Let $X$ be a smooth manifold (not necessarily compact) and let $G = Diff(X)$ be the group of diffeomorphisms. There are many meaningful topologies one can put on this group, and I would be happy to receive an answer concerning any of them. (As a starting point, I might demand that your topology is somewhere between the compact-open topology and the strong Whitney topology.)

Suppose $S$ is a compact Hausdorff space, and consider the natural injection

$$C^0(S, G) \to C^0(S \times X, X)$$

where $C^0(-,-)$ denotes the set of continuous maps.

Question 1: How close is the image of this map to the set of all functions $g: S \times X \to X$ for which $g$ is continuous, and $g|_{\{s\} \times X}$ is a diffeomorphism for all $s \in S$?

Regardless of what topology you put, let me make the following definition (for this post): If $S$ is a compact smooth manifold, then a function $f: S \to G$ is smooth if and only if the adjoint map

$g: S \times X \to X$

is smooth. The following is a smooth approximation question:

Question 2: Let $f_0 : S \to G$ be a continuous map. Does there exist a continuous map $f: S \times [0,1] \to G$ such that $f_1 = f|_{S \times \{1\}}$ is smooth, and $f|_{S \times \{0\}} = f_0$?

You can see my ignorance of Question 1 hampers my ability to address Question 2. Finally, my real interest:

Question 3: What if I further place a geometric structure on $X$ (such as a Riemannian metric or a symplectic structure) and I demand $G$ to be the group of isomorphisms, or of special kinds of isomorphisms (such as isometry, symplectomorphism, or Hamiltonian symplectomorphism). With the same definition of smoothness $f: S \to G$ as above, does Question 1 have the same answer?


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