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 4n + 2 dimensional almost-parallelizable smooth manifolds M, taking values in the 2-element group

Z/2Z.

It is equal to the Arf invariant of the quadratic form on the homology group

H_2n+1(M).

W. Browder (1969) proved that the Kervaire invariant vanishes unless n + 2 is a power of 2. It is nonzero for some manifold of dimension 2^k − 2 for

k = 1, 2, 3, 4, 5, 6 (Barratt, Jones & Mahowald 1984);

the question of in which dimensions there are manifolds of non-zero Kervaire invariant is called the Kervaire invariant problem. Kervaire (1960) used his invariant in 10 dimensions to find the first example of a PL manifold with no smooth structure.

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.

[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., MR0810962 
• Browder, W. B. (1969), “The Kervaire invariant of framed manifolds and its generalization”, Ann. of Math. 90: 157–186, <http://links.jstor.org/sici?sici=0003-486X%28196907%292%3A90%3A1%3C157%3ATKIOFM%3E2.0.CO%3B2-W> 
• Browder, W.B. (1972), Surgery on simply-connected manifolds, vol. 65, Ergebnisse der Mathematik und ihrer Grenzgebiete, New York-Heidelberg: Springer, MR0358813, ISBN 978-0387056296 
• Kervaire, M. (1960), “A manifold which does not admit any differentiable structure”, Comm. Math. Helv. 34: 257–270, <http://retro.seals.ch/digbib/view?did=c1:391766&sdid=c1:392119> 
• 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