Kervaire Invariant - History

History

The Kervaire invariant is a generalization of the Arf invariant of a framed surface (= 2-dimensional manifold with stably trivialized tangent bundle) which was used by Pontryagin in 1950 to compute of the homotopy group of maps (for ), which is the cobordism group of surfaces embedded in with trivialized normal bundle.

Kervaire (1960) used his invariant for n = 10 to construct the Kervaire manifold, a 10-dimensional PL manifold with no differentiable structure, the first example of such a manifold, by showing that his invariant does not vanish on this PL manifold, but vanishes on all smooth manifolds of dimension 10.

Kervaire & Milnor (1963) computes the group of exotic spheres (in dimension greater than 4), with one step in the computation depending on the Kervaire invariant problem. Specifically, they show that the set of exotic spheres of dimension n – specifically the monoid of smooth structures on the standard n-sphere – is isomorphic to the group Θ_n of h-cobordism classes of oriented homotopy n-spheres. They compute this latter in terms of a map

where is the cyclic subgroup of n-spheres that bound a parallelizable manifold of dimension n+1, is the nth stable homotopy group of spheres, and J is the image of the J-homomorphism, which is also a cyclic group. The and are easily understood cyclic factors, which are trivial or order two except in dimension in which case they are large, with order related to Bernoulli numbers. The quotients are the difficult parts of the groups. The map between these quotient groups is either an isomorphism or is injective and has an image of index 2. It is the latter if and only if there is an n-dimensional framed manifold of nonzero Kervaire invariant, and thus the classification of exotic spheres depends up to a factor of 2 on the Kervaire invariant problem.