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 X, and X is said to be the base of the covering, or simply the base. By abuse of terminology, \tilde X and p 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 first arose in the context of complex analysis (specifically, the technique of analytic continuation), where they were introduced by Riemann as domains on which naturally multivalued complex functions become single-valued. These spaces are now called Riemann surfaces. Covering spaces are an important tool in several areas of mathematics. In modern geometry, covering spaces (or branched coverings, which have slightly weaker conditions) are used in 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 S^1 by \mathbb{R} (see below). Under certain conditions, covering spaces also exhibit a Galois correspondence with the subgroups of the fundamental group.

Definition

Let X be a topological space. A covering of X is a continuous map such that for every x \in X there exists an open neighborhood U_x of x and a discrete space D_x such that and is a homeomorphism for every d \in D_x. The open sets V_{d} are called sheets, which are uniquely determined up to homeomorphism if U_x is connected. For each x \in X the discrete set \pi^{-1}(x) is called the fiber of x. If X is connected (and \tilde X is non-empty), it can be shown that \pi is surjective, and the cardinality of D_x is the same for all x \in X; this value is called the degree of the covering. If \tilde X is path-connected, then the covering is called a path-connected covering. This definition is equivalent to the statement that \pi is a locally trivial Fiber bundle. Some authors also require that \pi be surjective in the case that X is not connected.

Examples

Properties

Local homeomorphism

Since a covering maps each of the disjoint open sets of \pi^{-1}(U) homeomorphically onto U it is a local homeomorphism, i.e. \pi is a continuous map and for every e \in E there exists an open neighborhood V \subset E of e, such that is a homeomorphism. It follows that the covering space E and the base space X locally share the same properties.

Factorisation

Let X, Y and E be path-connected, locally path-connected spaces, and p,q and r be continuous maps, such that the diagram commutes.

Product of coverings

Let X and X' be topological spaces and and be coverings, then with is a covering. However, coverings of X\times X' are not all of this form in general.

Equivalence of coverings

Let X 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 E and E' isomorphic.

Lifting property

All coverings satisfy the lifting property, i.e.: Let I 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 F, i.e.. If X is a path-connected space, then for Y={0} it follows that the map \tilde F is a lift of a path in X and for Y=I it is a lift of a homotopy of paths in X. 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. Let X be a path-connected space and be a connected covering. Let x,y \in X be any two points, which are connected by a path \gamma, i.e. and. Let be the unique lift of \gamma, then the map is bijective. If X is a path-connected space and a connected covering, then the induced group homomorphism is injective and the subgroup of \pi_1(X) consists of the homotopy classes of loops in X, whose lifts are loops in E.

Branched covering

Definitions

Holomorphic maps between Riemann surfaces

Let X and Y be Riemann surfaces, i.e. one dimensional complex manifolds, and let be a continuous map. f is holomorphic in a point x \in X, if for any charts of x and of f(x), with, the map is holomorphic. If f is holomorphic at all x \in X, we say f is holomorphic. The map is called the local expression of f in x \in X. If is a non-constant, holomorphic map between compact Riemann surfaces, then f is surjective and an open map, i.e. for every open set U \subset X the image is also open.

Ramification point and branch point

Let be a non-constant, holomorphic map between compact Riemann surfaces. For every x \in X there exist charts for x and f(x) and there exists a uniquely determined, such that the local expression F of f in x is of the form. The number k_x is called the ramification index of f in x and the point x \in X is called a ramification point if k_x \geq 2. If k_x =1 for an x \in X, then x is unramified. The image point of a ramification point is called a branch point.

Degree of a holomorphic map

Let be a non-constant, holomorphic map between compact Riemann surfaces. The **degree ** of f is the cardinality of the fiber of an unramified point, i.e.. This number is well-defined, since for every y \in Y the fiber f^{-1}(y) is discrete and for any two unramified points, it is: It can be calculated by:

Branched covering

Definition

A continuous map is called a branched covering, if there exists a closed set with dense complement E \subset Y, such that is a covering.

Examples

Universal covering

Definition

Let be a simply connected covering. If is another simply connected covering, then there exists a uniquely determined homeomorphism, such that the diagram commutes. This means that p is, up to equivalence, uniquely determined and because of that universal property denoted as the universal covering of the space X.

Existence

A universal covering does not always exist, but the following properties guarantee its existence: Let X be a connected, locally simply connected topological space; then, there exists a universal covering. \tilde X is defined as and by. The topology on \tilde X is constructed as follows: Let be a path with. Let U be a simply connected neighborhood of the endpoint x=\gamma(1), then for every y \in U the paths \sigma_y inside U from x to y are uniquely determined up to homotopy. Now consider, then with is a bijection and \tilde U can be equipped with the final topology of. The fundamental group acts freely through on \tilde X and with is a homeomorphism, i.e..

Examples

G-coverings

Let G be a discrete group acting on the topological space X. This means that each element g of G is associated to a homeomorphism 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.

Smooth coverings

Let E and M be smooth manifolds with or without boundary. A covering is called a smooth covering if it is a smooth map and the sheets are mapped diffeomorphically onto the corresponding open subset of M . (This is in contrast to the definition of a covering, which merely requires that the sheets are mapped homeomorphically onto the corresponding open subset.)

Deck transformation

Definition

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 p:C \to X is a covering map and C (and therefore also X) 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 G-bundle, where is considered as a discrete topological group. Every universal cover p:D \to X is regular, with deck transformation group being isomorphic to the fundamental group \pi_1(X).

Examples

Properties

Let X be a path-connected space and be a connected covering. Since a deck transformation is bijective, it permutes the elements of a fiber p^{-1}(x) with x \in X and is uniquely determined by where it sends a single point. In particular, only the identity map fixes a point in the fiber. Because of this property every deck transformation defines a group action on E, i.e. let U \subset X be an open neighborhood of a x \in X and an open neighborhood of an, then is a group action.

Normal coverings

Definition

A covering is called normal, if. This means, that for every x \in X and any two there exists a deck transformation, such that d(e_0)=e_1.

Properties

Let X be a path-connected space and be a connected covering. Let be a subgroup of \pi_1(X), then p is a normal covering iff H is a normal subgroup of \pi_1(X). If is a normal covering and, then. If is a path-connected covering and, then , whereby N(H) is the normaliser of H. Let E be a topological space. A group \Gamma acts discontinuously on E, if every e \in E has an open neighborhood V \subset E with, such that for every with one has d_1 = d_2. If a group \Gamma acts discontinuously on a topological space E, then the quotient map with is a normal covering. Hereby is the quotient space and is the orbit of the group action.

Examples

Calculation

Let \Gamma be a group, which acts discontinuously on a topological space E and let be the normal covering.

Examples

Galois correspondence

Let X be a connected and locally simply connected space, then for every subgroup there exists a path-connected covering with. Let and be two path-connected coverings, then they are equivalent iff the subgroups and are conjugate to each other. Let X 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 d.

Classification

Definitions

Category of coverings

Let X be a topological space. The objects of**** the category**** **** are the coverings of**** X and the morphisms between two coverings **** and **** are continuous**** maps****,**** such**** that**** the diagram commutes.

G-Set

Let G be a topological group. The category is the category of sets which are G-sets. The morphisms are G-maps between G-sets. They satisfy the condition for every g \in G.

Equivalence

Let X be a connected and locally simply connected space, x \in X and be the fundamental group of X. Since G defines, by lifting of paths and evaluating at the endpoint of the lift, a group action on the fiber of a covering, the functor is an equivalence of categories.

Applications

[[Image:Rotating gimbal-xyz.gif|thumb|300px|[[Gimbal lock]] occurs because any map T3 → RP3 is not a covering map. In particular, the relevant map carries any element of T3, that is, an ordered triple (a,b,c) of angles (real numbers mod 2π), to the composition of the three coordinate axis rotations Rx(a)∘Ry(b)∘Rz(c) by those angles, respectively. Each of these rotations, and their composition, is an element of the rotation group SO(3), which is topologically RP3. This animation shows a set of three gimbals mounted together to allow three degrees of freedom. When all three gimbals are lined up (in the same plane), the system can only move in two dimensions from this configuration, not three, and is in gimbal lock. In this case it can pitch or yaw, but not roll (rotate in the plane that the axes all lie in).]] An important practical application of covering spaces occurs in charts on SO(3), the rotation group. This group occurs widely in engineering, due to 3-dimensional rotations being heavily used in navigation, nautical engineering, and aerospace engineering, among many other uses. Topologically, SO(3) is the real projective space RP3, with fundamental group Z/2, and only (non-trivial) covering space the hypersphere S3, which is the group Spin(3), and represented by the unit quaternions. Thus quaternions are a preferred method for representing spatial rotations – see quaternions and spatial rotation. However, it is often desirable to represent rotations by a set of three numbers, known as Euler angles (in numerous variants), both because this is conceptually simpler for someone familiar with planar rotation, and because one can build a combination of three gimbals to produce rotations in three dimensions. Topologically this corresponds to a map from the 3-torus T3 of three angles to the real projective space RP3 of rotations, and the resulting map has imperfections due to this map being unable to be a covering map. Specifically, the failure of the map to be a local homeomorphism at certain points is referred to as gimbal lock, and is demonstrated in the animation at the right – at some points (when the axes are coplanar) the rank of the map is 2, rather than 3, meaning that only 2 dimensions of rotations can be realized from that point by changing the angles. This causes problems in applications, and is formalized by the notion of a covering space.

Literature