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

TL;DR

This paper proves the 2-primary homotopy group _{61} of spheres is zero, establishing the last odd-dimensional sphere with a unique smooth structure and introducing a new technique for computing Adams differentials.

Contribution

It demonstrates the triviality of _{61} and introduces a novel method using algebraic and geometric Kahn-Priddy theorems to compute Adams differentials.

Findings

_{61} is zero

The 61-sphere has a unique smooth structure

New technique for Adams differential computation

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 $\langle 2, \theta_4, \theta_4, 2\rangle$. 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.