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

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 $X$, $q \gt n$, 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 ${\mathcal E}_{*}(A(Y))$ its subgroup formed of the elements inducing the identity on $H_{*} (Y,R)$, then we construct two short exact sequences:\begin{align*}& \underset{i} {\oplus} H_{q} (\Omega X,R) \rightarrowtail {\mathcal E} (A(Y)) \overset{} {\twoheadrightarrow} \Gamma^{q}_{n}, \\& \underset{i} {\oplus} H_{q} (\Omega X,R) \rightarrowtail {\mathcal E}_{*} (A(Y)) \overset{} {\twoheadrightarrow} \Pi^{q}_{n},\end{align*}where $i=\mathrm{rank} \,H_{q} (Y,X;R)$, $\Pi^{q}_{n}$ is a subgroup of ${\mathcal E}_{*} (A(X))$ and $\Gamma^{q}_{n}$ is a subgroup of $\mathrm{aut} (\mathrm{Hom} (H_{q}(Y, X;R))) \times {\mathcal E}(A(X))$.

Keywords

group of homotopy self-equivalences, Adams–Hilton model, Anick model, loop space

2010 Mathematics Subject Classification

55P10

Full Text (PDF format)

Received 9 June 2018

Received revised 14 November 2018

Accepted 29 March 2019

Published 5 June 2019