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Suppose given two manifolds $X$ and $Y$, both orientable of dimension $n$, and a map $f:X\to Y$. Is there a relationship between the degree of $f$ calculated with respect to homology (by which I mean the induced map on the top homology groups) and the degree of $f$ calculated with respect to cohomology (by which I mean the induced map on the top cohomology groups)? Thanks!

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They are the same. – Matt E Nov 11 '10 at 17:52
Yes, they are equal. I'm sure this follows from the universal coefficient theorem. – Robin Chapman Nov 11 '10 at 18:42
I assume this requires integral (co)homology, otherwise you only get a $G$-valued degree. But then what happens with non-orientable manifolds, e.g. $S^2 \rightarrow \mathbb{RP}^2$? – Aaron Mazel-Gee Nov 11 '10 at 20:43

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