On the Last Kervaire Invariant Problem

Abstract

We prove that the element $h_6^2$ is a permanent cycle in the Adams spectral sequence. As a result, we establish the existence of smooth framed manifolds with Kervaire invariant one in dimension 126, thereby resolving the final case of the Kervaire invariant problem. Combining this result with the theorems of Browder, Mahowald--Tangora, Barratt--Jones--Mahowald, and Hill--Hopkins--Ravenel, we conclude that smooth framed manifolds with Kervaire invariant one exist in and only in dimensions $2, 6, 14, 30, 62$, and $126$.