Subdivision of Simplicial Complex
Sanjay Mishra
TL;DR
The paper studies subdivisions of simplicial complexes, focusing on barycentric subdivision as a canonical refinement that preserves the geometric realization $|\mathcal{K}|$ while yielding finer combinatorial structure. It develops formal definitions, topological properties, and construction procedures (including starring from interior points) and establishes that repeated barycentric subdivision yields increasingly small mesh sizes through the standard barycentric and derived subdivisions. A central result is that $|\operatorname{Sd}(\mathcal{K})| = |\mathcal{K}|$ with a canonical homeomorphism $|\operatorname{Sd}(\mathcal{K})| \cong |\mathcal{K}|$, enabling reliable simplicial approximation and homological analysis. The work uses explicit examples and diagrams to demonstrate the interaction between subdivision, geometry, and topology, and discusses applications to metric compatibility and iterative refinement.
Abstract
This paper provides a self-contained exploration of subdivisions of simplicial complexes, with emphasis on barycentric subdivision. We present formal definitions of subdivisions, show how the realization of a complex is preserved under subdivision, and illustrate these concepts with explicit examples and detailed diagrams. The paper develops the general method of constructing subdivisions by starring from interior points, leading to the standard barycentric and derived subdivisions. We give precise statements and proofs demonstrating that repeated barycentric subdivision reduces the mesh below any prescribed scale, ensuring compatibility with given metrics and enabling applications such as simplicial approximation and homological analysis. Examples and TikZ illustrations clarify the structure of iterated subdivisions for finite complexes, highlighting their geometric and topological properties.