Let $\tau$ be a simplex of dimension $m$ and consider the first barycentric subdivision $\tau^{\prime}$ of $\tau$. For each vertex $v$ of $\tau$, we denote the star of $v$ in $\tau^{\prime}$ by $\Star(v)^{\prime}$ and define $\Gamma_{m-1}$ to be the polyhedral complex of dimension $m-1$ given by