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I wonder if we can embed a compact orientable surface of genus $g$ into another of genus $g'$, if $g < g'$. I already know that this is false if $g>g'$, because of the first homology groups. Any idea? I appreciate your hints. Best,

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up vote 2 down vote accepted

If $M$ is a submanifold of $M'$, where the manifolds are without boundary and of the same dimension, then $M$ is an open subset of $M'$, by the invariance of domain. If $M$ is also compact, then it's a closed subset of $M'$ as well. Therefore, if $M'$ is connected then $M=M'$.

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Thank you very much! –  math_failure Jan 15 '13 at 20:13

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