While my main question is in the title, I'm also trying to classify mappings between such manifolds. I'm new to the field which is why I'm looking for help regarding notation and precise proof-leading. I did not find any directly related question, but of course I'm happy for any reference to one. So here is my case:
Let $M$ be a topological $n$-manifold and let $\mathcal{D}_M$ be a differentiable structure defined as follows:
$\mathcal{D}_M$ is a family of continuous functions defined on open sets of $M$ such that
- $\forall p \in M$ there exists an open neighborhood $U_p$ and a homeomorphism $h: U \rightarrow P \subset \mathbb{R}^n$ such that for every open set $V \subset U$, $f:V \rightarrow \mathbb{R} \in \mathcal{D}_M$ if and only if $f\circ h^{-1} \in C^\infty (h(V))$.
- If $U = \bigcup_{i} U_i$ with $U_i$ open in $M$, then $f:U\rightarrow\mathbb{R}\in\mathcal{D}_M$, if and only if $f: U_i \rightarrow \mathbb{R} \in \mathcal{D}_M$ for all $i$.
Problem 1: A topological $n$-manifold $M$ equipped with such a differentiable structure can be equipped with a maximal Atlas and can hence be interpreted as a smooth manifold.
My ideas: Let $(U, \varphi)$ and $(V, \psi)$ be two charts on $M$. Given $f: U \cap V \rightarrow \mathbb{R} \in \mathcal{D}_M$ we can use that
$$ f\circ \varphi^{-1} \in C^\infty(\varphi(U\cap V)) \ \ \ \ \& \ \ \ \ f\circ \psi^{-1} \in C^\infty(\psi(U\cap V)) $$
to investigate $f \circ \varphi^{-1} = f \circ \psi^{-1} \circ (\psi \circ \varphi^{-1})$. Now as both $f \circ \varphi^{-1}$ and $f \circ \psi^{-1}$ are smooth, so is $\psi \circ \varphi^{-1}$. Hence the transition charts are smooth which allows us to define a maximal atlas on $M$.
Now while I intuitively know this should be true, I cannot find an explicit form of such a maximal atlas. Does the collection of all homeomorphisms $\varphi_a: U_a \rightarrow \mathbb{R}^n$ on $M$ with the respective sets $U_a$ suffice?
Problem 2: Let $F:(M, \mathcal{D}_M) \rightarrow (N, \mathcal{D}_N)$ be a continuous function. We call it differentiable if $f\circ F \in \mathcal{D}_M$ for all $f \in \mathcal{D}_N$. Now I want to show that $F$ being differentiable is equivalent to $F$ being smooth.
My ideas: While I think the main sketch of the following proof is correct, I'm not sure if it is precise regarding the sets the objects live on. Notice that $M$ and $N$ are taken to be smooth manifolds from here on.
$"\Leftarrow"$: Given the charts $(U, \varphi)$ on $M$ and $(V, \psi)$ on $N$, if $F$ is smooth, so is $\psi \circ F \circ \varphi^{-1}$ by definition. Now taking a function $f: V^* \subset V \rightarrow \mathbb{R} \in \mathcal{D}_M$, we have $f\circ \psi^{-1} \in C^\infty(\psi(V^*)) $. By concatenation we can now state that
$$ (f \circ \psi^{-1}) \circ (\psi \circ F \circ \varphi^{-1}) = (f \circ F) \circ \varphi^{-1} \in C^\infty (\varphi(F^{-1}(V^*) \cap U)) $$
So while this could render $f\circ F \in \mathcal{D}_M$, I'm unsure if the smaller set it is smooth on is a) correct and b) a problem regarding the definition of $\mathcal{D}_M$.
$"\Rightarrow"$: If $F$ is differentiable, then $f \circ F \in \mathcal{D}_M$ for all $f \in \mathcal{D}_M$. Now choosing fitting charts $(U,\varphi)$ on $M$ and $(V, \psi)$ on $N$, for an open subset $U^*$ of $U$, we have
$$ (f\circ F) \circ \varphi^{-1} \in C^\infty ( \varphi(U^*)) $$
By inserting $\psi^{-1} \circ \psi$ and hence restricting ourselves to $\varphi(F^{-1}(V)\cap U^*)$ we acquire
$$ (f\circ \psi^{-1}) \circ (\psi \circ F \circ \varphi^-1) \in C^\infty(\varphi(F^{-1}(V)\cap U^*)) $$
Now as $f\circ \psi^{-1}$ is smooth, due to $f\in \mathcal{D}_N$, the function in the right bracket should be smooth as well, right? This would lead to $F$ being smooth, but again I'm unsure if the set the function is smooth on is correct and suffices.
Also I'm kind of worried about not using the second property of functions in differentiable structures ...
While I apologize for this giant question, I highly value consistency and precision. Any criticism on the format, structure and length of my first question is welcome!