Postnikov decomposition and the group of self-equivalences of a rationalized space

Mahmoud Benkhalifa

Homology, homotopy, and applications, Tome 19 (2017) no. 1, pp. 209-224.

Résumé

Let $X$ be a simply connected rational CW complex of finite type. Write $X^{[n]}$ for the $n\text{th}$ Postnikov section of $X$. Let $\mathcal E(X^{[n+1]})$ denote the group of homotopy self-equivalences of $X^{[n+1]}$. We use Sullivan models in rational homotopy theory to construct two short exact sequences:\[\mathrm{Hom}\big(\pi_{n+1}(X);H^{n+1}(X^{[n]})\big) \rightarrowtail\mathcal{E}(X^{[n+1]}) \twoheadrightarrow D^{n+1}_{n},\]\[\mathrm{Hom}\big(\pi_{n+1}(X);H^{n+1}(X^{[n]})\big) \rightarrowtail\mathcal{E}_{\sharp}(X^{[n+1]}) \twoheadrightarrow G^{n+1}_{n},\]where $D^{n+1}_{n}$ is a subgroup of $\mathrm{aut}(\mathrm{Hom}(\pi_{q}(X) ;\Bbb Q))\times \mathcal{E}(X^{[n]})$ which is defined in terms of the Whitehead exact sequence of $X$ and where $G^{n+1}_{n}$ is a certain subgroup of $\mathcal E_{\sharp}(X^{[n]})$. Here $\mathcal E_{\sharp}(X^{[n]})$ is the subgroup of those elements inducing the identity on the homotopy groups. Moreover, we give an alternative proof of the Costoya–Viruel theorem: Every finite group occurs as $\mathcal E(X)$ where $X$ is rational.

Détail

DOI : 10.4310/HHA.2017.v19.n1.a11

Classification : 55P10
Keywords: group of homotopy self-equivalences, Whitehead exact sequence, Sullivan model, rational homotopy theory, Postnikov tower