Fibrations difféologiques et homotopie

Fibrations difféologiques et homotopie

Thèse d’état — November 29, 1985

In my thesis, I generalize the ordinary concept of fiber bundles, from the catgory of manifolds to the category of diffeological spaces. The main point is that, depending on the base space, diffeological fiber bundles are not necessarily locally trivial, even if, over manifolds, they are.

As examples of diffeological fibration, any quotient of Lie group by any subgroup, even if this subgroup is not closed and the quotient is not a manifold. This is the case for the quotient of the standard tori by irrational flows.
I define the higher homotopy groups for diffeological space, by recursion on the space of paths. I show the exactness of the homotopy sequence of diffeological fiber bundles. Applied to the case of irrational torus, for example, it tells us that the homotopy is the same as the homotopy of the 2-torus, since the fiber is contractible.
I generalized to diffeological spaces the notion of covering spaces, using diffeological fibration with discrete fiber, and I prove the existence and uniqueness of an universal covering for any diffeological space. The other covering spaces are then given by quotients. And, I show that the monodromy theorem remains true, applied to diffeological spaces.
This text is in french, I'm trying to include it into my current work about symplectic diffeology. I hope this will help people interested in diffeology and especially in the notion of fibration and homotopy over diffeological spaces.

This is the scan of my doctorat thesis, defended in 1985 at the University of Provence. It has been never published and it is practically inaccessible now, it's why I decided to scan it and publish it on the web.

I rencently noticed lot of misprints in the original text of my thesis. I recom-mend instead reading the chapter VIII of the book, even if it is unfinished.