In mathematics, the Arf invariant, named after Turkish mathematician Cahit Arf, who introduced it in 1941, is an element of F₂ associated to a non-singular quadratic form over the field F₂ with 2 elements, equal to the most common value of the quadratic form. Two such quadratic forms are isomorphic if and only if they have the same Arf invariant.

Structure of quadratic forms

Every non-singular quadratic form over F₂ can be written as an orthogonal sum A^m + Bⁿ of copies of the two 2-dimensional forms A and B, where A has 3 elements of norm 1, and B has one element of norm 1. The numbers m and n are not uniquely determined, because A + A is isomorphic to B + B. However m is uniquely determined mod 2, and the value of m mod 2 is the Arf invariant of the quadratic form.

If B is a quadratic form of dimension 2n, then it has 2²ⁿ⁻¹ + 2ⁿ⁻¹ elements of norm 1 if its Arf invariant is 1, and 2²ⁿ⁻¹ − 2ⁿ⁻¹ elements of norm 1 if its Arf invariant is 0.

The Arf invariant is additive; in other words, the Arf invariant of an orthogonal sum of two quadratic forms is the sum of their Arf invariants.

See also

• Arf-Kervaire invariant

References

•  | last = Kirby|first= Robion

   | authorlink = Robion Kirby

   | title = The topology of 4-manifolds

   | year = 1989

   | series = Lecture Notes in Mathematics|volume= 1374|publisher= Springer-Verlag

   |  ISBN =0-387-51148-2

  |doi=10.1007/BFb0089031}}