[Paper Review] Totally normal cellular stratified spaces and applications to the configuration space of graphs

Mizuki Furuse, Takashi Mukouyama+1 more|arXiv (Cornell University)|Dec 27, 2013

Topological and Geometric Data Analysis13 references3 citations

TL;DR

This paper introduces totally normal cellular stratified spaces as a generalization of regular cell complexes, using acyclic categories to model configuration spaces of graphs. It constructs a combinatorial model for the configuration space of graphs that avoids the need for fine subdivisions, leading to a simpler proof of Ghrist's theorem on homotopy dimension and enabling explicit computation of graph braid groups for small 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 by the third author 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.

Motivation & Objective

• To develop a combinatorial model for the configuration space of graphs that avoids the need for fine cell subdivisions required by Abrams' model.
• To generalize the framework of regular cell complexes using acyclic categories to handle spaces with incomplete cell boundaries.
• To provide a new, simpler proof of Ghrist's theorem on the homotopy dimension of configuration spaces of graphs.
• To enable explicit computation of the fundamental group of ordered and unordered configuration spaces for small graphs.
• To demonstrate the utility of totally normal cellular stratified spaces in topological combinatorics and configuration space theory.

Proposed method

• Introduce the notion of totally normal cellular stratified spaces by relaxing conditions on face posets, replacing them with acyclic categories and allowing cells with incomplete boundaries.
• Define the braid stratification on the configuration space of a graph, which partitions the space according to the relative positions of points.
• Construct a combinatorial model for configuration spaces using the braid stratification and the structure of totally normal cellular stratified spaces.
• Apply deformation retractions to simplify the model, particularly by removing leaves from the graph while preserving homotopy type.
• Use the acyclic category model to explicitly compute the fundamental group of configuration spaces for small graphs.
• Leverage the structure of the face category of a totally normal cellular stratified space to analyze homotopy and group-theoretic invariants.

Experimental results

Research questions

• RQ1 Can a combinatorial model for the configuration space of graphs be constructed without requiring fine cell subdivisions, as in Abrams' model?
• RQ2 How can the homotopy dimension of configuration spaces of graphs be re-proven using a more structured, category-theoretic framework?
• RQ3 What is the fundamental group of the ordered and unordered configuration space of two points on small graphs, and how can it be computed combinatorially?
• RQ4 How do totally normal cellular stratified spaces generalize regular cell complexes in the context of configuration spaces?
• RQ5 Can the symmetric group action on unordered configuration spaces be naturally incorporated into such a model?

Key findings

• The paper constructs a combinatorial model for the configuration space of graphs that avoids the need for fine subdivisions, providing a more efficient and explicit alternative to Abrams' model.
• The authors provide a simpler proof of Ghrist's theorem that configuration spaces of graphs are $K(/pi,1)$ spaces, using the framework of totally normal cellular stratified spaces.
• For small graphs, the fundamental group of the unordered configuration space of two points is computed explicitly using the model, yielding concrete group presentations.
• The model successfully captures the $Σ_k$-equivariant homotopy type of configuration spaces, addressing a limitation of Abrams' model.
• The deformation retraction used in the model is constructed explicitly via barycentric subdivision and regular neighborhood theory, ensuring topological control.
• The face category of a totally normal cellular stratified space is shown to be an acyclic category, enabling algebraic and homotopical analysis through category-theoretic tools.