Induced Cohomology maps

For a smooth map $F:N\rightarrow M$ between manifolds, $M,N$ there exists a pullback map of differential forms $F^*:\Omega(M) \rightarrow \Omega(N)$.

Pull back operator $F^*$ has a pleasant property. It commutes with d operator. For closed forms,

$d(F^* \omega)  = F^*(d\omega) = 0$

Thus it maps closed forms from $M$ to closed forms in $N$.

Similarly,

$F^*\omega=F^*d(\eta) = dF^*\eta$ for any exact form $\omega = d\eta$.

Thus it maps exact forms to exact forms.

$F^*$ induces a cohomology map

$F^{\#} : H^{k}(M) \rightarrow H^{k}(N)$ given by
$F^{\#}(\omega)=[F^*\omega]$.

What is nice about this is that diffeomorphism between manifolds $N \rightarrow M$ results in isomorphic vector spaces between $N$ and $M$.