Covering Spaces

The study of the fundamental group is intimately related to the concept of “covering space”. It provides a useful tool for the computation of the fundamental group of various spaces and plays an important role in the study of Riemann surfaces. Moreover, an extremely fruitful notion of “fiber bundle” can be viewed as a generalization of covering spaces. A covering space of a space X is defined by a map over X, called a “covering map”. We introduce this concept and discuss its elementary properties in Sect. 15.1. Section 15.2 is devoted to straightforward generalizations of the lifting properties of the exponential map $$p\!:{\mathbb {R}}^1\rightarrow {\mathbb {S}}^1$$ , $$p(t)=e^{2\pi \imath t}$$ , to covering maps. As a consequence of these generalizations, we obtain a general solution to the lifting problem for these maps in terms of fundamental groups (Theorem 15.2.6). In Sect. 15.3, an action of the fundamental group of the base space of a covering map on the “fiber” over the base point is defined and it is applied to prove the Borsuk–Ulam theorem for $${\mathbb {S}}^2$$ . The next three sections mainly concern with the problem of classification of covering spaces of a locally path-connected space. In Sect. 15.4, we define the notion of equivalence for covering spaces of a given space and study the group of covering transformations (self-equivalences) of a covering space. In Sect. 15.5, we shall treat “regular covering maps”. It will be seen that such coverings arise as the orbit map of “proper and free” actions of discrete groups. The last section deals with the existence of covering spaces of certain spaces. We first show that every space X satisfying a mild condition has a simply connected covering space, and then the results of the previous two sections are applied to show the existence of a covering space of X with the fundamental group isomorphic to a given subgroup of $$\pi (X)$$ .