Gottlieb groups of spheres
Let \(X\) be a connected space. The \(k\)-th Gottlieb group \(G_{k}(X)\) of \(X\) is the subgroup of the \(k\)-th homotopy group \(\pi_{k}(X)\) consisting of all elements which can be represented by a map \(f : S^{k} \rightarrow X\) such that \(f \vee id_{X}: S^{k} \vee X \rightarrow X\) extends (up to homotopy) to a map \(F:S^{k} \times X \rightarrow X\). This paper takes up a systematic study of the Gottlieb groups of spheres, \(G_{n+k}(S^{n})\), for \(k \leq 13\) using classical homotopy theoretic methods. The paper determines these groups except for the \(2\)-primary components in the cases: \(k=9, n= 53; k=11, n=115\). The main tools of the paper are the composition methods developed by Toda in his classic study of the homotopy groups of spheres, and builds on the work of Hoo, Mahowald, and Kervaire. The results are compiled into a series of tables for \(k=1\) up to \(k=13\) and \(n=2\) up to \(n=26\).
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