查看“︁算术拓扑”︁的源代码

'''算术拓扑'''（arithmetic topology）是结合了[[代數數論|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[[3-流形|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<ref>Sikora, Adam S. "Analogies between group actions on 3-manifolds and number fields." Commentarii Mathematici Helvetici 78.4 (2003): 832-844.
</ref>：

# 数域对应封闭、可定向的三维流形。
# 整数环的[[理想 (环论)|理想]]对应link，[[素理想]]对应扭结。
# 有理数域<math>\mathbb{Q}</math>对应[[三維球面|三维球面]]。

== 历史 ==
在1960年代，[[约翰·泰特]]基于[[伽羅瓦上同調|伽罗瓦上同调]]给出了[[類域論|类域论]]的拓扑解释<ref>J. Tate,  Duality theorems in Galois cohomology over number fields, (Proc. Intern. Cong. Stockholm, 1962, p. 288-295).
</ref>，[[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<ref>M. Artin and J.-L. Verdier,  Seminar on étale cohomology of number fields, Woods Hole Archived May 26, 2011, at the Wayback Machine, 1964.
</ref>。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<ref>[http://www.neverendingbooks.org/who-dreamed-up-the-primesknots-analogy Who dreamed up the primes=knots analogy?] {{Wayback|url=http://www.neverendingbooks.org/who-dreamed-up-the-primesknots-analogy |date=20110718061649 }} Archived July 18, 2011, at the Wayback Machine, neverendingbooks, lieven le bruyn's blog, may 16, 2011,</ref>，Barry Mazur作了进一步的研究<ref>[http://www.math.harvard.edu/~mazur/papers/alexander_polynomial.pdf Remarks on the Alexander Polynomial] {{Wayback|url=http://www.math.harvard.edu/~mazur/papers/alexander_polynomial.pdf |date=20191024034248 }}, Barry Mazur, c.1964</ref><ref>B. Mazur, [http://archive.numdam.org/item/ASENS_1973_4_6_4_521_0/ Notes on ´etale cohomology of number fields] {{Wayback|url=http://archive.numdam.org/item/ASENS_1973_4_6_4_521_0/ |date=20201128024138 }}, Ann. scient. ´Ec. Norm. Sup. 6 (1973), 521-552.</ref>。在1990年代Reznikov<ref>A. Reznikov, [https://link.springer.com/article/10.1007%2Fs000290050015 Three-manifolds class field theory (Homology of coverings for a nonvirtually b1-positive manifold)] {{Wayback|url=https://link.springer.com/article/10.1007%2Fs000290050015 |date=20190402074120 }}, Sel. math. New ser. 3, (1997), 361–399.</ref>与Kapranov<ref>M. Kapranov, [https://books.google.com.hk/books?hl=en&lr=&id=TOPa9irmsGsC&oi=fnd&pg=PA119&redir_esc=y Analogies between the Langlands correspondence and topological quantum field theory], Progress in Math., 131, Birkhäuser, (1995), 119–151.
</ref>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]]
* [[拓撲量子場論|拓扑量子场论]]
* [[朗蘭茲綱領|朗兰兹纲领]]

== 参考文献 ==
<references/>

== 延伸阅读 ==

* Masanori Morishita (2011), [https://www.springer.com/gp/book/9781447121572 Knots and Primes] {{Wayback|url=https://www.springer.com/gp/book/9781447121572 |date=20210122163622 }}, Springer, ISBN 978-1-4471-2157-2
* Masanori Morishita (2009), [[arxiv:0904.3399v1|Analogies Between Knots And Primes, 3-Manifolds And Number Rings]]
* Christopher Deninger (2002), [[arxiv:math/0204274v1|A note on arithmetic topology and dynamical systems]]
* Adam S. Sikora (2001), [[arxiv:math/0107210v2|Analogies between group actions on 3-manifolds and number fields]]
* [[柯蒂斯·麥克馬倫|柯蒂斯·麦克马伦]] (2003), [http://www.math.harvard.edu/~ctm/home/text/papers/fermat/fermat.pdf From dynamics on surfaces to rational points on curves] {{Wayback|url=http://www.math.harvard.edu/~ctm/home/text/papers/fermat/fermat.pdf |date=20160303180545 }}
* Chao Li and Charmaine Sia (2012), [http://www.julianlyczak.nl/seminar/knots2016-files/knots_and_primes.pdf Knots and Primes] {{Wayback|url=http://www.julianlyczak.nl/seminar/knots2016-files/knots_and_primes.pdf |date=20210201003221 }}

== 外部链接 ==

* [http://www.neverendingbooks.org/mazurs-dictionary Mazur's knotty dictionary] {{Wayback|url=http://www.neverendingbooks.org/mazurs-dictionary |date=20201203181858 }}

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