Séminaire des doctorants


by Mr Dimitri Le Meur (Toulouse)

318 (IMB)



"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 
Organized by

Nicolas Massin