# 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 (or more generally PL-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. It can be thought of as the simply-connected quadratic L-group and thus analogous to the other invariants from L-theory: the signature, a 4k-dimensional invariant (either symmetric or quadratic, ), and the De Rham invariant, a (4k+1)-dimensional symmetric invariant

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. The final case of dimension 126 remains open.

### Other articles related to "kervaire invariant, kervaire, invariant":

Kervaire Invariant - Kervaire–Milnor Invariant
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The Arf Invariant in Topology - Examples
... The Arf invariant of the framed surface is now defined Note that, so we had to stabilise, taking to be at least 4, in order to get an element of ... The Arf invariant of a framed surface detects whether there is a 3-manifold whose boundary is the given surface which extends the given framing ... structure on our surface.) Pontrjagin used the Arf invariant of framed surfaces to compute the 2-dimensional framed cobordism group, which is generated by the torus with the ...
Signature (topology) - Other Dimensions - Kervaire Invariant
... Such forms do not have a signature invariant if they are non-degenerate, any two such forms are equivalent ... forms need not be equivalent, being distinguished by the Arf invariant ... The resulting invariant of a manifold is called the Kervaire invariant ...
Homotopy Groups Of Spheres - Applications
... n is of the form 2k−2, in which case the image has index 1 or 2 (Kervaire Milnor 1963) ... The Kervaire invariant problem, about the existence of manifolds of Kervaire invariant 1 in dimensions 2k − 2 can be reduced to a question about stable homotopy groups of spheres ... groups of degree up to 48 has been used to settle the Kervaire invariant problem in dimension 26 − 2 = 62 (Barratt, Jones Mahowald 1984) ...