Simplicial approximation provides a framework for constructing simplicial complexes that are homotopy equivalent to a given manifold, provided a CW structure is explicitly known. However, its conventional implementation quickly becomes intractable on a computer: barycentric subdivision produces poorly shaped simplices, and the star condition introduces many vertices. To address these limitations, this article develops a subdivision scheme based on spherical Delaunay triangulations, which attains better refinement properties than barycentric subdivisions. Moreover, the star condition is reframed as two independent problems, one geometric and the other combinatorial, respectively tackled in the language of locally equiconnected spaces and the list homomorphism problem, allowing an exponential reduction in the number of vertices. Via a prototype implementation, we obtain simplicial complexes homotopy equivalent to Grassmannians and Stiefel manifolds up to dimension 5.

翻译：单纯逼近提供了一种框架，用于构造与给定流形同伦等价的单纯复形，前提是该流形的CW结构已明确给出。然而，其传统实现方法在计算机上迅速变得难以处理：重心剖分会产生形状不佳的单纯形，而星形条件会引入大量顶点。为解决这些局限性，本文发展了一种基于球面Delaunay三角剖分的细分方案，该方案获得了比重心剖分更优的细化性质。此外，星形条件被重新表述为两个独立问题——一个几何问题与一个组合问题，并分别通过局部等连通空间的语言与列表同态问题的语言加以处理，从而实现了顶点数量的指数级缩减。通过一个原型实现，我们得到了与维度不超过5的Grassmann流形及Stiefel流形同伦等价的单纯复形。