Topological realisations of absolute Galois groups

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.