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,
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 $M=M'$.