  We compute a lower bound for the number of simplices that are needed to
triangulate the Grassmann manifold $G_k(\mathbb{R}^n)$. In particular, we show
that the number of top-dimensional simplices grows exponentially with $n$. More
precise estimates are given for $k=2,3,4$. Our method can be used to estimate
the minimal size of triangulations for other spaces, like Lie groups, flag
manifolds, Stiefel manifolds etc.
