The homotopy theory of polyhedral products associated with flag complexes

TL;DR

This paper investigates the homotopy properties of polyhedral products associated with simplicial complexes, especially flag complexes, revealing conditions under which these spaces are co-H-spaces and analyzing the homotopy inverses of related maps.

Contribution

It establishes the existence of right homotopy inverses for certain maps between polyhedral products and characterizes when these products are co-H-spaces based on the combinatorial structure of the underlying complex.

Findings

Polyhedral products of the form $(\underline{CY},\underline Y)^K$ are co-H-spaces iff the 1-skeleton of $K$ is chordal.

The maps $f$ and $f\circ g$ have right homotopy inverses for flag complexes.

Homotopy properties are linked to the combinatorial structure of the simplicial complex.

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 $(\underline X,\underline A)^L\stackrel g\longrightarrow(\underline X,\underline A)^K\stackrel f\longrightarrow (\underline X,\underline A)^{K^f}$. 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.