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# Homology, Homotopy and Applications

## Volume 18 (2016)

### Number 2

### On nontriviality of certain homotopy groups of spheres

Pages: 337 – 344

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

#### Authors

#### 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.