Galois Symmetry of $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ on Topological Manifold Structures of Varieties

Abstract

We propose a definition of the profinite normal structure set for the set of all manifolds in a fixed profinite homotopy type. Using this framework, we prove that the Galois action of $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ on the underlying topological manifold structures of smooth, complete, simply-connected complex varieties defined over $\overline{\mathbb{Q}}$ of dimension at least $3$ factors through the abelianization of $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$. Moreover, this abelian action extends canonically to the entire profinite normal structure set. This result provides an answer to the question by Sullivan in the case of topological manifold structures of simply-connected varieties.