The set of smooth maps from exotic smooth manifolds to the reals Here a $M,N$ are topological manifolds and $\mathcal{A}$ and $\mathcal{B}$ are atlases. The brackets $[]$ denote the formation of the equivalence class of atlases.
Let $(M,[\mathcal{A}])$ and $(N,[\mathcal{B}])$ smooth manifolds exotic to each other ($M$ and $N$ homeomorphic, lets say $h(M)=N$, but not diffeomorphic with the smooth structures).
I wondered if the following statements are true.


*

*$f\in C^\infty (M,[\mathcal{A}])$ is not equivalent to  $f\circ h \in C^\infty (N,[\mathcal{B}])$

*There exists an $f\in C^\infty (M,[\mathcal{A}])$ such that there is no $g\in C^\infty (N,[\mathcal{B}])$ such that $f=g\circ h$ and the other way around: There exists an $g\in C^\infty (N,[\mathcal{B}])$ such that there is no $f\in C^\infty (M,[\mathcal{A}])$ such that $g=f\circ h^{-1}$.

*For all $f\in C^\infty (M,[\mathcal{A}])$ there is no $g\in C^\infty (N,[\mathcal{B}])$ such that $f=g\circ h$.


Or in more transparent version with atlases dropped from notation and $M=N$ as topological spaces, but still not diffeomorphic.


*

*$C^\infty M\neq C^\infty N$

*$C^\infty M\not\subset C^\infty N$ and $C^\infty N\not\subset C^\infty M$

*$C^\infty M\cap C^\infty N=\emptyset$


Thanks in advance and maybe the diffoelogy characterization of smoothness is helpful.
Kind regards
Mar
(corrected the error)
 A: *

*True, essentially by the definition. 

*False in general. Edit: There are examples of homeomorphic but not diffeomorphic manifolds $M, N$ so that there exists a smooth homeomorphism
$$
h: M\to N.
$$
(Milnor's exotic spheres satisfy this.) Note that, of course, the inverse of such $h$ is not smooth. Therefore, the pull-back via $h$ defines an embedding
$$
h^*: C^\infty(N)\to C^\infty(M), \quad h^*(\varphi)= \varphi \circ h.  
$$
Alternatively, if you take a homeomorphism $h$ whose inverse is smooth, you obtain the embedding
$$
h_*: C^\infty(M)\to C^\infty(N). 
$$

*Always false, take $f$ which is constant. 
Note that I was addressing here question in the first three numbered items, the last 3 numbered items make no sense to me. 
2nd Edit: One more thing, which answers a separate question by Jason (in comments). An isomorphism of ${\mathbb R}$-algebras 
$$
C^\infty(N)\to C^\infty(M)
$$
will imply diffeomorphism of the corresponding manifolds $M$ and $N$ (no need to assume a priori that they are homeomorphic). This is because there is a natural bijection 
$$
C^\infty(M,N)\to Hom(C^\infty(N), C^\infty(M))$$ 
see theorem 2.3 in the book 
Navarro Gonzalez and Sancho de Salas, "$C^\infty$-differentiable spaces", Springer Lecture Notes in Math. vol. 1824.  
