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 , 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.^{[1]}^{: 10 }
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).^{[2]}^{: 29 } Under certain conditions, covering spaces also exhibit a Galois correspondence with the subgroups of the fundamental group.
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.^{[2]}^{: 56 } For each the discrete set is called the fiber of . If is connected, it can be shown that is surjective, and 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.
Some authors also require that be surjective in the case that is not connected.^{[3]}
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.
Let and be path-connected, locally path-connected spaces, and and be continuous maps, such that the diagram
commutes.
Let and be topological spaces and and be coverings, then with is a covering.^{[6]}^{: 339 } However covering of are not all of this form in general.
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.
All coverings satisfy the lifting property, i.e.:
Let 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 , i.e. .^{[2]}^{: 60 }
If is a path-connected space, then for it follows that the map is a lift of a path in and for it is a lift of a homotopy of paths in .
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 .^{[2]}^{: 29 }
Let be a path-connected space and be a connected covering. Let be any two points, which are connected by a path , i.e. and . Let be the unique lift of , then the map
is bijective.^{[2]}^{: 69 }
If is a path-connected space and a connected covering, then the induced group homomorphism
is injective and the subgroup of consists of the homotopy classes of loops in , whose lifts are loops in .^{[2]}^{: 61 }
Let and be Riemann surfaces, i.e. one dimensional complex manifolds, and let be a continuous map. is holomorphic in a point , if for any charts of and of , with , the map is holomorphic.
If is holomorphic at all , we say is holomorphic.
The map is called the local expression of in .
If is a non-constant, holomorphic map between compact Riemann surfaces, then is surjective and an open map,^{[5]}^{: 11 } i.e. for every open set the image is also open.
Let be a non-constant, holomorphic map between compact Riemann surfaces. For every there exist charts for and and there exists a uniquely determined , such that the local expression of in is of the form .^{[5]}^{: 10 } The number is called the ramification index of in and the point is called a ramification point if . If for an , then is unramified. The image point of a ramification point is called a branch point.
Let be a non-constant, holomorphic map between compact Riemann surfaces. The degree of is the cardinality of the fiber of an unramified point , i.e. .
This number is well-defined, since for every the fiber is discrete^{[5]}^{: 20 } and for any two unramified points , it is:
It can be calculated by:
A continuous map is called a branched covering, if there exists a closed set with dense complement , such that is a covering.
Let be a simply connected covering. If is another simply connected covering, then there exists a uniquely determined homeomorphism , such that the diagram
commutes.^{[6]}^{: 482 }
This means that is, up to equivalence, uniquely determined and because of that universal property denoted as the universal covering of the space .
A universal covering does not always exist, but the following properties guarantee its existence:
Let be a connected, locally simply connected topological space; then, there exists a universal covering .
is defined as and by .^{[2]}^{: 64 }
The topology on is constructed as follows: Let be a path with . Let be a simply connected neighborhood of the endpoint , then for every the paths inside from to are uniquely determined up to homotopy. Now consider , then with is a bijection and can be equipped with the final topology of .
The fundamental group acts freely through on and with is a homeomorphism, i.e. .
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 H_{g} of X onto itself, in such a way that H_{g h} is always equal to H_{g} ∘ H_{h} 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.
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 is a covering map and (and therefore also ) is connected and locally path connected. The action of on each fiber is transitive. If this action is free on some fiber, then it is free on all fibers, and we call the cover regular (or normal or Galois). Every such regular cover is a principal -bundle, where is considered as a discrete topological group.
Every universal cover is regular, with deck transformation group being isomorphic to the fundamental group .
Let be a path-connected space and be a connected covering. Since a deck transformation is bijective, it permutes the elements of a fiber with and is uniquely determined by where it sends a single point. In particular, only the identity map fixes a point in the fiber.^{[2]}^{: 70 } Because of this property every deck transformation defines a group action on , i.e. let be an open neighborhood of a and an open neighborhood of an , then is a group action.
A covering is called normal, if . This means, that for every and any two there exists a deck transformation , such that .
Let be a path-connected space and be a connected covering. Let be a subgroup of , then is a normal covering iff is a normal subgroup of .
If is a normal covering and , then .
If is a path-connected covering and , then , whereby is the normaliser of .^{[2]}^{: 71 }
Let be a topological space. A group acts discontinuously on , if every has an open neighborhood with , such that for every with one has .
If a group acts discontinuously on a topological space , then the quotient map with is a normal covering.^{[2]}^{: 72 } Hereby is the quotient space and is the orbit of the group action.
Let be a group, which acts discontinuously on a topological space and let be the normal covering.
Let be a connected and locally simply connected space, then for every subgroup there exists a path-connected covering with .^{[2]}^{: 66 }
Let and be two path-connected coverings, then they are equivalent iff the subgroups and are conjugate to each other.^{[6]}^{: 482 }
Let 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 .
Let be a topological space. The objects of the category are the coverings of and the morphisms between two coverings and are continuous maps , such that the diagram
commutes.
Let 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 .
Let be a connected and locally simply connected space, and be the fundamental group of . Since 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.^{[2]}^{: 68–70 }
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 RP^{3}, with fundamental group Z/2, and only (non-trivial) covering space the hypersphere S^{3}, 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 T^{3} of three angles to the real projective space RP^{3} 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.
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