The Kervaire invariant is an invariant of a framed (4k+2)-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.
Contents
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
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 Z/2Z-coefficient homology group
q : H2m+1(M;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
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 the homotopy group
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 Θn of h-cobordism classes of oriented homotopy n-spheres. They compute this latter in terms of a map
where
Examples
For the standard embedded torus, the skew-symmetric form is given by
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 2k − 2 (two less than a power of two). The question is almost completely resolved; as of 2012 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 Browder (1969), which reduced the problem from differential topology to stable homotopy theory and showed that the only possible dimensions are 2k − 2, and Hill, Hopkins & Ravenel (2016), which showed that there were no such manifolds for
It is 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
History
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 Z/2Z, and a homomorphism from the 14th stable homotopy group of spheres onto Z/2Z. For n = 2, 6, 14 there is an exotic framing on Sn/2 x Sn/2 with Kervaire-Milnor invariant 1.