September 2017 The triviality of the 61-stem in the stable homotopy groups of spheres
Guozhen Wang, Zhouli Xu
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Ann. of Math. (2) 186(2): 501-580 (September 2017). DOI: 10.4007/annals.2017.186.2.3


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.


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Guozhen Wang. Zhouli Xu. "The triviality of the 61-stem in the stable homotopy groups of spheres." Ann. of Math. (2) 186 (2) 501 - 580, September 2017.


Published: September 2017
First available in Project Euclid: 21 December 2021

Digital Object Identifier: 10.4007/annals.2017.186.2.3

Primary: 55Q45

Keywords: Adams spectral sequences , Atiyah-Hirzebruch spectral sequences , cell diagrams , Smooth structures , stable homotopy groups of spheres

Rights: Copyright © 2017 Department of Mathematics, Princeton University


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Vol.186 • No. 2 • September 2017
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