Adams-Hilton models and higher Whitehead brackets for polyhedral products

TL;DR

This paper constructs explicit algebraic models for polyhedral products of spheres and Davis-Januszkiewicz spaces, linking Adams-Hilton models with higher Whitehead brackets to better understand their homotopy properties.

Contribution

It develops Adams-Hilton models that coincide with cobar constructions for specific polyhedral products, enabling explicit analysis of Whitehead products.

Findings

Constructed Adams-Hilton models matching cobar constructions.

Explicit chain representations of Whitehead products.

Enhanced understanding of homotopy invariants in polyhedral products.

Abstract

In this paper, we construct Adams-Hilton models for the polyhedral products of spheres $(\underline{S})^{\mathcal{K}}$ and Davis-Januszkiewicz spaces $\left(\mathbb{C} P^{\infty}\right)^{\mathcal{K}}$. We show that in these cases the Adams-Hilton model can be chosen so that it coincides with the cobar construction of the homology coalgebra. We apply the resulting models to the study of iterated higher Whitehead products in $\left(\mathbb{C} P^{\infty}\right)^{\mathcal{K}}$. Namely, we explicitly construct a chain in the cobar construction representing the homology class of the Hurewicz image of a Whitehead product.