Algebraic Topology - Computer Graphics and Geometric Modeling - page 430

Definition. A topological space X is said to be semi-locally simply connected if every
point x in X has a neighborhood U so that every closed curve in U that starts at x is
homotopic to a constant map in X .

Manifolds and CW complexes are semi-locally simply connected (use induction
on the number of cells for CW complexes).

7.4.2.13. Theorem. Let X be a path-connected, locally path-connected, and semi-
locally simply connected space. Let x 0 ΠX . If G is an arbitrary subgroup of p 1 ( X , x 0 ),
then there is a path-connected and locally path-connected space Y and covering space
( Y ,p) of X , so that for some point y 0 Πp -1 ( x 0 ), p * (p 1 ( Y , y 0 )) = G.

Proof. See [Mass67] or [Jäni84].

Definition. A universal cover or universal covering space for a space X is a covering
space ( Y ,p) for X , with the property that Y is path-connected, locally path-connected,
and simply connected.

By Theorem 7.4.2.12, the universal covering space of a space (if it exists) is unique
up to isomorphism. Therefore, if the projection p is obvious from the context, then
the common expression “the universal cover Y of X ” refers to the universal covering
space ( Y ,p).

The space R is the universal cover of the circle S 1 (see Example

7.4.2.14. Example.

7.4.2.2).

The sphere S n is the universal cover of projective space P n (see

7.4.2.15. Example.

Example 7.4.2.1).

7.4.2.16. Theorem. Let X be a path-connected, locally path-connected and semi-
locally simply connected space. Then X has a universal covering space and any two
are isomorphic.

Proof. Only the existence part of this theorem needs proving. See [Mass67] or

[Jäni84].

The reason that a universal covering space ( Y ,p) for a space X has the name it
has is that if ( Y ¢,p¢) is any other covering space for X , then there a unique (up to iso-
morphism) map ˜ : Y Æ Y ¢ making the following diagram commutative

ææ¢
¢

˜

Y Y

p p .

X

In fact, ( Y , ˜) will be a covering space for Y ¢. In other words, the universal covering
space of a space “covers” every other covering space of the space.
The covering transformations of a covering space are interesting. They obviously
form a group.