Let $X$ be a compact Riemann surface. I will denote by $H^{.}(X, \mathbb{C})$ (respectively $H^{.}(X, \mathbb{Z})$) the Cesh cohomology associated to the constant sheaf $\mathbb{C}$ (respectively $\mathbb{Z}$). The inclusion of sheaf $\mathbb{Z} \mapsto \mathbb{C}$ induces a map $H^{1}(X, \mathbb{Z}) \mapsto H^{1}(X, \mathbb{C})$.
Why is this map injective in the case where $X$ is a Riemann surface?
I use this to show $H^{1}(X, \mathbb{Z})$ is finitely generated over $\mathbb{Z}$ and torsion free so I can't use those facts to show the statement above.
I try to consider the exponential sequence $\{0\} \mapsto \mathbb{Z} \mapsto \mathbb{C} \mapsto \mathbb{C}^{*} \mapsto \{0\}$ but it doesn't help me.
Best Answer
The exponential sequence works fine here. We get a long exact sequence in cohomology
$$0 \to H^0(X, \mathbb{Z}) \to H^0(X, \mathbb{C}) \xrightarrow{\exp(2\pi i \cdot)} H^0(X, \mathbb{C}^{\times}) \xrightarrow{\partial} H^1(X, \mathbb{Z}) \xrightarrow{f} H^1(X, \mathbb{C}) \to \dots $$
and by exactness to show that $f$ is injective we need to show that $\partial = 0$. By exactness again we need to show that $\exp(2 \pi i \cdot)$ is surjective. But this is clear: the global sections of these sheaves correspond to locally constant functions to $\mathbb{C}$ and to $\mathbb{C}^{\times}$ respectively, so this follows from the fact that $\exp : \mathbb{C} \to \mathbb{C}^{\times}$ is surjective.
(Note that we use almost nothing about $X$ here; this argument applies to any space with reasonable local connectivity properties (although I'm not sure exactly what is necessary), e.g. any CW complex.)