A colimit decomposition for the loop homology of polyhedral products

TL;DR

This paper presents a colimit decomposition approach for the loop homology of polyhedral products, simplifying their study by reducing to 1-neighbourly complexes and providing explicit calculations for various classes.

Contribution

It introduces a colimit decomposition for loop homology algebras of polyhedral products, linking them to flagification and enabling explicit calculations for specific complexes.

Findings

Decomposition reduces analysis to 1-neighbourly complexes.

Explicit presentations for Davis–Januszkiewicz spaces.

Calculated Poincaré series for various polyhedral products.

Abstract

We show that the loop homology algebras of polyhedral products of the form $(\underline{X},\underline{*})^{\mathcal{K}}$ can be written as a colimit over the flagification of $\mathcal{K}$, and obtain a similar result for the Poincar\'e series. This effectively reduces the study of the algebras $H_*(\Omega(\underline{X},\underline{*})^{\mathcal{K}})$ to the case of 1-neighbourly simplicial complexes. We give presentations of the loop homology of Davis--Januszkiewicz spaces (i.e. Yoneda algebras of Stanley--Reisner rings) and calculate the Poincar\'e series of looped polyhedral products associated to various families of simplicial complexes, including HMF-presented complexes and skeleta of flag complexes.