The homotopy theory of polyhedral products associated with flag complexes

Abstract

If $K$ is a simplicial complex on $m$ vertices the flagification of $K$ is the minimal flag complex $K^f$ on the same vertex set that contains $K$. Letting $L$ be the set of vertices, there is a sequence of simplicial inclusions $L\to K\to K^f$. This induces a sequence of maps of polyhedral products $(\underline{X},\underline{A}{)}^L\stackrel{g}{⟶}(\underline{X},\underline{A}{)}^K\stackrel{f}{⟶}(\underline{X},\underline{A}{)}^{K^f}$. We show that $\Omega f$ and $\Omega f\circ \Omega g$ have right homotopy inverses and draw consequences. For a flag complex $K$ the polyhedral product of the form $(\underline{CY},\underline{Y}{)}^K$ is a co-$H$-space if and only if the $1$-skeleton of $K$ is a chordal graph, and we deduce that the maps $f$ and $f\circ g$ have right homotopy inverses in this case.

25 pages

Cited by in corpus (3)

• Pontryagin algebras and the LS-category of moment-angle complexes in the flag case
• Loop space decompositions of highly symmetric spaces with applications to polyhedral products
• Equivariant cohomology of moment-angle complexes with respect to coordinate subtori