First homology group for Riemann surfaces and for topological manifolds.

Question

I'm reading the book of Miranda "Algebraic Curves and Riemann Surfaces". Here he defines the first homology group $ H_1(X)$ of a Riemann surface $X$ as the quotients of the group of the closed chains modulo boundary chains. But at a certain point he says that there is a map associated to any differential form $\omega$ $$H_1(X)\to \mathbb{C}$$ $$\hspace{29pt}[\gamma]\mapsto \int_\gamma \omega$$ I think that this makes sense only if $\gamma$ is piecewise $C^1$. The group $H_1(X)$ for a topological surfaces is constructed starting from continuous paths. For a manifold which has also a differentiable structure (for example a Riemann Surface) one can surely do a very similar construction considering only piecewise $C^1$ paths. My question is: is it true that this two apparently different constructions give the same group? And why?