This article's lead sectionmay be too short to adequately summarize the key points. Please consider expanding the lead to provide an accessible overview of all important aspects of the article.(June 2023)
Let be a topological space. A covering of is a continuous map
such that there exists a discrete space and for every an open neighborhood, such that and is a homeomorphism for every . Often, the notion of a covering is used for the covering space as well as for the map . The open sets are called sheets, which are uniquely determined up to a homeomorphism if is connected.[1]: 56 For each the discrete subset is called the fiber of . The degree of a covering is the cardinality of the space . If is path-connected, then the covering is denoted as a path-connected covering.
ExamplesEdit
For every topological space , there is a covering map given by , which is called the trivial covering of
The map with is a covering of the unit circle. The base of the covering is and the covering space is . For any point such that , the set is an open neighborhood of . The preimage of under is
and the sheets of the covering are for The fiber of is
Another covering of the unit circle is the map with for some For an open neighborhood of an , one has:
.
A map which is a local homeomorphism but not a covering of the unit circle is with . There is a sheet of an open neighborhood of , which is not mapped homeomorphically onto .
PropertiesEdit
Local homeomorphismEdit
Since a covering maps each of the disjoint open sets of homeomorphically onto it is a local homeomorphism, i.e. is a continuous map and for every there exists an open neighborhood of , such that is a homeomorphism.
It follows that the covering space and the base space locally share the same properties.
If is a connected and non-orientable manifold, then there is a covering of degree , whereby is a connected and orientable manifold.[1]: 234
If is a graph, then it follows for a covering that is also a graph.[1]: 85
If is a connected manifold, then there is a covering , whereby is a connected and simply connected manifold.[3]: 32
If is a connected Riemann surface, then there is a covering which is also a holomorphic map[3]: 22 and is a connected and simply connected Riemann surface.[3]: 32
FactorisationEdit
Let and be path-connected, locally path-connected spaces, and and be continuous maps, such that the diagram
Let and be topological spaces and and be coverings, then with is a covering.[4]: 339
Equivalence of coveringsEdit
Let be a topological space and and be coverings. Both coverings are called equivalent, if there exists a homeomorphism , such that the diagram
commutes. If such a homeomorphism exists, then one calls the covering spaces and isomorphic.
Lifting propertyEdit
An important property of the covering is, that it satisfies the lifting property, i.e.:
Let be the unit interval and be a covering. Let be a continuous map and be a lift of , i.e. a continuous map such that . Then there is a uniquely determined, continuous map for which and which is a lift of , i.e. .[1]: 60
If is a path-connected space, then for it follows that the map is a lift of a path in and for it is a lift of a homotopy of paths in .
Because of that property one can show, that the fundamental group of the unit circle is an infinite cyclic group, which is generated by the homotopy classes of the loop with .[1]: 29
Let be a path-connected space and be a connected covering. Let be any two points, which are connected by a path , i.e. and . Let be the unique lift of , then the map
Let be a non-constant, holomorphic map between compact Riemann surfaces. For every there exist charts for and and there exists a uniquely determined , such that the local expression of in is of the form .[3]: 10 The number is called the ramification index of in and the point is called a ramification point if . If for an , then is unramified. The image point of a ramification point is called a branch point.
Degree of a holomorphic mapEdit
Let be a non-constant, holomorphic map between compact Riemann surfaces. The degree of is the cardinality of the fiber of an unramified point , i.e. .
This number is well-defined, since for every the fiber is discrete[3]: 20 and for any two unramified points , it is:
A continuous map is called a branched covering, if there exists a closed set with dense complement , such that is a covering.
ExamplesEdit
Let and , then with is branched covering of degree , whereby is a branch point.
Every non-constant, holomorphic map between compact Riemann surfaces of degree is a branched covering of degree .
Universal coveringEdit
DefinitionEdit
Let be a simply connected covering. If is another simply connected covering, then there exists a uniquely determined homeomorphism , such that the diagram
The topology on is constructed as follows: Let be a path with . Let be a simply connected neighborhood of the endpoint , then for every the paths inside from to are uniquely determined up to homotopy. Now consider , then with is a bijection and can be equipped with the final topology of .
The fundamental group acts freely through on and with is a homeomorphism, i.e. .
ExamplesEdit
with is the universal covering of the unit circle .
A topological space which has no universal covering is the Hawaiian earring:
One can show that no neighborhood of the origin is simply connected.[4]: 487, Example 1
G-coveringsEdit
Let G be a discrete groupacting on the topological spaceX. This means that each element g of G is associated to a homeomorphism Hg of X onto itself, in such a way that Hgh is always equal to Hg ∘ Hh 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 spaceX/G is a covering map. This is not always true since the action may have fixed points. An example for this is the cyclic group of order 2 acting on a product X × X by the twist action where the non-identity element acts by (x, y) ↦ (y, x). Thus the study of the relation between the fundamental groups 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.
Deck transformationEdit
DefinitionEdit
Let be a covering. A deck transformation is a homeomorphism , such that the diagram of continuous maps
commutes. Together with the composition of maps, the set of deck transformation forms a group, which is the same as .
Now suppose is a covering map and (and therefore also ) is connected and locally path connected. The action of 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 is considered as a discrete topological group.
Every universal cover is regular, with deck transformation group being isomorphic to the fundamental group.
ExamplesEdit
Let be the covering for some , then the map is a deck transformation and .
Let be the covering , then the map with is a deck transformation and .
As another important example, consider the complex plane and the complex plane minus the origin. Then the map with is a regular cover. The deck transformations are multiplications with -th roots of unity and the deck transformation group is therefore isomorphic to the cyclic group. Likewise, the map with is the universal cover.
PropertiesEdit
Let be a path-connected space and be a connected covering. Since a deck transformation is bijective, it permutes the elements of a fiber with and is uniquely determined by where it sends a single point. In particular, only the identity map fixes a point in the fiber.[1]: 70 Because of this property every deck transformation defines a group action on , i.e. let be an open neighborhood of a and an open neighborhood of an , then is a group action.
Normal coveringsEdit
DefinitionEdit
A covering is called normal, if . This means, that for every and any two there exists a deck transformation , such that .
PropertiesEdit
Let be a path-connected space and be a connected covering. Let be a subgroup of , then is a normal covering iff is a normal subgroup of .
If is a normal covering and , then .
If is a path-connected covering and , then , whereby is the normaliser of .[1]: 71
Let be a topological space. A group acts discontinuously on , if every has an open neighborhood with , such that for every with one has .
If a group acts discontinuously on a topological space , then the quotient map with is a normal covering.[1]: 72 Hereby is the quotient space and is the orbit of the group action.
ExamplesEdit
The covering with is a normal coverings for every .
Every simply connected covering is a normal covering.
CalculationEdit
Let be a group, which acts discontinuously on a topological space and let be the normal covering.
Let . The antipodal map with generates, together with the composition of maps, a group and induces a group action , which acts discontinuously on . Because of it follows, that the quotient map is a normal covering and for a universal covering, hence for .
Let be the special orthogonal group, then the map is a normal covering and because of , it is the universal covering, hence .
With the group action of on , whereby is the semidirect product, one gets the universal covering of the klein bottle, hence .
Let be the torus which is embedded in the . Then one gets a homeomorphism , which induces a discontinuous group action , whereby . It follows, that the map is a normal covering of the klein bottle, hence .
Let be embedded in the . Since the group action is discontinuously, whereby are coprime, the map is the universal covering of the lens space, hence .
Let and be two path-connected coverings, then they are equivalent iff the subgroups and are conjugate to each other.[4]: 482
Let be a connected and locally simply connected space, then, up to equivalence between coverings, there is a bijection:
September 01, 2023
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This article s lead section may be too short to adequately summarize the key points Please consider expanding the lead to provide an accessible overview of all important aspects of the article June 2023 A covering of a topological space X displaystyle X is a continuous map p E X displaystyle pi E rightarrow X with special properties Contents 1 Definition 2 Examples 3 Properties 3 1 Local homeomorphism 3 2 Factorisation 3 3 Product of coverings 3 4 Equivalence of coverings 3 5 Lifting property 4 Branched covering 4 1 Definitions 4 1 1 Holomorphic maps between Riemann surfaces 4 1 2 Ramification point and branch point 4 1 3 Degree of a holomorphic map 4 2 Branched covering 4 2 1 Definition 4 2 2 Examples 5 Universal covering 5 1 Definition 5 2 Existence 5 3 Examples 6 G coverings 7 Deck transformation 7 1 Definition 7 2 Examples 7 3 Properties 7 4 Normal coverings 7 4 1 Definition 7 4 2 Properties 7 4 3 Examples 7 5 Calculation 7 5 1 Examples 8 Galois correspondence 9 Classification 9 1 Definitions 9 1 1 Category of coverings 9 1 2 G Set 9 2 Equivalence 10 Applications 11 See also 12 Literature 13 ReferencesDefinition Edit nbsp Intuitively a covering locally projects a stack of pancakes above an open neighborhood U displaystyle U nbsp onto U displaystyle U nbsp Let X displaystyle X nbsp be a topological space A covering of X displaystyle X nbsp is a continuous map p E X displaystyle pi E rightarrow X nbsp such that there exists a discrete space D displaystyle D nbsp and for every x X displaystyle x in X nbsp an open neighborhood U X displaystyle U subset X nbsp such that p 1 U d D V d displaystyle pi 1 U displaystyle bigsqcup d in D V d nbsp and p V d V d U displaystyle pi V d V d rightarrow U nbsp is a homeomorphism for every d D displaystyle d in D nbsp Often the notion of a covering is used for the covering space E displaystyle E nbsp as well as for the map p E X displaystyle pi E rightarrow X nbsp The open sets V d displaystyle V d nbsp are called sheets which are uniquely determined up to a homeomorphism if U displaystyle U nbsp is connected 1 56 For each x X displaystyle x in X nbsp the discrete subset p 1 x displaystyle pi 1 x nbsp is called the fiber of x displaystyle x nbsp The degree of a covering is the cardinality of the space D displaystyle D nbsp If E displaystyle E nbsp is path connected then the covering p E X displaystyle pi E rightarrow X nbsp is denoted as a path connected covering Examples EditFor every topological space X displaystyle X nbsp there is a covering map p X X displaystyle pi X rightarrow X nbsp given by p x x displaystyle pi x x nbsp which is called the trivial covering of X displaystyle X nbsp nbsp The space Y 0 1 R displaystyle Y 0 1 times mathbb R nbsp is the covering space of X 0 1 S 1 displaystyle X 0 1 times S 1 nbsp The disjoint open sets S i displaystyle S i nbsp are mapped homeomorphically onto U displaystyle U nbsp The fiber of x displaystyle x nbsp consists of the points y i displaystyle y i nbsp The map r R S 1 displaystyle r mathbb R to S 1 nbsp with r t cos 2 p t sin 2 p t displaystyle r t cos 2 pi t sin 2 pi t nbsp is a covering of the unit circle S 1 displaystyle S 1 nbsp The base of the covering is S 1 displaystyle S 1 nbsp and the covering space is R displaystyle mathbb R nbsp For any point x x 1 x 2 S 1 displaystyle x x 1 x 2 in S 1 nbsp such that x 1 gt 0 displaystyle x 1 gt 0 nbsp the set U x 1 x 2 S 1 x 1 gt 0 displaystyle U x 1 x 2 in S 1 mid x 1 gt 0 nbsp is an open neighborhood of x displaystyle x nbsp The preimage of U displaystyle U nbsp under r displaystyle r nbsp is r 1 U n Z n 1 4 n 1 4 displaystyle r 1 U displaystyle bigsqcup n in mathbb Z left n frac 1 4 n frac 1 4 right nbsp and the sheets of the covering are V n n 1 4 n 1 4 displaystyle V n n 1 4 n 1 4 nbsp for n Z displaystyle n in mathbb Z nbsp The fiber of x displaystyle x nbsp isr 1 x t R cos 2 p t sin 2 p t x displaystyle r 1 x t in mathbb R mid cos 2 pi t sin 2 pi t x nbsp dd Another covering of the unit circle is the map q S 1 S 1 displaystyle q S 1 to S 1 nbsp with q z z n displaystyle q z z n nbsp for some n N displaystyle n in mathbb N nbsp For an open neighborhood U displaystyle U nbsp of an x S 1 displaystyle x in S 1 nbsp one has q 1 U i 1 n U displaystyle q 1 U displaystyle bigsqcup i 1 n U nbsp dd A map which is a local homeomorphism but not a covering of the unit circle is p R S 1 displaystyle p mathbb R to S 1 nbsp with p t cos 2 p t sin 2 p t displaystyle p t cos 2 pi t sin 2 pi t nbsp There is a sheet of an open neighborhood of 1 0 displaystyle 1 0 nbsp which is not mapped homeomorphically onto U displaystyle U nbsp Properties EditLocal homeomorphism Edit Since a covering p E X displaystyle pi E rightarrow X nbsp maps each of the disjoint open sets of p 1 U displaystyle pi 1 U nbsp homeomorphically onto U displaystyle U nbsp it is a local homeomorphism i e p displaystyle pi nbsp is a continuous map and for every e E displaystyle e in E nbsp there exists an open neighborhood V E displaystyle V subset E nbsp of e displaystyle e nbsp such that p V V p V displaystyle pi V V rightarrow pi V nbsp is a homeomorphism It follows that the covering space E displaystyle E nbsp and the base space X displaystyle X nbsp locally share the same properties If X displaystyle X nbsp is a connected and non orientable manifold then there is a covering p X X displaystyle pi tilde X rightarrow X nbsp of degree 2 displaystyle 2 nbsp whereby X displaystyle tilde X nbsp is a connected and orientable manifold 1 234 If X displaystyle X nbsp is a connected Lie group then there is a covering p X X displaystyle pi tilde X rightarrow X nbsp which is also a Lie group homomorphism and X g g is a path in X with g 0 1 X modulo homotopy with fixed ends displaystyle tilde X gamma gamma text is a path in X with gamma 0 boldsymbol 1 X text modulo homotopy with fixed ends nbsp is a Lie group 2 174 If X displaystyle X nbsp is a graph then it follows for a covering p E X displaystyle pi E rightarrow X nbsp that E displaystyle E nbsp is also a graph 1 85 If X displaystyle X nbsp is a connected manifold then there is a covering p X X displaystyle pi tilde X rightarrow X nbsp whereby X displaystyle tilde X nbsp is a connected and simply connected manifold 3 32 If X displaystyle X nbsp is a connected Riemann surface then there is a covering p X X displaystyle pi tilde X rightarrow X nbsp which is also a holomorphic map 3 22 and X displaystyle tilde X nbsp is a connected and simply connected Riemann surface 3 32 Factorisation Edit Let X Y displaystyle X Y nbsp and E displaystyle E nbsp be path connected locally path connected spaces and p q displaystyle p q nbsp and r displaystyle r nbsp be continuous maps such that the diagram nbsp commutes If p displaystyle p nbsp and q displaystyle q nbsp are coverings so is r displaystyle r nbsp If p displaystyle p nbsp and r displaystyle r nbsp are coverings so is q displaystyle q nbsp 4 485 Product of coverings Edit Let X displaystyle X nbsp and X displaystyle X nbsp be topological spaces and p E X displaystyle p E rightarrow X nbsp and p E X displaystyle p E rightarrow X nbsp be coverings then p p E E X X displaystyle p times p E times E rightarrow X times X nbsp with p p e e p e p e displaystyle p times p e e p e p e nbsp is a covering 4 339 Equivalence of coverings Edit Let X displaystyle X nbsp be a topological space and p E X displaystyle p E rightarrow X nbsp and p E X displaystyle p E rightarrow X nbsp be coverings Both coverings are called equivalent if there exists a homeomorphism h E E displaystyle h E rightarrow E nbsp such that the diagram nbsp commutes If such a homeomorphism exists then one calls the covering spaces E displaystyle E nbsp and E displaystyle E nbsp isomorphic Lifting property Edit An important property of the covering is that it satisfies the lifting property i e Let I displaystyle I nbsp be the unit interval and p E X displaystyle p E rightarrow X nbsp be a covering Let F Y I X displaystyle F Y times I rightarrow X nbsp be a continuous map and F 0 Y 0 E displaystyle tilde F 0 Y times 0 rightarrow E nbsp be a lift of F Y 0 displaystyle F Y times 0 nbsp i e a continuous map such that p F 0 F Y 0 displaystyle p circ tilde F 0 F Y times 0 nbsp Then there is a uniquely determined continuous map F Y I E displaystyle tilde F Y times I rightarrow E nbsp for which F y 0 F 0 displaystyle tilde F y 0 tilde F 0 nbsp and which is a lift of F displaystyle F nbsp i e p F F displaystyle p circ tilde F F nbsp 1 60 If X displaystyle X nbsp is a path connected space then for Y 0 displaystyle Y 0 nbsp it follows that the map F displaystyle tilde F nbsp is a lift of a path in X displaystyle X nbsp and for Y I displaystyle Y I nbsp it is a lift of a homotopy of paths in X displaystyle X nbsp Because of that property one can show that the fundamental group p 1 S 1 displaystyle pi 1 S 1 nbsp of the unit circle is an infinite cyclic group which is generated by the homotopy classes of the loop g I S 1 displaystyle gamma I rightarrow S 1 nbsp with g t cos 2 p t sin 2 p t displaystyle gamma t cos 2 pi t sin 2 pi t nbsp 1 29 Let X displaystyle X nbsp be a path connected space and p E X displaystyle p E rightarrow X nbsp be a connected covering Let x y X displaystyle x y in X nbsp be any two points which are connected by a path g displaystyle gamma nbsp i e g 0 x displaystyle gamma 0 x nbsp and g 1 y displaystyle gamma 1 y nbsp Let g displaystyle tilde gamma nbsp be the unique lift of g displaystyle gamma nbsp then the map L g p 1 x p 1 y displaystyle L gamma p 1 x rightarrow p 1 y nbsp with L g g 0 g 1 displaystyle L gamma tilde gamma 0 tilde gamma 1 nbsp is bijective 1 69 If X displaystyle X nbsp is a path connected space and p E X displaystyle p E rightarrow X nbsp a connected covering then the induced group homomorphism p p 1 E p 1 X displaystyle p pi 1 E rightarrow pi 1 X nbsp with p g p g displaystyle p gamma p circ gamma nbsp is injective and the subgroup p p 1 E displaystyle p pi 1 E nbsp of p 1 X displaystyle pi 1 X nbsp consists of the homotopy classes of loops in X displaystyle X nbsp whose lifts are loops in E displaystyle E nbsp 1 61 Branched covering EditDefinitions Edit Holomorphic maps between Riemann surfaces Edit Let X displaystyle X nbsp and Y displaystyle Y nbsp be Riemann surfaces i e one dimensional complex manifolds and let f X Y displaystyle f X rightarrow Y nbsp be a continuous map f displaystyle f nbsp is holomorphic in a point x X displaystyle x in X nbsp if for any charts ϕ x U 1 V 1 displaystyle phi x U 1 rightarrow V 1 nbsp of x displaystyle x nbsp and ϕ f x U 2 V 2 displaystyle phi f x U 2 rightarrow V 2 nbsp of f x displaystyle f x nbsp with ϕ x U 1 U 2 displaystyle phi x U 1 subset U 2 nbsp 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 nbsp is holomorphic If f displaystyle f nbsp is holomorphic at all x X displaystyle x in X nbsp we say f displaystyle f nbsp is holomorphic The map F ϕ f x f ϕ x 1 displaystyle F phi f x circ f circ phi x 1 nbsp is called the local expression of f displaystyle f nbsp in x X displaystyle x in X nbsp If f X Y displaystyle f X rightarrow Y nbsp is a non constant holomorphic map between compact Riemann surfaces then f displaystyle f nbsp is surjective and an open map 3 11 i e for every open set U X displaystyle U subset X nbsp the image f U Y displaystyle f U subset Y nbsp is also open Ramification point and branch point Edit Let f X Y displaystyle f X rightarrow Y nbsp be a non constant holomorphic map between compact Riemann surfaces For every x X displaystyle x in X nbsp there exist charts for x displaystyle x nbsp and f x displaystyle f x nbsp and there exists a uniquely determined k x N gt 0 displaystyle k x in mathbb N gt 0 nbsp such that the local expression F displaystyle F nbsp of f displaystyle f nbsp in x displaystyle x nbsp is of the form z z k x displaystyle z mapsto z k x nbsp 3 10 The number k x displaystyle k x nbsp is called the ramification index of f displaystyle f nbsp in x displaystyle x nbsp and the point x X displaystyle x in X nbsp is called a ramification point if k x 2 displaystyle k x geq 2 nbsp If k x 1 displaystyle k x 1 nbsp for an x X displaystyle x in X nbsp then x displaystyle x nbsp is unramified The image point y f x Y displaystyle y f x in Y nbsp of a ramification point is called a branch point Degree of a holomorphic map Edit Let f X Y displaystyle f X rightarrow Y nbsp be a non constant holomorphic map between compact Riemann surfaces The degree deg f displaystyle operatorname deg f nbsp of f displaystyle f nbsp is the cardinality of the fiber of an unramified point y f x Y displaystyle y f x in Y nbsp i e deg f f 1 y displaystyle operatorname deg f f 1 y nbsp This number is well defined since for every y Y displaystyle y in Y nbsp the fiber f 1 y displaystyle f 1 y nbsp is discrete 3 20 and for any two unramified points y 1 y 2 Y displaystyle y 1 y 2 in Y nbsp it is f 1 y 1 f 1 y 2 displaystyle f 1 y 1 f 1 y 2 nbsp It can be calculated by x f 1 y k x deg f displaystyle sum x in f 1 y k x operatorname deg f nbsp 3 29 Branched covering Edit Definition Edit A continuous map f X Y displaystyle f X rightarrow Y nbsp is called a branched covering if there exists a closed set with dense complement E Y displaystyle E subset Y nbsp 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 nbsp is a covering Examples Edit Let n N displaystyle n in mathbb N nbsp and n 2 displaystyle n geq 2 nbsp then f C C displaystyle f mathbb C rightarrow mathbb C nbsp with f z z n displaystyle f z z n nbsp is branched covering of degree n displaystyle n nbsp whereby z 0 displaystyle z 0 nbsp is a branch point Every non constant holomorphic map between compact Riemann surfaces f X Y displaystyle f X rightarrow Y nbsp of degree d displaystyle d nbsp is a branched covering of degree d displaystyle d nbsp Universal covering EditDefinition Edit Let p X X displaystyle p tilde X rightarrow X nbsp be a simply connected covering If b E X displaystyle beta E rightarrow X nbsp is another simply connected covering then there exists a uniquely determined homeomorphism a X E displaystyle alpha tilde X rightarrow E nbsp such that the diagram nbsp commutes 4 482 This means that p displaystyle p nbsp is up to equivalence uniquely determined and because of that universal property denoted as the universal covering of the space X displaystyle X nbsp Existence Edit A universal covering does not always exist but the following properties guarantee its existence Let X displaystyle X nbsp be a connected locally simply connected topological space then there exists a universal covering p X X displaystyle p tilde X rightarrow X nbsp X displaystyle tilde X nbsp is defined as X g g is a path in X with g 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 nbsp and p X X displaystyle p tilde X rightarrow X nbsp by p g g 1 displaystyle p gamma gamma 1 nbsp 1 64 The topology on X displaystyle tilde X nbsp is constructed as follows Let g I X displaystyle gamma I rightarrow X nbsp be a path with g 0 x 0 displaystyle gamma 0 x 0 nbsp Let U displaystyle U nbsp be a simply connected neighborhood of the endpoint x g 1 displaystyle x gamma 1 nbsp then for every y U displaystyle y in U nbsp the paths s y displaystyle sigma y nbsp inside U displaystyle U nbsp from x displaystyle x nbsp to y displaystyle y nbsp are uniquely determined up to homotopy Now consider U g s y y U homotopy with fixed ends displaystyle tilde U gamma sigma y y in U text homotopy with fixed ends nbsp then p U U U displaystyle p tilde U tilde U rightarrow U nbsp with p g s y g s y 1 y displaystyle p gamma sigma y gamma sigma y 1 y nbsp is a bijection and U displaystyle tilde U nbsp can be equipped with the final topology of p U displaystyle p tilde U nbsp The fundamental group p 1 X x 0 G displaystyle pi 1 X x 0 Gamma nbsp acts freely through g x g x displaystyle gamma tilde x mapsto gamma tilde x nbsp on X displaystyle tilde X nbsp and ps G X X displaystyle psi Gamma backslash tilde X rightarrow X nbsp with ps G x x 1 displaystyle psi Gamma tilde x tilde x 1 nbsp is a homeomorphism i e G X X displaystyle Gamma backslash tilde X cong X nbsp Examples Edit nbsp The Hawaiian earring Only the ten largest circles are shown r R S 1 displaystyle r mathbb R to S 1 nbsp with r t cos 2 p t sin 2 p t displaystyle r t cos 2 pi t sin 2 pi t nbsp is the universal covering of the unit circle S 1 displaystyle S 1 nbsp p S n R P n 1 1 S n displaystyle p S n to mathbb R P n cong 1 1 backslash S n nbsp with p x x displaystyle p x x nbsp is the universal covering of the projective space R P n displaystyle mathbb R P n nbsp for n gt 1 displaystyle n gt 1 nbsp q S U n R U n displaystyle q mathrm SU n ltimes mathbb R to U n nbsp with q A t exp 2 p i t 0 0 I n 1 x A displaystyle q A t begin bmatrix exp 2 pi it amp 0 0 amp I n 1 end bmatrix vphantom x A nbsp is the universal covering of the unitary group U n displaystyle U n nbsp 5 5 Theorem 1 Since S U 2 S 3 displaystyle mathrm SU 2 cong S 3 nbsp it follows that the quotient map f S U 2 S U 2 Z 2 S O 3 displaystyle f mathrm SU 2 rightarrow mathrm SU 2 backslash mathbb Z 2 cong mathrm SO 3 nbsp is the universal covering of the S O 3 displaystyle mathrm SO 3 nbsp A topological space which has no universal covering is the Hawaiian earring X n N x 1 x 2 R 2 x 1 1 n 2 x 2 2 1 n 2 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 nbsp One can show that no neighborhood of the origin 0 0 displaystyle 0 0 nbsp is simply connected 4 487 Example 1 G coverings EditLet G be a discrete group acting on the topological space X This means that each element g of G is associated to a homeomorphism Hg of X onto itself in such a way that Hg h is always equal to Hg Hh 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 X X by the twist action where the non identity element acts by x y y x Thus the study of the relation between the fundamental groups 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 Deck transformation EditDefinition Edit Let p E X displaystyle p E rightarrow X nbsp be a covering A deck transformation is a homeomorphism d E E displaystyle d E rightarrow E nbsp such that the diagram of continuous maps nbsp commutes Together with the composition of maps the set of deck transformation forms a group Deck p displaystyle operatorname Deck p nbsp which is the same as Aut p displaystyle operatorname Aut p nbsp Now suppose p C X displaystyle p C to X nbsp is a covering map and C displaystyle C nbsp and therefore also X displaystyle X nbsp is connected and locally path connected The action of Aut p displaystyle operatorname Aut p nbsp 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 nbsp bundle where G Aut p displaystyle G operatorname Aut p nbsp is considered as a discrete topological group Every universal cover p D X displaystyle p D to X nbsp is regular with deck transformation group being isomorphic to the fundamental group p 1 X displaystyle pi 1 X nbsp Examples Edit Let q S 1 S 1 displaystyle q S 1 to S 1 nbsp be the covering q z z n displaystyle q z z n nbsp for some n N displaystyle n in mathbb N nbsp then the map d k S 1 S 1 z z e 2 p i k n displaystyle d k S 1 rightarrow S 1 z mapsto z e 2 pi ik n nbsp is a deck transformation and Deck q Z n Z displaystyle operatorname Deck q cong mathbb Z mathbb nZ nbsp Let r R S 1 displaystyle r mathbb R to S 1 nbsp be the covering r t cos 2 p t sin 2 p t displaystyle r t cos 2 pi t sin 2 pi t nbsp then the map d k R R t t k displaystyle d k mathbb R rightarrow mathbb R t mapsto t k nbsp with k Z displaystyle k in mathbb Z nbsp is a deck transformation and Deck r Z displaystyle operatorname Deck r cong mathbb Z nbsp As another important example consider C displaystyle mathbb C nbsp the complex plane and C displaystyle mathbb C times nbsp the complex plane minus the origin Then the map p C C displaystyle p mathbb C times to mathbb C times nbsp with p z z n displaystyle p z z n nbsp is a regular cover The deck transformations are multiplications with n displaystyle n nbsp th roots of unity and the deck transformation group is therefore isomorphic to the cyclic group Z n Z displaystyle mathbb Z n mathbb Z nbsp Likewise the map exp C C displaystyle exp mathbb C to mathbb C times nbsp with exp z e z displaystyle exp z e z nbsp is the universal cover Properties Edit Let X displaystyle X nbsp be a path connected space and p E X displaystyle p E rightarrow X nbsp be a connected covering Since a deck transformation d E E displaystyle d E rightarrow E nbsp is bijective it permutes the elements of a fiber p 1 x displaystyle p 1 x nbsp with x X displaystyle x in X nbsp and is uniquely determined by where it sends a single point In particular only the identity map fixes a point in the fiber 1 70 Because of this property every deck transformation defines a group action on E displaystyle E nbsp i e let U X displaystyle U subset X nbsp be an open neighborhood of a x X displaystyle x in X nbsp and U E displaystyle tilde U subset E nbsp an open neighborhood of an e p 1 x displaystyle e in p 1 x nbsp then Deck p E E d U d U displaystyle operatorname Deck p times E rightarrow E d tilde U mapsto d tilde U nbsp is a group action Normal coverings Edit Definition Edit A covering p E X displaystyle p E rightarrow X nbsp is called normal if Deck p E X displaystyle operatorname Deck p backslash E cong X nbsp This means that for every x X displaystyle x in X nbsp and any two e 0 e 1 p 1 x displaystyle e 0 e 1 in p 1 x nbsp there exists a deck transformation d E E displaystyle d E rightarrow E nbsp such that d e 0 e 1 displaystyle d e 0 e 1 nbsp Properties Edit Let X displaystyle X nbsp be a path connected space and p E X displaystyle p E rightarrow X nbsp be a connected covering Let H p p 1 E displaystyle H p pi 1 E nbsp be a subgroup of p 1 X displaystyle pi 1 X nbsp then p displaystyle p nbsp is a normal covering iff H displaystyle H nbsp is a normal subgroup of p 1 X displaystyle pi 1 X nbsp If p E X displaystyle p E rightarrow X nbsp is a normal covering and H p p 1 E displaystyle H p pi 1 E nbsp then Deck p p 1 X H displaystyle operatorname Deck p cong pi 1 X H nbsp If p E X displaystyle p E rightarrow X nbsp is a path connected covering and H p p 1 E displaystyle H p pi 1 E nbsp then Deck p N H H displaystyle operatorname Deck p cong N H H nbsp whereby N H displaystyle N H nbsp is the normaliser of H displaystyle H nbsp 1 71 Let E displaystyle E nbsp be a topological space A group G displaystyle Gamma nbsp acts discontinuously on E displaystyle E nbsp if every e E displaystyle e in E nbsp has an open neighborhood V E displaystyle V subset E nbsp with V displaystyle V neq emptyset nbsp such that for every g G displaystyle gamma in Gamma nbsp with g V V displaystyle gamma V cap V neq emptyset nbsp one has d 1 d 2 displaystyle d 1 d 2 nbsp If a group G displaystyle Gamma nbsp acts discontinuously on a topological space E displaystyle E nbsp then the quotient map q E G E displaystyle q E rightarrow Gamma backslash E nbsp with q e G e displaystyle q e Gamma e nbsp is a normal covering 1 72 Hereby G E G e e E displaystyle Gamma backslash E Gamma e e in E nbsp is the quotient space and G e g e g G displaystyle Gamma e gamma e gamma in Gamma nbsp is the orbit of the group action Examples Edit The covering q S 1 S 1 displaystyle q S 1 to S 1 nbsp with q z z n displaystyle q z z n nbsp is a normal coverings for every n N displaystyle n in mathbb N nbsp Every simply connected covering is a normal covering Calculation Edit Let G displaystyle Gamma nbsp be a group which acts discontinuously on a topological space E displaystyle E nbsp and let q E G E displaystyle q E rightarrow Gamma backslash E nbsp be the normal covering If E displaystyle E nbsp is path connected then Deck q G displaystyle operatorname Deck q cong Gamma nbsp 1 72 If E displaystyle E nbsp is simply connected then Deck q p 1 G E displaystyle operatorname Deck q cong pi 1 Gamma backslash E nbsp 1 71 Examples Edit Let n N displaystyle n in mathbb N nbsp The antipodal map g S n S n displaystyle g S n rightarrow S n nbsp with g x x displaystyle g x x nbsp generates together with the composition of maps a group D g Z 2 Z displaystyle D g cong mathbb Z 2Z nbsp and induces a group action D g S n S n g x g x displaystyle D g times S n rightarrow S n g x mapsto g x nbsp which acts discontinuously on S n displaystyle S n nbsp Because of Z 2 S n R P n displaystyle mathbb Z 2 backslash S n cong mathbb R P n nbsp it follows that the quotient map q S n Z 2 S n R P n displaystyle q S n rightarrow mathbb Z 2 backslash S n cong mathbb R P n nbsp is a normal covering and for n gt 1 displaystyle n gt 1 nbsp a universal covering hence Deck q Z 2 Z p 1 R P n displaystyle operatorname Deck q cong mathbb Z 2Z cong pi 1 mathbb R P n nbsp for n gt 1 displaystyle n gt 1 nbsp Let S O 3 displaystyle mathrm SO 3 nbsp be the special orthogonal group then the map f S U 2 S O 3 Z 2 S U 2 displaystyle f mathrm SU 2 rightarrow mathrm SO 3 cong mathbb Z 2 backslash mathrm SU 2 nbsp is a normal covering and because of S U 2 S 3 displaystyle mathrm SU 2 cong S 3 nbsp it is the universal covering hence Deck f Z 2 Z p 1 S O 3 displaystyle operatorname Deck f cong mathbb Z 2Z cong pi 1 mathrm SO 3 nbsp With the group action z 1 z 2 x y z 1 1 z 2 x z 2 y displaystyle z 1 z 2 x y z 1 1 z 2 x z 2 y nbsp of Z 2 displaystyle mathbb Z 2 nbsp on R 2 displaystyle mathbb R 2 nbsp whereby Z 2 displaystyle mathbb Z 2 nbsp is the semidirect product Z Z displaystyle mathbb Z rtimes mathbb Z nbsp one gets the universal covering f R 2 Z Z R 2 K displaystyle f mathbb R 2 rightarrow mathbb Z rtimes mathbb Z backslash mathbb R 2 cong K nbsp of the klein bottle K displaystyle K nbsp hence Deck f Z Z p 1 K displaystyle operatorname Deck f cong mathbb Z rtimes mathbb Z cong pi 1 K nbsp Let T S 1 S 1 displaystyle T S 1 times S 1 nbsp be the torus which is embedded in the C 2 displaystyle mathbb C 2 nbsp Then one gets a homeomorphism a T T e i x e i y e i x p e i y displaystyle alpha T rightarrow T e ix e iy mapsto e i x pi e iy nbsp which induces a discontinuous group action G a T T displaystyle G alpha times T rightarrow T nbsp whereby G a Z 2 Z displaystyle G alpha cong mathbb Z 2Z nbsp It follows that the map f T G a T K displaystyle f T rightarrow G alpha backslash T cong K nbsp is a normal covering of the klein bottle hence Deck f Z 2 Z displaystyle operatorname Deck f cong mathbb Z 2Z nbsp Let S 3 displaystyle S 3 nbsp be embedded in the C 2 displaystyle mathbb C 2 nbsp Since the group action S 3 Z p Z S 3 z 1 z 2 k e 2 p i k p z 1 e 2 p i k q p z 2 displaystyle S 3 times mathbb Z pZ rightarrow S 3 z 1 z 2 k mapsto e 2 pi ik p z 1 e 2 pi ikq p z 2 nbsp is discontinuously whereby p q N displaystyle p q in mathbb N nbsp are coprime the map f S 3 Z p S 3 L p q displaystyle f S 3 rightarrow mathbb Z p backslash S 3 L p q nbsp is the universal covering of the lens space L p q displaystyle L p q nbsp hence Deck f Z p Z p 1 L p q displaystyle operatorname Deck f cong mathbb Z pZ cong pi 1 L p q nbsp Galois correspondence EditLet X displaystyle X nbsp be a connected and locally simply connected space then for every subgroup H p 1 X displaystyle H subseteq pi 1 X nbsp there exists a path connected covering a X H X displaystyle alpha X H rightarrow X nbsp with a p 1 X H H displaystyle alpha pi 1 X H H nbsp 1 66 Let p 1 E X displaystyle p 1 E rightarrow X nbsp and p 2 E X displaystyle p 2 E rightarrow X nbsp be two path connected coverings then they are equivalent iff the subgroups H p 1 p 1 E displaystyle H p 1 pi 1 E nbsp and H p 2 p 1 E displaystyle H p 2 pi 1 E nbsp are conjugate to each other 4 482 Let X displaystyle X nbsp be a connected and locally simply connected space then up to equivalence between coverings there is a bijection Subgroup of p 1 X path connected covering p E X H a X H X p p 1 E p normal subgroup of p 1 X mstyle s, wikipedia, wiki, book, books, library,