How to prove $H_1(\mathbb{C}/\Lambda, \mathbb{Z})$ to the lattice $\Lambda$ ,$\gamma \mapsto \int_\gamma dz$ is well defined?

Question

Let fix an arbitraly lattice of $\mathbb{C}$ and call it Λ.

How to prove $H_1(\mathbb{C}/\Lambda, \mathbb{Z})$ is isomorphic to the lattice $\Lambda$ via the map $\gamma \mapsto \int_\gamma dz$ ?

If once the map is proved to be well defined, the map is obviously group homomorphism and injection, and surjection may be managed.

So, I want to check the map is well defined .

Why $ \int_\gamma dz$ is in $Λ$?

Thank you for your help.

Answer 1

Take the loop $\gamma$ in $\mathbb{C}/\Lambda$ and lift it to a path $\bar{\gamma}$ in $\mathbb{C}$. Since $\gamma$ is a loop, the endpoints of $\bar{\gamma}$ must differ by an element of $\Lambda$. Since we're integrating $dz$, the integral will just be the difference between the two endpoints, and thus an element of $\Lambda$.