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

Lewis Stanton; Stephen Theriault

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

Algebraic Topology 2025-06-17 v2

Authors: Lewis Stanton , Stephen Theriault

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.

Keywords

algebraic topology cohomology simplicial complexes

Cite

@article{arxiv.2502.15385,
  title  = {Loop spaces of $n$-dimensional Poincar\'e duality complexes whose $(n-1)$-skeleton is a co-$H$-space},
  author = {Lewis Stanton and Stephen Theriault},
  journal= {arXiv preprint arXiv:2502.15385},
  year   = {2025}
}

Comments

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

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

Abstract

Keywords

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