- to every
*x*∈*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}to*U*for every*i*∈*I*.

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).

Table of contents |

2 Elementary properties 3 Universal covers 4 Deck transformation group, regular covers 5 Monodromy action |

Consider the unit circle *S*^{1} in **R**^{2}. Then the map *p* : **R** → *S*^{1} 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*) = *z*^{n} 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*.

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.

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 *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 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** → *S*^{1} given above is a universal cover. The map *S*^{3} → 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.

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*) = *z*^{n} 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}.

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*.