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 based on Galois cohomology, and also by Michael Artin and Jean-Louis Verdier based on Étale cohomology. Then David Mumford (and independently Yuri Manin) came up with an analogy between prime ideals and knots which was further explored by Barry Mazur. In the 1990s Reznikov and Kapranov began studying these analogies, coining the term arithmetic topology for this area of study.
- ^ J. Tate, Duality theorems in Galois cohomology over number fields, (Proc. Intern. Cong. Stockholm, 1962, p. 288-295).
- ^ M. Artin and J.-L. Verdier, Seminar on étale cohomology of number fields, Woods Hole, 1964.
- ^ Who dreamed up the primes=knots analogy?, neverendingbooks, lieven le bruyn's blog, may 16, 2011,
- ^ Remarks on the Alexander Polynomial, Barry Mazur, c.1964
- ^ B. Mazur, Notes on ´etale cohomology of number fields, Ann. scient. ´Ec. Norm. Sup. 6 (1973), 521-552.
- ^ A. Reznikov, Three-manifolds class field theory (Homology of coverings for a nonvirtually b1-positive manifold), Sel. math. New ser. 3, (1997), 361–399.
- ^ M. Kapranov, Analogies between the Langlands correspondence and topological quantum field theory, Progress in Math., 131, Birkhäuser, (1995), 119–151.