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 dim 𝐻^*(𝑋, ℤ) < ∞ or dim 𝜋_*(𝑋) < ∞, then the group of self-homotopy equivalences of 𝑋 is isomorphic to a linear algebraic group defined over ℚ.