Abstract:

Homotopy types of finite topological spaces have a simple combinatorial description. On the other hand, homotopy types of simplicial complexes are much harder to understand. Whitehead's simple homotopy types provide an approach to attack this problem.

After recalling the relationship between finite topological spaces, posets and simplicial complexes, I will show how problems of collapsibility of complexes can be studied from the optic of finite spaces. Then I will introduce the notion of strong homotopy types of simplicial complexes which can be defined by elementary moves (as in classical simple homotopy theory) or through contiguity classes of simplicial maps. This theory which is directly connected to Stong's homotopy theory of finite spaces, is simpler to handle than Whitehead's and has applications to the study of fixed points of simplicial actions and to evasiveness.

These results are part of joint works with Gabriel Minian.