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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?

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    $\begingroup$ You only need that every homology class has a piecewise $C^1$ representative $\gamma$, which I believe should be the case. The real meat of showing the map is well-defined is in showing that it is independent of the choice of such a representative. $\endgroup$ – Dan Rust Nov 6 '17 at 10:10
  • $\begingroup$ I think you are right. I think the strategy to make sense about all should be the following: 1) Prove that for any $[\gamma] \in H_1(X)$ there is a $\gamma'\in [\gamma]$ which is piecewise $C^1$. 2) if $\gamma\equiv \gamma'$ are piecewise $C^1$, then there is a $C^1$ homotopical equivalence between them. $\endgroup$ – Anselm Nov 6 '17 at 10:22

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