On the geometry of the space of fibrations

TL;DR

This paper explores the geometric structure of the space of fibrations between manifolds, revealing it as a principal bundle with diffeomorphism groups as structure groups, from a Frechet geometric perspective.

Contribution

It demonstrates that the space of fibrations forms a principal bundle structure with diffeomorphism groups, advancing the understanding of its geometric and topological properties.

Findings

Connected components form principal bundles with diffeomorphism groups.

The space of fibrations is a principal bundle over the base manifold.

The structure groups are the identity components of diffeomorphism groups.

Abstract

We study geometrical aspects of the space of fibrations between two given manifolds M and B, from the point of view of Frechet geometry. As a first result, we show that any connected component of this space is the base space of a Frechet-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 Frechet principal bundle with structure group the group of diffeomorphisms of the base B.