SoCG'21 - Simplicial approximation to CW-complexes in practice

Abstract: It is known that any topological manifold is homotopy equivalent to a CW-complex, and that any CW-complex is homotopy equivalent to a simplicial complex. Despite this fact, only a few of these simplicial complexes are available in practice. A striking example is given by the Grassmannians of d-planes in R^n, for which homotopy equivalent simplicial complexes are explicitly known only when d is 1 or n-1. I will describe an implementation of an algorithm that takes as an input a CW complex and returns a simplicial complex of same homotopy type. This algorithm, though well-known in the literature, requires some work to make it computationally tractable. A close attention is paid to triangulations of mapping cones, generalized subdivisions and weak simplicial approximations. I will illustrate the algorithm with the real projective spaces, the 3-dimensional lens spaces and the Grassmannian of 2-planes in R^4.