One obtains more information on covers of a topological space when it carries additional structure, for instance when it is a complex manifold. The complex manifolds of dimension 1 are called Riemann surfaces, and they already have a rich theory. The study of their covers creates a link between the Galois theory of fields and that of covers: finite étale algebras over the field of meromorphic functions on a connected compact Riemann surface correspond up to isomorphism to branched covers of the Riemann surface; by definition, the latter are topological covers outside a discrete exceptional set. As we shall see, all proper holomorphic surjections of Riemann surfaces define branched covers. The dictionary between branched covers and étale algebras over the function field has purely algebraic consequences: as an application, we shall prove that every finite group occurs as the Galois group of a finite Galois extension of the rational function field C(t).

Parts of this chapter were inspired by the expositions in [17] and [23].

Basic concepts

Let X be a Hausdorff topological space. A complex atlas on X is an open covering U = {Ui : i ∈ I} of X together with maps fi : Ui → C mapping Ui homeomorphically onto an open subset of C such that for each pair (i, j) ∈ I2 the map fj o f−1i : fi (Ui ∩ Uj) → C is holomorphic. The maps fi are called complex charts.

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 étale 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.

In the previous chapter the Riemann Existence Theorem created a link between the category of compact connected Riemann surfaces and that of finite extensions of C(t). This hints at a possibility of developing a theory of the fundamental group in a purely algebraic way. We shall now present such a theory for curves over an arbitrary perfect base field, using a modest amount of algebraic geometry. Over the complex numbers the results will be equivalent to those of the previous chapter, but a new and extremely important feature over an arbitrary base field k will be the existence of a canonical quotient of the algebraic fundamental group isomorphic to the absolute Galois group of k. In fact, over a subfield of C we shall obtain an extension of the absolute Galois group of the base field by the profinite completion of the topological fundamental group of the corresponding Riemann surface over C. This interplay between algebra and topology is a source for many powerful results in recent research. Among these we shall discuss applications to the inverse Galois problem, Belyi's theorem on covers of the projective line minus three points and some advanced results on ‘anabelian geometry’ of curves.

Reading this chapter requires no previous acquaintance with algebraic geometry. We shall, however, use some standard results from commutative algebra that we summarize in the first section. The next three sections contain foundational material, and the discussion of the fundamental group itself begins in Section 4.5.

The theory of the last chapter established an equivalence between the category of finite étale covers of a connected scheme and the category of finite continuous permutation representations of its algebraic fundamental group. We shall now study a linearization of this concept, also due to Grothendieck and developed in detail by Saavedra [81] and Deligne [14]. The origin is a classical theorem from the theory of topological groups due to Tannaka and Krein: they showed that one may recover a compact topological group from the category of its continuous unitary representations. In Grothendieck's algebraic context the group is a linear algebraic group, or more generally an affine group scheme, and one studies the category of its finite dimensional representations. The key features that enable one to reconstitute the group are the tensor structure on this category and the forgetful functor that sends a representation to its underlying vector space. Having abstracted the conditions imposed on the category of representations, one gets a theorem stating that a category with certain additional structure is equivalent to the category of finite dimensional representations of an affine group scheme. This can be applied in several interesting situations. We shall discuss in some detail the theory of differential Galois groups, and also Nori's fundamental group scheme that creates a link between the algebraic fundamental group and Tannakian theory.

We only treat so-called neutral Tannakian categories, but the reader familiar with Grothendieck's descent theory will have no particular difficulty afterwards in studying the general theory of [14]. Non-commutative generalizations have also been developed in connection with quantum groups; as samples of a vast literature we refer to the books of Chari–Pressley [9] and Majid [55].

Though the theory of the previous chapter is sufficient for many applications, a genuine understanding of the algebraic fundamental group only comes from Grothendieck's definition of the fundamental group for schemes. His theory encompasses the classification of finite covers of complex algebraic varieties of any dimension, Galois theory for extensions of arbitrary fields and even notions coming from arithmetic such as specialization modulo a prime. Moreover, it is completely parallel to the topological situation and clarifies the role of base points and universal covers – these have been somewhat swept under the carpet in the last chapter. In his original account in [29] Grothendieck adopted an axiomatic viewpoint and presented his constructions within the context of ‘Galois categories’. Here we choose a more direct approach, emphasizing the parallelism with topology. The background from algebraic geometry that is necessary for the basic constructions will be summarized in the first section. However, the proofs of some of the deeper results discussed towards the end of the chapter will require more refined techniques.

The vocabulary of schemes

In this section we collect the basic notions from the language of schemes that we shall need for the development of Grothendieck's theory of the fundamental group. Our intention is to summarize for the reader what will be needed; this concise overview can certainly not replace the study of standard references such as [34] or [64], let alone Grothendieck's magnum opus EGA.

Ever since the concepts of the Galois group and the fundamental group emerged in the course of the nineteenth century, mathematicians have been aware of the strong analogies between the two notions. In its early formulation Galois theory studied the effect of substitutions on roots of a polynomial equation; in the language of group theory this is a permutation action. On the other hand, the fundamental group made a first, if somewhat disguised, appearance in the study of solutions of differential equations in a complex domain. Given a local solution of the equation in the neighbourhood of a base point, one obtains another solution by analytic continuation along a closed loop: this is the monodromy action.

Leaving the naïve idea of substituting solutions, the next important observation is that the actions in question come from automorphisms of objects that do not depend on the equations any more but only on the base. In the context of Galois theory the automorphisms are those of a separable closure of the base field from which the coefficients of the equation are taken. For differential equations the analogous role is played by a universal cover of the base domain. The local solutions, which may be regarded as multi-valued functions in the neighbourhood of a base point, pull back to single-valued functions on the universal cover, and the monodromy action is the effect of composing with its topological automorphisms.