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

Guozhen Wang¹ ; Zhouli Xu²

¹ Shanghai Center for Mathematical Sciences, Fudan University, Shanghai, China, 200433 and Department of Mathematics, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark
² Department of Mathematics, The University of Chicago, Chicago, IL

Annals of mathematics, Tome 186 (2017) no. 2, pp. 501-580.

Résumé

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.

Détail

DOI : 10.4007/annals.2017.186.2.3