Covering Spaces and Discrete Groups

We have already met covering spaces in Chapter 2, where our discussion was geometric and basically limited to definitions and examples. In this chapter, we return to this topic from a more algebraic point of view, which will allow us to produce numerous examples coming from group actions and to classify all covering spaces with given base (provided the latter is “nice” enough). The main tools will be groups: • discrete groups acting on manifolds, as the source of numerous examples of covering spaces; • the fundamental groups of the spaces involved (and the homomorphisms induced by their maps), which will play the key role in the classification theorems of covering spaces.