Kervaire invariant

The Kervaire invariant is the Arf invariant of the quadratic form determined by the framing on the middle-dimensional ${\displaystyle \mathbb {Z} /2\mathbb {Z} }$-coefficient homology group

${\displaystyle q\colon H_{2m+1}(M;\mathbb {Z} /2\mathbb {Z} )\to \mathbb {Z} /2\mathbb {Z} ,}$

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 ${\displaystyle S^{2m+1}}$${\displaystyle \to }$ ${\displaystyle M^{4m+2}}$ determined by the framing, or by the triviality/non-triviality of the normal bundles of embeddings ${\displaystyle S^{2m+1}}$${\displaystyle \to }$ ${\displaystyle M^{4m+2}}$ (for ${\displaystyle m\neq 0,1,3}$) and the mod 2 Hopf invariant of maps ${\displaystyle S^{4m+2+k}\to S^{2m+1+k}}$ (for ${\displaystyle m=0,1,3}$).

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 ${\displaystyle \pi _{n+2}(S^{n})=\mathbb {Z} /2\mathbb {Z} }$ of maps ${\displaystyle S^{n+2}}$ ${\displaystyle \to }$ ${\displaystyle S^{n}}$ (for ${\displaystyle n\geq 2}$), which is the cobordism group of surfaces embedded in ${\displaystyle S^{n+2}}$ 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 ${\displaystyle \Theta _{n}}$ of h-cobordism classes of oriented homotopy n-spheres. They compute this latter in terms of a map

${\displaystyle \Theta _{n}/bP_{n+1}\to \pi _{n}^{S}/J,\,}$

where ${\displaystyle bP_{n+1}}$ is the cyclic subgroup of n-spheres that bound a parallelizable manifold of dimension ${\displaystyle n+1}$, ${\displaystyle \pi _{n}^{S}}$ is the nth stable homotopy group of spheres, and J is the image of the J-homomorphism, which is also a cyclic group. The groups ${\displaystyle bP_{n+1}}$ and ${\displaystyle J}$ have easily understood cyclic factors, which are trivial or order two except in dimension ${\displaystyle n=4k+3}$, 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.

For the standard embedded torus, the skew-symmetric form is given by ${\displaystyle {\begin{pmatrix}0&1\\-1&0\end{pmatrix}}}$ (with respect to the standard symplectic basis), and the skew-quadratic refinement is given by ${\displaystyle xy}$ with respect to this basis: ${\displaystyle Q(1,0)=Q(0,1)=0}$: the basis curves don't self-link; and ${\displaystyle Q(1,1)=1}$: 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.

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 ${\displaystyle 2^{k}-2}$ (two less than a power of two). The question is almost completely resolved; as of 2019 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 William Browder (1969), who reduced the problem from differential topology to stable homotopy theory and showed that the only possible dimensions are ${\displaystyle 2^{k}-2}$, and those of Michael A. Hill, Michael J. Hopkins, and Douglas C. Ravenel (2016), who showed that there were no such manifolds for ${\displaystyle k\geq 8}$ (${\displaystyle n\geq 254}$). 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 ${\displaystyle \mathbf {O} P^{2}}$ (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 ${\displaystyle \mathbb {Z} /2\mathbb {Z} }$, and a homomorphism from the 14th stable homotopy group of spheres onto ${\displaystyle \mathbb {Z} /2\mathbb {Z} }$. For n = 2, 6, 14 there is an exotic framing on ${\displaystyle S^{n/2}\times S^{n/2}}$ with Kervaire–Milnor invariant 1.

• Signature, a 4k-dimensional invariant
• De Rham invariant, a (4k + 1)-dimensional invariant

• Slides and video of lecture by Hopkins at Edinburgh, 21 April, 2009
• Arf-Kervaire home page of Doug Ravenel
• Harvard-MIT Summer Seminar on the Kervaire Invariant
• 'Kervaire Invariant One Problem' Solved, April 23, 2009, blog post by John Baez and discussion, The n-Category Café
• Exotic spheres at the manifold atlas