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).##