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

Abstract

We prove that the 2-primary 61 is zero.As a consequence, the Kervaire invariant element 5 is contained in the strictly defined 4-fold Toda bracket 2, 4 , 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.Contents 1 6.Two lemmas on Atiyah-Hirzebruch differentials 7. The cofiber of 8.The Adams spectral sequence of X 9.The Adams spectral sequence of X 10.The pull back 11.A homotopy relation 12. Another homotopy relation and the Adams differential d 5 (A ) = h 1 B 21 13.Appendix I 14.