A Simplicial Approach to Stratified Homotopy Theory

TL;DR

This paper develops a simplicial framework for stratified homotopy theory, introducing filtered homotopy groups and extending classical theorems to stratified spaces, with applications to conically stratified spaces.

Contribution

It introduces a model category of filtered simplicial sets, defines filtered homotopy groups, and extends Whitehead's theorem to stratified spaces, connecting homotopy types of strata and holinks.

Findings

Filtered homotopy groups characterize weak equivalences.

Stratified Whitehead's theorem established.

Homotopy types of conically stratified spaces depend on strata and holinks.

Abstract

In this article we consider the homotopy theory of stratified spaces through a simplicial point of view. We first consider a model category of filtered simplicial sets over some fixed poset $P$, and show that it is a simplicial combinatorial model category. We then define a generalization of the homotopy groups for any fibrant filtered simplicial set $X$ : the filtered homotopy groups $s\pi_n(X)$. They are diagrams of groups built from the homotopy groups of the different pieces of $X$. We then show that the weak equivalences are exactly the morphisms that induce isomorphisms on those filtered homotopy groups. Then, using filtered versions of the topological realisation of a simplicial set and of the simplicial set of singular simplices, we transfer those results to a category whose objects are topological spaces stratified over $P$. In particular, we get a stratified version of Whitehead's theorem. Specializing to the case of conically stratified spaces, a wide class of topological stratified spaces, we recover a theorem of Miller saying that to understand the homotopy type of conically stratified spaces, one only has to understand the homotopy type of strata and holinks. We then provide a family of examples of conically stratified spaces and of computations of their filtered homotopy groups.