On the group of self-homotopy equivalences of an elliptic space

Abstract:

Let $X$ be a simply connected rational elliptic space of formal dimension $n$ and let $\mathcal {E}(X)$ denote the group of homotopy classes of self-equivalences of $X$. If $X^{[k]}$ denotes the $k$th Postikov section of $X$ and $X^{k}$ denotes its $k$th skeleton, then making use of the models of Sullivan and Quillen we prove that $\mathcal {E}(X)\cong \mathcal {E}(X^{[n]})$ and if $n>m=\mathrm {max}\big \{k | \pi _{k}(X)\neq 0\big \}$, then $\mathcal {E}(X)\cong \mathcal {E}(X^{m+1})$. Moreover, in case when $X$ is 2-connected, we show that if $\pi _{n}(X)\neq 0$, then the group $\mathcal {E}(X)$ is infinite.