Definitions

# Arf invariant

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 Am + 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.

## 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}}

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