In the context of a mapping space, an important family of smooth maps are the evaluation maps. That is, for a sequentially compact Frölicher space and a manifold , we pick a point and consider the map given by . This is smooth (see smooth maps of mapping spaces?). At , the fibre is the space of smooth maps which take to . This is again a smooth manifold (as shown in manifold structure of mapping spaces). Providing has enough diffeomorphisms, this is the projection of a fibre bundle. That is to say, the sequence:
C^\infty(S,p;M,q) \to C^\infty(S,M) \to M
is a fibre bundle.
The remark about “enough diffeomorphisms” is the key to proving this. To prove that this is a fibre bundle, we need to show that if nearly takes to then we can deform to a map which takes to on the nose. Knowing that has the structure of a manifold, we can interpret the statement ” is close to ” as meaning that we have fixed a chart near and require that be in the codomain of that chart. We therefore have a good choice of deformation for itself: deform it along the “straight line” from to as defined by the chart. The problem with this is that it only tells us what to do with , not with the rest of . So we need to drag the rest of along with . This is where the diffeomorphisms come in: instead of moving along a path, we deform the entire manifold using a diffeomorphism so that is taken to . We do this using the methods of propagating flows.