In topology, a covering or covering projection is a map between topological spaces that, intuitively, locally acts like a projection of multiple copies of a space onto itself. In particular, coverings are special types of local homeomorphisms. If is a covering, is said to be a covering space or cover of , and is said to be the base of the covering, or simply the base. By abuse of terminology, and may sometimes be called covering spaces as well. Since coverings are local homeomorphisms, a covering space is a special kind of étale space.
Covering spaces 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 the construction of manifolds, orbifolds, and the morphisms between them. In algebraic topology, covering spaces are closely related to the fundamental group: for one, since all coverings have the homotopy lifting property, covering spaces are an important tool in the calculation of homotopy groups. A standard example in this vein is the calculation of the fundamental group of the circle by means of the covering of by (see below).[2]: 29 Under certain conditions, covering spaces also exhibit a Galois correspondence with the subgroups of the fundamental group.
Let be a topological space. A covering of is a continuous map
such that for every there exists an open neighborhood of and a discrete space such that and is a homeomorphism for every . The open sets are called sheets, which are uniquely determined up to homeomorphism if is connected.[2]: 56 For each the discrete set is called the fiber of . If is connected, it can be shown that is surjective, and the cardinality of is the same for all ; this value is called the degree of the covering. If is path-connected, then the covering is called a path-connected covering. This definition is equivalent to the statement that is a locally trivial Fiber bundle.
Some authors also require that be surjective in the case that is not connected.[3]
Examplesedit
For every topological space , the identity map is a covering. Likewise for any discrete space the projection taking is a covering. Coverings of this type are called trivial coverings; if has finitely many (say ) elements, the covering is called the trivial -sheeted 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.[2]: 234
If is a graph, then it follows for a covering that is also a graph.[2]: 85
If is a connected manifold, then there is a covering , whereby is a connected and simply connected manifold.[5]: 32
If is a connected Riemann surface, then there is a covering which is also a holomorphic map[5]: 22 and is a connected and simply connected Riemann surface.[5]: 32
Factorisationedit
Let and be path-connected, locally path-connected spaces, and and be continuous maps, such that the diagram
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. .[2]: 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 .
As a consequence, 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 .[2]: 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 .[5]: 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[5]: 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 , where by 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.[6]: 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 transitive. If this action is free on some fiber, then it is free 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.[2]: 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 .[2]: 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.[2]: 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.[6]: 482
Let be a connected and locally simply connected space, then, up to equivalence between coverings, there is a bijection:
For a sequence of subgroups one gets a sequence of coverings . For a subgroup with index, the covering has degree .
Classificationedit
Definitionsedit
Category of coveringsedit
Let be a topological space. The objects of the category are the coverings of and the
April 09, 2024
covering, space, topology, covering, covering, projection, between, topological, spaces, that, intuitively, locally, acts, like, projection, multiple, copies, space, onto, itself, particular, coverings, special, types, local, homeomorphisms, displaystyle, tild. In topology a covering or covering projection is a map between topological spaces that intuitively locally acts like a projection of multiple copies of a space onto itself In particular coverings are special types of local homeomorphisms If p X X displaystyle p tilde X to X is a covering X p displaystyle tilde X p is said to be a covering space or cover of X displaystyle X and X displaystyle X is said to be the base of the covering or simply the base By abuse of terminology X displaystyle tilde X and p displaystyle p may sometimes be called covering spaces as well Since coverings are local homeomorphisms a covering space is a special kind of etale space Intuitively a covering locally projects a stack of pancakes above an open neighborhood U displaystyle U onto U displaystyle U Covering spaces first arose in the context of complex analysis specifically the technique of analytic continuation where they were introduced by Riemann as domains on which naturally multivalued complex functions become single valued These spaces are now called Riemann surfaces 1 10 Covering spaces are an important tool in several areas of mathematics In modern geometry covering spaces or branched coverings which have slightly weaker conditions are used in the construction of manifolds orbifolds and the morphisms between them In algebraic topology covering spaces are closely related to the fundamental group for one since all coverings have the homotopy lifting property covering spaces are an important tool in the calculation of homotopy groups A standard example in this vein is the calculation of the fundamental group of the circle by means of the covering of S1 displaystyle S 1 by R displaystyle mathbb R see below 2 29 Under certain conditions covering spaces also exhibit a Galois correspondence with the subgroups of the fundamental group 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 editLet X displaystyle X nbsp be a topological space A covering of X displaystyle X nbsp is a continuous map p X X displaystyle pi tilde X rightarrow X nbsp such that for every x X displaystyle x in X nbsp there exists an open neighborhood Ux displaystyle U x nbsp of x displaystyle x nbsp and a discrete space Dx displaystyle D x nbsp such that p 1 Ux d DxVd displaystyle pi 1 U x displaystyle bigsqcup d in D x V d nbsp and p Vd Vd Ux displaystyle pi V d V d rightarrow U x nbsp is a homeomorphism for every d Dx displaystyle d in D x nbsp The open sets Vd displaystyle V d nbsp are called sheets which are uniquely determined up to homeomorphism if Ux displaystyle U x nbsp is connected 2 56 For each x X displaystyle x in X nbsp the discrete set p 1 x displaystyle pi 1 x nbsp is called the fiber of x displaystyle x nbsp If X displaystyle X nbsp is connected it can be shown that p displaystyle pi nbsp is surjective and the cardinality of Dx displaystyle D x nbsp is the same for all x X displaystyle x in X nbsp this value is called the degree of the covering If X displaystyle tilde X nbsp is path connected then the covering p X X displaystyle pi tilde X rightarrow X nbsp is called a path connected covering This definition is equivalent to the statement that p displaystyle p nbsp is a locally trivial Fiber bundle Some authors also require that p displaystyle pi nbsp be surjective in the case that X displaystyle X nbsp is not connected 3 Examples editFor every topological space X displaystyle X nbsp the identity map id X X displaystyle operatorname id X rightarrow X nbsp is a covering Likewise for any discrete space D displaystyle D nbsp the projection p X D X displaystyle pi X times D rightarrow X nbsp taking x i x displaystyle x i mapsto x nbsp is a covering Coverings of this type are called trivial coverings if D displaystyle D nbsp has finitely many say k displaystyle k nbsp elements the covering is called the trivial k displaystyle k nbsp sheeted covering of X displaystyle X nbsp nbsp The space Y 0 1 R displaystyle Y 0 1 times mathbb R nbsp is a covering space of X 0 1 S1 displaystyle X 0 1 times S 1 nbsp The disjoint open sets Si displaystyle S i nbsp are mapped homeomorphically onto U displaystyle U nbsp The fiber of x displaystyle x nbsp consists of the points yi displaystyle y i nbsp The map r R S1 displaystyle r mathbb R to S 1 nbsp with r t cos 2pt sin 2pt displaystyle r t cos 2 pi t sin 2 pi t nbsp is a covering of the unit circle S1 displaystyle S 1 nbsp The base of the covering is S1 displaystyle S 1 nbsp and the covering space is R displaystyle mathbb R nbsp For any point x x1 x2 S1 displaystyle x x 1 x 2 in S 1 nbsp such that x1 gt 0 displaystyle x 1 gt 0 nbsp the set U x1 x2 S1 x1 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 14 n 14 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 Vn 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 2pt sin 2pt 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 S1 S1 displaystyle q S 1 to S 1 nbsp with q z zn 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 S1 displaystyle x in S 1 nbsp one has q 1 U i 1nU 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 S1 displaystyle p mathbb R to S 1 nbsp with p t cos 2pt sin 2pt 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 2 234 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 1X 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 4 174 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 2 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 5 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 5 22 and X displaystyle tilde X nbsp is a connected and simply connected Riemann surface 5 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 6 485 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 6 339 However covering of X X displaystyle X times X nbsp are not all of this form in general 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 All coverings satisfy 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 2 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 As a consequence one can show that the fundamental group p1 S1 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 S1 displaystyle gamma I rightarrow S 1 nbsp with g t cos 2pt sin 2pt displaystyle gamma t cos 2 pi t sin 2 pi t nbsp 2 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 Lg p 1 x p 1 y displaystyle L gamma p 1 x rightarrow p 1 y nbsp with Lg g 0 g 1 displaystyle L gamma tilde gamma 0 tilde gamma 1 nbsp is bijective 2 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 p1 E p1 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 p1 E displaystyle p pi 1 E nbsp of p1 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 2 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 ϕx U1 V1 displaystyle phi x U 1 rightarrow V 1 nbsp of x displaystyle x nbsp and ϕf x U2 V2 displaystyle phi f x U 2 rightarrow V 2 nbsp of f x displaystyle f x nbsp with ϕx U1 U2 displaystyle phi x U 1 subset U 2 nbsp the map ϕf x f ϕx 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 ϕf x f ϕx 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 5 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 kx 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 zkx displaystyle z mapsto z k x nbsp 5 10 The number kx 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 kx 2 displaystyle k x geq 2 nbsp If kx 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 5 20 and for any two unramified points y1 y2 Y displaystyle y 1 y 2 in Y nbsp it is f 1 y1 f 1 y2 displaystyle f 1 y 1 f 1 y 2 nbsp It can be calculated by x f 1 y kx deg f displaystyle sum x in f 1 y k x operatorname deg f nbsp 5 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 zn displaystyle f z z n nbsp is branched covering of degree n displaystyle n nbsp where by 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 6 482 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 x0 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 2 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 x0 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 sy 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 sy 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 sy g sy 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 p1 X x0 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 Gx 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 S1 displaystyle r mathbb R to S 1 nbsp with r t cos 2pt sin 2pt displaystyle r t cos 2 pi t sin 2 pi t nbsp is the universal covering of the unit circle S1 displaystyle S 1 nbsp p Sn RPn 1 1 Sn 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 RPn displaystyle mathbb R P n nbsp for n gt 1 displaystyle n gt 1 nbsp q SU n R U n displaystyle q mathrm SU n ltimes mathbb R to U n nbsp with q A t exp 2pit 00In 1 xA 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 7 5 Theorem 1 Since SU 2 S3 displaystyle mathrm SU 2 cong S 3 nbsp it follows that the quotient map f SU 2 SU 2 Z2 SO 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 SO 3 displaystyle mathrm SO 3 nbsp A topological space which has no universal covering is the Hawaiian earring X n N x1 x2 R2 x1 1n 2 x22 1n2 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 6 487 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 transitive If this action is free on some fiber then it is free 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 p1 X displaystyle pi 1 X nbsp Examples edit Let q S1 S1 displaystyle q S 1 to S 1 nbsp be the covering q z zn displaystyle q z z n nbsp for some n N displaystyle n in mathbb N nbsp then the map dk S1 S1 z ze2pik 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 nZ displaystyle operatorname Deck q cong mathbb Z mathbb nZ nbsp Let r R S1 displaystyle r mathbb R to S 1 nbsp be the covering r t cos 2pt sin 2pt displaystyle r t cos 2 pi t sin 2 pi t nbsp then the map dk 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 zn 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 nZ 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 ez 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 2 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 e0 e1 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 e0 e1 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 p1 E displaystyle H p pi 1 E nbsp be a subgroup of p1 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 p1 X displaystyle pi 1 X nbsp If p E X displaystyle p E rightarrow X nbsp is a normal covering and H p p1 E displaystyle H p pi 1 E nbsp then Deck p p1 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 p1 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 2 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 d1 d2 G displaystyle d 1 d 2 in Gamma nbsp with d1V d2V displaystyle d 1 V cap d 2 V neq emptyset nbsp one has d1 d2 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 Ge displaystyle q e Gamma e nbsp is a normal covering 2 72 Hereby G E Ge e E displaystyle Gamma backslash E Gamma e e in E nbsp is the quotient space and Ge 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 S1 S1 displaystyle q S 1 to S 1 nbsp with q z zn 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 2 72 If E displaystyle E nbsp is simply connected then Deck q p1 G E displaystyle operatorname Deck q cong pi 1 Gamma backslash E nbsp 2 71 Examples edit Let n N displaystyle n in mathbb N nbsp The antipodal map g Sn Sn 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 2Z displaystyle D g cong mathbb Z 2Z nbsp and induces a group action D g Sn Sn g x g x displaystyle D g times S n rightarrow S n g x mapsto g x nbsp which acts discontinuously on Sn displaystyle S n nbsp Because of Z2 Sn RPn displaystyle mathbb Z 2 backslash S n cong mathbb R P n nbsp it follows that the quotient map q Sn Z2 Sn RPn 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 2Z p1 RPn 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 SO 3 displaystyle mathrm SO 3 nbsp be the special orthogonal group then the map f SU 2 SO 3 Z2 SU 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 SU 2 S3 displaystyle mathrm SU 2 cong S 3 nbsp it is the universal covering hence Deck f Z 2Z p1 SO 3 displaystyle operatorname Deck f cong mathbb Z 2Z cong pi 1 mathrm SO 3 nbsp With the group action z1 z2 x y z1 1 z2x z2 y displaystyle z 1 z 2 x y z 1 1 z 2 x z 2 y nbsp of Z2 displaystyle mathbb Z 2 nbsp on R2 displaystyle mathbb R 2 nbsp whereby Z2 displaystyle mathbb Z 2 nbsp is the semidirect product Z Z displaystyle mathbb Z rtimes mathbb Z nbsp one gets the universal covering f R2 Z Z R2 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 p1 K displaystyle operatorname Deck f cong mathbb Z rtimes mathbb Z cong pi 1 K nbsp Let T S1 S1 displaystyle T S 1 times S 1 nbsp be the torus which is embedded in the C2 displaystyle mathbb C 2 nbsp Then one gets a homeomorphism a T T eix eiy ei x p e iy displaystyle alpha T rightarrow T e ix e iy mapsto e i x pi e iy nbsp which induces a discontinuous group action Ga T T displaystyle G alpha times T rightarrow T nbsp whereby Ga Z 2Z displaystyle G alpha cong mathbb Z 2Z nbsp It follows that the map f T Ga 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 2Z displaystyle operatorname Deck f cong mathbb Z 2Z nbsp Let S3 displaystyle S 3 nbsp be embedded in the C2 displaystyle mathbb C 2 nbsp Since the group action S3 Z pZ S3 z1 z2 k e2pik pz1 e2pikq pz2 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 S3 Zp S3 Lp q displaystyle f S 3 rightarrow mathbb Z p backslash S 3 L p q nbsp is the universal covering of the lens space Lp q displaystyle L p q nbsp hence Deck f Z pZ p1 Lp 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 p1 X displaystyle H subseteq pi 1 X nbsp there exists a path connected covering a XH X displaystyle alpha X H rightarrow X nbsp with a p1 XH H displaystyle alpha pi 1 X H H nbsp 2 66 Let p1 E X displaystyle p 1 E rightarrow X nbsp and p2 E X displaystyle p 2 E rightarrow X nbsp be two path connected coverings then they are equivalent iff the subgroups H p1 p1 E displaystyle H p 1 pi 1 E nbsp and H p2 p1 E displaystyle H p 2 pi 1 E nbsp are conjugate to each other 6 482 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 p1 X path connected covering p E X H a XH Xp p1 E p normal subgroup of p1 X normal covering p E X H a XH Xp p1 E p displaystyle begin matrix qquad displaystyle text Subgroup of pi 1 X amp longleftrightarrow amp displaystyle text path connected covering p E rightarrow X H amp longrightarrow amp alpha X H rightarrow X p pi 1 E amp longleftarrow amp p displaystyle text normal subgroup of pi 1 X amp longleftrightarrow amp displaystyle text normal covering p E rightarrow X H amp longrightarrow amp alpha X H rightarrow X p pi 1 E amp longleftarrow amp p end matrix nbsp For a sequence of subgroups e H G p1 X displaystyle displaystyle text e subset H subset G subset pi 1 X nbsp one gets a sequence of coverings X XH H X XG G X X p1 X X displaystyle 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 nbsp For a subgroup H p1 X displaystyle H subset pi 1 X nbsp with index p1 X H d displaystyle displaystyle pi 1 X H d nbsp the covering a XH X displaystyle alpha X H rightarrow X nbsp has degree d displaystyle d nbsp Classification editDefinitions edit Category of coverings edit Let X displaystyle X nbsp be a topological space The objects of the category Cov X displaystyle boldsymbol Cov X nbsp are the coverings p E X displaystyle p E rightarrow X nbsp of X displaystyle X nbsp and the a, wikipedia, wiki, book, books, library,