integration of differential forms on covering space

Question

Let $M_1,M_2$ be $n$-dimensional oriented manifolds. Let $f: M_1\longrightarrow M_2$ be an orientation-preserving diffeomorphism. Then for any $\omega\in \Omega^n_c(M_2)$ we have(page 85 of {Madsen: from calculus to cohomology}) \begin{eqnarray*} \int_{M_2}\omega=\int_{M_1}f^*\omega. \end{eqnarray*} Generally, let $f: M_1\longrightarrow M_2$ be a $n$-sheeted covering map. Then whether is it true or not \begin{eqnarray*} n\int_{M_2}\omega=\int_{M_1}f^*\omega? \end{eqnarray*}

Answer 1

Yes, that's true. Check this first on an open set whose preimage is a disjoint union of sets, then reduce to that case by a partition of unity.