# Computing degree of map

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)?

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