Periodic problem on homotopy groups of Chang complexes \(C^{n+2,r}_r\) (Q2326322)

\textit{J.-P. Serre} [Ann. Math. (2) 54, 425--505 (1951; Zbl 0045.26003)] showed that, \(p\)-locally for \(p\) an odd prime, \(\Omega\mathbb{S}^{2n}\) has the homotopy type of the product of \(\mathbb{S}^{2n-1}\) and \(\Omega\mathbb{S}^{4n-1}\). This decomposition reduces the computation of the odd \(p\)-primary part of the homotopy groups of spheres to that of odd dimensional spheres. Further, \textit{J. Wu} [Topology 37, No. 5, 1011--1023 (1998; Zbl 1011.55008)] gave a decomposition of \(\Omega\Sigma X\) using the free Lie power functor \(L_n\). The class of elementary Chang complexes \(C^{n+2,r}_r\) for \(n\ge 3\), was found by \textit{S.-C. Chang} [Proc. R. Soc. Lond., Ser. A 202, 253--263 (1950; Zbl 0041.10201)] when he classified indecomposable homotopy types in \(\mathbf{A}^2\) with \(n\ge 3\), where \(\mathbf{A}^k_n\) is the homotopy category consisting of \((n-1)\)-connected finite \(CW\)-complexes with dimension less than or equal to \(n+k\) with \(n\ge k+1\). The first and third author [\textit{Z. Zhu} and \textit{J. Pan}, Homology Homotopy Appl. 19, No. 1, 293--318 (2017; Zbl 1383.55004)] studied the decomposability of smash products of indecomposable complexes in \(\mathbf{A}^2_n\) with \(n\ge 3\). Since the decomposition of the \(3\)-fold self smash product of any \(\mathbf{A}^2_n\)-complex has been completely determined and \(\Sigma^\ast C^{n+2,r}_r\) is a wedge summand of \(L_3(C^{n+2,r}_r)\), the authors concentrate on the study of \(C^{n+2,r}_r\). The main result stated in Theorem 1.1 describes a product decomposition of \(\Omega\Sigma C^{n+2,r}_r\) for \(n\ge 3\). As an application, it is shown that the stable homotopy group \(\pi^s_k(C^{n+3,r}_r)\) is a direct summand of \(\pi_{k+(p-1)(n+1)}(C^{n+3,r}_r)\) for \(n\ge 3\) provided \(p\ge 3\) is an odd integer with \(p\ge \frac{k-n+1}{n+1}\).