Self-closeness numbers of product spaces (Q6042773)

For connected based spaces \(X\) and \(Y\), let \([X,Y]\) denote the set of homotopy classes of based maps from \(X\) to \(Y\), and let \(\mathcal{E}(X)\subset [X,X]\) denote the group of homotopy classes of self-homotopy equivalences of \(X\). For each integer \(n\geq 0\), let \(\mathcal{A}^n_{\#}(X)\subset [X,X]\) denote the subset consisting of all \(f\in [X,X]\) such that \(f_*:\pi_k(X)\stackrel{\cong}{\longrightarrow}\pi_k(X)\) is an isomorphism for any \(k\leq n\). Let \(N\mathcal{E}(X)\) denote \textit{the self-closeness number} of \(X\) defined by \(N\mathcal{E}(X)=\min \{n:\mathcal{A}^n_{\#}(X)=\mathcal{E}(X)\}.\) From now on, let \(m\geq 2\) be a fixed integer and let \(X_1,\dots ,X_m\) be connected based CW complexes. For each \(1\leq k\leq m\), let \(p_k:X_1\times \dots \times X_m\) be the canonical projection. When \(f\in [X_1\times \dots \times X_m,X_1\times \dots \times X_m]\) and \(f_k=p_k\circ f\), we write \(f=(f_1,\dots ,f_m)\). Then \(\mathcal{A}^n_{\#}(X_1\times \dots \times X_m)\) is said to be \textit{reducible} if \(f=(f_1,\dots ,f_m)\in \mathcal{A}^n_{\#}(X_1\times \dots \times X_m)\) implies that the self-map \((f_1,\dots ,f_m)\) with one component (and by induction any number of components) \(f_k\) replaced by \(p_k\) belongs to \(\mathcal{A}^n_{\#}(X_1\times \dots \times X_m)\). In this paper, the author studies the the self-closeness number \(N\mathcal{E}(X)\) when \(X=X_1\times \dots \times X_m\). We denote by \(N=\max\{N\mathcal{A}^n_{\#}(X_k): 1\leq k\leq m\}\). Then he proves that if \(\mathcal{A}^N_{\#}(X_1\times \dots \times X_m)\) is reducible, then \(\mathcal{A}^N_{\#}(X_1\times \dots \times X_m) =\mathcal{E}(X_1\times \dots \times X_m)\) and \(N=N\mathcal{E}(X_1\times \dots \times X_m)\). As an application, he also determines self-closeness numbers of product spaces of some special spaces, such as Moore spaces, Eilenberg-MacLane spaces or atomic spaces. He gives a series of criteria for the reducibility and he obtains the above results by using them.