The triviality of the 61-stem in the stable homotopy groups of spheres

Abstract

We prove that the $2$-primary $\pi_{61}$ is zero. As a consequence, the Kervaire invariant element $\theta_5$ is contained in the strictly defined $4$-fold Toda bracket $(2,\theta_4\theta_4,2).

Our result has a geometric corollary: the $61$-sphere has a unique smooth structure, and it is the last odd dimensional case --- the only ones are $S^1$, $S^3$, $S^5$ and $S^{61}$.

Our proof is a computation of homotopy groups of spheres. A major part of this paper is to prove an Adams differential $d_3(D_3) = B_3$. We prove this differential by introducing a new technique based on the algebraic and geometric Kahn-Priddy theorems. The success of this technique suggests a theoretical way to prove Adams differentials in the sphere spectrum inductively by use of differentials in truncated projective spectra.