The decomposability of a smash product of \(\mathbf{A}^2_n\)-complexes (Q1688703)

Let \(\mathbf{A}^k_n\) \((n\geq k+1)\) be the homotopy category consisting of \((n-1)\)-connected finite CW complexes of dimension at most \(n+k\). Recall that any complex in \(\mathbf{A}^k_n\) is a suspension (and thus a co-\(H\)-space). In this paper, the authors study the classification problem of \(\mathbf{A}_n^k\) for the case \(k=2\). Since the suspension functor \(\Sigma :\mathbf{A}^2_n\to \mathbf{A}_{n+1}^2\) is an equivalence for \(n\geq 3\), this problem reduces to considering the case \(n=3\). In particular, the authors determine the decomposability of the smash product of two indecomposable \(\mathbf{A}^2_3\) complexes and they give the explicit decomposition whenever possible.