$\phi: \pi_1(S)\to \pi_1(S)$ is a homomorphism of fundamental group of closed orientable surface $S$ of genus >=2. If $\phi$ is not an epimorphism, then $\phi$ is induced by a non-surjective self map $f: S\to S$. Can $f(S)$ be a submanifold embedded in $S$?