In topology, a covering or covering projection is a surjective 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 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 by (see below). Under certain conditions, covering spaces also exhibit a Galois correspondance with the subgroups of the fundamental group.
Definition
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. For each the discrete set is called the fiber of . If is connected, it can be shown that 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.
Examples
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 .
Properties
Local homeomorphism
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. If is a connected Lie group, then there is a covering which is also a Lie group homomorphism and is a Lie group. If is a graph, then it follows for a covering that is also a graph. If is a connected manifold, then there is a covering , whereby is a connected and simply connected manifold. If is a connected Riemann surface, then there is a covering which is also a holomorphic map and is a connected and simply connected Riemann surface.
Factorisation
Let and be path-connected, locally path-connected spaces, and and be continuous maps, such that the diagram
commutes.
If and are coverings, so is . If and are coverings, so is .
Product of coverings
Let and be topological spaces and and be coverings, then with is a covering. However covering of are not all of this form in general.
Equivalence of coverings
Let 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 and isomorphic.