On the Homotopy Type of the Polyhedral Join over the Independence Complex of a Forest

We consider a certain class of simplicial complexes which includes the independence complexes of forests. We show that if a simplicial complex $K$ belongs to this class, then the polyhedral join ${\mathcal{Z}}_K^{\ast }(\underset{‾}{X},\mathrm{\varnothing })$ is homotopy equivalent to a wedge sum of CW complexes of the form ${\mathrm{\Sigma }}^r{X}_{i_1}\ast X_{i_2}\ast \cdots \ast X_{i_k}$, where $\underset{‾}{X}$ is a family $\{X_i{\}}_{i\in V(K)}$ of CW complexes and $\mathrm{\Sigma }$ denotes the unreduced suspension. This result is applied to study the homotopy type of the independence complex of the lexicographic product $G[H]$ of a graph $H$ over a forest $G$. We denote by $L_m$ a tree on $m$ vertices with no branches. We show that the geometric realization of the independence complex of $L_m[H]$ is homotopy equivalent to a wedge sum of spheres if $m\mathrm{\ne }2,3$ and the geometric realization of the independence complex of $H$ is homotopy equivalent to a wedge sum of same dimensional spheres.