Misplaced Pages

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62-507: [REDACTED] Look up covering in Wiktionary, the free dictionary. Covering may refer to: Window covering , material used to cover a window Cover (topology) , a collection of subsets of X {\displaystyle X} whose union is all of X {\displaystyle X} Covering space , a map in the mathematical field of topology that locally looks like

124-401: A base , in baseball Covering sickness , a disease of horses and other members of the family Equidae Coverage (disambiguation) Cover (disambiguation) Covering theorem (disambiguation) Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with the title Covering . If an internal link led you here, you may wish to change

186-399: A connected covering. Let H = p # ( π 1 ( E ) ) {\displaystyle H=p_{\#}(\pi _{1}(E))} be a subgroup of π 1 ( X ) {\displaystyle \pi _{1}(X)} , then p {\displaystyle p} is a normal covering iff H {\displaystyle H}

248-448: A connected covering. Since a deck transformation d : E → E {\displaystyle d:E\rightarrow E} is bijective , it permutes the elements of a fiber p − 1 ( x ) {\displaystyle p^{-1}(x)} with x ∈ X {\displaystyle x\in X} and is uniquely determined by where it sends

310-416: A connected, locally simply connected topological space; then, there exists a universal covering p : X ~ → X {\displaystyle p:{\tilde {X}}\rightarrow X} . X ~ {\displaystyle {\tilde {X}}} is defined as X ~ := { γ : γ  is

372-784: A continuous map. f {\displaystyle f} is holomorphic in a point x ∈ X {\displaystyle x\in X} , if for any charts ϕ x : U 1 → V 1 {\displaystyle \phi _{x}:U_{1}\rightarrow V_{1}} of x {\displaystyle x} and ϕ f ( x ) : U 2 → V 2 {\displaystyle \phi _{f(x)}:U_{2}\rightarrow V_{2}} of f ( x ) {\displaystyle f(x)} , with ϕ x ( U 1 ) ⊂ U 2 {\displaystyle \phi _{x}(U_{1})\subset U_{2}} ,

434-406: A deck transformation d : E → E {\displaystyle d:E\rightarrow E} , such that d ( e 0 ) = e 1 {\displaystyle d(e_{0})=e_{1}} . Let X {\displaystyle X} be a path-connected space and p : E → X {\displaystyle p:E\rightarrow X} be

496-417: 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 X × X by

558-459: A homeomorphism h : E → E ′ {\displaystyle h:E\rightarrow E'} , such that the diagram commutes. If such a homeomorphism exists, then one calls the covering spaces E {\displaystyle E} and E ′ {\displaystyle E'} isomorphic . All coverings satisfy the lifting property , i.e.: Let I {\displaystyle I} be

620-759: A lift of F | Y × { 0 } {\displaystyle F|_{Y\times \{0\}}} , i.e. a continuous map such that p ∘ F ~ 0 = F | Y × { 0 } {\displaystyle p\circ {\tilde {F}}_{0}=F|_{Y\times \{0\}}} . Then there is a uniquely determined, continuous map F ~ : Y × I → E {\displaystyle {\tilde {F}}:Y\times I\rightarrow E} for which F ~ ( y , 0 ) = F ~ 0 {\displaystyle {\tilde {F}}(y,0)={\tilde {F}}_{0}} and which

682-461: A non-constant, holomorphic map between compact Riemann surfaces. For every x ∈ X {\displaystyle x\in X} there exist charts for x {\displaystyle x} and f ( x ) {\displaystyle f(x)} and there exists a uniquely determined k x ∈ N > 0 {\displaystyle k_{x}\in \mathbb {N_{>0}} } , such that

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744-669: A path in  X  with  γ ( 0 ) = x 0 } /  homotopy with fixed ends {\displaystyle {\tilde {X}}:=\{\gamma :\gamma {\text{ is a path in }}X{\text{ with }}\gamma (0)=x_{0}\}/{\text{ homotopy with fixed ends}}} and p : X ~ → X {\displaystyle p:{\tilde {X}}\rightarrow X} by p ( [ γ ] ) := γ ( 1 ) {\displaystyle p([\gamma ]):=\gamma (1)} . The topology on X ~ {\displaystyle {\tilde {X}}}

806-628: A path-connected space and p : E → X {\displaystyle p:E\rightarrow X} be a connected covering. Let x , y ∈ X {\displaystyle x,y\in X} be any two points, which are connected by a path γ {\displaystyle \gamma } , i.e. γ ( 0 ) = x {\displaystyle \gamma (0)=x} and γ ( 1 ) = y {\displaystyle \gamma (1)=y} . Let γ ~ {\displaystyle {\tilde {\gamma }}} be

868-954: A single point. In particular, only the identity map fixes a point in the fiber. Because of this property every deck transformation defines a group action on E {\displaystyle E} , i.e. let U ⊂ X {\displaystyle U\subset X} be an open neighborhood of a x ∈ X {\displaystyle x\in X} and U ~ ⊂ E {\displaystyle {\tilde {U}}\subset E} an open neighborhood of an e ∈ p − 1 ( x ) {\displaystyle e\in p^{-1}(x)} , then Deck ⁡ ( p ) × E → E : ( d , U ~ ) ↦ d ( U ~ ) {\displaystyle \operatorname {Deck} (p)\times E\rightarrow E:(d,{\tilde {U}})\mapsto d({\tilde {U}})}

930-535: A studio album by American Christian heavy metal/hard rock band Stryper Covering: The Hidden Assault on Our Civil Rights , a 2006 book by Kenji Yoshini See also [ edit ] All pages with titles beginning with Covering All pages with titles containing Covering Covering a base , in baseball Covering sickness , a disease of horses and other members of the family Equidae Coverage (disambiguation) Cover (disambiguation) Covering theorem (disambiguation) Topics referred to by

992-479: Is holomorphic. The map F = ϕ f ( x ) ∘ f ∘ ϕ x − 1 {\displaystyle F=\phi _{f(x)}\circ f\circ \phi _{x}^{-1}} is called the local expression of f {\displaystyle f} in x ∈ X {\displaystyle x\in X} . If f : X → Y {\displaystyle f:X\rightarrow Y}

1054-497: Is surjective , and the cardinality of D x {\displaystyle D_{x}} is the same for all x ∈ X {\displaystyle x\in X} ; this value is called the degree of the covering. If X ~ {\displaystyle {\tilde {X}}} is path-connected , then the covering π : X ~ → X {\displaystyle \pi :{\tilde {X}}\rightarrow X}

1116-543: Is unramified . The image point y = f ( x ) ∈ Y {\displaystyle y=f(x)\in Y} of a ramification point is called a branch point. Let f : X → Y {\displaystyle f:X\rightarrow Y} be a non-constant, holomorphic map between compact Riemann surfaces. The degree deg ⁡ ( f ) {\displaystyle \operatorname {deg} (f)} of f {\displaystyle f}

1178-584: Is a group action . A covering p : E → X {\displaystyle p:E\rightarrow X} is called normal, if Deck ⁡ ( p ) ∖ E ≅ X {\displaystyle \operatorname {Deck} (p)\backslash E\cong X} . This means, that for every x ∈ X {\displaystyle x\in X} and any two e 0 , e 1 ∈ p − 1 ( x ) {\displaystyle e_{0},e_{1}\in p^{-1}(x)} there exists

1240-466: Is a homeomorphism for every d ∈ D x {\displaystyle d\in D_{x}} . The open sets V d {\displaystyle V_{d}} are called sheets , which are uniquely determined up to homeomorphism if U x {\displaystyle U_{x}} is connected . For each x ∈ X {\displaystyle x\in X}

1302-665: Is a normal subgroup of π 1 ( X ) {\displaystyle \pi _{1}(X)} . If p : E → X {\displaystyle p:E\rightarrow X} is a normal covering and H = p # ( π 1 ( E ) ) {\displaystyle H=p_{\#}(\pi _{1}(E))} , then Deck ⁡ ( p ) ≅ π 1 ( X ) / H {\displaystyle \operatorname {Deck} (p)\cong \pi _{1}(X)/H} . If p : E → X {\displaystyle p:E\rightarrow X}

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1364-1170: Is a bijection and U ~ {\displaystyle {\tilde {U}}} can be equipped with the final topology of p | U ~ {\displaystyle p_{|{\tilde {U}}}} . The fundamental group π 1 ( X , x 0 ) = Γ {\displaystyle \pi _{1}(X,x_{0})=\Gamma } acts freely through ( [ γ ] , [ x ~ ] ) ↦ [ γ . x ~ ] {\displaystyle ([\gamma ],[{\tilde {x}}])\mapsto [\gamma .{\tilde {x}}]} on X ~ {\displaystyle {\tilde {X}}} and ψ : Γ ∖ X ~ → X {\displaystyle \psi :\Gamma \backslash {\tilde {X}}\rightarrow X} with ψ ( [ Γ x ~ ] ) = x ~ ( 1 ) {\displaystyle \psi ([\Gamma {\tilde {x}}])={\tilde {x}}(1)}

1426-831: Is a continuous map such that for every x ∈ X {\displaystyle x\in X} there exists an open neighborhood U x {\displaystyle U_{x}} of x {\displaystyle x} and a discrete space D x {\displaystyle D_{x}} such that π − 1 ( U x ) = ⨆ d ∈ D x V d {\displaystyle \pi ^{-1}(U_{x})=\displaystyle \bigsqcup _{d\in D_{x}}V_{d}} and π | V d : V d → U x {\displaystyle \pi |_{V_{d}}:V_{d}\rightarrow U_{x}}

1488-529: Is a covering. However, coverings of X × X ′ {\displaystyle X\times X'} are not all of this form in general. Let X {\displaystyle X} be a topological space and p : E → X {\displaystyle p:E\rightarrow X} and p ′ : E ′ → X {\displaystyle p':E'\rightarrow X} be coverings. Both coverings are called equivalent , if there exists

1550-526: Is a covering. Let p : X ~ → X {\displaystyle p:{\tilde {X}}\rightarrow X} be a simply connected covering. If β : E → X {\displaystyle \beta :E\rightarrow X} is another simply connected covering, then there exists a uniquely determined homeomorphism α : X ~ → E {\displaystyle \alpha :{\tilde {X}}\rightarrow E} , such that

1612-483: Is a homeomorphism, i.e. Γ ∖ X ~ ≅ X {\displaystyle \Gamma \backslash {\tilde {X}}\cong X} . 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,

1674-508: Is a homeomorphism. It follows that the covering space E {\displaystyle E} and the base space X {\displaystyle X} locally share the same properties. Let X , Y {\displaystyle X,Y} and E {\displaystyle E} be path-connected, locally path-connected spaces, and p , q {\displaystyle p,q} and r {\displaystyle r} be continuous maps, such that

1736-443: Is a lift of F {\displaystyle F} , i.e. p ∘ F ~ = F {\displaystyle p\circ {\tilde {F}}=F} . If X {\displaystyle X} is a path-connected space, then for Y = { 0 } {\displaystyle Y=\{0\}} it follows that the map F ~ {\displaystyle {\tilde {F}}}

1798-414: Is a lift of a path in X {\displaystyle X} and for Y = I {\displaystyle Y=I} it is a lift of a homotopy of paths in X {\displaystyle X} . As a consequence, one can show that the fundamental group π 1 ( S 1 ) {\displaystyle \pi _{1}(S^{1})} of the unit circle

1860-470: Is a non-constant, holomorphic map between compact Riemann surfaces , then f {\displaystyle f} is surjective and an open map , i.e. for every open set U ⊂ X {\displaystyle U\subset X} the image f ( U ) ⊂ Y {\displaystyle f(U)\subset Y} is also open. Let f : X → Y {\displaystyle f:X\rightarrow Y} be

1922-410: Is a path-connected covering and H = p # ( π 1 ( E ) ) {\displaystyle H=p_{\#}(\pi _{1}(E))} , then Deck ⁡ ( p ) ≅ N ( H ) / H {\displaystyle \operatorname {Deck} (p)\cong N(H)/H} , whereby N ( H ) {\displaystyle N(H)}

Covering - Misplaced Pages Continue

1984-500: Is an infinite cyclic group , which is generated by the homotopy classes of the loop γ : I → S 1 {\displaystyle \gamma :I\rightarrow S^{1}} with γ ( t ) = ( cos ⁡ ( 2 π t ) , sin ⁡ ( 2 π t ) ) {\displaystyle \gamma (t)=(\cos(2\pi t),\sin(2\pi t))} . Let X {\displaystyle X} be

2046-479: Is called a branched covering , if there exists a closed set with dense complement E ⊂ Y {\displaystyle E\subset Y} , such that f | X ∖ f − 1 ( E ) : X ∖ f − 1 ( E ) → Y ∖ E {\displaystyle f_{|X\smallsetminus f^{-1}(E)}:X\smallsetminus f^{-1}(E)\rightarrow Y\smallsetminus E}

2108-502: Is called a path-connected covering . This definition is equivalent to the statement that π {\displaystyle \pi } is a locally trivial Fiber bundle . Some authors also require that π {\displaystyle \pi } be surjective in the case that X {\displaystyle X} is not connected. Since a covering π : E → X {\displaystyle \pi :E\rightarrow X} maps each of

2170-524: Is connected and locally path connected. The action of Aut ⁡ ( p ) {\displaystyle \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 G {\displaystyle G} -bundle , where G = Aut ⁡ ( p ) {\displaystyle G=\operatorname {Aut} (p)}

2232-506: Is considered as a discrete topological group. Every universal cover p : D → X {\displaystyle p:D\to X} is regular, with deck transformation group being isomorphic to the fundamental group π 1 ( X ) {\displaystyle \pi _{1}(X)} . Let X {\displaystyle X} be a path-connected space and p : E → X {\displaystyle p:E\rightarrow X} be

2294-568: Is constructed as follows: Let γ : I → X {\displaystyle \gamma :I\rightarrow X} be a path with γ ( 0 ) = x 0 {\displaystyle \gamma (0)=x_{0}} . Let U {\displaystyle U} be a simply connected neighborhood of the endpoint x = γ ( 1 ) {\displaystyle x=\gamma (1)} , then for every y ∈ U {\displaystyle y\in U}

2356-608: Is different from Wikidata All article disambiguation pages All disambiguation pages Covering 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 . Covering spaces are an important tool in several areas of mathematics. In modern geometry , covering spaces (or branched coverings , which have slightly weaker conditions) are used in

2418-449: Is the cardinality of the fiber of an unramified point y = f ( x ) ∈ Y {\displaystyle y=f(x)\in Y} , i.e. deg ⁡ ( f ) := | f − 1 ( y ) | {\displaystyle \operatorname {deg} (f):=|f^{-1}(y)|} . This number is well-defined, since for every y ∈ Y {\displaystyle y\in Y}

2480-1056: The paths σ y {\displaystyle \sigma _{y}} inside U {\displaystyle U} from x {\displaystyle x} to y {\displaystyle y} are uniquely determined up to homotopy . Now consider U ~ := { γ . σ y : y ∈ U } /  homotopy with fixed ends {\displaystyle {\tilde {U}}:=\{\gamma .\sigma _{y}:y\in U\}/{\text{ homotopy with fixed ends}}} , then p | U ~ : U ~ → U {\displaystyle p_{|{\tilde {U}}}:{\tilde {U}}\rightarrow U} with p ( [ γ . σ y ] ) = γ . σ y ( 1 ) = y {\displaystyle p([\gamma .\sigma _{y}])=\gamma .\sigma _{y}(1)=y}

2542-455: The unit interval and p : E → X {\displaystyle p:E\rightarrow X} be a covering. Let F : Y × I → X {\displaystyle F:Y\times I\rightarrow X} be a continuous map and F ~ 0 : Y × { 0 } → E {\displaystyle {\tilde {F}}_{0}:Y\times \{0\}\rightarrow E} be

Covering - Misplaced Pages Continue

2604-422: 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

2666-522: The composition of maps, the set of deck transformation forms a group Deck ⁡ ( p ) {\displaystyle \operatorname {Deck} (p)} , which is the same as Aut ⁡ ( p ) {\displaystyle \operatorname {Aut} (p)} . Now suppose p : C → X {\displaystyle p:C\to X} is a covering map and C {\displaystyle C} (and therefore also X {\displaystyle X} )

2728-418: 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

2790-420: The covering of S 1 {\displaystyle S^{1}} by R {\displaystyle \mathbb {R} } (see below ). Under certain conditions, covering spaces also exhibit a Galois correspondence with the subgroups of the fundamental group. Let X {\displaystyle X} be a topological space. A covering of X {\displaystyle X}

2852-444: The definition of a covering, which merely requires that the sheets are mapped homeomorphically onto the corresponding open subset.) Let p : E → X {\displaystyle p:E\rightarrow X} be a covering. A deck transformation is a homeomorphism d : E → E {\displaystyle d:E\rightarrow E} , such that the diagram of continuous maps commutes. Together with

2914-890: The diagram commutes. Let X {\displaystyle X} and X ′ {\displaystyle X'} be topological spaces and p : E → X {\displaystyle p:E\rightarrow X} and p ′ : E ′ → X ′ {\displaystyle p':E'\rightarrow X'} be coverings, then p × p ′ : E × E ′ → X × X ′ {\displaystyle p\times p':E\times E'\rightarrow X\times X'} with ( p × p ′ ) ( e , e ′ ) = ( p ( e ) , p ′ ( e ′ ) ) {\displaystyle (p\times p')(e,e')=(p(e),p'(e'))}

2976-424: The diagram commutes. This means that p {\displaystyle p} is, up to equivalence, uniquely determined and because of that universal property denoted as the universal covering of the space X {\displaystyle X} . A universal covering does not always exist, but the following properties guarantee its existence: Let X {\displaystyle X} be

3038-428: The discrete set π − 1 ( x ) {\displaystyle \pi ^{-1}(x)} is called the fiber of x {\displaystyle x} . If X {\displaystyle X} is connected (and X ~ {\displaystyle {\tilde {X}}} is non-empty), it can be shown that π {\displaystyle \pi }

3100-752: The disjoint open sets of π − 1 ( U ) {\displaystyle \pi ^{-1}(U)} homeomorphically onto U {\displaystyle U} it is a local homeomorphism, i.e. π {\displaystyle \pi } is a continuous map and for every e ∈ E {\displaystyle e\in E} there exists an open neighborhood V ⊂ E {\displaystyle V\subset E} of e {\displaystyle e} , such that π | V : V → π ( V ) {\displaystyle \pi |_{V}:V\rightarrow \pi (V)}

3162-639: The fiber f − 1 ( y ) {\displaystyle f^{-1}(y)} is discrete and for any two unramified points y 1 , y 2 ∈ Y {\displaystyle y_{1},y_{2}\in Y} , it is: | f − 1 ( y 1 ) | = | f − 1 ( y 2 ) | . {\displaystyle |f^{-1}(y_{1})|=|f^{-1}(y_{2})|.} It can be calculated by: A continuous map f : X → Y {\displaystyle f:X\rightarrow Y}

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3224-519: The free dictionary. Covering may refer to: Window covering , material used to cover a window Cover (topology) , a collection of subsets of X {\displaystyle X} whose union is all of X {\displaystyle X} Covering space , a map in the mathematical field of topology that locally looks like the projection of multiple copies of a space onto itself Covering (martial arts) , an act of protecting against an opponent's strikes The Covering ,

3286-401: The fundamental group of the symmetric square of a space. Let E and M be smooth manifolds with or without boundary . A covering π : E → M {\displaystyle \pi :E\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 M . (This is in contrast to

3348-420: The homotopy classes of loops in X {\displaystyle X} , whose lifts are loops in E {\displaystyle E} . Let X {\displaystyle X} and Y {\displaystyle Y} be Riemann surfaces , i.e. one dimensional complex manifolds , and let f : X → Y {\displaystyle f:X\rightarrow Y} be

3410-451: The link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Covering&oldid=1235941726 " Category : Disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages covering [REDACTED] Look up covering in Wiktionary,

3472-500: The local expression F {\displaystyle F} of f {\displaystyle f} in x {\displaystyle x} is of the form z ↦ z k x {\displaystyle z\mapsto z^{k_{x}}} . The number k x {\displaystyle k_{x}} is called the ramification index of f {\displaystyle f} in x {\displaystyle x} and

3534-485: The map ϕ f ( x ) ∘ f ∘ ϕ x − 1 : C → C {\displaystyle \phi _{f(x)}\circ f\circ \phi _{x}^{-1}:\mathbb {C} \rightarrow \mathbb {C} } is holomorphic . If f {\displaystyle f} is holomorphic at all x ∈ X {\displaystyle x\in X} , we say f {\displaystyle f}

3596-400: The point x ∈ X {\displaystyle x\in X} is called a ramification point if k x ≥ 2 {\displaystyle k_{x}\geq 2} . If k x = 1 {\displaystyle k_{x}=1} for an x ∈ X {\displaystyle x\in X} , then x {\displaystyle x}

3658-467: The projection of multiple copies of a space onto itself Covering (martial arts) , an act of protecting against an opponent's strikes The Covering , a studio album by American Christian heavy metal/hard rock band Stryper Covering: The Hidden Assault on Our Civil Rights , a 2006 book by Kenji Yoshini See also [ edit ] All pages with titles beginning with Covering All pages with titles containing Covering Covering

3720-413: The same term [REDACTED] This disambiguation page lists articles associated with the title Covering . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Covering&oldid=1235941726 " Category : Disambiguation pages Hidden categories: Short description

3782-477: The twist action where the non-identity element acts by ( 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

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3844-623: The unique lift of γ {\displaystyle \gamma } , then the map is bijective . If X {\displaystyle X} is a path-connected space and p : E → X {\displaystyle p:E\rightarrow X} a connected covering, then the induced group homomorphism is injective and the subgroup p # ( π 1 ( E ) ) {\displaystyle p_{\#}(\pi _{1}(E))} of π 1 ( X ) {\displaystyle \pi _{1}(X)} consists of

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