Topological realisations of absolute Galois groups

TL;DR

This paper constructs a topological space associated with a field of characteristic zero that captures its absolute Galois group as its fundamental group, providing a geometric perspective on Galois theory.

Contribution

It introduces a functorial construction of a compact Hausdorff space whose fundamental group matches the absolute Galois group, using rational Witt vectors.

Findings

The space's fundamental group is isomorphic to the absolute Galois group of the field.

In the cyclotomic case, the classical fundamental group is dense in the Galois group.

A variant construction encodes descent data via Frobenius automorphisms.

Abstract

Let $F$ be a field of characteristic $0$ containing all roots of unity. We construct a functorial compact Hausdorff space $X_F$ whose profinite fundamental group agrees with the absolute Galois group of $F$, i.e. the category of finite covering spaces of $X_F$ is equivalent to the category of finite extensions of $F$. The construction is based on the ring of rational Witt vectors of $F$. In the case of the cyclotomic extension of $\mathbb{Q}$, the classical fundamental group of $X_F$ is a (proper) dense subgroup of the absolute Galois group of $F$. We also discuss a variant of this construction when the field is not required to contain all roots of unity, in which case there are natural Frobenius-type automorphisms which encode the descent along the cyclotomic extension.