Basically, I want to know if the following is true:
Given an orientantion-preserving homeomorphism $f:R\to S$ between Riemann surfaces $R$ and $S$, does exist an orientation-preserving diffeomorphism $\tilde{f}:R\to S$ which is homotopic to $f$?
The context of the question: I am reading a book (An Introduction to Teichmüller Spaces, by Imayoshi & Taniguchi) where, in a specific proof, they seem to implicitly use this fact, without making it clear. I don't know if this is really what they're using, but I've figure out that this would be suffice condition for the conclusion. (In the specific case, $R$ (and hence $S$) is a closed (i.e. compact) Riemann surface of genus $g$.)
If someone has the book and wants to look into this more closely, it is at the page 15, right at the second paragraph: "To prove the surjectivity, it is sufficient to show (...)"
Furthermore, if it is true, do you guys recommend some references to it? Is this a "well-known" fact? I'm a beginner ^_^
The mapping class group can be defined with diffeomorphisms.
https://en.wikipedia.org/wiki/Mapping_class_group_of_a_surface#Definition_and_examples
Since you can assume that $g:S\rightarrow R$ is a diffeomorphism. If $f:R\rightarrow S$ is an homeomorphism, $g\circ f$ is homotopic to a diffeomorphism $h:R\rightarrow R$ and $g^{-1}\circ h$ is homotopic to $f$.