Arf invariant

In mathematics, the Arf invariant of a nonsingular quadratic form over a field of characteristic 2 was defined by Turkish mathematician 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 , 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 -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 4"k"-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 2"k"-dimensional manifolds.
Definitions.
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 formula_1 is nondegenerate. The form formula_2 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 formula_3 with formula_4 in "K". The Arf invariant is defined to be the product formula_5. If the form formula_6 is equivalent to formula_7, then the products formula_5 and formula_9 differ by an element of the form formula_10 with formula_11 in "K". These elements form an additive subgroup "U" of "K". Hence the coset of formula_5 modulo "U" is an invariant of formula_13, which means that it is not changed when formula_13 is replaced by an equivalent form.
Every nonsingular quadratic form formula_13 over "K" is equivalent to a direct sum formula_16 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(formula_13) is defined to be the sum of the Arf invariants of the formula_18. By definition, this is a coset of "K" modulo "U". Arf showed that indeed formula_19 does not change if formula_13 is replaced by an equivalent quadratic form, which is to say that it is an invariant of formula_13.
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 "H"1("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 "H"1("F", F2) by Kummer theory.
Arf's main results.
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 particular, this holds over the field F2. 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 "K"2 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": "K"2] is at most 2, then every quadratic form over "K" is completely characterized by its dimension, its Arf invariant and its Clifford algebra. Examples of such fields are function fields (or power series fields) of one variable over perfect base fields.
Quadratic forms over F2.
Over F2, the Arf invariant is 0 if the quadratic form is equivalent to a direct sum of copies of the binary form formula_22, and it is 1 if the form is a direct sum of formula_23 with a number of copies of formula_22.
William Browder has called the Arf invariant the "democratic invariant" because it is the value which is assumed most often by the quadratic form. Another characterization: "q" has Arf invariant 0 if and only if the underlying 2"k"-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 2"k" 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).
The Arf invariant in topology.
Let "M" be a compact, connected 2"k"-dimensional manifold with a boundary formula_25
such that the induced morphisms in formula_26-coefficient homology
are both zero (e.g. if formula_28 is closed). The intersection form 
is non-singular. (Topologists usually write F2 as formula_26.) A quadratic refinement for formula_31 is a function formula_32 which satisfies
Let formula_34 be any 2-dimensional subspace of formula_35, such that formula_36. Then there are two possibilities. Either all of formula_37 are 1, or else just one of them is 1, and the other two are 0. Call the first case formula_38, and the second case formula_39. Since every form is equivalent to a symplectic form, we can always find subspaces formula_34 with "x" and "y" being formula_41-dual. We can therefore split formula_35 into a direct sum of subspaces isomorphic to either formula_39 or formula_38. Furthermore, by a clever change of basis, formula_45 We therefore define the Arf invariant 