Let $M$ and $N$ be compact orientable and connected smooth $n$-manifolds and $F:M \to N$ a smooth map. Suppose $$\int_M F^* \eta \ne 0$$ for some $\eta \in \Omega^n(N)$. Then $F$ is surjective. Give an example that shows the converse is not true.
A non-surjective map has degree $0$ so the first part is clear. I could not think of an example for the converse, however. I want to find two compact oriented connected manifolds such that $F$ is surjective but $\int_M F^* \eta = 0$ for all $\eta \in \Omega^n(N)$.
Best Answer
Here is a concrete realization of Mike Miller's comment. Think of $S^1$ as sitting in $\mathbb{C}$ and consider the map \begin{align*} \varphi: S^1 & \to S^1 \\ x+iy & \mapsto e^{2\pi i x}. \end{align*} Then $\varphi$ is both surjective and null-homotopic and thus serves as a counterexample.