Simplicial approximation to CW complexes in practice

We describe an algorithm that takes as an input a CW complex and returns a simplicial complex of the same homotopy type. This algorithm, although well-known in the literature, requires some work to make it computationally tractable. We pay close attention to weak simplicial approximation, which we implement for generalized barycentric and edgewise subdivisions. We also propose a new subdivision process, based on Delaunay complexes. In order to facilitate the computation of a simplicial approximation, we introduce a simplification step, based on edge contractions. We define a new version of simplicial mapping cone, which requires less simplices. Last, we illustrate the algorithm with the real projective spaces, the 3-dimensional lens spaces and the Grassmannian of 2-planes in $\mathbb{R}^4$.