"Mister... WHAT is a manifold ?! (Part I ?)" Every PhD student know a little about manifolds. Nevertheless, its formal definition hides difficulties. For instance, it is not explained how the open sets join together (i.e the topological nature of their multiple intersections, and how the open sets and their intersections are linked together) ; this prevents to build explicitly manifold, and prevents the comprehension and their classification. We will talk here about the classification of topological manifolds. In dimension 1, we show that the 2 connex manifolds possible are the circle and the line. In dimension 2, all surfaces are triangulable ; this enables to classify compact surfaces (actually, it enables to classify all the surfaces, but we will not show this). For the dimensions greater than 4, it is possible to use the complexity of the fundamental groups to show that it impossible to classify manifolds (so it is impossible answer the question of the title ! ). We might talk in some other seminar of the Ricci flow, which enable to classify manifolds in dimension 3 ; but it is a bit more complicated, so it will be impossible to talk in detail of this topic (indeed, with 3 remunerated hardworking years it will be not certain we manage to make it).
Nicolas Massin
