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''''Arithmetic topology''' is an area of [[mathematics]] that is a combination of [[algebraic number theory]] and [[topology]]. It establishes an analogy between [[number field]]s and closed, orientable [[3-manifold]]s. ==Analogies== The following are some of the analogies used by mathematicians between number fields and 3-manifolds:<ref>Sikora, Adam S. "Analogies between group actions on 3-manifolds and number fields." Commentarii Mathematici Helvetici 78.4 (2003): 832-844.</ref> #A number field corresponds to a closed, orientable 3-manifold #[[Ideal (ring theory)|Ideals]] in the ring of integers correspond to [[link (knot theory)|links]], and [[prime ideal]]s correspond to knots. #The field '''Q''' of [[rational number]]s corresponds to the [[3-sphere]]. Expanding on the last two examples, there is an analogy between [[knot (mathematics)|knots]] and [[prime number]]s 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 symbol]]s are all 1). Therefore these primes have been called a "proper Borromean triple modulo 2"<ref>{{Citation |last=Vogel |first=Denis |date=13 February 2004 |title=Massey products in the Galois cohomology of number fields |url=http://www.ub.uni-heidelberg.de/archiv/4418 |id={{URN|nbn|de:bsz:16-opus-44188}}}}</ref> or "mod 2 Borromean primes".<ref>{{Citation |last=Morishita |first=Masanori |date=22 April 2009 |title=Analogies between Knots and Primes, 3-Manifolds and Number Rings |arxiv=0904.3399}}</ref> ==History== In the 1960s topological interpretations of [[class field theory]] were given by [[John Tate]]<ref>J. Tate, Duality theorems in Galois cohomology over number fields, (Proc. Intern. Cong. Stockholm, 1962, p. 288-295).</ref> based on [[Galois cohomology]], and also by [[Michael Artin]] and [[Jean-Louis Verdier]]<ref>M. Artin and J.-L. Verdier, [http://www.jmilne.org/math/Documents/WoodsHole3.pdf Seminar on étale cohomology of number fields, Woods Hole], 1964.</ref> based on [[Étale cohomology]]. Then [[David Mumford]] (and independently [[Yuri Manin]]) came up with an analogy between [[prime ideals]] and [[Knot (mathematics)|knots]]<ref>[http://www.neverendingbooks.org/index.php/who-dreamed-up-the-primesknots-analogy.html Who dreamed up the primes=knots analogy?], neverendingbooks, lieven le bruyn's blog, may 16, 2011,</ref> which was further explored by [[Barry Mazur]].<ref>[http://www.math.harvard.edu/~mazur/papers/alexander_polynomial.pdf Remarks on the Alexander Polynomial], Barry Mazur, c.1964</ref><ref>B. 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.</ref> In the 1990s Reznikov<ref>A. Reznikov, [http://www.springerlink.com/content/v9jc215brrhl4mxf/ Three-manifolds class field theory (Homology of coverings for a nonvirtually b1-positive manifold)], Sel. math. New ser. 3, (1997), 361–399.</ref> and Kapranov<ref>M. Kapranov, [http://books.google.co.uk/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.</ref> began studying these analogies, coining the term '''arithmetic topology''' for this area of study. ==See also== *[[Arithmetic geometry]] *[[Arithmetic dynamics]] *[[Topological quantum field theory]] *[[Langlands program]] ==Notes== <references/> ==Further reading== *Masanori Morishita (2011), [http://www.springer.com/mathematics/numbers/book/978-1-4471-2157-2 Knots and Primes], Springer, ISBN 978-1-4471-2157-2 *Masanori Morishita (2009), [http://arxiv.org/abs/0904.3399v1 Analogies Between Knots And Primes, 3-Manifolds And Number Rings] *Christopher Deninger (2002), [http://arxiv.org/abs/math/0204274v1 A note on arithmetic topology and dynamical systems] *Adam S. Sikora (2001), [http://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), [http://www.math.harvard.edu/~sia/notes/knots_and_primes.pdf Knots and Primes] ==External links== *[http://www.neverendingbooks.org/mazurs-dictionary Mazur’s knotty dictionary] {{Number theory-footer}} [[Category:Number theory]] [[Category:3-manifolds]] [[Category:Knot theory]]'