算术拓扑

算术拓扑(arithmetic topology)是结合了代数数论拓扑学的数学领域。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[1]

  1. 数域对应封闭、可定向的三维流形。
  2. 整数环的理想对应link,素理想对应扭结。
  3. 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[2]迈克尔·阿廷让-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

参考文献

  1. Sikora, Adam S. "Analogies between group actions on 3-manifolds and number fields." Commentarii Mathematici Helvetici 78.4 (2003): 832-844.
  2. J. Tate, Duality theorems in Galois cohomology over number fields, (Proc. Intern. Cong. Stockholm, 1962, p. 288-295).
  3. M. Artin and J.-L. Verdier, Seminar on étale cohomology of number fields, Woods Hole Archived May 26, 2011, at the Wayback Machine, 1964.
  4. Who dreamed up the primes=knots analogy? 页面存档备份,存于 Archived July 18, 2011, at the Wayback Machine, neverendingbooks, lieven le bruyn's blog, may 16, 2011,
  5. Remarks on the Alexander Polynomial, Barry Mazur, c.1964
  6. B. Mazur, Notes on ´etale cohomology of number fields, Ann. scient. ´Ec. Norm. Sup. 6 (1973), 521-552.
  7. A. Reznikov, Three-manifolds class field theory (Homology of coverings for a nonvirtually b1-positive manifold), Sel. math. New ser. 3, (1997), 361–399.
  8. M. Kapranov, Analogies between the Langlands correspondence and topological quantum field theory, Progress in Math., 131, Birkhäuser, (1995), 119–151.

延伸阅读

外部链接

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.