The homotopy theory of polyhedral products associated with flag complexes

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 $(\underline X,\underline A)^L\stackrel g\longrightarrow(\underline X,\underline A)^K\stackrel f\longrightarrow (\underline X,\underline A)^{K^f}$. We show that $\mathrm{\Omega }f$ and $\mathrm{\Omega }f\circ \mathrm{\Omega }g$ have right homotopy inverses and draw consequences. For a flag complex $K$ the polyhedral product of the form $(\underset{‾}{CY},\underset{‾}{Y}{)}^K$ 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.