Formality on rationalizations of simply connected CW complexes

TL;DR

This paper proves that simply connected CW complexes with rational cohomology concentrated in degrees up to 3 are formal, extending to their Sullivan algebra realizations, which simplifies their rational homotopy type analysis.

Contribution

It establishes formality for a class of simply connected CW complexes based on cohomology degree constraints, linking topological and algebraic formality.

Findings

Rationalization of certain CW complexes is formal.

Formality of Sullivan algebra realizations with low-degree homology.

Provides criteria for formality based on cohomology degrees.

Abstract

In this paper, we show that for a simply connected CW complex $Y$ with $H^{*}(Y;\mathbb{Q})$ of finite dimension, if $H^{*}(Y;\mathbb{Q})$ is concentrated in degrees $\leq 3$, 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 $\leq 3$ are formal.