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Anybody could help me with this exercise, please? If $M$ is a compact, connected, orientable and smooth $n$-manifold:

1) Show that there is a one-to-one correspondence between orientations of $M$ and orientations of the vector space of its de Rham cohomology, under which the cohomology class of a smooth orientation form is an oriented basis for $H_{dR}(M)$.

2) Now suppose $M$ and $N$ are smooth $n$-manifolds with given orientations. Show that a diffeomorphism $F\colon M \rightarrow N$ is orientation preserving if and only if the pullback between their rham cohomologies is orientation preserving.

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  • $\begingroup$ What do you know / have you tried already? Helping is hard without knowing where to start. $\endgroup$ – T'x Dec 10 '17 at 18:01
  • $\begingroup$ Actually, I am lost, I don't know how to start. $\endgroup$ – Irene Gil Dec 10 '17 at 18:04
  • $\begingroup$ Do you know the fact that an orientation on $M$ is equivalent to a nowhere vanishing $n$-form on $M$? $\endgroup$ – T'x Dec 10 '17 at 18:07
  • $\begingroup$ Yes, I do, but how can I relate this fact with the cohomology? $\endgroup$ – Irene Gil Dec 10 '17 at 18:10
  • $\begingroup$ $H^n(M)$ consists of equivalence classes of $n$-forms. $\endgroup$ – T'x Dec 10 '17 at 18:16

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