nLab
evaluation fibration of mapping spaces

Contents

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 SS and a manifold MM, we pick a point pSp \in S and consider the map ev p:C (S,M)Mev_p \colon C^\infty(S,M) \to M given by ev p(α)=α(p)ev_p(\alpha) = \alpha(p). This is smooth (see smooth maps of mapping spaces?). At qMq \in M, the fibre is the space of smooth maps SMS \to M which take pp to qq. This is again a smooth manifold (as shown in manifold structure of mapping spaces). Providing MM has enough diffeomorphisms, this is the projection of a fibre bundle. That is to say, the sequence:

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

Revised on June 3, 2011 08:43:39 by Urs Schreiber (89.204.137.115)