Arf invariant

In mathematics, the Arf invariant of a nonsingular quadratic form over a field of characteristic 2 was defined by Turkish mathematician 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, for 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 mathematics, the Arf invariant of a nonsingular quadratic form over a field of characteristic 2 was defined by Turkish mathematician 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, for 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 Leonard Dickson (1901), even for any finite field of characteristic 2, and Arf proved it for an arbitrary perfect field. 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-dimensional 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. The Arf invariant is defined for a quadratic form q over a field K of characteristic 2 such that q is nonsingular, in the sense that the associated bilinear form b ( u , v ) = q ( u + v ) − q ( u ) − q ( v ) {displaystyle b(u,v)=q(u+v)-q(u)-q(v)} is nondegenerate. The form b {displaystyle b} is alternating since K has characteristic 2; it follows that a nonsingular quadratic form in characteristic 2 must have even dimension. Any binary (2-dimensional) nonsingular quadratic form over K is equivalent to a form q ( x , y ) = a x 2 + x y + b y 2 {displaystyle q(x,y)=ax^{2}+xy+by^{2}} with a , b {displaystyle a,b} in K. The Arf invariant is defined to be the product a b {displaystyle ab} . If the form q ′ ( x , y ) = a ′ x 2 + x y + b ′ y 2 {displaystyle q'(x,y)=a'x^{2}+xy+b'y^{2}} is equivalent to q ( x , y ) {displaystyle q(x,y)} , then the products a b {displaystyle ab} and a ′ b ′ {displaystyle a'b'} differ by an element of the form u 2 + u {displaystyle u^{2}+u} with u {displaystyle u} in K. These elements form an additive subgroup U of K. Hence the coset of a b {displaystyle ab} modulo U is an invariant of q {displaystyle q} , which means that it is not changed when q {displaystyle q} is replaced by an equivalent form. Every nonsingular quadratic form q {displaystyle q} over K is equivalent to a direct sum q = q 1 + … + q r {displaystyle q=q_{1}+ldots +q_{r}} of nonsingular binary forms. This was shown by Arf, but it had been earlier observed by Dickson in the case of finite fields of characteristic 2. The Arf invariant Arf( q {displaystyle q} ) is defined to be the sum of the Arf invariants of the q i {displaystyle q_{i}} . By definition, this is a coset of K modulo U. Arf showed that indeed Arf( q {displaystyle q} ) does not change if q {displaystyle q} is replaced by an equivalent quadratic form, which is to say that it is an invariant of q {displaystyle q} .