Loop space decompositions of $(2n-2)$-connected $(4n-1)$-dimensional Poincar\'{e} Duality complexes

TL;DR

This paper simplifies and extends the homotopy decomposition of loop spaces of certain high-dimensional Poincaré Duality complexes, building on prior work by Beben and Wu.

Contribution

It provides a simplified, more general framework for decomposing loop spaces of these complexes, extending previous results.

Findings

Simplified the decomposition process of loop spaces.

Extended the class of complexes for which decomposition applies.

Enhanced understanding of the homotopy types of these spaces.

Abstract

Beben and Wu showed that if $M$ is a $(2n-2)$-connected $(4n-1)$-dimensional Poincar\'{e} Duality complex such that $n\geq 3$ and $H^{2n}(M;\mathbb{Z})$ consists only of odd torsion, then $\Omega M$ can be decomposed up to homotopy as a product of simpler, well studied spaces. We use a result from \cite{BT2} to greatly simplify and enhance Beben and Wu's work and to extend it in various directions.