Kervaire invariant
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In mathematics, the Kervaire invariant, named for Michel Kervaire, is defined in geometric topology. It is an invariant of an n-dimensional framed differentiable manifold M for n=4m+2, taking values in the 2-element group
- Z/2Z = {0,1}.
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)
Z/2Z.
The quadratic form q can be defined by algebraic topology using functional Steenrod squares, and geometrically via the self-intersections of immersions S2m + 1 M4m + 2 determined by the framing, or by the triviality/non-triviality of the normal bundles of embeddings S2m + 1 M4m + 2 (for 
) and the mod 2 Hopf invariant of maps 
(for m = 0,1,3).
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 πn + 2(Sn) = Z / 2Z of maps Sn + 2 Sn (for 
), which is the cobordism group of surfaces embedded in Sn + 2 with trivialized normal bundle.
Kervaire (1960) used his invariant for n=10 to construct a 10-dimensional PL manifold with no differentiable structure, the first example of such a manifold.
William Browder (1969) proved that the Kervaire invariant is zero for manifolds of dimension n
2k − 2. The Kervaire invariant is nonzero for some manifold of dimension n=2k − 2 for
- n = 2, 6, 14, 30, 62 (Barratt, Jones & Mahowald 1984).
The question of in which dimensions n there are n-dimensional framed manifolds of non-zero Kervaire invariant is called the Kervaire invariant problem. On 21 April 2009 during the Edinburgh conference to celebrate the 80th birthday of Michael Atiyah, Michael Hopkins announced that in joint work with Mike Hill and Doug Ravenel he obtained a solution to the problem, except possibly for n=126: there are no n-dimensional framed manifolds with Kervaire invariant 1 for n > 126.
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.
[edit] 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:, MR0810962
- Browder, W. (1969), "The Kervaire invariant of framed manifolds and its generalization", Ann. of Math. 90: 157–186, doi:, http://links.jstor.org/sici?sici=0003-486X%28196907%292%3A90%3A1%3C157%3ATKIOFM%3E2.0.CO%3B2-W
- Browder, W. (1972), Surgery on simply-connected manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete, 65, New York-Heidelberg: Springer, MR0358813, ISBN 978-0387056296
- A.V. Chernavskii (2001), "Arf invariant", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104
- Kervaire, M. (1960), "A manifold which does not admit any differentiable structure", Comm. Math. Helv. 34: 257–270, doi:, http://retro.seals.ch/digbib/view?did=c1:391766&sdid=c1:392119
- 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, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104
- Shtan'ko, M.A. (2001), "Kervaire-Milnor invariant", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104
- Snaith, Victor P. (2009), Stable homotopy around the Arf-Kervaire invariant, Progress in Mathematics, 273, Birkhäuser Verlag, MR2498881, ISBN 978-3-7643-9903-0, http://www.springerlink.com/content/978-3-7643-9903-0
