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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∈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 to U for every i∈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∈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).

Table of contents
1 Examples
2 Elementary properties
3 Universal covers
4 Deck transformation group, regular covers
5 Monodromy action

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

Every cover p : C → X is a local homeomorphism (i.e. to every c∈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.

For every x∈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 c∈C 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.

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.

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:

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.

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 Cn.

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.