Cellular Stratified Spaces I: Face Categories and Classifying Spaces

TL;DR

This paper develops a theory of cellular stratified spaces, introducing face categories and classifying spaces, and shows how these structures relate to cell complexes and Morse theory, with potential applications in topology.

Contribution

It introduces cylindrically normal cellular stratified spaces and constructs their face categories, extending classical concepts like barycentric subdivision and linking to Morse theory.

Findings

Classifying space BC(X) embeds into X for cellular stratified spaces.

When X is a cell complex, BC(X) is homeomorphic to X.

BC(X) deformation retracts onto X when stratification is locally polyhedral.

Abstract

The notion of cellular stratified spaces was introduced in a joint work of the author with Basabe, Gonz\'alez, and Rudyak [1009.1851] with the aim of constructing a cellular model of the configuration space of a sphere. In particular, it was shown that the classifying space (order complex) of the face poset of a totally normal regular cellular stratified space $X$ can be embedded in $X$ as a strong deformation retract. Here we elaborate on this idea and develop the theory of cellular stratified spaces. We introduce the notion of cylindrically normal cellular stratified spaces and associate a topological category $C(X)$, called the face category, to such a stratified space $X$. We show that the classifying space $BC(X)$ of $C(X)$ can be naturally embedded into $X$. When $X$ is a cell complex, the embedding is a homeomorphism and we obtain an extension of the barycentric subdivision of regular cell complexes. Furthermore, when the cellular stratification on $X$ is locally polyhedral, we show that $BC(X)$ is a deformation retract of $X$. We discuss possible applications at the end of the paper. In particular, the results in this paper can be regarded as a common framework for the Salvetti complex for the complement of a complexified hyperplane arrangement and a version of Morse theory due to Cohen, Jones, and Segal.