Kervaire invariant

In mathematics, the Kervaire invariant is an invariant of a framed ${\displaystyle (4k+2)}$-dimensional manifold that measures whether the manifold could be surgically converted into a sphere. This invariant evaluates to 0 if the manifold can be converted to a sphere, and 1 otherwise. This invariant was named after Michel Kervaire who built on work of Cahit Arf.

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 ${\displaystyle L_{4k+2}}$, and thus analogous to the other invariants from L-theory: the signature, a ${\displaystyle 4k}$-dimensional invariant (either symmetric or quadratic, ${\displaystyle L^{4k}\cong L_{4k}}$), and the De Rham invariant, a ${\displaystyle (4k+1)}$-dimensional symmetric invariant ${\displaystyle L^{4k+1}}$.

In any given dimension, there are only two possibilities: either all manifolds have Arf–Kervaire invariant equal to 0, or half have Arf–Kervaire invariant 0 and the other half have Arf–Kervaire invariant 1.

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. On May 30, 2024, Zhouli Xu (in collaboration with Weinan Lin and Guozhen Wang), announced during a seminar at Princeton University that the final case of dimension 126 has been settled. Xu stated that ${\displaystyle h_{6}^{2}}$ survives so that there exists a manifold of Kervaire invariant 1 in dimension 126. Xu, Zhouli (May 30, 2024). "Computing differentials in the Adams spectral sequence".. (https://www.math.princeton.edu/events/computing-differentials-adams-spectral-sequence-2024-05-30t170000)