Kervaire invariant

In mathematics, the Kervaire invariant is an invariant of a framed formula_1-dimensional manifold that measures whether the manifold could be surgically converted into a sphere. This invariant evaluates to 0 if the manifold can be converted to a sphere, and 1 otherwise. This invariant was named after Michel Kervaire who built on work of Cahit Arf.
The Kervaire invariant is defined as the Arf invariant of the skew-quadratic form on the middle dimensional homology group. It can be thought of as the simply-connected "quadratic" L-group formula_2, and thus analogous to the other invariants from L-theory: the signature, a formula_3-dimensional invariant (either symmetric or quadratic, formula_4), and the De Rham invariant, a formula_5-dimensional "symmetric" invariant formula_6.
In any given dimension, there are only two possibilities: either all manifolds have Arf–Kervaire invariant equal to 0, or half have Arf–Kervaire invariant 0 and the other half have Arf–Kervaire invariant 1.
The Kervaire invariant problem is the problem of determining in which dimensions the Kervaire invariant can be nonzero. For differentiable manifolds, this can happen in dimensions 2, 6, 14, 30, 62, and possibly 126, and in no other dimensions. The final case of dimension 126 remains open.
Definition.
The Kervaire invariant is the Arf invariant of the quadratic form determined by the framing on the middle-dimensional formula_7-coefficient homology group
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 formula_9formula_10 formula_11 determined by the framing, or by the triviality/non-triviality of the normal bundles of embeddings formula_9formula_10 formula_11 (for formula_15) and the mod 2 Hopf invariant of maps formula_16
(for formula_17).
History.
The Kervaire invariant is a generalization of the Arf invariant of a framed surface (that is, a 2-dimensional manifold with stably trivialized tangent bundle) which was used by Lev Pontryagin in 1950 to compute the homotopy group formula_18 of maps formula_19 formula_10 formula_21 (for formula_22), which is the cobordism group of surfaces embedded in formula_19 with trivialized normal bundle.
 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.
 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 formula_24 of "h"-cobordism classes of oriented homotopy "n"-spheres. They compute this latter in terms of a map
where formula_26 is the cyclic subgroup of "n"-spheres that bound a parallelizable manifold of dimension formula_27, formula_28 is the "n"th stable homotopy group of spheres, and "J" is the image of the J-homomorphism, which is also a cyclic group. The groups formula_26 and formula_30 have easily understood cyclic factors, which are trivial or order two except in dimension formula_31, in which case they are large, with order related to the 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.
Examples.
For the standard embedded torus, the skew-symmetric form is given by formula_32 (with respect to the standard symplectic basis), and the skew-quadratic refinement is given by formula_33 with respect to this basis: formula_34: the basis curves don't self-link; and formula_35: a (1,1) self-links, as in the Hopf fibration. This form thus has Arf invariant 0 (most of its elements have norm 0; it has isotropy index 1), and thus the standard embedded torus has Kervaire invariant 0.
Kervaire invariant problem.
The question of in which dimensions "n" there are "n"-dimensional framed manifolds of nonzero Kervaire invariant is called the Kervaire invariant problem. This is only possible if "n" is 2 mod 4, and indeed one must have "n" is of the form formula_36 (two less than a power of two). The question is almost completely resolved; only the case of dimension 126 is open: there are manifolds with nonzero Kervaire invariant in dimension 2, 6, 14, 30, 62, and none in all other dimensions other than possibly 126.
The main results are those of , who reduced the problem from differential topology to stable homotopy theory and showed that the only possible dimensions are formula_36, and those of , who showed that there were no such manifolds for formula_38 (formula_39). Together with explicit constructions for lower dimensions (through 62), this leaves open only dimension 126.
It was conjectured by Michael Atiyah that there is such a manifold in dimension 126, and that the higher-dimensional manifolds with nonzero Kervaire invariant are related to well-known exotic manifolds two dimension higher, in dimensions 16, 32, 64, and 128, namely the Cayley projective plane formula_40 (dimension 16, octonionic projective plane) and the analogous Rosenfeld projective planes (the bi-octonionic projective plane in dimension 32, the quateroctonionic projective plane in dimension 64, and the octo-octonionic projective plane in dimension 128), specifically that there is a construction that takes these projective planes and produces a manifold with nonzero Kervaire invariant in two dimensions lower.
Kervaire–Milnor invariant.
The Kervaire–Milnor invariant is a closely related invariant of framed surgery of a 2, 6 or 14-dimensional framed manifold, that gives isomorphisms from the 2nd and 6th stable homotopy group of spheres to formula_7, 
and a homomorphism from the 14th stable homotopy group of spheres onto formula_7. For "n" = 2, 6, 14 there is an
exotic framing on formula_43 with Kervaire–Milnor invariant 1.