Kervaire Invariant - Definition

Definition

The Kervaire invariant is the Arf invariant of the quadratic form determined by the framing on the middle-dimensional Z/2Z-coefficient homology group

q : H_2m+1(M;Z/2Z) Z/2Z,

and is thus sometimes called the Arf–Kervaire invariant. The quadratic form (properly, skew-quadratic form) is a quadratic refinement of the usual ε-symmetric form on the middle dimensional homology of an (unframed) even-dimensional manifold; the framing yields the quadratic refinement.

The quadratic form q can be defined by algebraic topology using functional Steenrod squares, and geometrically via the self-intersections of immersions determined by the framing, or by the triviality/non-triviality of the normal bundles of embeddings (for ) and the mod 2 Hopf invariant of maps (for ).