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So I could barely understand the problem statement ("oriented boundary given by a surface $F$ having the map $f$"), nor how to proceed. Can I get some hints? Thank you.

Consider the smooth map $f: F \to S^2$ of degree $n$. For which values of $n$ does there exist a compact, oriented 3-manifold $X$ with oriented boundary given by a surface $F$ having the map $f$, and a closed 2-form $\eta \in \Omega^2(X)$ whose restriction to $F$ is $f^*\omega$? Prove that your answer is complete.

1LiterTears
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    Dear Jellyfish, Do you know any facts relating differential forms on $X$ and boundaries of regions in $X$? Especially one that would fit together nicely with assumption that $\eta$ is closed? Regards, – Matt E Aug 17 '13 at 03:35

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