Kervaire Invariant - Kervaire Invariant Problem

The question of in which dimensions n there are n-dimensional framed manifolds of nonzero Kervaire invariant is called the Kervaire invariant problem. This is only possible if n is 2 mod 4, and indeed one must have n is 2k − 2 (two less than a power of two). The question is almost completely resolved; as of 2012 only the case of dimension 126 is open: there are manifolds with nonzero Kervaire invariant in dimension 2, 6, 14, 30, 62, and none in all other dimensions other than possibly 126.

The main results are Browder (1969), which reduced the problem from differential topology to stable homotopy theory and showed that the only possible dimensions are 2k − 2, and Hill, Hopkins & Ravenel (2009), which showed that there were no such manifolds for . Together with explicit constructions for lower dimensions (through 62), this leaves open only dimension 126.

It is conjectured by Michael Atiyah that there is such a manifold in dimension 126, and that the higher-dimensional manifolds with nonzero Kervaire invariant are related to well-known exotic manifolds two dimension higher, in dimensions 16, 32, 64, and 128, namely the Cayley projective plane (dimension 16, octonionic projective plane) and the analogous Rosenfeld projective planes (the bi-octonionic projective plane in dimension 32, the quater-octonionic projective plane in dimension 64, and the octo-octonionic projective plane in dimension 128), specifically that there is a construction that takes these projective planes and produces a manifold with nonzero Kervaire invariant in two dimensions lower.