On the homotopy links of stratified cell complexes

Abstract

Homotopy links have proven to be one of the most powerful tools of stratified homotopy theory. In previous work, we described combinatorial models for the generalized homotopy links of a stratified simplicial set. For many purposes, in particular to investigate the stratified homotopy hypothesis, a more general version of this result pertaining to stratified cell complexes is needed. Here we prove that, given a stratified cell complex $X$, the generalized homotopy links can be computed in terms of certain subcomplexes of a subdivision of $X$. As a consequence, it follows that homotopy links map certain pushout diagrams of stratified cell complexes into homotopy pushout diagrams. This result is crucial to the development of (semi-)model structures for stratified homotopy theory in which geometric examples of stratified spaces, such as Whitney stratified spaces, are bifibrant.

Figures (1)

• Figure 1: The upper left corner shows a stratified cell structure for the pinched torus, stratified over the poset $\{ {\color{red} 0} < {\color{green} 1} < {\color{NewBlue} 2} \}$. Vertices with the same name, and edges with the same markings are being identified and the stratification is indicated by the coloring. To its right, a barycentric subdivision $\Psi$ of this cell structure is shown. In the following row there are illustrations of the subcomplexes ${\mathcal{U}}^{\Psi}_{\mathcal{X}}(\mathcal{I})$ for $\mathcal{I} = \{{\color{red} 0} < {\color{NewBlue} 2}\}, \{ {\color{red} 0} < {\color{green} 1} < {\color{NewBlue} 2} \}, \{{\color{green} 1} < {\color{NewBlue} 2} \}$.