Probably the most studied invariant in Topological Data Analysis is the homology of a space. The usual approach is to triangulate the space and try to reduce it in order to make the computations more feasible. A common reduction technique is that of collapsing. In a collapsing process we perform a sequence of elementary collapses, where at each step we delete a free face and the unique facet containing it. If we are able to reduce a complex to one of its vertices then we say it is collapsible and its homology is trivial. Collapsibility implies that the space is contractible but the converse is not always true, probably the best known example is the Dunce Hat.
We are going to explore the difference between these two concepts and look for minimal examples of contractible non collapsible complexes in each dimension and how often they arise.