Stratified Homotopy Theory and a Whitehead Group for Stratified Spaces

TL;DR

This thesis extends classical simple homotopy theory to stratified spaces, establishing a Whitehead group and torsion for filtered simplicial sets, and connecting stratified and classical homotopy theories.

Contribution

It introduces a combinatorial Whitehead group and torsion for filtered simplicial sets, generalizing classical concepts to stratified spaces and establishing their formal properties.

Findings

Fully faithful embedding of filtered homotopy categories.

Characterization of morphisms as stratified homotopy classes.

Generalization of classical simple homotopy theory to filtered spaces.

Abstract

In this master thesis, we extend results from classical simple homotopy theory to the world of stratified homotopy theory. To obtain a well-established framework to work in, we prove a series of results on two model categories of simplicial sets and topological spaces, both equipped with a notion of filtration, introduced by Sylvain Douteau in his PHD thesis. In particular, we show that there is a fully faithful embedding of homotopy categories from the (finite) filtered simplicial into the filtered topological setting. We also use these results to characterize the morphisms in the topological filtered homotopy category between filtered spaces that are triangulable and stratified in some very general sense as stratified homotopy classes of stratum preserving maps. Then, moving on to simple homotopy theory, we propose a class of combinatorial elementary expansions for filtered simplicial sets that generalize both the classical ones as well as a class of stratified expansions suggested by Banagl et al. We show that they fulfill a series of axioms, suggested by Eckmann and Siebenmann for the construction of simple homotopy theory. In doing so, we obtain a combinatorially defined Whitehead group and torsion for filtered simplicial sets (and hence also for triangulable stratified spaces). We then begin a detailed investigation of their formal properties, proving for example that a Mayer-Vietoris formula holds. Next, we apply our results on stratified homotopy theory to obtain a more topological description of the Whitehead group and to generalize the Whitehead torsion to arbitrary stratum preserving maps of triangulated filtered spaces. Finally, we prove that our simple homotopy theory is a generalization of the classical one, in the sense that it agrees with the latter when one considers trivially filtered simplicial sets as CW-complexes.