4 - Covering Spaces and the Fundamental Group

Summary

The material in the first section is standard, and can be found in most introductory books on algebraic topology, for example, [Mas]. The goal is to provide a shortcut to the results needed here. We supply full proofs, except for the very first facts on homotopy.

The material in the second section is less standard, but certainly elementary. The first subsection (about coverings of a disc minus center) is preparatory. Then we turn to our main topic, the coverings of the punctured sphere. We study the behavior near a ramified point, in particular, we introduce the class of distinguished inertia group generators. This allows us to define the ramification type, in analogy to the definition in Chapter 2 for fields. The existence of coverings of prescribed ramification type (a topological version of Riemann's existence theorem) is the main result of this chapter. The proof is in the spirit of classical Riemann surface theory: It consists of a glueing process that uses the Galois group to index the sheets of the covering. As a by-product, this allows us to determine the fundamental group of a punctured sphere.

In this chapter, R and S denote topological spaces. From Section 4.1.2 on, S is a topological manifold.

The General Theory

Homotopy

For real t, s we let [t, s], [t, s[etc. denote the closed interval{t′: t ≤ t′ ≤ s}, the half-open interval {t′ : t ≤ t′ < s}, etc. The unit interval [0, 1] is also denoted by I.