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.

67 pages, minor changes, accepted version

References in corpus (3)

• Classical and motivic Adams charts
• Classical and motivic Adams-Novikov charts
• The algebraic Atiyah-Hirzebruch spectral sequence of real projective spectra