Covering space

Overlays are studied in the mathematical branch of topology. An overlay of a topological space consists of another topological space, the overlay space, and a continuous map that maps from the superposition of space in the main room and has certain properties.

Clearly, one can imagine a Überlagung so that you rolls the output space on the overlay space or the output space wrapping with the overlay space.

Definition

Be a topological space. An overlay of a topological space, together with a continuous image surjective

So that, for every point in an environment for which there is the archetype of an association among pairs of disjoint open sets, which are represented respectively by p homeomorphic to.

Often the concept of superposition is used for both the overlay area and for the overlay image. With respect to a is, the fiber of. It consists of a finite or infinite number of discrete points. It is said that the elements of the fiber lying about. The open sets are called leaves.

Examples

Consider the unit circle in. The real line is then superimposed on the overlay image

The line is so infinitely often wrapped around the circle. The leaves on an interval of the circle are intervals on the number line that repeats with period. Each fiber has infinitely many elements (). The isomorphism between the fundamental group and of the additive group over the integers can be proven very clearly with the help of this overlay.

The complex plane without the origin, is of itself overshadowed by the figure

Each fiber has elements here.

An example from quantum mechanics refers to the group SO (3) of rotations of three-dimensional real space. To it a part of a " double " overlay the SU (2), that is the group of the " complex twists " of the so-called Spinorgruppe. In contrast to the SO (3) is simply connected.

Properties

Each overlay is a local homeomorphism, ie the restriction of the overlay image on a small environment is a homeomorphism onto an open subset. Therefore, and have the same local properties:

• If a manifold is, as well as any superposition of
• If a Riemann surface, so this is also any superposition of and is then holomorphic.
• If a Lie group, as well as any superposition of, and is then a Lie group homomorphism.
• If a CW is complex, including superposition of each.

For each connected component of the number of elements of a fiber via a point (and therefore the number of blades on an environment ) is always the same. Does each fiber elements, it is called a multiple - overlay.

It is the high elevation property: If an overlay, one way in and one point above the starting point ( ie ), then there is a unique path in about (ie ) with initial point. Paths can be clearly raise after so when specifying a start point from the fiber.

Are and two points in which are connected by a path, the path mediated by the high elevation property is a bijective mapping between the fibers and over.

Universal covering

A superposition is called universal covering, if simply connected.

In general, there is a topological space many different overlays. Is, for example, superimposing and superimposing, it is also an overlay of. The name " universal cover " comes from the fact that it is also overlay each other coherent superposition of.

All of the above universal property follows that the universal covering is unique up to a homeomorphism ( two universal overlays are in fact due to this characteristic in each case the superposition of the other, it follows that they must be homeomorphic ).

If and only has a universal covering, if connected, locally path-connected and semi- locally simply connected. One can construct the universal covering by fixing a point and considered at any point in the set of homotopy classes of paths from to. The topology obtained locally because an environment has, the loops are globally contractible and which therefore said homotopy classes everywhere must be the same, so you can provide the cross product of the environment with the (discrete topologisierten ) Amount of homotopy classes with the product topology. Subject to these conditions, this construct is then a universal covering.

The universal cover of is usually referred to.

The above example is a universal overlay. Another example is the universal covering the projective space through the sphere

For n> 1

The group of deck transformations, regular overlays

A deck transformation of a superposition is a homeomorphism which is compatible with the projection, ie. The set of transformations of the overlay covering is a group of the combination of the sequential execution. The deck transformation group is designated.

Of the compatibility with the projection follows that each cover a transformation point reflects back to a point in the same fiber. Since the deck transformations beyond homeomorphisms, so are bijective, the elements of a fiber can be permuted. This defines a group operation of the deck transformation group on each fiber.

If an overlay image, and (and thus ) is continuous and local path connected, the operation of each fiber is free. If the operation is also transitive on a fiber, it is this on all fibers. In this case, the superposition is called normal, regular or Galois field.

For example, each universal covering is regular. As the example. Here there are the deck transformations of multiplications by roots of unity, the group is isomorphic to the cyclic group of order.

The group of deck transformations of the universal cover is isomorphic to the fundamental group of the base space; the universal covering of a principal bundle.

Classification

Possess a universal covering, and be a point of. The following two constructions provide an equivalence of categories between the category of superpositions of the category of sets and co- operation:

• An overlay is assigned to the fiber.
• One set is assigned the associated bundle; it is a bundle of fibers with discrete fiber, which is a superposition.

Coherent superpositions corresponding quantities with transitive operation, and up to isomorphism, these are classified by subgroups of. A coherent superposition corresponds to the subgroup.