I got stuck in the following problem. Let $M$ be the manifold defined by the equation $x^2+y^2+z^4=1$ and $F: M \to S^2$ defined as $F(x,y,z)=(x,y,z^2)$. I have to calculate the integral $\int_M F^* \omega$ for an arbitrary $\omega$ of degree two in $S^2$.
I believe that the result must be zero. What leaded me to think this is: the homology of the sphere is $\mathbb{R}$, so there is only one non-zero class of two-forms in the sphere, the one defined by the volume form (I'm not completely sure about this).
If $\omega$ belongs to the null class, then it is an exact form and applying Stokes' theorem (and using that the pullback commutes with the exterior derivative) the integral is equal to 0.
So, the only thing to check is that the integral is 0 for the class of the volume form. To do this I thought that I might apply some kind of symmetry argument because I've already proven that $F$ is a local diffeomorphism for $z \neq 0$ and it preserves the orientation in one side an inverts it in the other one. I haven't been able to conclude my argument.
How should I proceed? Is there any easier approach (maybe without using homological arguments)?
$F:M\rightarrow S^2$ is a differentiable map between two compact manifolds. Given a $2$-form $\omega\in \Omega(S^2)$, $\int_MF^*\omega= degree(F)\int_{S^2}\omega$. Here it is easy to see that the degree of $F$ is 0 since $F$ is not surjective.