A Neukirch-Uchida Theorem for 3-Manifolds

TL;DR

This paper establishes a topological analogue of the Neukirch-Uchida theorem for 3-manifolds, linking the homeomorphism class of certain covers to isomorphisms of their absolute Galois groups.

Contribution

It introduces the concept of absolute Galois groups for 3-manifolds using Chebotarev-like links and proves their role in classifying branched covers.

Findings

Two branched covers over a stably Chebotarev link are homeomorphic iff their Galois groups are isomorphic.

Defines the absolute Galois group of a 3-manifold as an inverse limit of profinite completions.

Provides a topological framework paralleling number-theoretic Galois theory.

Abstract

The classical Neukirch-Uchida theorem states that the absolute Galois group determines a number field up to isomorphism. We prove an analogue of this theorem for 3-manifolds in the framework of arithmetic topology. We study infinite links in 3-manifolds that behave like the set of primes, satisfying a Chebotarev density property. Relative to such a stably Chebotarev link, we define the absolute Galois group of a 3-manifold as the inverse limit of profinite completions of finite sublink complements. Our main result shows that two branched covers of the three-sphere over a stably Chebotarev link are homeomorphic if and only if their absolute Galois groups are isomorphic via a characteristic-preserving isomorphism. The proof translates the key ideas from the number-theoretic argument into topology, relying on Hilbert ramification theory for infinite covers and local-global principles. In doing so, it also provides a systematic justification for viewing Chebotarev links as the precise topological analogue of prime numbers in anabelian geometry. In addition, we discuss further conditions for links to play the role of prime numbers