Intuitively, a covering locally projects a "stack of pancakes" above an open neighborhood onto
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 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 (and is non-empty), 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]
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 space is a covering space of . The disjoint open sets are mapped homeomorphically onto . The fiber of consists of the points .
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 .
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.[2]: 234
If is a graph, then it follows for a covering that is also a graph.[2]: 85
If is a connected manifold, then there is a covering , whereby is a connected and simply connected manifold.[5]: 32
If is a connected Riemann surface, then there is a covering which is also a holomorphic map[5]: 22 and is a connected and simply connected Riemann surface.[5]: 32
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
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 simply connected covering. If is another simply connected covering, then there exists a uniquely determined homeomorphism , such that the diagram
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. .
with is the universal covering of the unitary group.[7]: 5, Theorem 1
Since , it follows that the quotient map is the universal covering of the .
A topological space which has no universal covering is the Hawaiian earring: One can show that no neighborhood of the origin is simply connected.[6]: 487, Example 1
Let G be a discrete groupacting on the topological spaceX. This means that each element g of G is associated to a homeomorphism Hg of X onto itself, in such a way that Hgh 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 spaceX/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 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.)
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 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 -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 the covering for some , then the map for is a deck transformation and .
Let be the covering , then the map for is a deck transformation and .
As another important example, consider the complex plane and the complex plane minus the origin. Then the map with is a regular cover. The deck transformations are multiplications with -th roots of unity and the deck transformation group is therefore isomorphic to the cyclic group. Likewise, the map with is the universal cover.
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.
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 . The antipodal map with generates, together with the composition of maps, a group and induces a group action , which acts discontinuously on . Because of it follows, that the quotient map is a normal covering and for a universal covering, hence for .
Let be the special orthogonal group, then the map is a normal covering and because of , it is the universal covering, hence .
With the group action of on , whereby is the semidirect product, one gets the universal covering of the klein bottle, hence .
Let be the torus which is embedded in the . Then one gets a homeomorphism , which induces a discontinuous group action , whereby . It follows, that the map is a normal covering of the klein bottle, hence .
Let be embedded in the . Since the group action is discontinuously, whereby are coprime, the map is the universal covering of the lens space, hence .
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
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
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).
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.
^Forster, Otto (1981). "Chapter 1: Covering Spaces". Lectures on Riemann Surfaces. GTM. Translated by Bruce Gillian. New York: Springer. ISBN9781461259633.