arithmetic topology

The following are some of the analogies used by mathematicians between number fields and 3-manifolds:Sikora, Adam S. "Analogies between group actions on 3-manifolds and number fields." Commentarii Mathematici Helvetici 78.4 (2003): 832-844.

1. A number field corresponds to a closed, orientable 3-manifold
2. Ideals in the ring of integers correspond to links, and prime ideals correspond to knots.
3. The field Q of rational numbers corresponds to the 3-sphere.

Expanding on the last two examples, there is an analogy between knots and prime numbers in which one considers "links" between primes. The triple of primes {{nowrap|(13, 61, 937)}} are "linked" modulo 2 (the Rédei symbol is −1) but are "pairwise unlinked" modulo 2 (the Legendre symbols are all 1). Therefore these primes have been called a "proper Borromean triple modulo 2"{{Citation |last=Vogel |first=Denis |date=February 13, 2004 |title=Massey products in the Galois cohomology of number fields |doi=10.11588/heidok.00004418 |url=http://www.ub.uni-heidelberg.de/archiv/4418 |id={{URN|nbn|de:bsz:16-opus-44188}}}} or "mod 2 Borromean primes".{{Citation |last=Morishita |first=Masanori |date=April 22, 2009 |title=Analogies between Knots and Primes, 3-Manifolds and Number Rings |arxiv=0904.3399|bibcode=2009arXiv0904.3399M }}

In the 1960s topological interpretations of class field theory were given by John TateJ. Tate, Duality theorems in Galois cohomology over number fields, (Proc. Intern. Cong. Stockholm, 1962, p. 288-295). based on Galois cohomology, and also by Michael Artin and Jean-Louis VerdierM. Artin and J.-L. Verdier, [http://www.jmilne.org/math/Documents/WoodsHole3.pdf Seminar on étale cohomology of number fields, Woods Hole] {{webarchive |url=https://web.archive.org/web/20110526230017/http://www.jmilne.org/math/Documents/WoodsHole3.pdf |date=May 26, 2011 }}, 1964. based on Étale cohomology. Then David Mumford (and independently Yuri Manin) came up with an analogy between prime ideals and knots[http://www.neverendingbooks.org/who-dreamed-up-the-primesknots-analogy Who dreamed up the primes=knots analogy?] {{webarchive |url=https://web.archive.org/web/20110718061649/http://www.neverendingbooks.org/index.php/who-dreamed-up-the-primesknots-analogy.html |date=July 18, 2011 }}, neverendingbooks, lieven le bruyn's blog, May 16, 2011, which was further explored by Barry Mazur.[http://www.math.harvard.edu/~mazur/papers/alexander_polynomial.pdf Remarks on the Alexander Polynomial], Barry Mazur, c.1964B. Mazur, [http://archive.numdam.org/ARCHIVE/ASENS/ASENS_1973_4_6_4/ASENS_1973_4_6_4_521_0/ASENS_1973_4_6_4_521_0.pdf Notes on ´etale cohomology of number fields], Ann. scient. ´Ec. Norm. Sup. 6 (1973), 521-552. In the 1990s ReznikovA. Reznikov, [https://doi.org/10.1007%2Fs000290050015 Three-manifolds class field theory (Homology of coverings for a nonvirtually b1-positive manifold)], Sel. math. New ser. 3, (1997), 361–399. and KapranovM. Kapranov, [https://books.google.com/books?hl=en&lr=&id=TOPa9irmsGsC&oi=fnd&pg=PA119 Analogies between the Langlands correspondence and topological quantum field theory], Progress in Math., 131, Birkhäuser, (1995), 119–151. began studying these analogies, coining the term arithmetic topology for this area of study.

• Arithmetic geometry
• Arithmetic dynamics
• Topological quantum field theory
• Langlands program

• Masanori Morishita (2011), [https://www.springer.com/mathematics/numbers/book/978-1-4471-2157-2 Knots and Primes], Springer, {{ISBN|978-1-4471-2157-2}}
• Masanori Morishita (2009), [https://arxiv.org/abs/0904.3399v1 Analogies Between Knots And Primes, 3-Manifolds And Number Rings]
• Christopher Deninger (2002), [https://arxiv.org/abs/math/0204274v1 A note on arithmetic topology and dynamical systems]
• Adam S. Sikora (2001), [https://arxiv.org/abs/math/0107210v2 Analogies between group actions on 3-manifolds and number fields]
• Curtis T. McMullen (2003), [http://www.math.harvard.edu/~ctm/home/text/papers/fermat/fermat.pdf From dynamics on surfaces to rational points on curves]
• Chao Li and Charmaine Sia (2012), [https://www.math.columbia.edu/~chaoli/tutorial2012/knots-and-primes.pdf Knots and Primes]

• [http://www.neverendingbooks.org/mazurs-dictionary Mazur’s knotty dictionary]

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{{Areas of mathematics}}

Category:Algebraic number theory

Category:3-manifolds

Category:Knot theory