Kervaire invariant

In mathematics, 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.

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 $L_{4k+2}$, and thus analogous to the other invariants from L-theory: the signature, a $4k$-dimensional invariant (either symmetric or quadratic, $L^{4k}\cong L_{4k}$), and the De Rham invariant, a $(4k+1)$-dimensional symmetric invariant $L^{4k+1}$.

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. 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. Xu stated that $h_{6}^{2}$ survives so that there exists a manifold of Kervaire invariant 1 in dimension 126. Xu, Zhouli (May 30, 2024). "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). "On the Last Kervaire Invariant Problem".).

Definition

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

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

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

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

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

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

Examples

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

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 $2^{k}-2$ (two less than a power of two). The question is almost completely resolved: there are manifolds with nonzero Kervaire invariant in dimension 2, 6, 14, 30, 62, and none in all other dimensions other than possibly 126. 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 $2^{k}-2$, and those of Michael A. Hill, Michael J. Hopkins, and Douglas C. Ravenel (2016), who showed that there were no such manifolds for $k\geq 8$ ($n\geq 254$). 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 $\mathbf {O} P^{2}$ (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.

History

• Kervaire (1960) proved that the Kervaire invariant is zero for manifolds of dimension 10, 18
• Kervaire & Milnor (1963) proved that the Kervaire invariant can be nonzero for manifolds of dimension 6, 14
• Anderson, Brown & Peterson (1966) proved that the Kervaire invariant is zero for manifolds of dimension 8n+2 for n>1
• Mahowald & Tangora (1967) proved that the Kervaire invariant can be nonzero for manifolds of dimension 30
• Browder (1969) proved that the Kervaire invariant is zero for manifolds of dimension n not of the form 2^k − 2.
• Barratt, Jones & Mahowald (1984) showed that the Kervaire invariant is nonzero for some manifold of dimension 62. An alternative proof was given later by Xu (2016).
• Hill, Hopkins & Ravenel (2016) showed that the Kervaire invariant is zero for n-dimensional framed manifolds for n = 2^k− 2 with k ≥ 8. They constructed a cohomology theory Ω with the following properties from which their result follows immediately:
  • The coefficient groups Ωⁿ(point) have period 2⁸ = 256 in n
  • The coefficient groups Ωⁿ(point) have a "gap": they vanish for n = -1, -2, and -3
  • The coefficient groups Ωⁿ(point) can detect non-vanishing Kervaire invariants: more precisely if the Kervaire invariant for manifolds of dimension n is nonzero then it has a nonzero image in Ω⁻ⁿ(point)

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 $\mathbb {Z} /2\mathbb {Z}$, and a homomorphism from the 14th stable homotopy group of spheres onto $\mathbb {Z} /2\mathbb {Z}$. For n = 2, 6, 14 there is an exotic framing on $S^{n/2}\times S^{n/2}$ with Kervaire–Milnor invariant 1.

See also

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

References

• Anderson, Donald W.; Brown, Edgar H. Jr.; Peterson, Franklin P. (January 1966). "SU-corbodism, KO-characteristic Numbers, and the Kervaire Invariant". Annals of Mathematics. Second Series. 83 (1). Mathematics Department, Princeton University: 54–67. doi:10.2307/1970470. JSTOR 1970470.
• Barratt, Michael G.; Jones, J. D. S.; Mahowald, Mark E. (1984). "Relations amongst Toda brackets and the Kervaire invariant in dimension 62". Journal of the London Mathematical Society. 2. 30 (3): 533–550. CiteSeerX 10.1.1.212.1163. doi:10.1112/jlms/s2-30.3.533. MR 0810962.
• Browder, William (1969). "The Kervaire invariant of framed manifolds and its generalization". Annals of Mathematics. 90 (1): 157–186. doi:10.2307/1970686. JSTOR 1970686.
• Browder, William (1972), Surgery on simply-connected manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 65, New York-Heidelberg: Springer, pp. ix+132, ISBN 978-0-387-05629-6, MR 0358813
• Chernavskii, A.V. (2001) [1994], "Arf invariant", Encyclopedia of Mathematics, EMS Press
• Hill, Michael A.; Hopkins, Michael J.; Ravenel, Douglas C. (2016). "On the nonexistence of elements of Kervaire invariant one". Annals of Mathematics. 184 (1): 1–262. arXiv:0908.3724. doi:10.4007/annals.2016.184.1.1.
• Kervaire, Michel A. (1960). "A manifold which does not admit any differentiable structure". Commentarii Mathematici Helvetici. 34: 257–270. doi:10.1007/bf02565940. MR 0139172. S2CID 120977898.
• Kervaire, Michel A.; Milnor, John W. (1963). "Groups of homotopy spheres: I" (PDF). Annals of Mathematics. 77 (3): 504–537. doi:10.2307/1970128. JSTOR 1970128. MR 0148075.
• Mahowald, Mark E.; Tangora, Martin (1967). "Some differentials in the Adams spectral sequence". Topology. 6 (3): 349–369. doi:10.1016/0040-9383(67)90023-7. MR 0214072.
• Miller, Haynes (2012) [2011], Kervaire Invariant One (after M. A. Hill, M. J. Hopkins, and D. C. Ravenel), Seminaire Bourbaki, arXiv:1104.4523, Bibcode:2011arXiv1104.4523M
• Milnor, John W. (2011), "Differential topology forty-six years later" (PDF), Notices of the American Mathematical Society, 58 (6): 804–809
• Rourke, Colin P.; Sullivan, Dennis P. (1971), "On the Kervaire obstruction", Annals of Mathematics, (2), 94 (3): 397–413, doi:10.2307/1970764, JSTOR 1970764
• Shtan'ko, M.A. (2001) [1994], "Kervaire invariant", Encyclopedia of Mathematics, EMS Press
• Shtan'ko, M.A. (2001) [1994], "Kervaire-Milnor invariant", Encyclopedia of Mathematics, EMS Press
• Snaith, Victor P. (2009), Stable homotopy around the Arf-Kervaire invariant, Progress in Mathematics, vol. 273, Birkhäuser Verlag, doi:10.1007/978-3-7643-9904-7, ISBN 978-3-7643-9903-0, MR 2498881
• Snaith, Victor P. (2010), The Arf-Kervaire Invariant of framed manifolds, arXiv:1001.4751, Bibcode:2010arXiv1001.4751S
• Xu, Zhouli (2016), "The Strong Kervaire invariant problem in dimension 62", Geometry & Topology, 20 (3): 1611–1624, arXiv:1410.6199, doi:10.2140/gt.2016.20.1611, MR 3523064
• Lin, Weinan; Wang, Guozhen; Xu, Zhouli, On the Last Kervaire Invariant Problem, arXiv:2412.10879

External links

• 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

Popular news stories

• Hypersphere Exotica: Kervaire Invariant Problem Has a Solution! A 45-year-old problem on higher-dimensional spheres is solved–probably, by Davide Castelvecchi, August 2009 Scientific American
• Ball, Philip (2009). "Hidden riddle of shapes solved". Nature. doi:10.1038/news.2009.427.
• Mathematicians solve 45-year-old Kervaire invariant puzzle, Erica Klarreich, 20 Jul 2009