I am studying for a topics exam and the reference I'm using seems very sparse on the topic of Poincare duality. A sample exam question: let $M = \mathbb{R}^3/\mathbb{Z}^3$ be a three-dimensional torus, and $C = \pi(L)$, where $L \subset \mathbb{R}^3$ is the oriented line segment from $(0,1,1)$ to $(1,3,5)$ and $\pi: \mathbb{R}^3 \to M$ is the quotient map. Find a differential form on $M$ which represents the Poincare dual of $C$.
As far as I understand, the Poincare dual is the unique $2$-form $\omega$ so that $$ \int_C \eta = \int_M \omega \wedge \eta $$ for every $1$-form $\eta$ on $M$. I'm also unsure I understand exactly how integration on this quotient manifold works, so detailed solutions/corrections would be appreciated -- if $(x,y,z)$ are the coordinates on $\mathbb{R}^3$, then intuitively I believe $\int_M dx \wedge dy \wedge dz$ should equal $1$, and I think I can justify this because $dx \wedge dy \wedge dz$ is a scalar multiple of the volume form $d \theta \wedge d \phi \wedge d \mu$ on $T^3 = S^1 \times S^1 \times S^1$. I also think then for a $1$-form $\eta = a dx + b dy + c dz$, $a, b, c \in \mathbb{R}$, $\int_{C} \eta = a + 2b + 4c$ since $C$ is a loop that wraps around once in the $x$ direction, twice in the $y$ direction, etc., but I'm unsure how to justify this more rigorously than that, and I'm not sure how the calculation proceeds if $a, b, c$ are smooth real-valued functions on $C$ instead.
One proposed solution I found to this question so far writes $\omega$ as $A dx \wedge dy + B dx \wedge dz + C dy \wedge dz$, $\eta = a dx + b dy + c dz$, and computes $$ \int_M \omega \wedge \eta = \int_M (Ac - Bb + Ca) dx \wedge dy \wedge dz $$ and then pulls $Ac - Bb + Ca$ out of the integral, which seems to imply that they took all of these to be constant functions, and then solved C = 1, A = 4, B = -2 for the Poincare dual. If this solution is correct, why is it enough to only consider constant functions when finding the Poincare dual?