Arf invariant

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For the closely related knot invariant, see Arf invariant (knot).

In mathematics, the Arf invariant, named after 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.

[edit] Structure of quadratic forms

Every non-singular quadratic form over F₂ can be written as an orthogonal sum A^m + 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, becuse 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.

[edit] References

• Arf, Cahit (1941). "Untersuchungen über quadratische Formen in Körpern der Karacteristic 2, I". J. Reine Angew. Math 183: 148–167.

• Kirby, Robion (1989). The topology of 4-manifolds. Lecture Notes in Mathematics, no. 1374, Springer-Verlag. ISBN 0-387-51148-2.
• A.V. Chernavskii, "Arf invariant" SpringerLink Encyclopaedia of Mathematics (2001)