Consider the $3$-torus $Y=T^3$, a subset $\Sigma\subset Y$, and $\Sigma^*=Y\setminus\overline\Sigma$.
We assume both $\Sigma$ and $\Sigma^*$ to be open, **connected**, and smoothly bounded.

I am concerned with the subsets $$ A = \ker (H^1(Y)\to H^1(\Sigma)), \quad B=\ker (H^1(Y)\to H^1(\Sigma^*))$$ of the first de Rham cohomology of $Y$. The maps are induced by the restriction maps.

My conjecture is that one of the following three statements must hold true:

- $A=0$ or
- $B=0$ or
- $A=B$ and $\dim A=1$.

Can you help me prove or disprove this conjecture?

**Why I expect the conjecture to be true:**
Honestly, I have only very crude intuitive arguments for this. If $a\in A\subset\mathbb R^3$ is nonzero, then $a$ can be written as a gradient in $\Sigma$. This means that there cannot be a closed loop $\gamma$ in $\Sigma$ "going in the direction of $a$", meaning that the fundamental group class of that loop in $Y$ (considered as an element in $\mathbb R^3$) is non-orthogonal to $a$. But the existence of such a loop is obstructed only by $\Sigma^*$, which means that there must be some sort of two-dimensional plane inside $\Sigma^*$ which is "orthogonal" to $a$. But then, given any direction $b\in\mathbb R^3$ orthogonal to $a$, we can find a closed loop in that plane (thus in $\Sigma^*$) "going in the direction of $b$". This shows that $b\notin A$.

**What I have already done:**
I have already looked at the Mayer Vietoris sequence, but it does not seem to yield enough information. But it helps me to draw conclusions in case I already know the conjecture to be true. Indeed, denoting by $k$ the number of connected components of $\partial\Sigma$, we then know that
$$\begin{align*}
1 &\ge\dim A\cap B = \dim\ker(H^1(Y)\to H^1(\Sigma)\oplus H^1(\Sigma^*)) \\
&= \dim \operatorname{im} (H^0(\partial \Sigma)\to H^1(Y)) \\
&= k - \dim\ker(H^0(\partial\Sigma)\to H^1(Y)) \\
&= k - \dim\operatorname{im}(H^0(\Sigma)\oplus H^0(\Sigma^*)\to H^0(\partial\Sigma) = k-1,
\end{align*}
$$
showing that $\partial\Sigma$ can have at most $2$ connected components.
I have no independet proof of this result, so my second question would be if this statement is true.