Integral along different loops on torus gives a full lattice in $\mathbb C$

Question

Let $S$ be a compact Riemann surface we $g=1$, $\alpha,\beta$ be the generator of $\pi_1(S)$, $\omega\neq 0$ be a fixed holomorphic $1$-form. How can we know $\int_\alpha \omega$ and $\int_\beta \omega$ are $\mathbb R$-linearly independent?

Answer 1

This is by de Rham's theorem. (Which is used in the proof that the Abel-Jacobi map is an isomorphism.)

Answer 2

Since $g=1$, $\omega$ is a basis of $H^{1,0}(S)$. Fix a point $x_0$, and let $\Lambda$ be the lattice spanned by $\int_\alpha \omega$ and $\int_\beta \omega$. Then the Abelian-Jacobian map $$S \to \mathbb C /\Lambda \\ x \mapsto(\int_{x_0}^x \omega,\int_{x_0}^x \omega)$$ is an isomorphism. In particular $\int_\alpha \omega$ and $\int_\beta \omega$ form a $\mathbb R$-basis of $\mathbb C$.