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

TL;DR

This paper investigates how the Galois group acts on the topological manifold structures of certain complex varieties, showing that the action factors through an abelian quotient and extends to a profinite structure set.

Contribution

It introduces a profinite normal structure set for manifolds within a fixed profinite homotopy type and proves the Galois action factors through its abelianization for specific varieties.

Findings

Galois action factors through the abelianization for certain varieties

Extension of the Galois action to the profinite normal structure set

Provides an answer to Sullivan's question for simply-connected 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.