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
[编辑]以下是数域和三维流形之间的一些类比[1]：
历史
[编辑]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[2]，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[3]。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[4]，Barry Mazur作了进一步的研究[5][6]。在1990年代Reznikov[7]与Kapranov[8]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
[编辑]参考文献
[编辑]- ^ Sikora, Adam S. "Analogies between group actions on 3-manifolds and number fields." Commentarii Mathematici Helvetici 78.4 (2003): 832-844.
- ^ J. Tate, Duality theorems in Galois cohomology over number fields, (Proc. Intern. Cong. Stockholm, 1962, p. 288-295).
- ^ M. Artin and J.-L. Verdier, Seminar on étale cohomology of number fields, Woods Hole Archived May 26, 2011, at the Wayback Machine, 1964.
- ^ Who dreamed up the primes=knots analogy? （页面存档备份，存于互联网档案馆） Archived July 18, 2011, at the Wayback Machine, neverendingbooks, lieven le bruyn's blog, may 16, 2011,
- ^ Remarks on the Alexander Polynomial （页面存档备份，存于互联网档案馆）, Barry Mazur, c.1964
- ^ B. Mazur, Notes on ´etale cohomology of number fields （页面存档备份，存于互联网档案馆）, Ann. scient. ´Ec. Norm. Sup. 6 (1973), 521-552.
- ^ A. Reznikov, Three-manifolds class field theory (Homology of coverings for a nonvirtually b1-positive manifold) （页面存档备份，存于互联网档案馆）, Sel. math. New ser. 3, (1997), 361–399.
- ^ M. Kapranov, Analogies between the Langlands correspondence and topological quantum field theory, Progress in Math., 131, Birkhäuser, (1995), 119–151.
延伸阅读
[编辑]- Masanori Morishita (2011), Knots and Primes （页面存档备份，存于互联网档案馆）, 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 （页面存档备份，存于互联网档案馆）
- Chao Li and Charmaine Sia (2012), Knots and Primes （页面存档备份，存于互联网档案馆）