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.
The Kervaire invariant problem is the problem of determining in which dimensions the Kervaire invariant can be nonzero.
On May 30, 2024, Zhouli Xu (in collaboration with Weinan Lin and Guozhen Wang), announced during a seminar at Princeton University that the final case of dimension 126 has been settled.
survives so that there exists a manifold of Kervaire invariant 1 in dimension 126.
"Computing differentials in the Adams spectral sequence".. (https://www.math.princeton.edu/events/computing-differentials-adams-spectral-sequence-2024-05-30t170000) A preprint has been published on the ArXiv (See Lin, Weinan; Wang, Guozhen; Xu, Zhouli (December 14, 2024).
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
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
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
of h-cobordism classes of oriented homotopy n-spheres.
is the cyclic subgroup of n-spheres that bound a parallelizable manifold of dimension
have easily understood cyclic factors, which are trivial or order two except in dimension
, in which case they are large, with order related to the Bernoulli numbers.
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.
For the standard embedded torus, the skew-symmetric form is given by
(with respect to the standard symplectic basis), and the skew-quadratic refinement is given by
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.
The question of in which dimensions n there are n-dimensional framed manifolds of nonzero Kervaire invariant is called the Kervaire invariant problem.
However, Zhouli Xu (in collaboration with Weinan Lin and Guozhen Wang) announced on May 30, 2024 that there exists a manifold with nonzero Kervaire invariant in dimension 126.
The main results are those of William Browder (1969), who reduced the problem from differential topology to stable homotopy theory and showed that the only possible dimensions are
This dimension now appears to be closed by the December 14, 2024 ArXiv post by Weinan Lin, Guozhen Wang and Zhouli Xu, which still needs to be peer reviewed.
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
(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.
[1] 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
, and a homomorphism from the 14th stable homotopy group of spheres onto