Some results and Algorithms on matroids, simplicial complexes and Alexandroff spaces

In 1950 when JHC Whitehead introduced the idea of elementary collapse of simplicial complexes and the simple homotopy type. In 2012 Barmar and Minian return to the topic and develop the theory of strong collapse of simplicial complexes, which has interesting applications to collapsibility problems. In this thesis we first review both concepts and a third one - edge contraction- and explore their consequences on matroids (a special kind of simplicial complexes). Secondly, we study a generalization of the idea of strong collapse to (non-finite) Alexandroff spaces. Finally, we present several algorithms to facilitate the exploration of all these concepts in the case of finite simplicial complexes and directed graphs.