Stellar Stratifications on Classifying Spaces

TL;DR

This paper extends the understanding of stratified spaces by characterizing their face posets and establishing that their exit-path simplicial sets form quasicategories, linking combinatorial and topological structures.

Contribution

It introduces cylindrically normal stellar complexes, providing a discrete analogue of Morse theory cell decompositions and proving the exit-path set forms a quasicategory.

Findings

Characterization of face posets for cylindrically normal stellar complexes

Discrete Morse theory analogue via classifying spaces

Proof that exit-path sets are quasicategories

Abstract

We extend Bj\"orner's characterization of the face poset of finite CW complexes to a certain class of stratified spaces, called cylindrically normal stellar complexes. As a direct consequence, we obtain a discrete analogue of cell decompositions in smooth Morse theory, by using the classifying space model introduced in arXiv:1612.08429. As another application, we show that the exit-path simplicial set $\mathrm{Exit}(X)$ of a finite cylindrically normal CW stellar complex $X$ is a quasicategory.