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Suppose that $M$ and $N$ are both closed compact connected orientable $n$-manifolds and $f : M \to N$ is a map of with degree > 1. Does this imply that $f$ restricted to any codimension 0 submanifold is surjective?

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No. Take $f:S^1 \to S^1$ to be $f(z)=z^2$, and $U=\{x \in S^1 \mid x_1,x_2 >0\}$. $U$ is a codimension $0$ submanifold, but $f|U$ is not surjective.

If you restrict $f$ to codimension $0$ compact submanifolds, the result is true, but you are talking about a triviality, since the inclusion will be a homeomorphism (being open and closed in a connected manifold).

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