On the geometry of the space of fibrations

Abstract

We study geometrical aspects of the space of fibrations between two given manifolds $M$ and $B$, from the point of view of Fréchet geometry. As a first result, we show that any connected component of this space is the base space of a Fréchet-smooth principal bundle with the identity component of the group of diffeomorphisms of $M$ as total space. Second, we prove that the space of fibrations is also itself the total space of a smooth Fréchet principal bundle with structure group the group of diffeomorphisms of the base $B$ .