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

Runjie Hu

TL;DR

The authors define the profinite normal structure set $S(X)^{\wedge}_N$ to capture finite-information liftings of the Spivak normal fibration, enabling a Galois action on manifold structures of varieties defined over $\overline{\mathbb{Q}}$. They show that for simply-connected complex varieties of dimension at least $3$, the Galois action of $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ on the $\overline{\mathbb{Q}}$-algebraic elements of $\mathbf{S}(Y)^{\wedge}_N$ factors through the abelianization $\widehat{\mathbb{Z}}^{\times}$ via the map $\omega'$, and this abelian action extends to the entire profinite normal structure set. Moreover, this action agrees with the abelianized Galois action on the corresponding profinite topological structure set through the root-of-unity map $\omega$, effectively answering Sullivan’s question in this higher-dimensional, simply-connected setting. A key technical piece is Lemma ['GaloisMainII'], which relates the Galois action to Adams operations at odd primes and to the $2$-adic $L$- and $k^{\sigma_2}$-classes, with Section 5 providing the construction and computation of the $k^{\sigma_2}$ class. Overall, the work isolates an algebraic facet of Galois symmetry on manifold structures and lays groundwork for a geometric/combinatorial interpretation of this symmetry.

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.

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

TL;DR

The authors define the profinite normal structure set to capture finite-information liftings of the Spivak normal fibration, enabling a Galois action on manifold structures of varieties defined over . They show that for simply-connected complex varieties of dimension at least , the Galois action of on the -algebraic elements of factors through the abelianization via the map , and this abelian action extends to the entire profinite normal structure set. Moreover, this action agrees with the abelianized Galois action on the corresponding profinite topological structure set through the root-of-unity map , effectively answering Sullivan’s question in this higher-dimensional, simply-connected setting. A key technical piece is Lemma ['GaloisMainII'], which relates the Galois action to Adams operations at odd primes and to the -adic - and -classes, with Section 5 providing the construction and computation of the class. Overall, the work isolates an algebraic facet of Galois symmetry on manifold structures and lays groundwork for a geometric/combinatorial interpretation of this symmetry.

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 on the underlying topological manifold structures of smooth, complete, simply-connected complex varieties defined over of dimension at least factors through the abelianization of . 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.
Paper Structure (5 sections, 11 theorems, 12 equations)

This paper contains 5 sections, 11 theorems, 12 equations.

Table of Contents

  1. Introduction
  2. profinite normal structure set and abelianized Galois symmetry
  3. Main Theorems
  4. Proof of Lemma \ref{['GaloisMainII']}
  5. The characteristic class $k^{\sigma_2}$

Key Result

Theorem 1.1

For a simply-connected, compact, smooth complex variety $X$ defined over $\overline{\mathbb{Q}}$ of dimension at least $3$, the Galois action of $\mathop{\mathrm{Gal}}\nolimits(\overline{\mathbb{Q}}/\mathbb{Q})$ on the underlying topological manifold structures of the $\overline{\mathbb{Q}}$-varieti

Theorems & Definitions (21)

  • Theorem 1.1
  • Definition 2.1
  • Proposition 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • Definition 3.1
  • Theorem 3.2
  • Lemma 3.3
  • Definition 3.4
  • ...and 11 more