  We introduce the theory of strong homotopy types of simplicial complexes.
Similarly to classical simple homotopy theory, the strong homotopy types can be
described by elementary moves. An elementary move in this setting is called a
strong collapse and it is a particular kind of simplicial collapse. The
advantage of using strong collapses is the existence and uniqueness of cores
and their relationship with the nerves of the complexes. From this theory we
derive new results for studying simplicial collapsibility with a different
point of view. We analyze vertex-transitive simplicial $G$-actions and prove a
particular case of the Evasiveness conjecture for simplicial complexes.
Moreover, we reduce the general conjecture to the class of minimal complexes.
We also strengthen a result of V. Welker on the barycentric subdivision of
collapsible complexes. We obtain this and other results on collapsibility of
polyhedra by means of the characterization of the different notions of
collapses in terms of finite topological spaces.
