Arf Invariant

In mathematics, the Arf invariant of a nonsingular quadratic form over a field of characteristic 2 was defined by Cahit Arf (1941) when he started the systematic study of quadratic forms over arbitrary fields of characteristic 2. The Arf invariant is the substitute, in characteristic 2, of the discriminant for quadratic forms in characteristic not 2. Arf used his invariant, among others, in his endeavor to classify quadratic forms in characteristic 2.

In the special case of the 2-element field F₂ the Arf invariant can be described as the element of F₂ that occurs most often among the values of the form. Two nonsingular quadratic forms over F₂ are isomorphic if and only if they have the same dimension and the same Arf invariant. This fact was essentially known to Dickson (1901), even for any finite field of characteristic 2, and it follows from Arf's results for an arbitrary perfect field. An assessment of Arf's results in the framework of the theory of quadratic forms can be found in,

The Arf invariant is particularly applied in geometric topology, where it is primarily used to define an invariant of (4k+2)-dimensional manifolds (singly even-dimension manifolds: surfaces (2-manifolds), 6-manifolds, 10-manifolds, etc.) with certain additional structure called a framing, and thus the Arf–Kervaire invariant and the Arf invariant of a knot. The Arf invariant is analogous to the signature of a manifold, which is defined for 4k-dimensional manifolds (doubly even-dimensional); this 4-fold periodicity corresponds to the 4-fold periodicity of L-theory. The Arf invariant can also be defined more generally for certain 2k-dimensional manifolds.