# Formality criteria in terms of higher Whitehead brackets

## TL;DR

This paper introduces criteria based on higher Whitehead brackets to determine the non-formality of differential graded Lie algebras, and explores their relation to spectral sequences and rational homotopy theory.

## Contribution

It provides new criteria for non-formality using higher Whitehead brackets and connects formality with spectral sequence behavior and $L_infty$ algebra techniques.

## Key findings

- Higher Whitehead brackets can detect non-formality.
- Formality does not imply spectral sequence collapse.
- Applications to rational homotopy theory and minimal models.

## Abstract

We provide two criteria for discarding the formality of a differential graded Lie algebra in terms of higher Whitehead brackets, which are the Lie analogue of the Massey products of a differential graded associative algebra. We also show that formality of a differential graded Lie algebra is not equivalent to the collapse of the Quillen spectral sequence. Finally, we use $L_\infty$ algebras and Quillen's formulation of rational homotopy theory to recover and improve a classical theorem for detecting higher Whitehead products in Sullivan minimal models and give some applications.