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 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.)

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

in K. The Arf invariant is defined to be the product

in K. These elements form an additive subgroup U of K. Hence the coset of

This was shown by Arf, but it had been earlier observed by Dickson in the case of finite fields of characteristic 2.

) is defined to be the sum of the Arf invariants of the

By definition, this is a coset of K modulo U. Arf[1] showed that indeed

is replaced by an equivalent quadratic form, which is to say that it is an invariant of

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.

For a field K of characteristic 2, Artin–Schreier theory identifies the quotient group of K by the subgroup U above with the Galois cohomology group H1(K, F2).

In other words, the nonzero elements of K/U are in one-to-one correspondence with the separable quadratic extension fields of K. So the Arf invariant of a nonsingular quadratic form over K is either zero or it describes a separable quadratic extension field of K. This is analogous to the discriminant of a nonsingular quadratic form over a field F of characteristic not 2.

In that case, the discriminant takes values in F*/(F*)2, which can be identified with H1(F, F2) by Kummer theory.

If the field K is perfect, then every nonsingular quadratic form over K is uniquely determined (up to equivalence) by its dimension and its Arf invariant.

In this case, the subgroup U above is zero, and hence the Arf invariant is an element of the base field F2; it is either 0 or 1.

If the field K of characteristic 2 is not perfect (that is, K is different from its subfield K2 of squares), then the Clifford algebra is another important invariant of a quadratic form.

A corrected version of Arf's original statement is that if the degree [K: K2] is at most 2, then every quadratic form over K is completely characterized by its dimension, its Arf invariant and its Clifford algebra.

Over F2, the Arf invariant is 0 if the quadratic form is equivalent to a direct sum of copies of the binary form

William Browder has called the Arf invariant the democratic invariant[3] because it is the value which is assumed most often by the quadratic form.

[4] Another characterization: q has Arf invariant 0 if and only if the underlying 2k-dimensional vector space over the field F2 has a k-dimensional subspace on which q is identically 0 – that is, a totally isotropic subspace of half the dimension.

In other words, a nonsingular quadratic form of dimension 2k has Arf invariant 0 if and only if its isotropy index is k (this is the maximum dimension of a totally isotropic subspace of a nonsingular form).

Let M be a compact, connected 2k-dimensional manifold with a boundary

into a direct sum of subspaces isomorphic to either