A colimit decomposition for the loop homology of polyhedral products

We show that the loop homology algebras of polyhedral products of the form $(\underset{‾}{X},\underset{‾}{\ast }{)}^{\mathcal{K}}$ can be written as a colimit over the flagification of $\mathcal{K}$, and obtain a similar result for the Poincaré series. This effectively reduces the study of the algebras $H_{\ast }(\mathrm{\Omega }(\underset{‾}{X},\underset{‾}{\ast }{)}^{\mathcal{K}})$ 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 Poincaré series of looped polyhedral products associated to various families of simplicial complexes, including HMF-presented complexes and skeleta of flag complexes.