$F:M\to N$ is surjective if $\int_M F^* \eta \ne 0$ for some $\eta \in \Omega^n(N)$

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 $$\eta \in \Omega^n(N)$$.

• @MoisheCohen Yes I'm looking for a counterexample – mysatellite Dec 31 '18 at 2:01
• Choose $F$ surjective but null-homotopic. – user98602 Dec 31 '18 at 2:20
• I don't know if you read the deleted answer and all the comments therein before it was deleted. Are you still interested in an answer? – Amitai Yuval Dec 31 '18 at 16:24

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.