Poincaré duals and intersections [closed]

If I have $A$, $B$ two submanifolds of dimension n each included in a $2n-$manifold $M$ whose n-cohomology group is free of rank 1 and generator $\alpha$ .denote $\epsilon_{A}$ and $\epsilon_{B}$ both poincaré duals. $\epsilon_{A}=d_{1} \alpha$ and $\epsilon_{B}=d_{2} \alpha$. What can I say about the coefficients $d_{i}$?

a subquestion is: when could I tell that two submanifolds intersect transversally without having to go through the whole machinery that defines such an intersection?