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 $\E(X)$ denote the group of homotopy classes of self-equivalences of $X$. If $X^{[k]}$ denotes the $k^{\text{th}}$ Postikov section of $X$ and $X^k$ denotes its $k^{\text{th}}$ skeleton, then making use of the models of Sullivan and Quillen we prove that $\E(X)\cong\E(X^{[n]})$ and if $n>m=max\{k|{\pi }_k(X)\ne 0\}$ and $\E(X)$ is finite, then $\E(X)\cong\E(X^{m+1})$. Moreover, in case when $X$ is 2-connected, we show that if ${\pi }_n(X)\ne 0$, then the group $\E(X)$ is infinite.