3维流形的Neukirch-Uchida定理
Rigidity and Galois theory of 3-manifolds branched over Chebotarev links
AI总结：
本文在算术拓扑框架下，证明了3维流形的绝对伽罗华群决定其分支覆盖结构，类比数论中的Neukirch-Uchida定理。
摘要：
经典的Neukirch-Uchida定理指出，绝对伽罗华群决定了数域的同构类。本文在算术拓扑框架下，证明了3维流形的类似定理。我们研究了在3维流形中行为类似素数的无限链，满足Chebotarev密度性质。相对于此类稳定Chebotarev链，我们定义3维流形的绝对伽罗华群为有限子链补集的完备余有限群的逆极限。主要结果表明，若两个分支覆盖的绝对伽罗华群通过保持特征的同构映射，则它们是同胚的。证明将数论论证的核心思想转化为拓扑学，依赖于无限覆盖的Hilbert分层理论和局部-整体原理。此外，我们还讨论了进一步的条件，以使链在算术几何中扮演素数的角色。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
