Kervaire invariant

In mathematics, the Kervaire invariant, named for Michel Kervaire, is defined in geometric topology. It is an invariant of 4"n" + 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"_2"n"+1("M").

W. harvtxt|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 harv|Barratt|Jones|Mahowald|1984;

the question of in which dimensions there are manifolds of non-zero Kervaire invariant is called the Kervaire invariant problem. harvtxt|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 anexotic framing on "S"^n/2 x "S"^n/2 with Kervaire-Milnor invariant 1.

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