Self-pair homotopy equivalences related to co-variant functors (Q6545074)

This manuscript explores self-homotopy equivalences within the framework of the standard arrow category of pointed topological spaces, employing the definition of homotopy equivalences as presented in [\textit{K.-Y. Lee}, J. Korean Math. Soc. 43, No. 3, 491--506 (2006; Zbl 1098.55005)]. The authors focus on analyzing subgroups of self-homotopy equivalences of specific maps and organized according to ordered families of covariant group-valued functors.\N\NWithin this context, given a pointed map \(\alpha\colon X\to Y\), an integer \(n,\) and an ordered family of covariant group-valued functors \(\mathcal{F}=\{ F_k\colon HoTop_*\to Gr\,\mid\, k\in\mathbb{Z}\}\), the authors define the subgroups \(\mathcal{E}_\mathcal{F}^n(\alpha; Id_X)\leq\mathcal{E}_\mathcal{F}^n(\alpha)\leq \mathcal{E}(\alpha)\) as follows (Definiton 3.5):\N\[\N\mathcal{E}_\mathcal{F}^n(\alpha)=\{ [f_X,f_Y]\in \mathcal{E}(\alpha)\,\mid\, F_k(f_X)=F_k(Id_X),\text{ and } F_k(f_Y)=F_k(Id_Y),\text{ for } 0\leq k\leq n\},\N\]\N\[\N\mathcal{E}_\mathcal{F}^n(\alpha, Id_X)=\{ [f_X,f_Y]\in \mathcal{E}(\alpha)\,\mid\,f_X=Id_X,\text{ and } F_k(f_Y)=F_k(Id_Y),\text{ for } 0\leq k\leq n\}.\N\]\NThese subgroups generalize familiar concepts from the study of self-homotopy equivalences. For instance:\N\begin{itemize}\N\item When \(\mathcal{F}\) consists of homotopy group functors, \(\mathcal{E}_\mathcal{F}^n(X \to *)\) corresponds to the subgroup of self-homotopy equivalences of \(X\) that induce the identity on homotopy groups up to dimension \(n\), denoted as \(\mathcal{E}_\sharp^n(X)\).\N\N\item When \(\mathcal{F}\) consists of homology group functors, \(\mathcal{E}_\mathcal{F}^n(X \to *)\) becomes \(\mathcal{E}_*^n(X)\), the subgroup preserving homology groups up to dimension \(n\).\N\end{itemize}\N\N\NA key structural result is the exact sequence (Corollary 3.6):\N\[\N1\to \mathcal{E}_\mathcal{F}^n(\alpha, Id_X) \to \mathcal{E}_\mathcal{F}^n(\alpha) \to \mathcal{E}_\mathcal{F}^n(X\to *).\tag{1}\N\]\NBuilding on this, the authors derive a split short exact sequence (Theorem 3.7) for the standard inclusion \(i: X \to X \diamond Y\), where \(X \diamond Y\) denotes either the product or wedge of pointed spaces, and the functors satisfy \(F_k(X \diamond Y) = F_k(X) \times F_k(Y)\):\N\[\N1 \to \mathcal{E}_\mathcal{F}^n(i, Id_X) \to \mathcal{E}_\mathcal{F}^n(i) \to \mathcal{E}_\mathcal{F}^n(X \to *) \to 1.\N\]\NIn the final section, the manuscript uses this splitting sequence to compute specific examples of \(\mathcal{E}_\mathcal{F}^n(i)\) for cases where \(i\) is the standard inclusion of a factor into a wedge of Moore spaces, and \(\mathcal{F}\) consists of homology group functors.