SERGEI O. IVANOV, ROMAN MIKHAILOV, AND JIE WU have recently(2nd June 2015) published a paper in arxive giving a proof that for for $n\geq2$, $\pi_n(S^2)$ is non-zero. You can look at it in the following link.
http://arxiv.org/pdf/1506.00952.pdf Sergei O. Ivanov, Roman Mikhailov, Jie Wu, On nontriviality of homotopy groups of spheres, arXiv:1506.00952