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

  title={The triviality of the 61-stem in the stable homotopy groups of spheres},
  author={Guozhen Wang and Zhouli Xu},
  journal={arXiv: Algebraic Topology},
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… 

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