Jacobi identity in polyhedral products

Abstract

We show that a relation among minimal non-faces of a fillable complex K yields an identity of iterated (higher) Whitehead products in a polyhedral product over K. In particular, for the (n−1)-skeleton of a simplicial n-sphere, we always have such an identity, and for the (n−1)-skeleton of a (n+1)-simplex, the identity is the Jacobi identity of Whitehead products (n=1) and Hardie's identity for higher Whitehead products (n≥2).