Arithmetic topology explained

Arithmetic topology is an area of mathematics that is a combination of algebraic number theory and topology. It establishes an analogy between number fields and closed, orientable 3-manifolds.

Analogies

The following are some of the analogies used by mathematicians between number fields and 3-manifolds:^[1]

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 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" or "mod 2 Borromean primes".

History

In the 1960s topological interpretations of class field theory were given by John Tate^[2] based on Galois cohomology, and also by Michael Artin and Jean-Louis Verdier^[3] based on Étale cohomology. Then David Mumford (and independently Yuri Manin) came up with an analogy between prime ideals and knots^[4] which was further explored by Barry Mazur.^{[5] [6]} In the 1990s Reznikov^[7] and Kapranov^[8] 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

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, 1964.
4. http://www.neverendingbooks.org/who-dreamed-up-the-primesknots-analogy Who dreamed up the primes=knots analogy?
5. http://www.math.harvard.edu/~mazur/papers/alexander_polynomial.pdf Remarks on the Alexander Polynomial
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.

Further reading

• Masanori Morishita (2011), Knots and Primes, Springer,
• 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
• Curtis T. McMullen (2003), From dynamics on surfaces to rational points on curves
• Chao Li and Charmaine Sia (2012), Knots and Primes

External links

• Mazur’s knotty dictionary