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?

Question

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?

Answer 1

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 $$\eta=\sum_{1\leq i_1<\cdots<i_k\leq n}\eta_{i_1,\dots,i_k}\theta_{i_1}\wedge\cdots\wedge\theta_{i_k}$$ 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 $1\leq i_1<\cdots<i_k\leq n$ 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.