I'm trying to understand the definition of the moduli space of genus $g$ Riemann surfaces as the quotient space of the Teichmuller space by the action of the mapping class group. I think I have an understanding, but I believe it would need to rely on this claim:
Given any Riemann surface $\Sigma$ and any element $[f]\in MCG(\Sigma)$ (where $MCG(\Sigma)$ is the mapping class group of $\Sigma$, i.e. homeomorphisms $f:\Sigma\to\Sigma$ up to isotopy) there exists a representative $f:\Sigma\to \Sigma$ of $[f]$ such that $f$ is also a biholomorphism.
I have no idea how I could possibly go about proving this. However, if this were true, I could understand the above definition of the moduli space as saying that two Riemann surfaces $\Sigma$ and $\Sigma'$ are related by an element of the mapping class group if and only if they were in fact related by a biholomorphism. Is this claim true? If so, is there a good reference for a proof? If not, how else could I interpret the above definition of the moduli space of genus $g$ curves?
The claim is not true. A homeomorphism has degree $\pm 1$ but a biholomorphism has degree $1$ as biholomorphisms are orientation-preserving. So any $f : \Sigma \to \Sigma$ which has degree $-1$ will not be homotopic to a biholomorphism. For example, take the antipodal map on $\mathbb{CP}^1$.