算术拓扑

算术拓扑（arithmetic topology）是结合了代数数论与拓扑学的数学领域。它在代数数域和封闭可定向的三维流形之间建立起类比。

 类比 
以下是数域和三维流形之间的一些类比Sikora, Adam S. "Analogies between group actions on 3-manifolds and number fields." Commentarii Mathematici Helvetici 78.4 (2003): 832-844.：

 数域对应封闭、可定向的三维流形。
 整数环的理想对应link，素理想对应扭结。
 有理数域对应三维球面。

 历史 
在1960年代，约翰·泰特基于伽罗瓦上同调给出了类域论的拓扑解释J. Tate,  Duality theorems in Galois cohomology over number fields, (Proc. Intern. Cong. Stockholm, 1962, p. 288-295).，迈克尔·阿廷与让-路易·韦迪耶基于平展上同调也给出了类似解释M. Artin and J.-L. Verdier,  Seminar on étale cohomology of number fields, Woods Hole Archived May 26, 2011, at the Wayback Machine, 1964.。之后戴维·芒福德与尤里·马宁各自独立地提出素理想与扭结的类比Who dreamed up the primes=knots analogy?  Archived July 18, 2011, at the Wayback Machine, neverendingbooks, lieven le bruyn's blog, may 16, 2011,，Barry Mazur作了进一步的研究Remarks on the Alexander Polynomial , Barry Mazur, c.1964B. Mazur, Notes on ´etale cohomology of number fields , Ann. scient. ´Ec. Norm. Sup. 6 (1973), 521-552.。在1990年代ReznikovA. Reznikov, Three-manifolds class field theory (Homology of coverings for a nonvirtually b1-positive manifold) , Sel. math. New ser. 3, (1997), 361–399.与KapranovM. Kapranov, Analogies between the Langlands correspondence and topological quantum field theory, Progress in Math., 131, Birkhäuser, (1995), 119–151.开始研究这些类比，并首创术语“算术拓扑”来称呼这一研究领域。

 另见 

 算术几何
 算术动力学
 拓扑量子场论
 朗兰兹纲领

 参考文献 

 延伸阅读 

 Masanori Morishita (2011), Knots and Primes , Springer, ISBN 978-1-4471-2157-2
 Masanori Morishita (2009), Analogies Between Knots And Primes, 3-Manifolds And Number Rings
 Christopher Deninger (2002), A note on arithmetic topology and dynamical systems
 Adam S. Sikora (2001), Analogies between group actions on 3-manifolds and number fields
 柯蒂斯·麦克马伦 (2003), From dynamics on surfaces to rational points on curves 
 Chao Li and Charmaine Sia (2012), Knots and Primes 

 外部链接 

 Mazur's knotty dictionary 

Category:数论
Category:三维流形
Category:纽结理论