm_Decktrafo.png

commutes. Together with the composition of maps, the set of deck transformation forms a group $\backslash operatorname\{Deck\}(p)$, which is the same as $\backslash operatorname\{Aut\}(p)$.

Now suppose $p:C\; \backslash to\; X$ is a covering map and $C$ (and therefore also $X$) is connected and locally path connected. The action of $\backslash operatorname\{Aut\}(p)$ on each fiber is free. If this action is 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, where $G\; =\; \backslash operatorname\{Aut\}(p)$ is considered as a discrete topological group.

Every universal cover $p:D\; \backslash to\; X$ is regular, with deck transformation group being isomorphic to the fundamental group {{nowrap|$\backslash pi\_1(X)$.}}

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

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

A covering $p:E\; \backslash rightarrow\; X$ is called normal, if $\backslash operatorname\{Deck\}(p)\; \backslash backslash\; E\; \backslash cong\; X$. This means, that for every $x\; \backslash in\; X$ and any two $e\_0,e\_1\; \backslash in\; p^\{-1\}(x)$ there exists a deck transformation $d:E\; \backslash rightarrow\; E$, such that $d(e\_0)=e\_1$.

Let $X$ be a path-connected space and $p:E\; \backslash rightarrow\; X$ be a connected covering. Let $H=p\_\{\backslash \#\}(\backslash pi\_1(E))$ be a subgroup of $\backslash pi\_1(X)$, then $p$ is a normal covering iff $H$ is a normal subgroup of $\backslash pi\_1(X)$.

If $p:E\; \backslash rightarrow\; X$ is a normal covering and $H=p\_\{\backslash \#\}(\backslash pi\_1(E))$, then $\backslash operatorname\{Deck\}(p)\; \backslash cong\; \backslash pi\_1(X)/H$.

If $p:E\; \backslash rightarrow\; X$ is a path-connected covering and $H=p\_\{\backslash \#\}(\backslash pi\_1(E))$, then $\backslash operatorname\{Deck\}(p)\; \backslash cong\; N(H)/H$, whereby $N(H)$ is the normaliser of $H$.{{r|Hatcher|p=71}}

Let $E$ be a topological space. A group $\backslash Gamma$ acts discontinuously on $E$, if every $e\; \backslash in\; E$ has an open neighborhood $V\; \backslash subset\; E$ with $V\; \backslash neq\; \backslash empty$, such that for every $d\_1,\; d\_2\; \backslash in\; \backslash Gamma$

with $d\_1\; V\; \backslash cap\; d\_2\; V\; \backslash neq\; \backslash empty$

one has $d\_1\; =\; d\_2$.

If a group $\backslash Gamma$ acts discontinuously on a topological space $E$, then the quotient map $q:\; E\; \backslash rightarrow\; \backslash Gamma\; \backslash backslash\; E$ with $q(e)=\backslash Gamma\; e$ is a normal covering.{{r|Hatcher|p=72}} Hereby $\backslash Gamma\; \backslash backslash\; E\; =\; \backslash \{\backslash Gamma\; e:\; e\; \backslash in\; E\backslash \}$ is the quotient space and $\backslash Gamma\; e\; =\; \backslash \{\backslash gamma(e):\backslash gamma\; \backslash in\; \backslash Gamma\backslash \}$ is the orbit of the group action.

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

Let $\backslash Gamma$ be a group, which acts discontinuously on a topological space $E$ and let $q:\; E\; \backslash rightarrow\; \backslash Gamma\; \backslash backslash\; E$ be the normal covering.

• If $E$ is path-connected, then $\backslash operatorname\{Deck\}(q)\; \backslash cong\; \backslash Gamma$.{{r|Hatcher|p=72}}

• If $E$ is simply connected, then $\backslash operatorname\{Deck\}(q)\backslash cong\; \backslash pi\_1(\backslash Gamma\; \backslash backslash\; E)$.{{r|Hatcher|p=71}}

• Let $n\; \backslash in\; \backslash mathbb\{N\}$. The antipodal map $g:S^n\; \backslash rightarrow\; S^n$ with $g(x)=-x$ generates, together with the composition of maps, a group $D(g)\; \backslash cong\; \backslash mathbb\{Z/2Z\}$ and induces a group action $D(g)\; \backslash times\; S^n\; \backslash rightarrow\; S^n,\; (g,x)\backslash mapsto\; g(x)$, which acts discontinuously on $S^n$. Because of $\backslash mathbb\{Z\_2\}\; \backslash backslash\; S^n\; \backslash cong\; \backslash mathbb\{R\}P^n$ it follows, that the quotient map $q\; :\; S^n\; \backslash rightarrow\; \backslash mathbb\{Z\_2\}\backslash backslash\; S^n\; \backslash cong\; \backslash mathbb\{R\}P^n$ is a normal covering and for $n\; >\; 1$ a universal covering, hence $\backslash operatorname\{Deck\}(q)\backslash cong\; \backslash mathbb\{Z/2Z\}\backslash cong\; \backslash pi\_1(\{\backslash mathbb\{R\}P^n\})$ for $n\; >\; 1$.
• Let $\backslash mathrm\{SO\}(3)$ be the special orthogonal group, then the map $f\; :\; \backslash mathrm\{SU\}(2)\; \backslash rightarrow\; \backslash mathrm\{SO\}(3)\; \backslash cong\; \backslash mathbb\{Z\_2\}\; \backslash backslash\; \backslash mathrm\{SU\}(2)$ is a normal covering and because of $\backslash mathrm\{SU\}(2)\; \backslash cong\; S^3$, it is the universal covering, hence $\backslash operatorname\{Deck\}(f)\; \backslash cong\; \backslash mathbb\{Z/2Z\}\; \backslash cong\; \backslash pi\_1(\backslash mathrm\{SO\}(3))$.
• With the group action $(z\_1,z\_2)*(x,y)=(z\_1+(-1)^\{z\_2\}x,z\_2+y)$ of $\backslash mathbb\{Z^2\}$ on $\backslash mathbb\{R^2\}$, whereby $(\backslash mathbb\{Z^2\},*)$ is the semidirect product $\backslash mathbb\{Z\}\; \backslash rtimes\; \backslash mathbb\{Z\}$, one gets the universal covering $f:\; \backslash mathbb\{R^2\}\; \backslash rightarrow\; (\backslash mathbb\{Z\}\; \backslash rtimes\; \backslash mathbb\{Z\})\; \backslash backslash\; \backslash mathbb\{R^2\}\; \backslash cong\; K$ of the klein bottle $K$, hence $\backslash operatorname\{Deck\}(f)\; \backslash cong\; \backslash mathbb\{Z\}\; \backslash rtimes\; \backslash mathbb\{Z\}\; \backslash cong\; \backslash pi\_1(K)$.
• Let $T\; =\; S^1\; \backslash times\; S^1$ be the torus which is embedded in the $\backslash mathbb\{C^2\}$. Then one gets a homeomorphism $\backslash alpha:\; T\; \backslash rightarrow\; T:\; (e^\{ix\},e^\{iy\})\; \backslash mapsto\; (e^\{i(x+\backslash pi)\},e^\{-iy\})$, which induces a discontinuous group action $G\_\{\backslash alpha\}\; \backslash times\; T\; \backslash rightarrow\; T$, whereby $G\_\{\backslash alpha\}\; \backslash cong\; \backslash mathbb\{Z/2Z\}$. It follows, that the map $f:\; T\; \backslash rightarrow\; G\_\{\backslash alpha\}\; \backslash backslash\; T\; \backslash cong\; K$ is a normal covering of the klein bottle, hence $\backslash operatorname\{Deck\}(f)\; \backslash cong\; \backslash mathbb\{Z/2Z\}$.
• Let $S^3$ be embedded in the $\backslash mathbb\{C^2\}$. Since the group action $S^3\; \backslash times\; \backslash mathbb\{Z/pZ\}\; \backslash rightarrow\; S^3:\; ((z\_1,z\_2),[k])\; \backslash mapsto\; (e^\{2\; \backslash pi\; i\; k/p\}z\_1,e^\{2\; \backslash pi\; i\; k\; q/p\}z\_2)$ is discontinuously, whereby $p,q\; \backslash in\; \backslash mathbb\{N\}$ are coprime, the map $f:S^3\; \backslash rightarrow\; \backslash mathbb\{Z\_p\}\; \backslash backslash\; S^3\; =:\; L\_\{p,q\}$ is the universal covering of the lens space $L\_\{p,q\}$, hence $\backslash operatorname\{Deck\}(f)\; \backslash cong\; \backslash mathbb\{Z/pZ\}\; \backslash cong\; \backslash pi\_1(L\_\{p,q\})$.

Let $X$ be a connected and locally simply connected space, then for every subgroup $H\backslash subseteq\; \backslash pi\_1(X)$ there exists a path-connected covering $\backslash alpha:X\_H\; \backslash rightarrow\; X$ with $\backslash alpha\_\{\backslash \#\}(\backslash pi\_1(X\_H))=H$.{{r|Hatcher|p=66}}

Let $p\_1:E\; \backslash rightarrow\; X$ and $p\_2:\; E\text{'}\; \backslash rightarrow\; X$ be two path-connected coverings, then they are equivalent iff the subgroups $H\; =\; p\_\{1\backslash \#\}(\backslash pi\_1(E))$ and $H\text{'}=p\_\{2\backslash \#\}(\backslash pi\_1(E\text{'}))$ are conjugate to each other.{{r|Munkres|p=482}}

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

\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}

For a sequence of subgroups $$

\displaystyle \{\text{e}\} \subset H \subset G \subset \pi_1(X)

one gets a sequence of coverings $$

\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

. For a subgroup $$

H \subset \pi_1(X)

with index $$

\displaystyle[\pi_1(X):H] = d

, the covering $$

\alpha:X_H \rightarrow X

has degree $d$.

Let $X$ be a topological space. The objects of the category $\backslash boldsymbol\{Cov(X)\}$ are the coverings $p:E\; \backslash rightarrow\; X$ of $X$ and the morphisms between two coverings $p:E\; \backslash rightarrow\; X$ and $q:F\backslash rightarrow\; X$ are continuous maps $f:E\; \backslash rightarrow\; F$, such that the diagram

File:Kommutierendes_Diagramm_Cov.png

commutes.

Let $G$ be a topological group. The category $\backslash boldsymbol\{G-Set\}$ is the category of sets which are G-sets. The morphisms are G-maps $\backslash phi:X\; \backslash rightarrow\; Y$ between G-sets. They satisfy the condition $\backslash phi(gx)=g\; \backslash ,\; \backslash phi(x)$ for every $g\; \backslash in\; G$.

Let $X$ be a connected and locally simply connected space, $x\; \backslash in\; X$ and $G\; =\; \backslash pi\_1(X,x)$ be the fundamental group of $X$. Since $G$ defines, by lifting of paths and evaluating at the endpoint of the lift, a group action on the fiber of a covering, the functor $F:\backslash boldsymbol\{Cov(X)\}\; \backslash longrightarrow\; \backslash boldsymbol\{G-Set\}:\; p\; \backslash mapsto\; p^\{-1\}(x)$ is an equivalence of categories.{{r|Hatcher|pp=68-70}}

Image:Rotating gimbal-xyz.gif occurs because any map {{nowrap|T³ → RP³}} is not a covering map. In particular, the relevant map carries any element of T³, that is, an ordered triple (a,b,c) of angles (real numbers mod 2{{pi}}), to the composition of the three coordinate axis rotations R_x(a)∘R_y(b)∘R_z(c) by those angles, respectively. Each of these rotations, and their composition, is an element of the rotation group SO(3), which is topologically RP³.

This animation shows a set of three gimbals mounted together to allow three degrees of freedom. When all three gimbals are lined up (in the same plane), the system can only move in two dimensions from this configuration, not three, and is in gimbal lock. In this case it can pitch or yaw, but not roll (rotate in the plane that the axes all lie in).]]

An important practical application of covering spaces occurs in charts on SO(3), the rotation group. This group occurs widely in engineering, due to 3-dimensional rotations being heavily used in navigation, nautical engineering, and aerospace engineering, among many other uses. Topologically, SO(3) is the real projective space RP³, with fundamental group Z/2, and only (non-trivial) covering space the hypersphere S³, which is the group Spin(3), and represented by the unit quaternions. Thus quaternions are a preferred method for representing spatial rotations – see quaternions and spatial rotation.

However, it is often desirable to represent rotations by a set of three numbers, known as Euler angles (in numerous variants), both because this is conceptually simpler for someone familiar with planar rotation, and because one can build a combination of three gimbals to produce rotations in three dimensions. Topologically this corresponds to a map from the 3-torus T³ of three angles to the real projective space RP³ of rotations, and the resulting map has imperfections due to this map being unable to be a covering map. Specifically, the failure of the map to be a local homeomorphism at certain points is referred to as gimbal lock, and is demonstrated in the animation at the right – at some points (when the axes are coplanar) the rank of the map is 2, rather than 3, meaning that only 2 dimensions of rotations can be realized from that point by changing the angles. This causes problems in applications, and is formalized by the notion of a covering space.

• 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)

• {{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}}

Category:Algebraic topology

Category:Homotopy theory

Category:Fiber bundles

Category:Topological graph theory