Abstract:
The geometric version of the Andrews-Curtis conjecture states that any contractible 2-complex can be deformed into a point by means of simplicial expansions and collapses in such a way that all the complexes involved in the deformation are of dimension not greater than 3. This long standing problem is closely related to the Zeeman's conjecture and therefore to Poincaré's.

In joint work with G. Minian we introduce the class of quasi constructible complexes which contains the class of constructible 2-complexes and we prove using techniques of finite spaces that contractible quasi constructible complexes satisfy the AC conjecture.

I will recall the relationship between simple homotopy theory of finite spaces and of complexes and I will show how this is used to prove the above result.