Let $\phi: M \rightarrow N$ and $\psi: N \rightarrow P $ be differentiable applications of differentiable varietes. Prove:
a) $(\psi∘\phi)^*=\phi^* ∘ \psi^*$
What I tried:
$(\psi∘\phi)^*(\omega^*)(u)= (\omega^*)(\psi∘\phi(u))^*= \psi^*(\omega^*(\phi(u)))= (\psi^*(\omega^*))(\phi(u))= \psi^*(\phi^*(\omega^*(u))) =\phi^* ∘ \psi^*(\omega^*)(u)$
b) If $\phi$ is a diffeomorphism, then the vector spaces $H^r(M)$ and $H^r(N)$ are isomorphic for all r.
For the b) I know only the definition of diffeomorphism and isomorphism, but I don't know how to "connect" them.