Kervaire invariant

Summary

The Kervaire invariant is an invariant in $\Z/2$ of a closed framed smooth manifold of dimension $4k + 2$. It is the Arf invariant of a quadratic refinement, built from the framing, of the mod $2$ intersection form on the middle cohomology. Michel Kervaire used it in 1960 to give a closed topological manifold of dimension $10$ with no smooth structure. Through the Pontryagin–Thom construction it becomes a homomorphism on the stable homotopy groups of spheres. The question of where it is nonzero is the Kervaire invariant problem.