3
$\begingroup$

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*}

$\endgroup$
2
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.