Arithmetic topology

Arithmetic topology is an area of mathematics that is a combination of algebraic number theory and topology. In the 1960s topological interpretations of class field theory were given by John Tate^[1] based on Galois cohomology, and also by Michael Artin and Jean-Louis Verdier^[2] based on Étale cohomology. Then David Mumford (and independently Yuri Manin) came up with an analogy between prime ideals and knots^[3] which was further explored by Barry Mazur^[4][5]. In the 1990s Reznikov^[6] and Kapranov^[7] 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. ^ J. Tate, Duality theorems in Galois cohomology over number fields, (Proc. Intern. Cong. Stockholm, 1962, p. 288-295).
2. ^ M. Artin and J.-L. Verdier, Seminar on étale cohomology of number fields, Woods Hole, 1964.
3. ^ Who dreamed up the primes=knots analogy?, neverendingbooks, lieven le bruyn's blog, may 16, 2011,
4. ^ Remarks on the Alexander Polynomial, Barry Mazur, c.1964
5. ^ B. Mazur, Notes on ´etale cohomology of number fields, Ann. scient. ´Ec. Norm. Sup. 6 (1973), 521-552.
6. ^ A. Reznikov, Three-manifolds class field theory (Homology of coverings for a nonvirtually b1-positive manifold), Sel. math. New ser. 3, (1997), 361–399.
7. ^ 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, 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
• Curtis T. McMullen (2003), From dynamics on surfaces to rational points on curves

External links

• Mazur’s knotty dictionary