Homology, Homotopy and Applications

Volume 18 (2016)

Number 2

On nontriviality of certain homotopy groups of spheres

Pages: 337 – 344

DOI: https://dx.doi.org/10.4310/HHA.2016.v18.n2.a18

Authors

Sergei O. Ivanov (Chebyshev Laboratory, St. Petersburg State University, St. Petersburg, Russia)

Roman Mikhailov (Chebyshev Laboratory, St. Petersburg State University, St. Petersburg, Russia; and Steklov Mathematical Institute, St. Petersburg, Russia)

Jie Wu (Department of Mathematics, National University of Singapore)

Abstract

We provide an alternative proof of Gray’s result that, for an odd prime $p$, there is a non-trivial $\mathbb{Z} / p$-component in the homotopy group $\pi_{(2p-2)n+1}(S^3)$. As a corollary, it follows that, for $n \geq 2$, the homotopy groups $\pi_n(S^2)$ are non-zero.

Keywords

homotopy group, Lambda algebra, Toda element

2010 Mathematics Subject Classification

Primary 55Q40. Secondary 55T15.

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Published 29 November 2016