In some cases, there is an easy method available to study the topology. If the target manifold, say $M$, embeds as a submanifold of some convenient vector space, say $V$, then $C^\infty(S,M)$ embeds as a submanifold of $C^\infty(S,V)$. 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 $C^\infty(S,V)$ to $C^\infty(S,M)$. Furthermore, if $M$ is a deformation retract of a neighbourhood of its image in $V$, then so is $C^\infty(S,M)$ in $C^\infty(S,V)$.

Embeddings

Theorem

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