Arithmetic topology

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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.

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  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.

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