The group of self-homotopy equivalences of a rational space cannot be a free abelian group

Abstract

In this paper, we prove that a free abelian group cannot occur as the group of self-homotopy equivalences of a rational CW-complex of finite type. Thus, we generalize a result due to Sullivan–Wilkerson showing that if $X$ is a rational CW-complex of finite type such that $\mathrm{dim}\,H^{*}(X, \mathbb{Z}) < \infty$ or $\mathrm{dim}\,\pi_{*}(X) < \infty$, then the group of self-homotopy equivalences of $X$ is isomorphic to a linear algebraic group defined over $\mathbb{Q}$.