# transversal intersection and Poincaré duals

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?

Your initial question still makes little sense to me. The middle-dimensional homology group is apparently $\mathbb Z$, so your coefficients tell you what multiple of the generator your classes are. If I get rid of the isomorphism, I could say: I have two integers $\alpha$ and $\beta$, and I want to represent $\alpha = n \cdot 1$ and $\beta = m \cdot 1$, what is $n$ and $m$? Or am I badly misunderstanding your question? –  Ryan Budney Jun 1 '11 at 0:37