Given a $4$-dimensional simply connected manifold $M$ and open sets $U,V\subseteq M$ such that $U\cup V=M$ we can compute the deRham cohomology in terms of the Mayer-Vietoris sequence: \begin{align*} 0\rightarrow H^1(U)\oplus H^1(V) \rightarrow H^1(U\cap V) \xrightarrow{\delta} H^2(X) \rightarrow H^2(U)\oplus H^2(V) \end{align*}
Now assume that the intersection $U\cap V$ is homotopy equivalent to a $2$-dimensional surface $S\subseteq M$.
My Question: Given the Poincare dual $\omega_S\in H^2(X)$ of $S$, is there a relation between $\omega_S$ and the image of $\delta:H^1(U\cap V)\rightarrow H^2(X)$?
