2011/06/19 by Dai Tamaki, Tamaki, Dai · 2 citations
Mathematics · #14N20 #57N80 #Algebraic Topology (math.AT) #Combinatorics (math.CO) #FOS: Mathematics #Geometric Topology (math.GT) #math.AT #math.CO #math.GT #msc:14N20 #msc:57N80
paper · pdf · doi:10.48550/arxiv.1106.3772
This paper and the second part arXiv:1111.4774 are now combined as a single article arXiv:1609.04500
arxiv created 2016/09/16 · arxiv updated 2016/09/19
The notion of cellular stratified spaces was introduced in a joint work of the author with Basabe, González, 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.