Citation: Fagan, Lucas, Yang Qiu, and Zhenghan Wang. "Classification of Cellular Fake Surfaces." arXiv preprint arXiv:2406.09439 (2024).
Web: https://arxiv.org/abs/2406.09439#
Tags: Mathematical
This paper studies generic polyhedra as interesting mathematical objects in their own right, as opposed to studying them for any direct application. This is motivated by the fact that generic polyhedra have shown up in many different areas (including error correcting codes from manifolds in work by Zhenghan Wang and Mike Freedman).
A polyhedron is a collection of vertices, edges, faces, and cells of higher dimension (a CW complex). The idea is that some singularities can be continuously deformed into other by small perturbations. A 4-valent vertex of a graph could easily be broken up into two 3-valent vertices connected by an edge. In general, perturbations will turn a polyhedron into some irreducible type which we call generic. Generic 1-complexes are 3-valent graphs. The next level after that is generic 2-complexes.
This paper gives some general treatment about how these generic 2-complexes work. The bulk of the novel work from this paper comes from a numerical search in low dimensions. They compute all of the examples with a small number of singularities, find the patterns, and verify some conjectures.