In this post, it is outlined how to find a differential $n$-form on $U_0 = \mathbb{R}^n\backslash\{\text{pt}\}$ whose exterior derivative is zero but which is not the exterior derivative of an $(n-1)$-form.
Suppose I have $U_1 = \mathbb{R}^3\backslash(S^1 \times \{0\})$, so that there is a torus (which has empty boundary, and so is closed [as a manifold]) which does not bound in $U_1$; how do I find a vector field $\vec{F}$ (resp. a 2-form $\omega$) on $U_1$ with $\vec{\nabla} \cdot \vec{F} = 0$ (resp. $d\omega = 0$) but which does not admit a vector potential function $\vec{A}$ on $U_1$ with $\vec{\nabla} \times \vec{A} = \vec{F}$ (resp. a 1-form $\theta$ on $U_1$ with $d\theta = \omega$)? This should exist by de Rham's Theorem and should have its integral over the torus, $\unicode{x222F}_{T^2} \vec{F} \cdot d\vec{S}$ (resp. $\unicode{x222F}_{T^2} \omega$), non-zero.
In principle, how do I extend this to $U_2 = \mathbb{R}^3\backslash(\text{Wedge of Two Circles})$ (which has a 2-holed-torus that has empty boundary but does not bound in $U_2$), $U_3 = \mathbb{R}^3\backslash(\text{Wedge of Three Circles})$ (which has a 3-holed-torus that has empty boundary but does not bound in $U_3$), etc.?
Thanks much in advance.