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Let $\Gamma$ be a curve. What are explicit elements in $H^1(\Gamma, \mathbb{C})$? For example, let $\Gamma$ be the plane curve $y=x^2$. What is $H^1(\Gamma, \mathbb{C})$? Thank you very much.

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up vote 3 down vote accepted

Singular cohomology depends only on the homotopy type of the curve, so the cohomology is the same as if you collapse a "maximal tree" (which more properly is a notion belonging to graph theory, but I'll appropriate it anyways). In any case, it's a pretty lousy invariant of curves.

Anyways, $H^0\cong \mathbb{C}^c$ and $H^1\cong \mathbb{C}^z$, where $c$ is the number of path-components and $z$ is the number of cycles. So in the case of your example $y=x^2$, $H^1$ is trivial since the curve has no self-intersections. More generally, $H^1(\Gamma;\mathbb{C})$ is generated by the duals of the images of the fundamental class in $H_1(S^1;\mathbb{C})$ coming from various maps $S^1 \rightarrow \Gamma$, by the above and by the universal coefficient theorem.

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