Relations in the 24-th homotopy groups of spheres

The main purpose of this note is to give a proof of the fact that the Toda brackets $$and$$ are not trivial. This is an affirmative answer of the second author's Conjecture (Determination of the $P$-image by Toda brackets, Geometry and Topology Monographs 13(2008), 355-383). The second purpose is to show the relation $\bar{\nu}_7\omega_{15}=\nu_7\sigma_{10}\kappa_{17}$ in $\pi^7_{31}$.Comment: 27 page

Determination of the order of the P-image by Toda brackets

The present paper gives a proof of the author's paper in the Proceeding of the International Conference on Homotopy Theory and Related Topics at Korea University (2005), 109--113, on the orders of Whitehead products of iota_n with alpha in pi^n_{n+k}, (n > k+1, k < 25) and improves and extends it. The method is to use composition methods in the homotopy groups of spheres and rotation groups.Comment: This is the version published by Geometry & Topology Monographs on 19 March 200

Gottlieb groups of spheres

This paper takes up the systematic study of the Gottlieb groups G(n+k)(S-n) of spheres for k = 4.ArticleTopology. 47(6):399-430 (2008)journal articl