In mathematics, specifically topology, a covering map is a continuous surjective map p : C → X, with C and X being topological spaces, which has the following property: Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with the study of topological spaces. ... In topology and related areas of mathematics a continuous function is a morphism between topological spaces; that is, a mapping which preserves the topological structure. ... In mathematics, a surjective function (or onto function or surjection) is a function with the property that all possible output values of the function are generated when the input ranges over all the values in the domain. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...

to every x in X there exists an open neighborhood U such that p ^-1(U) is a union of mutually disjoint open sets S_i (where i ranges over some index set I) in C such that p restricted to S_i yields a homeomorphism from S_i onto U for every i in I.

A covering map is also simply called a cover; we say C is a covering space of X or C covers X. For each x in X, the set p ^-1(x) is called the fiber over x; the sets S_i are called the sheets over U. One generally pictures C as "hovering above" X, with p mapping "downwards", the sheets over U being horizontally stacked above each other and above U, and the fiber over x consisting of those points of C that lie "vertically above" x. In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... This is a glossary of some terms used in the branch of mathematics known as topology. ... In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ... In mathematics, two sets are said to be disjoint if they have no element in common. ... In the mathematical field of topology a homeomorphism or topological isomorphism (from the Greek words homeos = identical and morphe = shape) is a special isomorphism between topological spaces which respects topological properties. ...

A special case, called an open cover (or just cover) is when C is the disjoint union of a collection of open sets X_i, with union X. A cover of any set S is the special case of this idea, when S carries the discrete topology (so that any subset is open). In mathematics, a cover of a set X is a collection of subsets C of X whose union is X. In symbols, if C = {Uα : α ∈ A} is an indexed family of subsets of X, then C is a cover if More generally, if Y is a subset of X... In mathematics, a cover of a set X is a collection of subsets C of X whose union is X. In symbols, if C = {Uα : α ∈ A} is an indexed family of subsets of X, then C is a cover if More generally, if Y is a subset of X... In set theory, a disjoint union (or discriminated union) is a union of a collection of sets whose members are pairwise disjoint. ... In topology and related fields of mathematics, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are isolated from each other in a certain sense. ...

Examples

Consider the unit circle S¹ in R². Then the map p : R → S¹ with Illustration of a unit circle. ...

p(t) = (cos(t),sin(t))

is a cover.

Consider the complex plane with the origin removed, denoted by C^×, and pick a non-zero integer n. Then p : C^× → C^× given by The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ... The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ...

p(z) = zⁿ

is a cover. Here every fiber has n elements.

If G is group (considered as a discrete topological group), then every principal G-bundle is a covering map. Here every fiber can be identified with G. In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In topology and related fields of mathematics, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are isolated from each other in a certain sense. ... In mathematics, a topological group G is a group that is also a topological space such that the group multiplication G × G → G and the inverse operation G → G are continuous maps. ... In mathematics, a principal G-bundle is a special kind of fiber bundle for which the fibers are all G-torsors (also known as principal homogeneous spaces) for the action of a topological group G. Principal G-bundles are G-bundles in the sense that the group G also serves...

Elementary properties

Common local properties: Every cover p : C → X is a local homeomorphism (i.e. to every there exists an open set A in C containing c and an open set B in X such that the restriction of p to A yields a homeomorphism between A and B). This implies that C and X share all local properties. If X is simply connected, then this holds globally as well, and the covering p is a homeomorphism. In topology, a local homeomorphism is a map from one topological space to another that respects locally the topological structure of the two spaces. ... In the mathematical field of topology a homeomorphism or topological isomorphism (from the Greek words homeos = identical and morphe = shape) is a special isomorphism between topological spaces which respects topological properties. ... A geometrical object is called simply connected if it consists of one piece and doesnt have any circle-shaped holes or handles. Higher-dimensional holes are allowed. ...

Cardinality: For every , the fiber over x is a discrete subset of C. On every connected component of X, the cardinality of the fibers is the same (possibly infinite). If every fiber has 2 elements, we speak of a double cover. In topology and related fields of mathematics, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are isolated from each other in a certain sense. ... In topology and related branches of mathematics, a topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. ... In mathematics, the cardinality of a set is a measure of the number of elements of the set. There are two approaches to cardinality – one which compares sets directly using bijections, injections, and surjections, and another which uses cardinal numbers. ...

The lifting property: If p : C → X is a cover and γ is a path in X (i.e. a continuous map from the unit interval [0,1] into X) and is a point "lying over" γ(0) (i.e. p(c) = γ(0)), then there exists a unique path ρ in C lying over γ (i.e. p o ρ = γ) and with ρ(0) = c. The curve ρ is called the lift of γ. If x and y are two points in X connected by a path, then that path furnishes a bijection between the fiber over x and the fiber over y via the lifting property. In mathematics, in particular in homotopy theory within algebraic topology, the homotopy lifting property (also known as the right lifting property) is a technical condition on a continuous function from a topological space E to another one, B. It is designed to support the picture of E above B, by... In mathematics, the unit interval is the interval [0,1], that is the set of all real numbers x such that zero is less than or equal to x and x is less than or equal to one. ... In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one-to-one and onto. ...

Equivalance: Let and be two coverings. One then says that the two coverings (p₁,C₁) and (p₂,C₂) are equivalent if there exists a homeomorphism and . Equivalence classes of coverings correspond to conjugacy classes, as discussed below. If p₂₁ is a covering rather than a homeomorphism, then one says that (p₂,C₂) dominates (p₁,C₁) (given that ).

Universal covers

A cover q : D → X is a universal cover iff D is simply connected. The name comes from the following important property: if p : C → X is any cover of X with C connected, then there exists a covering map f : D → C such that p o f = q. This can be phrased as "The universal cover of X covers all connected covers of X." ↔ ⇔ ≡ For other possible meanings of iff, see IFF. In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. Although P iff Q is most standard, common alternative phrases include Q is necessary and sufficient for P and P... A geometrical object is called simply connected if it consists of one piece and doesnt have any circle-shaped holes or handles. Higher-dimensional holes are allowed. ... In various branches of mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique morphism under certain conditions. ...

The map f is unique in the following sense: if we fix x∈X and d∈D with q(d) = x and c∈C with p(c) = x, then there exists a unique covering map f : D → C such that p o f = q and f(d) = c.

If X has a universal cover, then that universal cover is essentially unique: if q₁ : D₁ → X and q₂ : D₂ → X are two universal covers of X, then there exists a homeomorphism f : D₁ → D₂ such that q₂ o f = q₁.

The space X has a universal cover if and only if it is path-connected, locally path-connected and semi-locally simply connected. The universal cover of X can be constructed as a certain space of paths in X. In topology and related branches of mathematics, a topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. ... In topology and related branches of mathematics, a topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. ... In mathematics, in particular topology, a topological space X is called semi-locally simply connected if every point x in X has a neighborhood U such that the homomorphism from the fundamental group of U to the fundamental group of X, induced by the inclusion map of U into X...

The example R → S¹ given above is a universal cover. The map S³ → SO(3) from unit quaternions to rotations of 3D space described in quaternions and spatial rotation is also a universal cover. In mathematics, the quaternions are a non-commutative extension of the complex numbers. ... Rotation is the movement of a body in such a way that the distance between a certain fixed point and any given point of that body remains constant. ... Quaternions are used in computer graphics and related fields because they allow for compact representations of rotations in 3D space. ...

If the space X carries some additional structure, then its universal cover normally inherits that structure:

• if X is a manifold, then so is its universal cover C
• if X is a Riemann surface, then so is its universal cover C, and p is a holomorphic map
• if X is a Lie group (as in the two examples above), then so is its universal cover C, and p is a homomorphism of Lie groups.

The universal cover first arose in the theory of analytic functions as the natural domain of an analytic continuation. This is the current mathematics collaboration of the week! Please help improve it to featured article standard at manifold/rewrite. ... In mathematics, particularly in complex analysis, a Riemann surface is a one-dimensional complex manifold. ... In mathematics, a Lie group is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ... In mathematics, an analytic function is one that is locally given by a convergent power series. ... In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. ...

Deck transformation group, regular covers

A deck transformation or automorphism of a cover p : C → X is a homeomorphism f : C → C such that p o f = p. The set of all deck transformations of p forms a group under composition, the deck transformation group Aut(p). Every deck transformation permutes the elements of each fiber. This defines a group action of the deck transformation group on each fiber. Note that if f is not the identity, then f has no fixed points. In the mathematical field of topology a homeomorphism or topological isomorphism (from the Greek words homeos = identical and morphe = shape) is a special isomorphism between topological spaces which respects topological properties. ... In mathematics, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite. ... In mathematics, especially in abstract algebra and related areas, a permutation is a bijection from a finite set X onto itself. ... In mathematics, groups are often used to describe symmetries of objects. ... In mathematics, a fixed point of a function is a point that is mapped to itself by the function. ...

Now suppose p : C → X is a covering map and C (and therefore also X) is connected and locally path connected. The action of 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. Every such regular cover is a principal G-bundle, where G = Aut(p) is considered as a discrete topological group. In mathematics, groups are often used to describe symmetries of objects. ... In mathematics, groups are often used to describe symmetries of objects. ... In mathematics, a principal G-bundle is a special kind of fiber bundle for which the fibers are all G-torsors (also known as principal homogeneous spaces) for the action of a topological group G. Principal G-bundles are G-bundles in the sense that the group G also serves...

Every universal cover p : D → X is regular, with deck transformation group being isomorphic to the opposite of the fundamental group π(X). In category theory, an abstract branch of mathematics, the dual of a category C is the category formed by reversing all the morphisms of C. That is, we take Cop to be the category with objects that are those of C, but with the morphisms from X to Y in... In mathematics, the fundamental group is one of the basic concepts of algebraic topology. ...

The example p : C^× → C^× with p(z) = zⁿ from above is a regular cover. The deck transformations are multiplications with n-th roots of unity and the deck transformation group is therefore isomorphic to the cyclic group C_n. In mathematics, the n-th roots of unity or de Moivre numbers, named after Abraham de Moivre (1667 - 1754), are complex numbers located on the unit circle. ... In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that, when written multiplicatively, every element of the group is a power of a. ...

Monodromy action

Again suppose p : C → X is a covering map and C (and therefore also X) is connected and locally path connected. If x∈X and c belongs to the fiber over x (i.e. p(c) = x), and γ:[0,1]→X is a path with γ(0)=γ(1)=x, then this path lifts to a unique path in C with starting point c. The end point of this lifted path need not be c, but it must lie in the fiber over x. It turns out that this end point only depends on the class of γ in the fundamental group π(X,x), and in this fashion we obtain a right group action of π(X,x) on the fiber over x. This is known as the monodromy action. In mathematics, in particular in homotopy theory within algebraic topology, the homotopy lifting property is a technical condition on a continuous function from a topological space X to another one, Y. It is designed to support the picture of X above Y, by allowing a homotopy taking place in Y... In mathematics, the fundamental group is one of the basic concepts of algebraic topology. ... In mathematics, groups are often used to describe symmetries of objects. ... In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology and differential geometry behave as they run round a singularity. ...

So there are two actions on the fiber over x: Aut(p) acts on the left and π(X,x) acts on the right. These two actions are compatible in the following sense:

f.(c.γ) = (f.c).γ

for all f∈Aut(p), c∈p ^-1(x) and γ∈π(X,x).

If p is a universal cover, then the monodromy action is regular; if we identify Aut(p) with the opposite group of π(X,x), then the monodromy action coincides with the action of Aut(p) on the fiber over x. In category theory, an abstract branch of mathematics, the dual of a category C is the category formed by reversing all the morphisms of C. That is, we take Cop to be the category with objects that are those of C, but with the morphisms from X to Y in...

Group structure redux

The deck transformation group and the monodromy action can be understood to relate the normal subgroups of the fundamental group π₁(X) of X and the fundamental group π₁(C) of the cover. Furthermore, these equate the conjugacy classes of subgroups of π₁(X) and equivalence classes of coverings. As a result, one can conclude that X=C/Aut(p), that is, the manifold X is given as the quotient of the covering manifold under the action of the deck transformation group. These inter-relationships are explored below. In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element g−1ng is still in N. The statement N is a normal subgroup of G is written: . There are... In mathematics, the fundamental group is one of the basic concepts of algebraic topology. ... In mathematics, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes reveals many important features of a groups structure. ...

Let γ be a curve in X. Denote by γ_C the lift of γ to C. Consider the set In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ... In mathematics, in particular in homotopy theory within algebraic topology, the homotopy lifting property is a technical condition on a continuous function from a topological space X to another one, Y. It is designed to support the picture of X above Y, by allowing a homotopy taking place in Y...

Note that Γ_p(c) is a group, and that is is a subgroup of π₁(X,p(c)). Note also that it depends on c, and that different values of c in the same fiber yield different subgroups. Each such subgroups is conjugate to another by the monodromy action. To see this, pick two points c₁,c₂ in the same fiber: p(c₁) = p(c₂) = x and let g be a curve in C connecting c₁ to c₂. Then p(g) is a closed curve in X. If γ_C is a closed curve in C passing through c₁, then gγ_Cg ^{− 1} is a closed curve in C passing through c₂. Thus, we have shown In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In mathematics, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes reveals many important features of a groups structure. ...

Γ_p(c₂) = gΓ_p(c₁)g ^{− 1}

and so we have the result that Γ_p(c₁) and Γ_p(c₂) are conjugate subgroups of π₁(X,x). All of the conjugate subgroups may be obtained in this way.

It should be clear that two equivalent coverings lead to the same conjugacy class of subgroups of π₁(X,x); there is a bijective correspondence between equivalence classes of coverings and conjugacy classes of subgroups of π₁(X). In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one_to_one and onto. ...

Note that this implies that the fundamental group π₁(C) is isomorphic to Γ_p. Let N(Γ_p) be the normalizer of Γ_p in π₁(X). The deck transformation group Aut(p) is isomorphic to N(Γ_p) / Γ_p. If p is a universal covering, then Γ_p is the trivial group, and Aut(p) is isomorphic to π₁(X). In group theory, the centralizer and normalizer of a subset S of a group G are subgroups of G which have a restricted action on the elements of S and S as a whole, respectively. ... The following list in mathematics contains the finite groups of small order up to group isomorphism. ...

As a corollary, let us reverse this argument. Let Γ be a normal subgroup of π₁(X,x). By the above arguments, this defines a (regular) covering . Let c₁ in C be in the fiber of x. Then for every other c₂ in the fiber of x, there is precisely one deck transformation that takes c₁ to c₂. This deck transformation corresponds to a curve g in C connecting c₁ to c₂. In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element g−1ng is still in N. The statement N is a normal subgroup of G is written: . There are...

Note that Aut(p) operates properly discontinuously on C, and so we have that X=C/Aut(p), that is, X is the manifold given by the quotient of the covering manifold by the deck transformation group. In mathematics, a free regular set is a subset of a topological space that is acted upon disjointly under a given group action. ...

References

• Hershel M. Farkas and Irwin Kra, Riemann Surfaces (1980), Springer-Verlag, New York. ISBN 0-387-90465-4 (See Chapter 1 for a simple review)
• Jurgen Jost, Compact Riemann Surfaces (2002), Springer-Verlag, New York. ISBN 3-540-43299-X (See Section 1.3)