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arXiv · 2604.25344

A colimit decomposition for the loop homology of polyhedral products

Lewis Stanton·Fedor Vylegzhanin
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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.

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Lewis Stanton, Fedor Vylegzhanin. 2026-04-28. A colimit decomposition for the loop homology of polyhedral products. https://arxiv.org/abs/2604.25344

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