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The homotopy theory of polyhedral products associated with flag complexes
- Part of
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- Journal:
- Compositio Mathematica / Volume 155 / Issue 1 / January 2019
- Published online by Cambridge University Press:
- 23 November 2018, pp. 206-228
- Print publication:
- January 2019
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- Article
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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 
$L\stackrel{}{\longrightarrow }K\stackrel{}{\longrightarrow }K^{f}$. This induces a sequence of maps of polyhedral products 
$(\text{}\underline{X},\text{}\underline{A})^{L}\stackrel{g}{\longrightarrow }(\text{}\underline{X},\text{}\underline{A})^{K}\stackrel{f}{\longrightarrow }(\text{}\underline{X},\text{}\underline{A})^{K^{f}}$. We show that 
$\unicode[STIX]{x1D6FA}f$ and 
$\unicode[STIX]{x1D6FA}f\circ \unicode[STIX]{x1D6FA}g$ have right homotopy inverses and draw consequences. For a flag complex 
$K$ the polyhedral product of the form 
$(\text{}\underline{CY},\text{}\underline{Y})^{K}$ 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.
