Kervaire invariant

In mathematics, the Kervaire invariant, named for Michel Kervaire, is defined in geometric topology. It is an invariant of a (4k+2)-dimensional (singly even-dimensional) framed differentiable manifold M, taking values in the 2-element group Z/2Z = {0,1}. 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: 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 : H_2m+1(M;Z/2Z) $\to$ 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 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 (= 2-dimensional manifold with stably trivialized tangent bundle) which was used by Pontryagin in 1950 to compute of the homotopy group π_{n + 2}(Sⁿ) = Z / 2Z of maps S^{n + 2} $\to$ Sⁿ (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.

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 non-zero Kervaire invariant is called the Kervaire invariant problem. This is only possible if n is 2 mod 4.

• 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 zero 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.
• Hill, Hopkins & Ravenel (2009) 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, 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)

Together these results imply that there are manifolds with nonzero Kervaire invariant manifolds in dimension 2, 6, 14, 30, 62, and none in all other dimensions other than 126. As of 2010 the case of dimension 126 is still open.

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 S^n/2 x S^n/2 with Kervaire-Milnor invariant 1.

References

• Barratt, M. G.; Jones, J. D. S.; Mahowald, M. E. (1984), "Relations amongst Toda brackets and the Kervaire invariant in dimension 62", J. London Math. Soc. (2) 30 (3): 533–550., doi:10.1112/jlms/s2-30.3.533, MR0810962 
• Browder, W. (1969), "The Kervaire invariant of framed manifolds and its generalization", Ann. Of Math. 90 (1): 157–186, doi:10.2307/1970686, JSTOR 1970686 
• Browder, W. (1972), Surgery on simply-connected manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete, 65, New York-Heidelberg: Springer, pp. ix+132, ISBN 978-0387056296, MR0358813 
• Chernavskii, A.V. (2001), "Arf invariant", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1556080104, http://www.encyclopediaofmath.org/index.php?title=A/a013230 
• Hill, Michael A.; Hopkins, Michael J.; Ravenel, Douglas C. (2009). "On the non-existence of elements of Kervaire invariant one". arXiv:0908.3724 [math.AT]. 
• Kervaire, M. (1960), "A manifold which does not admit any differentiable structure", Comm. Math. Helv. 34: 257–270, doi:10.1007/BF02565940, http://retro.seals.ch/digbib/view?did=c1:391766&sdid=c1:392119 
• Mahowald, Mark; Tangora, Martin (1967), "Some differentials in the Adams spectral sequence", Topology 6 (3): 349–369, doi:10.1016/0040-9383(67)90023-7, MR0214072 
• Milnor, John W. (2011), "Differential topology forty-six years later", Notices of the American Mathematical Society 58 (6): 804–809, http://www.ams.org/notices/201106/rtx110600804p.pdf 
• Rourke, C. P. and Sullivan, D. P., On the Kervaire obstruction, Ann. of Math. (2) 94, 397—413 (1971)
• Shtan'ko, M.A. (2001), "Kervaire invariant", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1556080104, http://www.encyclopediaofmath.org/index.php?title=K/k055350 
• Shtan'ko, M.A. (2001), "Kervaire-Milnor invariant", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1556080104, http://www.encyclopediaofmath.org/index.php?title=k/k055360 
• Snaith, Victor P. (2009), Stable homotopy around the Arf-Kervaire invariant, Progress in Mathematics, 273, Birkhäuser Verlag, ISBN 978-3-7643-9903-0, MR2498881 
• Snaith, Victor P. (2010), The Arf-Kervaire Invariant of framed manifolds, arXiv:1001.4751 

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