Covering map |
In onto U by p .
A covering map is also simply called a cover; the domain (mathematics) C of a covering map on X is called a covering space of X , and C is said to cover X . For each x in X , the inverse image of x under a given covering map is called the fiber over x. The mutually disjoint components of the inverse image of an open 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 .
A special case, called an open cover (or just cover (topology)) is when C is the disjoint union of a collection of open sets X i , with union X . A cover of any set S is the special case of this idea, when S carries the discrete topology (so that any subset is open).
= Examples =
Consider the unit circle S 1 in R2. Then the map p : R → S 1 with
: p ( t ) = (cos( t ),sin( t ))
is a cover where each point of S 1 is covered infinitely often.
Consider the complex plane with the origin removed, denoted by C×, and pick a non-zero integer n . Then p : C× → C× given by
: p ( z ) = z n
is a cover. Here every fiber has n elements.
If G is group (mathematics) (considered as a discrete space topological group), then every principal bundle is a covering map. Here every fiber can be identified with G .
= Elementary properties =
Common local properties: Every cover p : C → X is a local homeomorphism (i.e. to every cin C 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, then this holds globally as well, and the covering p is a homeomorphism.
Cardinality: For every xin X, the fiber over x is a discrete space subset of C . On every connected space 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 between the fiber over x and the fiber over y via the lifting property.
Equivalance: Let p_1:C_1 ightarrow X and p_2:C_2 ightarrow X be two coverings. One then says that the two coverings (p_1,C_1) and (p_2,C_2) are equivalent if there exists a homeomorphism p_{21}:C_2 ightarrow C_1 and p_2 = p_1 circ p_{21}. Equivalence classes of coverings correspond to conjugacy classes, as discussed below. If p_{21} is a covering rather than a homeomorphism, then one says that (p_2,C_2) dominates (p_1,C_1) (given that p_2 = p_1 circ p_{21}).
= Universal covers =
A cover q : D → X is a universal cover iff D is simply connected. The name comes from the following important universal property: if p : C → X is any cover of X with C connected, then there exists a covering map f : D → C such that p o f = q . This can be phrased as The universal cover of X covers all connected covers of X .
The map f is unique in the following sense: if we fix x ∈ X and d ∈ D with q ( d ) = x and c ∈ C with p ( c ) = x , then there exists a unique covering map f : D → C such that p o f = q and f ( d ) = c .
If X has a universal cover, then that universal cover is essentially unique: if q 1 : D 1 → X and q 2 : D 2 → X are two universal covers of X , then there exists a homeomorphism f : D 1 → D 2 such that q 2 o f = q 1.
The space X has a universal cover if and only if it is connectedness, connectedness and semi-locally simply connected. The universal cover of X can be constructed as a certain space of paths in X .
The example R → S 1 given above is a universal cover. The map S 3 → SO(3) from quaternion to rotations of 3D space described in quaternions and spatial rotation is also a universal cover.
If the space X carries some additional structure, then its universal cover normally inherits that structure:
The universal cover first arose in the theory of analytic functions as the natural domain of an analytic continuation.
= Deck transformation group, regular covers =
A deck transformation or automorphism of a cover p : C → X is a .
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 group action. If this action is group action on some fiber, then it is transitive on all fibers, and we call the cover regular. Every such regular cover is a principal 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 Dual_(category_theory) of the fundamental group π( X ).
The example p : C× → C× with p ( z ) = z n from above is a regular cover. The deck transformations are multiplications with n -th root of unity and the deck transformation group is therefore isomorphic to the cyclic group C n .
= Monodromy action =
Again suppose p : C → X is a covering map and C (and therefore also X ) is connected and locally path connected. If x ∈ 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 Homotopy lifting property 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 π( X , x ), and in this fashion we obtain a right group action of π( X , x ) 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 π( X , x ) acts on the right. These two actions are compatible in the following sense: : f .( c .γ) = ( f . c ).γ for all f ∈Aut( p ), c ∈ p -1( x ) and γ∈π( X , x ).
If p is a universal cover, then the monodromy action is regular; if we identify Aut( p ) with the Dual_(category_theory) group of π( X , x ), then the monodromy action coincides with the action of Aut( p ) on the fiber over x .
=Group structure redux=
The deck transformation group and the monodromy action can be understood to relate the normal subgroups of the fundamental group pi_1(X) of X and the fundamental group pi_1(C) of the cover. Furthermore, these equate the conjugacy classes of subgroups of pi_1(X) 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.
Let γ be a curve in X . Denote by gamma_C the Homotopy lifting property of γ to C . Consider the set
:Gamma_p(c) = { gamma : gamma_C mbox{ is a closed curve in } C mbox { passing through } cin C }
Note that Gamma_p(c) is a group (mathematics), and that is is a subgroup of pi_1(X,p(c)). Note also that it depends on c , and that different values of c in the same fiber yield different subgroups. Each such subgroups is conjugacy class to another by the monodromy action. To see this, pick two points c_1, c_2 in the same fiber: p(c_1)=p(c_2)=x and let g be a curve in C connecting c_1 to c_2. Then p(g) is a closed curve in X . If gamma_C is a closed curve in C passing through c_1, then ggamma_C g^{-1} is a closed curve in C passing through c_2. Thus, we have shown
:Gamma_p(c_2) = g Gamma_p(c_1) g^{-1}
and so we have the result that Gamma_p(c_1) and Gamma_p(c_2) are conjugate subgroups of pi_1(X,x). 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 pi_1(X,x); there is a bijective correspondence between equivalence classes of coverings and conjugacy classes of subgroups of pi_1 (X).
Note that this implies that the fundamental group pi_1(C) is isomorphic to Gamma_p. Let N(Gamma_p) be the normalizer of Gamma_p in pi_1(X). The deck transformation group Aut( p ) is isomorphic to N(Gamma_p)/Gamma_p. If p is a universal covering, then Gamma_p is the trivial group, and Aut( p ) is isomorphic to pi_1(X).
As a corollary, let us reverse this argument. Let Γ be a normal subgroup of pi_1(X,x). By the above arguments, this defines a (regular) covering p:X ightarrow C. Let c_1 in C be in the fiber of x . Then for every other c_2 in the fiber of x , there is precisely one deck transformation that takes c_1 to c_2. This deck transformation corresponds to a curve g in C connecting c_1 to c_2.
Note that Aut( p ) operates free regular set 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.
=References=
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