# Smooth approximations to continuous families of diffeomorphisms

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?