Totally normal cellular stratified spaces and applications to the configuration space of graphs

Abstract

The notion of regular cell complexes plays a central role in topological combinatorics because of its close relationship with posets. A generalization, called totally normal cellular stratified spaces, was introduced in [I. Basabe, J. González, Y.B. Rudyak and D. Tamaki, Higher topological complexity and homotopy dimension of configuration spaces on spheres, arXiv: 1009.1851v5], [D. Tamaki, Cellular Stratified Spaces I. Face Categories and Classifying Spaces, arXiv:1106.3772] by relaxing two conditions; face posets are replaced by acyclic categories and cells with incomplete boundaries are allowed. The aim of this article is to demonstrate the usefulness of totally normal cellular stratified spaces by constructing a combinatorial model for the configuration space of graphs. As an application, we obtain a simpler proof of Ghrist's theorem on the homotopy dimension of the configuration space of graphs. We also make sample calculations of the fundamental group of ordered and unordered configuration spaces of two points for small graphs.