# Is true that a closed $k-$form on $T^n$, the torus, is exact if, and only if, the integral of $\omega$ over every compact submanifold is zero?

Is true that a closed $k-$form on $T^n$, the torus, is exact if, and only if, the integral of $\omega$ over every compact and oriented submanifold is zero?

Let $$\omega$$ be a closed $$k$$-form on the torus. Let $$\theta_1,\dots,\theta_n$$ be the $$n$$ angle $$1$$-forms on $$T^n$$. We know that these generate the cohomology of $$T^n$$, so we know that $$\omega$$ is cohomologous to a (unique!) form of the form $$\eta=\sum_{1\leq i_1<\cdots with all the coefficients $$\eta_{i_1,\dots,i_k}$$ constant. Moreover, the form $$\omega$$ is exact iff all these coefficients are zero.
Suppose now the integral of $$\omega$$ along all closed submanifolds of dimension $$k$$ is zero.
Pick indices $$i_1,\dots,i_k$$ such that $$1\leq i_1<\cdots and consider a $$k$$-torus $$T^k$$ contained in $$T^n$$ in which the coordinates with indices not in $$\{i_1,\dots,i_k\}$$ are constant. The integral of $$\omega$$ over this subtorus is the same as the integral over $$T^k$$ of $$\eta$$. Compute this last integral, and see that it is a non-zero multiple of the number $$\eta_{i_1,\dots,i_k}$$. It follows that all coefficients of $$\eta$$ are zero.