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

Guozhen Wang Shanghai Center for Mathematical Sciences; University of Copenhagen Zhouli Xu Massachusetts Institute of Technology

Algebraic Topology and General Topology mathscidoc:1705.02002

Annals of Mathematics, 186, (2), 501-580, 2017
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.
Adams spectral sequences, Atiyah-Hirzebruch spectral sequences, Stable homotopy groups, cell diagrams, smooth structures
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@inproceedings{guozhen2017the,
  title={The triviality of the 61-stem in the stable homotopy groups of spheres},
  author={Guozhen Wang, and Zhouli Xu},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170530085415416484755},
  booktitle={Annals of Mathematics},
  volume={186},
  number={2},
  pages={501-580},
  year={2017},
}
Guozhen Wang, and Zhouli Xu. The triviality of the 61-stem in the stable homotopy groups of spheres. 2017. Vol. 186. In Annals of Mathematics. pp.501-580. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170530085415416484755.
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