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Suppose 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 (the induced map on the top homology groups) and the degree of $f$ calculated with respect to cohomology (the induced map on the top cohomology groups)?

Thanks in advance!

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    They are the same.2010-11-11
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    Yes, they are equal. I'm sure this follows from the universal coefficient theorem.2010-11-11
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    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. $S^2 \rightarrow \mathbb{RP}^2$?2010-11-11

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