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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?

Thank you in advance

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Crossposted to MO. –  t.b. Jun 1 '11 at 0:22
You have a history of asking homework-type problems, but not giving us much context as to what you understand, what you've tried, or why you care about the question. In this, you haven't described your map. Your second question does not appear to be a sub-question of the first, but then it's not clear what your first question is. –  Ryan Budney Jun 1 '11 at 0:27
@Ryan I first thank you for taking the time to read my questions. I can ensure you that I have NEVER posted a homework question on this forum. I ask questions that would help me better understand my course so that I could tackle serious exercices afterward. I don't like this discussion to be honest, it is the second time that I had to explain this –  El Moro Jun 1 '11 at 0:33
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
@El Moro: No it certainly doesn't physically harm anyone :) Look, I don't mind whatever you choose to do, but I do think that you could get better answers by asking fewer questions. However, I do think that cross-posting isn't very nice as users of one forum might not know about the question on the other one. That much should be obvious. –  t.b. Jun 1 '11 at 0:48

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