On the nonexistence of elements of Kervaire invariant one

Abstract

We show that the Kervaire invariant one elements $\theta_{j}\in\pi_{2^{j+1}-2}S^{0}$ exist only for $j\le 6$. By Browder’s Theorem, this means that smooth framed manifolds of Kervaire invariant one exist only in dimensions $2$, $6$, $14$, $30$, $62$, and possibly $126$. Except for dimension $126$ this resolves a longstanding problem in algebraic topology.