Simplicial approximation to CW complexes with spherical Delaunay triangulations

TL;DR

This paper introduces a new subdivision scheme based on spherical Delaunay triangulations that improves the construction of simplicial complexes homotopy equivalent to manifolds, reducing complexity and enabling practical computations.

Contribution

It develops a spherical Delaunay-based subdivision method and reframes the star condition as two problems, significantly reducing vertices needed for homotopy-equivalent complexes.

Findings

Successfully constructed complexes for Grassmannians and Stiefel manifolds up to dimension 5.

Achieved exponential reduction in vertices compared to traditional barycentric subdivision.

Demonstrated practical implementation with improved refinement properties.

Abstract

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.