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.