Formality of certain CW complexes

TL;DR

This paper investigates conditions under which a topological space formed by attaching a disk to a formal space remains formal, providing corrected theorems, examples, and discussing the validity of previous results in rational homotopy theory.

Contribution

The paper corrects earlier errors in formality criteria for CW complexes and offers new conditions and examples for the formality of spaces obtained by attaching cells.

Findings

Provides a corrected version of a key theorem on formality.

Offers examples of formality in CW complexes with even-dimensional cells.

Discusses the validity of previous results on Schubert varieties and formality.

Abstract

Let $X$ be a simply connected path connected topological space which is formal in the sense of rational homotopy theory. Let $Y=X\cup_\alpha\mathbb{D}^{n}$ where $\alpha:\mathbb{S}^{n-1}\to X$ is a non-torsion element. Then we obtain a condition on $\alpha$ for the formality of $Y$. We give several illustrative examples concerning the formality of a finite CW complex having only even dimensional cells. This is the corrected version of the earlier version which contained a serious error in Theorem 1.4. This theorem, which now Theorem 1.1 of this version, has now been corrected. The proofs of Theorems 1.1, 1.2, and 1.3 of the first version are not valid as they used the erroneous result. In fact, we provide here a counterexample to the assertion of Theorem 1.1. (See Example 3.1 below.) We do not know if the statement of Theorem 1.2, which asserted the formality of Schubert varieties in a generalized flag variety $G/B$, is valid. Theorem 1.3 is correct as stated as it had been proved previously by Panov and Ray using entirely different techniques.