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A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c0269001.png" /> of a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c0269002.png" /> onto a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c0269003.png" /> such that each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c0269004.png" /> has a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c0269005.png" /> the pre-image of which under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c0269006.png" /> is a union of open subsets that are mapped homeomorphically onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c0269007.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c0269008.png" />. Equivalently: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c0269009.png" /> is a [[Locally trivial fibre bundle|locally trivial fibre bundle]] with discrete fibre.
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Coverings are usually considered on the assumption that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c02690010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c02690011.png" /> are connected; it is also usually assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c02690012.png" /> is locally connected and locally simply-connected. Under these assumptions one can establish a relationship between the fundamental groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c02690013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c02690014.png" />: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c02690015.png" />, then the induced homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c02690016.png" /> maps <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c02690017.png" /> isomorphically onto a subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c02690018.png" /> and, by varying the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c02690019.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c02690020.png" />, one obtains exactly all subgroups in the corresponding class of conjugate subgroups. If this class consists of a single subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c02690021.png" /> (i.e. if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c02690022.png" /> is a normal divisor), the covering is said to be regular. In that case one obtains a free action of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c02690023.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c02690024.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c02690025.png" /> playing the role of the quotient mapping onto the orbit space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c02690026.png" />. This action is generated by lifting loops: If one associates with a loop <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c02690027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c02690028.png" />, the unique path <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c02690029.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c02690030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c02690031.png" />, then the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c02690032.png" /> will depend only on the class of the loop in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c02690033.png" /> and on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c02690034.png" />. Thus, each element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c02690035.png" /> corresponds to a permutation of points in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c02690036.png" />. This permutation has no fixed points if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c02690037.png" />, and it depends continuously on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c02690038.png" />. One obtains a homeomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c02690039.png" />.
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A mapping $p:X\rightarrow Y$ of a space $X$ onto a space $Y$ such that each point $y\in Y$ has a neighbourhood $U(y)$ the pre-image of which under $p$ is a union of open subsets that are mapped homeomorphically onto $U(y)$ by $p$. Equivalently: $p$ is a [[Locally trivial fibre bundle|locally trivial fibre bundle]] with discrete fibre.
  
In the general case this construction defines only a permutation in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c02690040.png" />, i.e. there is an action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c02690041.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c02690042.png" />, known as the monodromy of the covering. A special case of a regular covering is a [[Universal covering|universal covering]], for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c02690043.png" />. In general, given any subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c02690044.png" />, one can construct a unique covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c02690045.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c02690046.png" />. The points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c02690047.png" /> are the classes of paths <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c02690048.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c02690049.png" />: Two paths <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c02690050.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c02690051.png" /> are identified if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c02690052.png" /> and if the loop <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c02690053.png" /> lies in an element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c02690054.png" />. The point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c02690055.png" /> for the paths of one class is taken as the image of this class; this defines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c02690056.png" />. The topology in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c02690057.png" /> is uniquely determined by the condition that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c02690058.png" /> be a covering; it is here that the local simple-connectedness of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c02690059.png" /> is essential. For any mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c02690060.png" /> of an arcwise-connected space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c02690061.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c02690062.png" />, its lifting into a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c02690063.png" /> exists if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c02690064.png" />. A partial order relation can be defined on the coverings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c02690065.png" /> (a covering of a covering is a covering); this relation is dual to the inclusion of subgroups in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c02690066.png" />. In particular, the universal covering is the unique maximal element.
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Coverings are usually considered on the assumption that $X$ and $Y$ are connected; it is also usually assumed that $Y$ is locally connected and locally simply-connected. Under these assumptions one can establish a relationship between the fundamental groups $\pi_1(X,x_0)$ and $\pi_1(Y,y_0)$: If $p(x_0)=y_0$, then the induced homomorphism $p^*$ maps $\pi_1(X,x_0)$ isomorphically onto a subgroup of $\pi_1(Y,y_0)$ and, by varying the point $x_0$ in $p^{-1}(y)$, one obtains exactly all subgroups in the corresponding class of conjugate subgroups. If this class consists of a single subgroup $H$ (i.e. if $H$ is a normal divisor), the covering is said to be regular. In that case one obtains a free action of the group $G=\pi_1(Y,y_0)/H$ on $X$, with $p$ playing the role of the quotient mapping onto the orbit space $Y$. This action is generated by lifting loops: If one associates with a loop $q:[0,1]\rightarrow Y$, $q(0)=q(1)=y_0$, the unique path $\bar{q}:[0,1]\rightarrow X$ such that $\bar{q}(0)=x_0$ and $p\bar{q}=q$, then the point $\bar{q}(1)$ will depend only on the class of the loop in $G$ and on $x_0$. Thus, each element of $G$ corresponds to a permutation of points in $p^{-1}(y_0)$. This permutation has no fixed points if $\gamma\neq 1$, and it depends continuously on $y_0$. One obtains a homeomorphism of $X$.
  
Examples. The parametrization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c02690067.png" /> of the circle defines a covering of the circle by the real line, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c02690068.png" />, often described in the complex form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026900/c02690069.png" /> and called the exponential covering. Similarly, the torus is covered by the plane. Identification of antipodal points on a sphere yields a covering by the sphere of a projective space of corresponding dimension. In general, free actions of discrete groups are a source of regular coverings (over the orbit space); not every such action yields a covering (the orbit space may be non-separable), but finite groups do.
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In the general case this construction defines only a permutation in $p^{-1}(y)$, i.e. there is an action of $\pi_1(Y,y_0)$ on $p^{-1}(y)$, known as the monodromy of the covering. A special case of a regular covering is a [[Universal covering|universal covering]], for which $G=\pi_1(Y,y_0)$. In general, given any subgroup $H\subset\pi_1(Y,y_0)$, one can construct a unique covering $p:(X,x_0)\rightarrow (Y,y_0)$ for which $p^*(\pi_1(X,x_0))=H$. The points of $X$ are the classes of paths $q:[0,1]\rightarrow Y$, $q(0)=x_0$: Two paths $q_1$ and $q_2$ are identified if $q_1(1)=q_2(1)$ and if the loop $q_1q_2^{-1}$ lies in an element of $H$. The point $q(1)$ for the paths of one class is taken as the image of this class; this defines $p$. The topology in $X$ is uniquely determined by the condition that $p$ be a covering; it is here that the local simple-connectedness of $Y$ is essential. For any mapping $f$ of an arcwise-connected space $(Z,z_0)$ into $(Y,y_0)$, its lifting into a mapping $\bar{f}:(Z,z_0)\rightarrow (X,x_0)$ exists if and only if $f^*(\pi_1(Z,z_0))\subset H$. A partial order relation can be defined on the coverings of $Y$ (a covering of a covering is a covering); this relation is dual to the inclusion of subgroups in $\pi_1(Y,y_0)$. In particular, the universal covering is the unique maximal element.
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Examples. The parametrization $(\cos \phi,\sin \phi)$ of the circle defines a covering of the circle by the real line, $\phi\in \mathbb{R}$, often described in the complex form $e^{i\phi}$ and called the exponential covering. Similarly, the torus is covered by the plane. Identification of antipodal points on a sphere yields a covering by the sphere of a projective space of corresponding dimension. In general, free actions of discrete groups are a source of regular coverings (over the orbit space); not every such action yields a covering (the orbit space may be non-separable), but finite groups do.
  
  

Revision as of 10:40, 16 May 2017


A mapping $p:X\rightarrow Y$ of a space $X$ onto a space $Y$ such that each point $y\in Y$ has a neighbourhood $U(y)$ the pre-image of which under $p$ is a union of open subsets that are mapped homeomorphically onto $U(y)$ by $p$. Equivalently: $p$ is a locally trivial fibre bundle with discrete fibre.

Coverings are usually considered on the assumption that $X$ and $Y$ are connected; it is also usually assumed that $Y$ is locally connected and locally simply-connected. Under these assumptions one can establish a relationship between the fundamental groups $\pi_1(X,x_0)$ and $\pi_1(Y,y_0)$: If $p(x_0)=y_0$, then the induced homomorphism $p^*$ maps $\pi_1(X,x_0)$ isomorphically onto a subgroup of $\pi_1(Y,y_0)$ and, by varying the point $x_0$ in $p^{-1}(y)$, one obtains exactly all subgroups in the corresponding class of conjugate subgroups. If this class consists of a single subgroup $H$ (i.e. if $H$ is a normal divisor), the covering is said to be regular. In that case one obtains a free action of the group $G=\pi_1(Y,y_0)/H$ on $X$, with $p$ playing the role of the quotient mapping onto the orbit space $Y$. This action is generated by lifting loops: If one associates with a loop $q:[0,1]\rightarrow Y$, $q(0)=q(1)=y_0$, the unique path $\bar{q}:[0,1]\rightarrow X$ such that $\bar{q}(0)=x_0$ and $p\bar{q}=q$, then the point $\bar{q}(1)$ will depend only on the class of the loop in $G$ and on $x_0$. Thus, each element of $G$ corresponds to a permutation of points in $p^{-1}(y_0)$. This permutation has no fixed points if $\gamma\neq 1$, and it depends continuously on $y_0$. One obtains a homeomorphism of $X$.

In the general case this construction defines only a permutation in $p^{-1}(y)$, i.e. there is an action of $\pi_1(Y,y_0)$ on $p^{-1}(y)$, known as the monodromy of the covering. A special case of a regular covering is a universal covering, for which $G=\pi_1(Y,y_0)$. In general, given any subgroup $H\subset\pi_1(Y,y_0)$, one can construct a unique covering $p:(X,x_0)\rightarrow (Y,y_0)$ for which $p^*(\pi_1(X,x_0))=H$. The points of $X$ are the classes of paths $q:[0,1]\rightarrow Y$, $q(0)=x_0$: Two paths $q_1$ and $q_2$ are identified if $q_1(1)=q_2(1)$ and if the loop $q_1q_2^{-1}$ lies in an element of $H$. The point $q(1)$ for the paths of one class is taken as the image of this class; this defines $p$. The topology in $X$ is uniquely determined by the condition that $p$ be a covering; it is here that the local simple-connectedness of $Y$ is essential. For any mapping $f$ of an arcwise-connected space $(Z,z_0)$ into $(Y,y_0)$, its lifting into a mapping $\bar{f}:(Z,z_0)\rightarrow (X,x_0)$ exists if and only if $f^*(\pi_1(Z,z_0))\subset H$. A partial order relation can be defined on the coverings of $Y$ (a covering of a covering is a covering); this relation is dual to the inclusion of subgroups in $\pi_1(Y,y_0)$. In particular, the universal covering is the unique maximal element.

Examples. The parametrization $(\cos \phi,\sin \phi)$ of the circle defines a covering of the circle by the real line, $\phi\in \mathbb{R}$, often described in the complex form $e^{i\phi}$ and called the exponential covering. Similarly, the torus is covered by the plane. Identification of antipodal points on a sphere yields a covering by the sphere of a projective space of corresponding dimension. In general, free actions of discrete groups are a source of regular coverings (over the orbit space); not every such action yields a covering (the orbit space may be non-separable), but finite groups do.


Comments

A covering is also a termed a covering projection. Every covering has the homotopy lifting property (cf. Covering homotopy) and hence is a Hurewicz fibre space or fibration.

References

[a1] E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966) pp. Chapt. 2
How to Cite This Entry:
Covering. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Covering&oldid=13627
This article was adapted from an original article by A.V. Chernavskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article