The theory of covering spaces is deeply intertwined with the study of fundamental groups. It is nearly impossible to take up a study of one without encountering the other.
The special open neighborhoods U of x given in the definition are called evenly-covered neighborhoods. The evenly-covered neighborhoods form an open cover of the space X. The homeomorphic copies in C of an evenly-covered neighborhood U are called the sheets over U. One generally pictures C as "hovering above" X, with p mapping "downwards", the sheets over U being horizontally stacked above each other and above U, and the fiber over x consisting of those points of C that lie "vertically above" x.
Warning: Many authors impose some connectivity conditions on the spaces X and C in the definition of a covering map. In particular, many authors require both spaces to be path-connected and locally path-connected. When studying results about covering spaces, one should always be careful to check the connectivity assumptions the author is making. Also some authors do not require covering maps to be surjective. However, if X is connected and C is nonempty then surjectivity of the covering map actually follows from the other axioms.
Consider the unit circle S1 in R2. Then the map p : R → S1 with
is a cover where each point of S1 is covered infinitely often.
is a cover. Here every fiber has n elements.
A further example, originating from physics (see quantum mechanics), is the special orthogonal group SO(3) of rotations of , which has the "double" covering group SU(2) of unitary rotations of (in quantum mechanics acting as the group of spinor rotations). Both groups have identical Lie algebras, but only SU(2) is simply connected.
Common local properties: Every cover p : C → X is a local homeomorphism (i.e. to every there exists an open set A in C containing c and an open set B in X such that the restriction of p to A yields a homeomorphism between A and B). This implies that C and X share all local properties. If X is simply connected and C is connected, then this holds globally as well, and the covering p is a homeomorphism.
Cardinality: For every x in X, the fiber over x is a discrete subset of C. On every connected component of X, the cardinality of the fibers is the same (possibly infinite). If every fiber has 2 elements, we speak of a double cover.
The lifting property: If p : C → X is a cover and γ is a path in X (i.e. a continuous map from the unit interval [0,1] into X) and is a point "lying over" γ(0) (i.e. p(c) = γ(0)), then there exists a unique path ρ in C lying over γ (i.e. p o ρ = γ) and with ρ(0) = c. The curve ρ is called the lift of γ. If x and y are two points in X connected by a path, then that path furnishes a bijection between the fiber over x and the fiber over y via the lifting property.
Equivalence: Let and be two coverings. One then says that the two coverings and are equivalent if there exists a homeomorphism and . Equivalence classes of coverings correspond to conjugacy classes, as discussed below. If is a covering rather than a homeomorphism, then one says that dominates (given that ).
A connected covering space is a universal cover if it is simply connected. The name universal cover comes from the following important property: if the mapping is a universal cover of the space and the mapping is any cover of the space where the covering space is connected, then there exists a covering map such that . This can be phrased as
The universal cover of the space covers all connected covers of the space .
The map is unique in the following sense: if we fix a point in the space and a point in the space with and a point in the space with , then there exists a unique covering map such that and .
If the space has a universal cover then that universal cover is essentially unique: if the mappings and are two universal covers of the space then there exists a homeomorphism such that .
The space has a universal cover if it is connected, locally path-connected and semi-locally simply connected. The universal cover of the space can be constructed as a certain space of paths in the space .
If the space carries some additional structure, then its universal cover normally inherits that structure:
A deck transformation or automorphism of a cover p : C → X is a homeomorphism f : C → C such that p o f = p. The set of all deck transformations of p forms a group under composition, the deck transformation group Aut(p). Deck transformations are also called covering transformations. Every deck transformation permutes the elements of each fiber. This defines a group action of the deck transformation group on each fiber. Note that by the unique lifting property, if f is not the identity and C is path connected, then f has no fixed points.
Now suppose p : C → X is a covering map and C (and therefore also X) is connected and locally path connected. The action of Aut(p) 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. Every such regular cover is a principal G-bundle, where G = Aut(p) is considered as a discrete topological group.
Every universal cover p : D → X is regular, with deck transformation group being isomorphic to the fundamental group .
The example p : C× → C× with p(z) = zn from above is a regular cover. The deck transformations are multiplications with n-th roots of unity and the deck transformation group is therefore isomorphic to the cyclic group Cn.
Another example: with from above is regular. Here one has a hierarchy of deck transformation groups. In fact Cx! is a subgroup of Cy!, for .
Again suppose p : C → X is a covering map and C (and therefore also X) is connected and locally path connected. If x in X and c belongs to the fiber over x (i.e. p(c) = x), and γ:[0,1]→X is a path with γ(0)=γ(1)=x, then this path lifts to a unique path in C with starting point c. The end point of this lifted path need not be c, but it must lie in the fiber over x. It turns out that this end point only depends on the class of γ in the fundamental group , and in this fashion we obtain a right group action of on the fiber over x. This is known as the monodromy action.
So there are two actions on the fiber over x: Aut(p) acts on the left and acts on the right. These two actions are compatible in the following sense:
If p is a universal cover, then the monodromy action is regular; if we identify Aut(p) with the opposite group of π(X,x), then the monodromy action coincides with the action of Aut(p) on the fiber over x.
The deck transformation group and the monodromy action can be understood to relate the normal subgroups of the fundamental group of X and the fundamental group of the cover. Furthermore, these equate the conjugacy classes of subgroups of and equivalence classes of coverings. As a result, one can conclude that X=C/Aut(p), that is, the manifold X is given as the quotient of the covering manifold under the action of the deck transformation group. These inter-relationships are explored below.
Note that is a group, and that it is a subgroup of . Note also that it depends on c, and that different values of c in the same fiber yield different subgroups. Each such subgroups is conjugate to another by the monodromy action. To see this, pick two points in the same fiber: and let g be a curve in C connecting to . Then p(g) is a closed curve in X. If is a closed curve in C passing through , then is a closed curve in C passing through . Thus, we have shown
and so we have the result that and are conjugate subgroups of . All of the conjugate subgroups may be obtained in this way.
It should be clear that two equivalent coverings lead to the same conjugacy class of subgroups of ; there is a bijective correspondence between equivalence classes of coverings and conjugacy classes of subgroups of .
Note that this implies that the fundamental group is isomorphic to . Let be the normalizer of in . The deck transformation group Aut(p) is isomorphic to . If p is a universal covering, then is the trivial group, and Aut(p) is isomorphic to .
As a corollary, let us reverse this argument. Let Γ be a normal subgroup of . By the above arguments, this defines a (regular) covering . Let in C be in the fiber of x. Then for every other in the fiber of x, there is precisely one deck transformation that takes to . This deck transformation corresponds to a curve g in C connecting to .
Note that Aut(p) operates properly discontinuously on C, and so we have that X=C/Aut(p), that is, X is the manifold given by the quotient of the covering manifold by the deck transformation group.
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