  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.
