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

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Authors

Guozhen Wang

Shanghai Center for Mathematical Sciences, Fudan University, Shanghai, China, 200433 and Department of Mathematics, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark

Zhouli Xu

Department of Mathematics, The University of Chicago, Chicago, IL

Current address:

Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA