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Wikipedia / Covering space

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 is said to be a covering space or cover is said to be the base of the covering, or simply the base. By abuse of terminology may sometimes be called covering spaces as well. Since coverings are local homeomorphisms, a covering space is a special kind of étalé 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

be a topological space. A covering is a continuous map

there exists an open neighborhood and a discrete space is the disjoint union is a homeomorphism are called sheets is connected. is called the fiber is surjective, and the cardinality; this value is called the degree is path-connected is called a path-connected covering is a locally trivial fiber bundle.

is not connected.

Examples

  • , the identity map is a covering. Coverings of this type are called trivial coverings) elements, the covering is called the trivial -sheeted covering.

    is a covering of the unit circle is

  • is

  • , one has:

    1. .

    A map which is a local homeomorphism.

    is a homomorphic double covering.

Properties

Local homeomorphism

is a homeomorphism.

locally share the same properties.

Factorisation

be continuous maps, such that the diagram commutes.

  • .

  • .

Product of coverings

is a covering. are not all of this form in general.

Equivalence of coverings

be coverings. Both coverings are called equivalent, such that the diagram isomorphic.

Lifting property

All coverings satisfy the lifting property, i.e.:

be the unit interval.

is a lift of a path it is a lift of a homotopy.

As a consequence, one can show that the fundamental group of the unit circle is an infinite cyclic group.

, then the map

is bijective.

a connected covering, then the induced group homomorphism

  1. ,

is injective and the subgroup.

Branched covering

Definitions

Holomorphic maps between Riemann surfaces

be Riemann surfaces, i.e. one dimensional complex manifolds is holomorphic in a point, if for any charts is holomorphic.

is holomorphic.

is called the local expression.

is a non-constant, holomorphic map between compact Riemann surfaces is surjective and an open map, i.e. for every open set the image is also open.

Ramification point and branch point

. is called the ramification index is called a ramification point is unramified of a ramification point is called a branch point.

Degree of a holomorphic map

be a non-constant, holomorphic map between compact Riemann surfaces. The .

is discrete

It can be calculated by:

Branched covering

Definition

is called a branched covering, if there exists a closed set with dense is a covering.

Examples

Universal covering

Definition

be a simply connected, such that the diagram commutes.

is, up to equivalence, uniquely determined and because of that universal property denoted as the universal covering.

Existence

A universal covering does not always exist. The following theorem guarantees its existence for a certain class of base spaces.

be a connected, locally simply connected

The topology there is a path that is unique up to homotopy can be equipped with the final topology

acts freely

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 by the twist action where the non-identity element acts by . 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 and be smooth manifolds with or without boundary is called a smooth covering if it is a smooth map and the sheets are mapped diffeomorphically onto the corresponding open subset of . (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

be a covering. A deck transformation, such that the diagram of continuous maps commutes. Together with the composition of maps, the set of deck transformation forms a group.

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 is considered as a discrete topological group.

is regular, with deck transformation group being isomorphic to the fundamental group

Examples

Properties

is bijective 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 is a group action.

Normal coverings

Definition

.

Properties

be a subgroup is a normal subgroup.

.

is the normaliser.

acts discontinuously.

, then the quotient map is a normal covering. is the quotient space is the orbit of the group action.

Examples

  • .

  • Every simply connected covering is a normal covering.

Calculation

be the normal covering.

  • .

    .

Examples

Galois correspondence

be a connected and locally simply connected space, then for every subgroup.

are conjugate to each other.

be a connected and locally simply connected space, then, up to equivalence between coverings, there is a bijection:

with index.

Classification

Definitions

Category of coverings

be a topological space. The objects of the category and the morphisms, such that the diagram commutes.

G-Set

be a topological group. The category is the category of sets which are G-sets. The morphisms are G-maps.

Equivalence

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

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.

See also

Further reading

References