nLab
topology of mapping spaces

Contents

Idea

As shown in manifolds of mapping spaces, the space of smooth maps from a sequentially compact Frölicher (or diffeological or Chen) space is again a smooth manifold. As such, its topology can be determined by considering how the charts are glued together.

In some cases, there is an easy method available to study the topology. If the target manifold, say MM, embeds as a submanifold of some convenient vector space, say VV, then C (S,M)C^\infty(S,M) embeds as a submanifold of C (S,V)C^\infty(S,V). If the original embedding is as a closed submanifold, so is the embedding of mapping spaces. This means that it is possible to propagate topological results down from C (S,V)C^\infty(S,V) to C (S,M)C^\infty(S,M). Furthermore, if MM is a deformation retract of a neighbourhood of its image in VV, then so is C (S,M)C^\infty(S,M) in C (S,V)C^\infty(S,V).

Embeddings

Theorem

Let g:MNg \colon M \to N be an embedding of smooth manifolds. Let SS be a sequentially compact Frölicher space. Then C (S,g):C (S,M)C (S,N)C^\infty(S,g) \colon C^\infty(S,M) \to C^\infty(S,N) is an embedding of smooth manifolds.

Revised on June 3, 2011 08:40:16 by Urs Schreiber (89.204.137.115)