Cellular Stratified Spaces I: Face Categories and Classifying Spaces

Details Cellular Stratified Spaces I: Face Categories and Classifying Spaces Cellular Stratified Spaces I: Face Categories and Classifying Spaces

2011-06-19

arxiv.org/abs/1106.3772v3

Mathematics

Algebraic Topology

73 pages, many figures. Version 2: A minor update. Version 3: The title is changed and the paper is split into two parts. The

Scientific paper

The notion of cellular stratified spaces was introduced in a joint work of the author with Basabe, Gonz{\'a}lez, 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.

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