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

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