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Suppose I have to smooth oriented manifolds, $M$ and $N$ and a local diffeomorphism $f : M \rightarrow N$. Let $\omega$ be a differential form of maximum degree on $N$, let's say, $r$. How can I rewrite $$\int_N \omega$$ in terms of the integral of the pullback of $\omega$ by $f$, $f^*\omega$?

So, I know that when $f$ is a diffeomorphism, then $$\int_N\omega = \pm \int_M f^*\omega$$ depending on whether $f$ preserves the orientation or not. But that's in part due to the fact that $f$ is bijective, but that condition is removed when assuming $f$ is a local diffeomorphism. So is there a nice way to write that integral in terms of the pullback?

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Yeah, and you can make it work even if $f$ is not a local diffeomorphism, only a proper map. The relevant notion is that of the degree of a smooth proper map between oriented manifolds of the same dimension.

Assume for simplicity that $M,N$ are closed (compact, without boundary), non-empty and connected and let $f \colon M \rightarrow N$ be an arbitrary smooth map. Since $M,N$ are oriented, we can a generator $[\omega_1] \in H^{\text{top}}(M)$ such that $\omega_1$ is consistent with the orientation on $M$ and $\int_M \omega_1 = 1$. Choose $[\omega_2]$ similarly for $N$. The map $f$ induces a map $f^{*} \colon H^{\text{top}}(N) \rightarrow H^{\text{top}}(M)$ on cohomology and since the top cohomology groups are one-dimensional, we must have $f^{*}([\omega_2]) = c [\omega_1]$ for some $c \in \mathbb{R}$. This $c = \deg(f)$ is called the degree of $f$ and is a priori a real number. However, it can be shown that $c$ is in fact an integer which can be computed by counting the number of preimages of a regular value $p \in N$ with appropriate signs which take the orientations of $M,N$ into consideration.

Knowing that, given $\omega \in \Omega^{\text{top}}(N)$, write $[\omega] = c[\omega_2]$ for some $c \in \mathbb{R}$. Then

$$ [f^{*}(\omega)] = f^{*}([\omega]) = f^{*}(c[\omega_2]) = c \deg(f) [\omega_1] $$

so

$$ \int_M f^{*}(\omega) = \deg(f) c \int_N \omega_1 = \deg(f) c = \deg(f) c\int_N \omega_2 = \deg(f) \int_N \omega. $$

In particular, if $f$ is a diffeomorphism, $\deg(f) = \pm 1$ so this generalizes your starting point. If $f$ is a local diffeomorphism then $f \colon M \rightarrow N$ is a covering map and $\deg(f)$ will be the number of points in an arbirary fiber, counted with appropriate signs. For much more details and proofs, see the book "Differential Forms in Algebraic Topology".

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    $\begingroup$ For your particular case, you can avoid mentioning cohomology by showing first (assuming $M,N$ are compact) that $f$ is a covering map. Then, over a basic neighborhood $U \subseteq N$, we have $f^{-1}(U) = \sqcup {U_i}$ where $f|_{U_i} \colon U_i \rightarrow U$ is a diffeomorphism which reverses or preserves orientation. Hence $\int_{f^{-1}(U)} f^{*}(\omega) = \sum \pm \int_{N} \omega$. Then you can get the global result by a partition of unity after showing that "the signs stay constant" on the intersections so there isn't any unwanted cancellations. $\endgroup$ – levap Jan 25 '17 at 21:19

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