Loop spaces of $n$-dimensional Poincar\'e duality complexes whose $(n-1)$-skeleton is a co-$H$-space

TL;DR

This paper proves a loop space decomposition for certain Poincaré Duality complexes with co-H-space skeletons, unifying known results and discovering new examples, with implications for their homology and retract properties.

Contribution

It establishes a general loop space decomposition for Poincaré Duality complexes under specific conditions, unifying previous results and introducing new classes of examples.

Findings

Loop space decompositions for Poincaré Duality complexes.

Retract properties of these complexes off loops of their skeletons.

Homology described as a one-relator algebra.

Abstract

Under certain hypotheses, we prove a loop space decomposition for simply-connected Poincar\'e Duality complexes of dimension $n$ whose $(n-1)$-skeleton is a co-$H$-space. This unifies many known decompositions obtained in different contexts and establishes many new families of examples. As consequences, we show that such a looped Poincar\'{e} Duality complex retracts off the loops of its $(n-1)$-skeleton and describe its homology as a one-relator algebra.