nLab
evaluation fibration of mapping spaces

Idea

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 $S$ and a manifold $M$, we pick a point $p\in S$ and consider the map ${\mathrm{ev}}_p:C^{\mathrm{\infty }}(S,M)\to M$ given by ${\mathrm{ev}}_p(\alpha )=\alpha (p)$. This is smooth (see smooth maps of mapping spaces?). At $q\in M$, the fibre is the space of smooth maps $S\to M$ which take $p$ to $q$. This is again a smooth manifold (as shown in manifold structure of mapping spaces). Providing $M$ has enough diffeomorphisms, this is the projection of a fibre bundle. That is to say, the sequence:

(1)$$C^{\mathrm{\infty }}(S,p;M,q)\to C^{\mathrm{\infty }}(S,M)\to M$$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 $\beta :S\to M$ nearly takes $p$ to $q$ then we can deform $\beta$ to a map which takes $p$ to $q$ on the nose. Knowing that $M$ has the structure of a manifold, we can interpret the statement ”$\beta (p)$ is close to $q$” as meaning that we have fixed a chart near $q$ and require that $\beta (p)$ be in the codomain of that chart. We therefore have a good choice of deformation for $\beta (p)$ itself: deform it along the “straight line” from $q$ to $\beta (p)$ as defined by the chart. The problem with this is that it only tells us what to do with $\beta (p)$, not with the rest of $\beta$. So we need to drag the rest of $\beta$ along with $\beta (p)$. This is where the diffeomorphisms come in: instead of moving $\beta (p)$ along a path, we deform the entire manifold using a diffeomorphism so that $\beta (p)$ is taken to $q$. We do this using the methods of propagating flows.