The Adams-Hilton model and the group of self-homotopy equivalences of a simply connected CW-complex (Q2326321)

First, recall the classical work due to Adams-Hilton [\textit{J. F. Adams} and \textit{P. J. Hilton}, Comment. Math. Helv. 30, 305--330 (1956; Zbl 0071.16403)]. If \(Y\) is a simply connected CW-complex, it follows from Adams-Hilton that there is a morphism of differential graded algebras (which is called the Adams-Hilton model of \(Y\)) \[ \Theta_Y:(\mathbb{T}V,d)\to C_*(\Omega Y) \] such that \(\Theta_Y\) is a quasi-isomorphism (i.e. it induces an isomorphism on homology) and that it restricts to a quasi-isomorphism \((\mathbb{T}V^{\otimes\leq m},d)\to C_*(\Omega Y^{m+1})\) for each \(n\), where \(Y^{m+1}\) denotes the \((m+1)\)-skeleton of \(Y\), and \(\mathbb{T}V\) denotes the free tensor algebra on a free graded \(\mathbb{Z}\)-module \(V\). We denote by \(A(Y)\) the Adams-Hilton model of \(Y\). Note that there is a reasonable concept of homotopy among chain algebra morphisms and let \(\mathcal{E}(A(Y))\) denote the group of homotopy self-equivalences of the chain algebra morphism \(A(Y)\). Similarly, let \(\mathcal{E}_*(A(Y))\) denote its subgroup whose elements induce the identity on the homology \(H_*(Y;\mathbb{Z})\). Now let \(X\) be a simply connected \(n\)-dimensional CW-complex and let \(Y=X\cup_{\alpha}\big(\bigcup_{j\in J}e_j^q\big)\) be the space obtained by attaching cells of dimension \(q\) to \(X\) by a map \(\alpha :\vee_{j\in J}S^{q-1}\to X\) (\(n<q)\). In this situation, since \(n<q\), \(Y\) is simply connected and the group \(\mathcal{E}(A(Y))\) is well-defined. Then the author considers the group \(\mathcal{E}(A(Y))\) and proves that there are two Barcus-Barratt type short exact sequences \[ \begin{tikzcd} 1 \ar[r] &H_q(\Omega X;R)^{\oplus i}\ar[r] &\mathcal{E}(A(Y)) \ar[r] &\Gamma^q_n\ar[r] &1 \\ 1 \ar[r] &H_q(\Omega X;R)^{\oplus i} \ar[r] &\mathcal{E}_*(A(Y)) \ar[r] &\Pi^q_n \ar[r] &1 \end{tikzcd} \] where \(R\) is a principal ideal domain, \(i=\operatorname{rank}H_q(Y,X;R)=\operatorname{card}(J)\), \(\Pi^q_n\) is a subgroup of \(\Gamma^q_n\), and \(\Gamma^q_n\) is a certain subgroup of \(\mathrm{aut}(\mathrm{Hom}(H_q(Y,X;R)))\times \mathcal{E}(A(X))\).