The aim of this article is clearly expressed in the title. The authors prove lower bounds for the number of vertices or top-dimensional simplices (facets) for a triangulation of the Grassmannian \(G_k(\mathbb{R}^n)\). The estimates are based on the knowledge of the maximal height of cup-powers in the cohomology. This method was developed by the authors [Discrete Comput. Geom. 63, No. 1, 31–48 (2020; Zbl 1431.55003)]. For the projective space \(\mathbb{R}P^{n-1}\) (the case \(k=1\)) a lower bound for the number of vertices is \(\binom{n+1}{2}\). For \(G_2(\mathbb{R}^n)\) the authors obtain \((n-2)^2+ 2^s(2n-2^s- 1)\) if \(2^s<n\le 2^{s+1}\). Similar bounds are given for \(k=3\) and \(k=4\), for \(G_3(\mathbb{R}^8)\) the bound is \(117\).
Another tool is the Lower Bound Theorem (LBT) together with its generalizations. This implies in Thm. 4.7 that the number of facets in a triangulated \(G_k(\mathbb{R}^{n+k})\) is at least \(2^{nk}+k(n^3+ n^2)/2+ k(k+2)(k-1)n/2\).
Surprisingly, this bound is sharp for \(k=1\), \(n=2\) since the smallest triangulation of \(\mathbb{R}P^2\) has \(10\) triangles. Unfortunately, for \(k\ge 2\) explicit examples do not seem to be known. So the question remains open how sharp these bounds are.
The authors remark that their method can be used also for other spaces, like Lie groups, flag manifolds, Stiefel manifolds etc. For Lie groups see the recent paper by H. Duan et al. [Topology Appl. 293, Article ID 107559, 13 p. (2021; Zbl 1468.57022)].