Formality criteria in terms of higher Whitehead brackets

Abstract

We provide two criteria for establishing the non-formality of a differential graded Lie algebra in terms of higher Whitehead brackets, which are the Lie analogues 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 its associated Quillen spectral sequence. Finally, we use L∞ 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.