Arf Invariant - The Arf Invariant in Topology

The Arf Invariant in Topology

Let M be a compact, connected 2k-dimensional manifold with a boundary such that the induced morphisms in -coefficient homology

,

are both zero (e.g. if is closed). The intersection form

is non-singular. (Topologists usually write F₂ as .) A quadratic refinement for is a function which satisfies

Let be any 2-dimensional subspace of, such that . Then there are two possibilities. Either all of are 1, or else just one of them is 1, and the other two are 0. Call the first case, and the second case . Since every form is equivalent to a symplectic form, we can always find subspaces with x and y being -dual. We can therefore split into a direct sum of subspaces isomorphic to either or . Furthermore, by a clever change of basis, . We therefore define the Arf invariant

= (number of copies of in a decomposition Mod 2) .