Formality on rationalizations of simply connected CW complexes

Abstract

In this paper, we show that for a simply connected CW complex $Y$ with $H^{\ast }(Y;\mathbb{Q})$ of finite dimension, if $H^{\ast }(Y;\mathbb{Q})$ is concentrated in degrees $\le 3$, then the rationalization $Y_{\mathbb{Q}}$ is formal. As an application, we show that the spatial realizations of simply connected Sullivan algebras with rational homology concentrated in degrees $\le 3$ are formal.