POTW Smooth proper self-maps on Rn

[Euge]

Let $f : \mathbb{R}^n \to \mathbb{R}^n$ be a smooth proper map that is not surjective. If $\omega$ is a generator of $H^n_c(\mathbb{R}^n)$ (the $n$th de Rham cohomology of $\mathbb{R}^n$ with compact supports), show that $$\int_{\mathbb{R}^n} f^*\omega = 0$$

 

[Infrared]

I haven't learned (or remember?) compactly supported cohomology well, so please let me know if there are errors here.

Spoiler

If we consider $\mathbb{R}^n$ as $S^n-\{N\}$ we can view $\omega$ as a top form on $S^n$ that vanishes in a neighborhood of the north pole and $f$ as a smooth map $S^n\to S^n$ that fixes the north pole. Since $f:S^n\to S^n$ misses a point it is homotopic to a constant map as $S^n-\{\text{point}\}$ is contractible. So, an application of Stokes' theorem gives
$\int_{\mathbb{R}^n} f^*\omega=\int_{S^n} f^*\omega=\int_{S^n}(\text{constant map})^*\omega=0.$

Stokes' theorem is used to say that integrating pullbacks of a closed form by homotopic maps on a closed manifold give the same answer. This is seen by considering a homotopy $H(x,t)$ and applying Stokes' theorem to the integral $\int_{S^n\times [0,1]} d(H^*\omega).$