I'm working on the following question from Lee's Smooth Manifolds:
11-17. Let $\mathbb{T}^n = \mathbb{S}^1 \times \cdots \times \mathbb{S}^1 \subseteq \mathbb{C}^n$ denote the $n$-torus. For each $j = 1, \ldots,n$, let $\gamma_j : [0,1] \to \mathbb{T}^n$ be the curve segment
$$\gamma_j(t) = \big ( 1,\ldots,e^{2\pi i t},\ldots,1 \big ) \quad \text{(with $e^{2 \pi i t}$ in the $j$th place)}.$$
Show that a closed covector field $\omega$ on $\mathbb{T}^n$ is exact if and only if $\int_{\gamma_j} \omega = 0$ for $j = 1, \ldots, n$. [Hint: consider first $(\epsilon^n)^*\omega$, where $\epsilon^n : \mathbb{R}^n \to \mathbb{T}^n$ is the smooth covering map $\epsilon^n \big ( x^1, \ldots, x^n \big ) = \big ( e^{2 \pi i {x^1}}, \ldots, e^{2 \pi i {x^n}} \big )$.]
I've used the hint to find a potential function $f$ for the pullback $(\varepsilon^n)^* \omega$ and I'm trying to use this $f$ to define a potential function for $\omega$, however the conditions we are given do not seem strong enough? Using integration over $\gamma_j$, we can get that $f$ must agree at all integer coordinates, however to make $f$ well-defined on the torus, don't we also need it to agree on the "edges"? For example, in the case $n=2$ shouldn't we also have for $t \in [0,1]$ that $$f(0, t) = f(1, t)$$ This doesn't seem to follow from the integration condition we are given.
Thanks to the comment above, I have realized why the "edges" agree: $$f(1,t) - f(0,t) = \int_{(0,t)}^{(0,0)} (\varepsilon^n)^* \omega + \int_{(0,0)}^{(1,0)} (\varepsilon^n)^* \omega +\int_{(1,0)}^{(1,t)} (\varepsilon^n)^* \omega = 0$$ The first and the third integral are mapped to paths that are precisely inverses of each other, so they cancel out. The middle integral is $0$ by the assumption. My problem is solved.