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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

A simplicial approach to stratified homotopy theory


Author: Sylvain Douteau
Journal: Trans. Amer. Math. Soc. 374 (2021), 955-1006
MSC (2010): Primary 55U35, 57N80, 18G30, 18G55
DOI: https://doi.org/10.1090/tran/8264
Published electronically: November 25, 2020
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Abstract: In this article we consider the homotopy theory of stratified spaces from 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.


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Additional Information

Sylvain Douteau
Affiliation: LAMFA CNRS UMR 7352 - Université de Picardie Jules Verne, Amiens, France

DOI: https://doi.org/10.1090/tran/8264
Received by editor(s): August 4, 2018
Received by editor(s) in revised form: March 20, 2020
Published electronically: November 25, 2020
Article copyright: © Copyright 2020 American Mathematical Society