Jacobi identity in polyhedral products

TL;DR

This paper establishes identities among higher Whitehead products in polyhedral products derived from fillable complexes, revealing connections to classical identities like the Jacobi identity in specific simplicial structures.

Contribution

It links combinatorial properties of simplicial complexes to algebraic identities among Whitehead products, including classical identities like the Jacobi identity.

Findings

Identifies relations among Whitehead products in fillable complexes.

Shows the Jacobi identity arises in the context of the $(n-1)$-skeleton of a simplex.

Provides a framework connecting combinatorics of complexes to algebraic topology identities.

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 of higher Whitehead products ($n\ge 2$).