Covering map

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In mathematics, specifically topology, a covering map is a continuous surjective map p : CX, with C and X being topological spaces, which has the following property:

to every x in X there exists an open neighborhood U such that p -1(U) is a union of mutually disjoint open sets Si (where i ranges over some index set I) such that p restricted to Si yields a homeomorphism from Si onto U for every i in I.

A covering map is also simply called a cover; we say C is a covering space of X or C covers X. For each x in X, the set p -1(x) is called the fiber over x; the sets Si 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) is when C is the disjoint union of a collection of open sets Xi, 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).

Contents

Examples

Consider the unit circle S1 in R2. Then the map p : RS1 with

p(t) = (cos(t),sin(t))

is a cover.

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) = zn

is a cover. Here every fiber has n elements.

If G is group (considered as a discrete topological group), then every principal G-bundle is a covering map. Here every fiber can be identified with G.

Elementary properties

Common local properties: Every cover p : CX is a local homeomorphism (i.e. to every <math>c\in C<math> 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 <math>x\in X<math>, 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 : CX is a cover and γ is a path in X (i.e. a continuous map from the unit interval [0,1] into X) and <math>c\in C<math> 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.

Equivalance: Let <math>p_1:C_1\rightarrow X<math> and <math>p_2:C_2\rightarrow X<math> be two coverings. One then says that the two coverings <math>(p_1,C_1)<math> and <math>(p_2,C_2)<math> are equivalent if there exists a homeomorphism <math>p_{21}:C_2\rightarrow C_1<math> and <math>p_2 = p_1 \circ p_{21}<math>. Equivalence classes of coverings correspond to conjugacy classes, as discussed below. If <math>p_{21}<math> is a covering rather than a homeomorphism, then one says that <math>(p_2,C_2)<math> dominates <math>(p_1,C_1)<math> (given that <math>p_2 = p_1 \circ p_{21}<math>).

Universal covers

A cover q : DX is a universal cover iff D is simply connected. The name comes from the following important property: if p : CX is any cover of X with C connected, then there exists a covering map f : DC 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 xX and dD with q(d) = x and cC with p(c) = x, then there exists a unique covering map f : DC such that p o f = q and f(d) = c.

If X has a universal cover, then that universal cover is essentially unique: if q1 : D1X and q2 : D2X are two universal covers of X, then there exists a homeomorphism f : D1D2 such that q2 o f = q1.

The space X has a universal cover if and only if it is path-connected, locally path-connected and semi-locally simply connected. The universal cover of X can be constructed as a certain space of paths in X.

The example RS1 given above is a universal cover. The map S3 → SO(3) from unit quaternions 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:

  • if X is a manifold, then so is its universal cover C
  • if X is a Riemann surface, then so is its universal cover C, and p is a holomorphic map
  • if X is a Lie group (as in the two examples above), then so is its universal cover C, and p is a homomorphism of Lie groups.

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 : CX is a homeomorphism f : CC 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). Every deck transformation permutes the elements of each fiber. This defines a group action of the deck transformation group on each fiber. Note that if f is not the identity, then f has no fixed points.

Now suppose p : CX 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 : DX is regular, with deck transformation group being isomorphic to the opposite of the fundamental group π(X).

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.

Monodromy action

Again suppose p : CX is a covering map and C (and therefore also X) is connected and locally path connected. If xX 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 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), cp -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 opposite 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 <math>\pi_1(X)<math> of X and the fundamental group <math>\pi_1(C)<math> of the cover. Furthermore, these equate the conjugacy classes of subgroups of <math>\pi_1(X)<math> 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 <math>\gamma_C<math> the lift of γ to C. Consider the set

<math>\Gamma_p(c) = \{ \gamma : \gamma_C \mbox{ is a closed curve in } C

\mbox { passing through } c\in C \}<math>

Note that <math>\Gamma_p(c)<math> is a group, and that is is a subgroup of <math>\pi_1(X,p(c))<math>. 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 <math>c_1, c_2<math> in the same fiber: <math>p(c_1)=p(c_2)=x<math> and let g be a curve in C connecting <math>c_1<math> to <math>c_2<math>. Then p(g) is a closed curve in X. If <math>\gamma_C<math> is a closed curve in C passing through <math>c_1<math>, then <math>g\gamma_C g^{-1}<math> is a closed curve in C passing through <math>c_2<math>. Thus, we have shown

<math>\Gamma_p(c_2) = g \Gamma_p(c_1) g^{-1}<math>

and so we have the result that <math>\Gamma_p(c_1)<math> and <math>\Gamma_p(c_2)<math> are conjugate subgroups of <math>\pi_1(X,x)<math>. 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 <math>\pi_1(X,x)<math>; there is a bijective correspondence between equivalence classes of coverings and conjugacy classes of subgroups of <math>\pi_1 (X)<math>.

Note that this implies that the fundamental group <math>\pi_1(C)<math> is isomorphic to <math>\Gamma_p<math>. Let <math>N(\Gamma_p)<math> be the normalizer of <math>\Gamma_p<math> in <math>\pi_1(X)<math>. The deck transformation group Aut(p) is isomorphic to <math>N(\Gamma_p)/\Gamma_p<math>. If p is a universal covering, then <math>\Gamma_p<math> is the trivial group, and Aut(p) is isomorphic to <math>\pi_1(X)<math>.

As a corollary, let us reverse this argument. Let Γ be a normal subgroup of <math>\pi_1(X,x)<math>. By the above arguments, this defines a (regular) covering <math>p:X\rightarrow C<math>. Let <math>c_1<math> in C be in the fiber of x. Then for every other <math>c_2<math> in the fiber of x, there is precisely one deck transformation that takes <math>c_1<math> to <math>c_2<math>. This deck transformation corresponds to a curve g in C connecting <math>c_1<math> to <math>c_2<math>.

Note that Aut(p) operates properly discontinously 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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