Finding a basis for the cohomology vector space of 1-forms in the 2-torus, $H^1 (T^2)$

Question

I would like help in understanding where I am going wrong here:

If I consider the 2-torus $T^2 = S^1 \times S^1$ with an atlas $(\theta_1,\theta_2)$, I can define 2 closed 1-forms

$\omega_1 = d\theta_1, \omega_2 = d\theta_2$

which are closed, $d\omega_i = 0$ for $i=1,2$ since $d^2 = 0$ i.e. nilpotent. What I do not understand is why they are inequivalent (and hence used as the basis which span $H^1(T^2)$). From the definition of the cohomology equivalence classes, these two 1-forms are inequivalent if they do not differ by an exact 1-form i.e.

$\omega_1 - \omega_2 = d\alpha$

has no solution for $\alpha$. However if I set

$\alpha = \theta_1 - \theta_2$

then in the coordinate basis

$d\alpha = \frac{\partial \alpha}{\partial \theta_i} d\theta_i = d\theta_1 - d\theta_2$

as required. Where is the problem here?

Answer 1

Neither $\theta_{1}$ nor $\theta_{2}$ is a (global, continuous) function on the torus, only on the universal cover. Integrating "$d\theta_{i}$" over a closed path parallel to the $\theta_{i}$ axis gives $1$, while the integral of an exact form would be $0$.