Covering map

In mathematics, specifically topology, a covering map is a continuous surjective map p : C → X, 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 S_i (where i ranges over some index set I) such that p restricted to S_i yields a homeomorphism from S_i 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 S_i 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 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 S¹ in R². Then the map p : R → S¹ 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 : 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, then this holds globally as well, and the covering p is a homeomorphism.

Cardinality: For every , 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.

Equivalance: Let and be two coverings. One then says that the two coverings (p₁,C₁) and (p₂,C₂) are equivalent if there exists a homeomorphism and . Equivalence classes of coverings correspond to conjugacy classes, as discussed below. If p₂₁ is a covering rather than a homeomorphism, then one says that (p₂,C₂) dominates (p₁,C₁) (given that ).

Universal covers

A cover q : D → X is a universal cover iff D is simply connected. The name comes from the following important 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 q₁ : D₁ → X and q₂ : D₂ → X are two universal covers of X, then there exists a homeomorphism f : D₁ → D₂ 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 R → S¹ 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 : 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). 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 : 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 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 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 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), 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 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 π₁(X) of X and the fundamental group π₁(C) of the cover. Furthermore, these equate the conjugacy classes of subgroups of π₁(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 γ_C the lift of γ to C. Consider the set

Note that Γ_p(c) is a group, and that is is a subgroup of π₁(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 conjugate to another by the monodromy action. To see this, pick two points c₁,c₂ in the same fiber: p(c₁) = p(c₂) = x and let g be a curve in C connecting c₁ to c₂. Then p(g) is a closed curve in X. If γ_C is a closed curve in C passing through c₁, then gγ_Cg ^{- 1} is a closed curve in C passing through c₂. Thus, we have shown

Γ_p(c₂) = gΓ_p(c₁)g ^{- 1}

and so we have the result that Γ_p(c₁) and Γ_p(c₂) are conjugate subgroups of π₁(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 π₁(X,x); there is a bijective correspondance between equivalence classes of coverings and conjugacy classes of subgroups of π₁(X).

Note that this implies that the fundamental group π₁(C) is isomorphic to Γ_p. Let N(Γ_p) be the normalizer of Γ_p in π₁(X). The deck transformation group Aut(p) is isomorphic to N(Γ_p) / Γ_p. If p is a universal covering, then Γ_p is the trivial group, and Aut(p) is isomorphic to π₁(X).

As a corollary, let us reverse this argument. Let Γ be a normal subgroup of π₁(X,x). By the above arguments, this defines a (regular) covering . Let c₁ in C be in the fiber of x. Then for every other c₂ in the fiber of x, there is precisely one deck transformation that takes c₁ to c₂. This deck transformation corresponds to a curve g in C connecting c₁ to c₂.

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

• Hershel M. Farkas and Irwin Kra, Riemann Surfaces (1980), Springer-Verlag, New York. ISBN 0-387-90465-4 (See Chapter 1 for a simple review)
• Jurgen Jost, Compact Riemann Surfaces (2002), Springer-Verlag, New York. ISBN 3-540-43299-X (See Section 1.3)