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, \indent China, 200433 {\rm and}
Department of Mathematics, University of Copenhagen,\hfill\break\indent 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