Covering map

to every x∈X there exists an open neighborhood U such that p ^-1(U) is a union of 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 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 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.

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.

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, 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 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.

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 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

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.