Simplicial approximation to CW complexes with spherical Delaunay triangulations
This addresses a computational bottleneck in topology for researchers working with manifold approximations, offering a more efficient method.
The paper tackles the computational intractability of conventional simplicial approximation for manifolds by developing a subdivision scheme based on spherical Delaunay triangulations and reframing the star condition, resulting in an exponential reduction in vertices and enabling homotopy-equivalent simplicial complexes for Grassmannians and Stiefel manifolds up to dimension 5.
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.