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

TY - JOUR

T1 - The triviality of the 61-stem in the stable homotopy groups of spheres

AU - Wang, Guozhen

AU - Xu, Zhouli

PY - 2017/9/1

Y1 - 2017/9/1

N2 - We prove that the 2-primary π61 is zero. As a consequence, the Kervaire invariant element θ5 is contained in the strictly defined 4-fold Toda bracket 〈2, θ4, θ4, 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 S1, S3, S5 and S61. Our proof is a computation of homotopy groups of spheres. A major part of this paper is to prove an Adams differential d3(D3) = B3. 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 in- ductively by use of differentials in truncated projective spectra.

AB - We prove that the 2-primary π61 is zero. As a consequence, the Kervaire invariant element θ5 is contained in the strictly defined 4-fold Toda bracket 〈2, θ4, θ4, 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 S1, S3, S5 and S61. Our proof is a computation of homotopy groups of spheres. A major part of this paper is to prove an Adams differential d3(D3) = B3. 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 in- ductively by use of differentials in truncated projective spectra.

U2 - 10.4007/annals.2017.186.2.3

DO - 10.4007/annals.2017.186.2.3

M3 - Journal article

SN - 0003-486X

VL - 186

SP - 501

EP - 580

JO - Annals of Mathematics

JF - Annals of Mathematics

IS - 2

ER -