Class in $H^1(T^2)$ that is not dual to closed 1-dim submanifold of $T^2$

Question

The definition for Poincaré dual $\omega$ of $N\subset M$ a $k$-dimensional submanifold is that given $i:N\to M$ we have $\int_N i^*\theta=\int_M \theta\wedge\omega$ for every $k$-form $\theta$. I know how to find the dual to a given differential form, but I am confused about the reverse process here. Thanks!