• fundamental groups
• galois theory

we are going to present a new tool in the study of absolute Galois groups.

Given a field $ F $, it is of primary interest to classify its finite extensions. That is, given a separable closure $ \bar{F}/F $, to understand the structure of the absolute Galois group $ \Gal(\bar{F}/F) $. This is a very hard task. So much so that, at the present, a lot of problems remain open, even for the field $ F = \Q $ of rational numbers.

We present a construction that translates this algebraic problem into a geometrical setting. Namely, we assign to each field $ F $ an Hausdorff, compact and connected topological space $ X_F $, whose finite covering spaces are in one-to-one correspondence with finite field extensions of $ F $. This is possible under the assumption that $ F $ contains $ \Q(\mu_\infty) $: i.e. $ F $ is of characteristic zero and all roots of unity $ \mu_\infty $ already belong to this field. We are going to prove the following.

Theorem. Let $ F \supseteq \Q(\mu_\infty) $ be a field. There is a functor from the category of finite étale algebras over $ F $ to the category of finite covering spaces over $ X_F $
\[ \FEA /F \to \FC /X_F, \]
which is a category equivalence.

In particular, it follows that the étale fundamental group of $ X_F $ over a point $ \chi\in X_F $ is isomorphic to the absolute Galois group of $ F $: that is, we have an isomorphism $$ \pi^{\text{ét}}(X_F, \chi) \simeq \Gal(\bar{F}/F). $$

The construction is due to P. Scholze and R. Kucharczyk, who presented the idea in their recent paper "Topological realsations of absolute Galois groups". This work is essentially a restructuring of their work, together with a record of some attempts to extend the limits of this construction. This thesis is not going to present a self-contained theory, but rather the pedant dissection of a mathematical object. As such, the techniques used will vary largely in complexity during the exposition. We aim to present every step in the most accessible way possible, trying to relegate harder techniques into the latest sections. Despite our efforts, a sound familiarity with algebraic tools is required.

In the first chapter, the Galois theories of finite extensions and topological covering spaces are going to be briefly introduced. That settled, we are going to present the main construction. We build the topological space $ X_{\bar{F}} $ as the product of an infinite numbers of copies of the solenoid
\[ \Q^\vee = \Hom(\Q, \S^1), \]
hence we obtain a compact Hausdorff connected topological space. We then discuss a natural action of the absolute Galois group $ G $ on $ \XF $, to then define the desired space $ X_F $ as the quotient $ G \backslash \XF $. The action being proper, $ X_F $ retains all topological properties of $ \XF $. In the last section the main theorem is proven.

In the second chapter, the topological space $ X_F $ is further studied. Namely, we concentrate on the relation between the number of path components of $ X_F $ and the multiplicative structure of $ F $. We say that a field $ F $ is multiplicatively free if its multiplicative group $ F^\times $ is free.

Theorem. If $ F $ is countable and all its finite extensions are multiplicatively free,
then $ X_F $ is path connected.

A field satisfying the hypotheses of the above theorem is $ F = \Q(\mu_\infty). $ \\

We then compute sheaf cohomology groups of $ X_F $ with coefficients in a locally constant sheaf $ A $. In particular, when $ A $ is a torsion abelian group we find out that cohomology groups are related to the cohomology of the absolute Galois group $ G $; that is, there are isomorphisms
\[ \HH^p(X_F, A) \simeq \HH^p(G, A) \qquad\forall p > 0. \]
Cohomology groups with integral coefficients do not admit such an explicit description, nonetheless they fit in short exact sequences
\[ 0 \to \HH^p(X_F, \Z) \to \HH^p(X_F, \Q) \to \HH^p(X_F, \Q/\Z) \to 0 \qquad \forall p > 0.\]
In these sequences, we also know the central term, i.e. $ \HH^p(X_F, \Q) = \bigwedge^p \bar{F}^\times\otimes \Q. $\\

A discussion on the necessity of the hypothesis on roots of unity follows, in which we suggest a different construction which does not require said hypothesis: for every characteristic zero field $ F $ we build a topological space $ Z_F $, which is compact and Hausdorff but not connected. We then show that when $ Z_F $ is connected, it is homeomorphic to $ X_{F(\mu_\infty)} $, hence this new construction ultimately reduces to the initial setting. \\

In the last chapter we investigate an analogous construction in the realm of algebraic geometry. That is, we present a connected complex scheme $ \Xs_F $, whose finite étale covering maps are in correspondence with finite étale algebras over $ F $.

Theorem. Let $ F \supseteq \Q(\mu_\infty) $ be a field. There is a functor from the category of finite étale covers over $ \Spec F $ to the category of finite étale covers over $ \Xs_F $
\[ \FEC/\Spec F \to \FEC/\mathcal{X}_F \]
which is a category equivalence.

In particular, it follows that the étale fundamental group of $ \Xs_F $ over a point $ \bar{x}\in \Xs_F $ is isomorphic to the absolute Galois group of $ F $: that is, we have an isomorphism $$ \pi^{\text{ét}}(\Xs_F, \bar{x}) \simeq \Gal(\bar{F}/F). $$