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

By Guozhen Wang, Zhouli Xu. 2016

49 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/eb5e22aad30edfaadc23464768dfc665059abf1b

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

  1. Bogdan Gheorghe
  2. Zhouli Xu
  3. Guozhen Wang

In Mathematical Physics

  1. Bogdan Gheorghe
  2. Zhouli Xu
  3. Guozhen Wang

In Statistical and Nonlinear Physics

  1. Bogdan Gheorghe
  2. Zhouli Xu
  3. Guozhen Wang

Citations in the corpus