Rigidity and Galois theory of 3-manifolds branched over Chebotarev links

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 $\mathcal{L}$ in 3-manifolds that satisfy a Chebotarev density property, and hence behave like the set of primes. We define the absolute Galois group of a 3-manifold relative to such a link as the inverse limit of profinite completions of finite sublink complements. Our main result shows that two covers of $S^3$ branched over finite sublinks of $\mathcal{L}$ are isomorphic if and only if their absolute Galois groups relative to $\mathcal{L}$ are isomorphic via a characteristic-preserving isomorphism. The proof translates the key ideas from the number-theoretic argument into 3-manifolds, developing Hilbert ramification theory for infinite covers and local-global principles. Doing so also provides a systematic justification for viewing Chebotarev links as topological analogues of the set of prime numbers. In addition, we propose further conditions on Chebotarev links which would further strengthen their connection to the prime numbers.

arXiv commentv1: minor revisions with improved exposition. 37 pages, comments welcome!