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

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.

Keywords

group of homotopy self-equivalences, Whitehead exact sequence, Sullivan model, rational homotopy theory, Postnikov tower

2010 Mathematics Subject Classification

55P10

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Published 6 June 2017