Arithmetic topology is an area of mathematics that is a combination of algebraic number theory and

topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ho ...

. It establishes an analogy between

number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a ...

s and closed, orientable

3-manifold In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds ...

Analogies

The following are some of the analogies used by mathematicians between number fields and 3-manifolds: #A number field corresponds to a closed, orientable 3-manifold #

Ideals Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considered ...

in the ring of integers correspond to links, and prime ideals correspond to knots. #The field Q of

rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...

s corresponds to the

3-sphere In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It may be embedded in 4-dimensional Euclidean space as the set of points equidistant from a fixed central point. Analogous to how the boundary of a ball in three dimens ...

. Expanding on the last two examples, there is an analogy between

knots A knot is a fastening in rope or interwoven lines. Knot may also refer to: Places * Knot, Nancowry, a village in India Archaeology * Knot of Isis (tyet), symbol of welfare/life. * Minoan snake goddess figurines#Sacral knot Arts, entertainmen ...

and

prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only way ...

s in which one considers "links" between primes. The triple of primes are "linked" modulo 2 (the

Rédei symbol Rédei is a Hungarian surname. Notable people with the surname include: *George Rédei (1921–2008), Hungarian plant biologist *László Rédei László Rédei (15 November 1900 – 21 November 1980) was a Hungarian mathematician. Rédei gradu ...

is −1) but are "pairwise unlinked" modulo 2 (the

Legendre symbol In number theory, the Legendre symbol is a multiplicative function with values 1, −1, 0 that is a quadratic character modulo an odd prime number ''p'': its value at a (nonzero) quadratic residue mod ''p'' is 1 and at a non-quadratic residue ...

s 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 based on

Galois cohomology In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups. A Galois group ''G'' associated to a field extension ''L''/''K'' acts in a nat ...

, and also by

Michael Artin Michael Artin (; born 28 June 1934) is a German-American mathematician and a professor emeritus in the Massachusetts Institute of Technology mathematics department, known for his contributions to algebraic geometry.Jean-Louis Verdier Jean-Louis Verdier (; 2 February 1935 – 25 August 1989) was a French mathematician who worked, under the guidance of his doctoral advisor Alexander Grothendieck, on derived categories and Verdier duality. He was a close collaborator of Grothe ...

based on

Étale cohomology In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectu ...

. Then

David Mumford David Bryant Mumford (born 11 June 1937) is an American mathematician known for his work in algebraic geometry and then for research into vision and pattern theory. He won the Fields Medal and was a MacArthur Fellow. In 2010 he was awarded ...

(and independently

Yuri Manin Yuri Ivanovich Manin (russian: Ю́рий Ива́нович Ма́нин; born 16 February 1937) is a Russian mathematician, known for work in algebraic geometry and diophantine geometry, and many expository works ranging from mathematical lo ...

) came up with an analogy between

prime ideals In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together ...

and

which was further explored by

Barry Mazur Barry Charles Mazur (; born December 19, 1937) is an American mathematician and the Gerhard Gade University Professor at Harvard University. His contributions to mathematics include his contributions to Wiles's proof of Fermat's Last Theorem in ...

. In the 1990s Reznikov and KapranovM. Kapranov
Analogies between the Langlands correspondence and topological quantum field theory
Progress in Math., 131, Birkhäuser, (1995), 119–151. began studying these analogies, coining the term arithmetic topology for this area of study.

Notes

External links

Mazur’s knotty dictionary
{{Areas of mathematics Number theory 3-manifolds Knot theory

Analogies

History

See also

Notes

Further reading

External links