Loop spaces of $n$-dimensional Poincaré duality complexes whose $(n-1)$-skeleton is a co-$H$-space

Abstract

Under certain hypotheses, we prove a loop space decomposition for simply-connected Poincaré 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 Poincaré Duality complex retracts off the loops of its $(n-1)$-skeleton and describe its homology as a one-relator algebra.

35 pages, changes to main theorem thanks to referee comments. Version accepted by Transactions of the AMS