# Approximate Triangulations of Grassmann Manifolds
## 1 Introduction  
Smooth manifolds admit piecewise-linear triangulations [1]. However, there are many subsequent questions one might ask: How many simplices are required? What is the minimal number of vertices? Is there an algorithm to construct a triangulation? A great deal of work in algebraic topology has been devoted to these topics. The question of the number of simplices required to triangulate a given manifold is often attacked by sophisticated cohomological methods involving characteristic classes (such arguments also often yield estimates on the minimal embedding dimension for the manifold). Surprisingly, much of this work is very recent [2, 3]. A main result in [3] is the following. **Theorem 1** ([3], Theorem 3.10). : _Every triangulation of the Grassmann manifold \(G_{k}(\mathbb{R}^{n+k})\) must have at least_  
\[[(n+k)(n+k+1)-2 kn]\cdot(2^{kn+1}-1)\]  
_simplices._  
For example, any triangulation of the manifold \(G_{2}(\mathbb{R}^{4})\) must have at least 372 simplices. The Grassmann manifolds will be defined in Section 2.1 below. These are important spaces to study because of their utility in algebraic topology, especially with respect to the study of characteristic classes [4]. Moreover, while the theory of these spaces is rich, there are still many unsolved questions about them. For example, if we pass to the limit as \(n\to\infty\), we obtain an infinite Grassmannian \(G_{k}(\mathbb{R}^{\infty})\); it is still unknown [5] if these spaces are triangulable for \(k\geq 3\). If one could construct compatible triangulations of the various \(G_{k}(\mathbb{R}^{n})\), then one might be able to answer this question. Our methods here do not apply, however. Unfortunately, most results along these lines are _not_ constructive; that is, the proofs do not yield an explicit triangulation of the manifold. In fact, if one seeks a triangulation of a Grassmannian \(G_{k}(\mathbb{R}^{n})\), the end result is usually disappointment. For the smallest nontrivial space, \(G_{1}(\mathbb{R}^{3})=\mathbb{R}P^{2}\), there are many well-known small triangulations, and even an algorithm to generate a triangulation from any collection of points in general position [6]. Beyond that, however, results are sparse. In this paper, we develop a procedure to find what we call an _approximate triangulation_ of the manifold \(G_{k}(\mathbb{R}^{n})\) (Definition 3). The basic idea is to first generate a sample of points on \(G_{k}(\mathbb{R}^{n})\)This already leads to technical difficulties involving embeddings of these spaces into a Euclidean space \(\mathbb{R}^{N}\), but we are able to solve this. We then build a nested family of simplicial complexes on the point cloud, parametrized by the positive real numbers. The persistent homology of this family is then computed and we identify an interval of parameters for which the mod 2 homology of the complexes in that range agrees with that of \(G_{k}(\mathbb{R}^{n})\). Such a complex is then a viable model for the manifold: its vertices lie in \(G_{k}(\mathbb{R}^{n})\subset\mathbb{R}^{N}\) and it has the correct mod 2 homology. (As a further check, we can compute the mod 3 homology to ensure it is correct as well if we wish.) We then implement this procedure for the following spaces: \(\mathbb{R}P^{2}\subset\mathbb{R}^{4}\), \(\mathbb{R}P^{2}\subset\mathbb{R}^{5}\), \(\mathbb{R}P^{3}\subset\mathbb{R}^{9}\), and \(G_{2}(\mathbb{R}^{4})\subset\mathbb{R}^{16}\). Computational limitations have so far prohibited further calculations; we discuss this in Section 4. In effect, what we are doing is taking a tool developed for examining unknown objects (persistent homology modules built from point clouds) and using it on spaces that are well-understood (Grassmann manifolds) to build new structures associated with those spaces (approximate triangulations). We have chosen the Grassmannians because their homology is relatively easy to compute and they arise in numerous areas in mathematics (topology, algebraic geometry, combinatorics).