算术拓扑
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。
参考文献
- 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 Archived July 18, 2011, at the Wayback Machine, neverendingbooks, lieven le bruyn's blog, may 16, 2011,
- Remarks on the Alexander Polynomial, Barry Mazur, c.1964
- B. Mazur, Notes on ´etale cohomology of number fields, 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), 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
- 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
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