Order of the identity of the stable summands of \(\Omega^{2k}S^{2n+1}\) (Q1891181)

Let \(D_ j (\Omega ^{2k} S^{2n+1})\) denote summands in Snaith's stable decomposition of \(\Omega ^{2k} S^{2n+1}\). The main result is that \(p^ e\) times the stable identity map of the \(p\)-localization of \(D_{p^ m} (\Omega^{2k} S^{2n+1})\) is null, where \(e\) is approximately \(m+k\) if \(p\) is odd or \(k=2\), and \(m+ {3\over 2} k\) if \(p=2\) and \(k>2\). Bounds for the stable order of \(D_ j (\Omega^{2k} S^{2n+1})\) when \(j\) is not a \(p\)-power follow by a smash product decomposition. The proof of the main result is obtained by stabilizing factorizations through \(\Omega^{2k-2} S^{2n-1}\) of power maps of \(\Omega^{2k} S^{2n+1}\), and using induction.