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

• 成果类型: Article
• 署名作者: Wang, Guozhen; Xu, Zhouli
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2017.186.2.3
• 发表日期: 2017
• 页码: 501-580
• 关键词: kervaire invariant; ELEMENTS; THEOREM

摘要

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.