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 dimH∗(X,Z) < ∞ or dimπ∗(X) < ∞, then the group of self-homotopy equivalences of X is isomorphic to a linear algebraic group defined over Q.