Polyhedral products for shifted complexes and higher Whitehead products

Abstract

This paper studies the map between polyhedral products $\mathcal{Z}_K(C\underline{X},\underline{X})\to\mathcal{Z}_K(\Sigma\underline{X},*)$ induced from the pinch maps $(CX_i,X_i)\to(\Sigma X_i,*)$, which is the higher order Whitehead product if $K$ is the boundary of a simplex. When $K$ is a shifted complex, a wedge decomposition of $\mathcal{Z}_K(C\underline{X},\underline{X})$ is given by the authors. Based on this decomposition, when $K$ is shifted, the induced pinch map is explicitly described as a wedge of iterated Whitehead products each of which includes at most one higher product. As a corollary, the Jacobi identity of Whitehead products including higher products due to Hardie is generalized.