Fundamental groups in topology

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In the last section we saw that when studying extensions of some field it is plausible to conceive the base field as a point and a finite separable extension (or, more generally, a finite etale algebra) as a finite discrete set of points mapping to this base point. Galois theory then equips the situation with a continuous action of the absolute Galois group which leaves the base point fixed. It is natural to try to extend this situation by taking as a base not just a point but a more general topological space. The role of field extensions would then be played by certain continuous surjections, called covers, whose fibres are finite (or, even more generally, arbitrary discrete) spaces. We shall see in this chapter that under some restrictions on the base space one can develop a topological analogue of the Galois theory of fields, the part of the absolute Galois group being taken by the fundamental group of the base space. In the second half of the chapter we give a reinterpretation of the main theorem of Galois theory for covers in terms of locally constant sheaves. Esoteric as these objects may seem to the novice, they stem from reformulating in a modern language very classical considerations from analysis, such as the study of local solutions of holomorphic differential equations. In fact, the whole concept of the fundamental group arose from Riemann's study of the monodromy representation for hypergeometric differential equations, a topic we shall briefly discuss at the end of the chapter. Our exposition therefore traces history backwards, but hopefully reflects the intimate connection between differential equations and the fundamental group.