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.

Read more about Kervaire InvariantDefinition, History, Examples, Kervaire Invariant Problem, Kervaire–Milnor Invariant

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Kervaire Invariant - Kervaire–Milnor Invariant
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The Arf Invariant in Topology - Examples
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Signature (topology) - Other Dimensions - Kervaire Invariant
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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) ...