Pullbacks of volume forms on the circle fail to extend to closed forms

Question

I am working through this problem:

Let $M$ be a compact oriented 3-manifold with boundary, where the boundary is $\partial M=S^1\times S^1$. Let $\theta_i\in\Omega^1(\partial M)$, $i = 1, 2$ be the 1-forms obtained by pulling back the standard volume form $\theta\in\Omega^1(S^1)$ by the first and second projections $\pi_i:S^1\times S^1\to S^1$, $i = 1, 2$. Prove that it is not possible to extend both $\theta_1$ and $\theta_2$ to closed 1-forms on M (i.e. one of $\theta_1$ must fail to extend in this way).

My progress so far: Assume both $\theta_i$ could be extended to closed forms, then their wedge product is also closed. Then I would like to use Stokes to reach a contradiction - the volume of the torus is obviously non zero so I hope I can write $vol(nonzero)=\int_M d\theta_1\wedge d\theta_2=\int_{\partial M} d(d\theta_1\wedge d\theta_2)=0$ Am I correct? Any help and/or correction would be appreciated.

Answer 1

You're on the right track. Elaborating on the hints by Arctic Char and Ted Shifrin ...

Suppose both $\theta_1$ and $\theta_2$ extend to closed one-forms $\omega_1$, $\omega_2$ on $M$. Notice this means that the three-form $d(\omega_1\wedge\omega_2) = d\omega_1\wedge\omega_2 \pm \omega_1\wedge (d\omega_2) = 0$ is identically zero. Now compute: $$ 0 = \int_M d(\omega_1\wedge\omega_2) = \int_{\partial M} \omega_1\wedge \omega_2 = \int_{\partial M} \theta_1\wedge\theta_2 \neq 0 $$