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I was told that the statement in the title is true, since if we have such $f : M \to \mathbb S^1$ and $M'$, we can defined $g : M \# M' \to \mathbb S^1$ to be constant on the attaching ball; that is, $M' \setminus B$ where $\partial B$ is the attached boundary in $M'$ and be $f$ on others, but not sure how to justify rigorously.

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