Adams-Hilton model and the group of self-homotopy equivalences of a simply connected cw-complex

Abstract

Let $R$ be a principal ideal domain (PID). For a simply connected CW-complex $X$ of dimension $n$, let $Y$ be a space obtained by attaching cells of dimension $q$ to $X$, $q>n$, and let $A(Y)$ denote an Adams-Hilton model of $Y$. If $\mathcal{E}(A(Y))$ denotes the group of homotopy self-equivalences of $A(Y)$ and ${\mathcal{E}}_{\ast }(A(Y))$ its subgroup of the elements inducing the identity on $H_{\ast }(Y,R)$, then we construct two short exact sequences: $$\underset{i}{\oplus}\,H_{q}(ΩX,R)\rightarrowtail \mathcal{E}(A(Y))\overset{}{ \twoheadrightarrow}Γ^{q}_{n}\,\,\,\,\,\,\,\,\,\,\,\,,\,\,\,\,\,\,\,\,\,\,\,\,\underset{i}{\oplus}\,H_{q}(ΩX,R) \rightarrowtail \E_{*}(A(Y))\overset{}{ \twoheadrightarrow}Π^{q}_{n} $$ where $i=\mathrm{rank}{H}_q(Y,X;R)$, ${\Gamma }_n^q$ is a subgroup of $\aut(\mathrm{Hom}_{}(H_{q}( Y,X;R))\times \E(A(X))$ and ${\Pi }_n^q$ is a subgroup of ${\mathcal{E}}_{\ast }(A(X))$.