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$\phi: \pi_1(S)\to \pi_1(S)$ is a homomorphism of fundamental group of closed orientable surface $S$ of genus >=2. If $\phi$ is not an epimorphism, can we find a non-surjective self map $f: S\to S$ induces $\phi$ such that $f(S)$ be a submanifold embedded in $S$?

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    $\begingroup$ Could the people voting to close please explain why? $\endgroup$
    – Mark Grant
    Commented Nov 8, 2013 at 7:03
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    $\begingroup$ Taking $\mathbb{T}^2= S^1 \times S^1$, the real $2$-torus, one has $\pi_1(\mathbb{T}^2)=\mathbb{Z} \oplus \mathbb{Z}$. Taking as $\phi$ the projection onto one of the factors, it seems to me that the image of the corresponding self-map $\mathbb{T}^2 \to \mathbb{T}^2$ is a copy of $S^1$. $\endgroup$
    – Francesco Polizzi
    Commented Nov 8, 2013 at 7:19
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    $\begingroup$ @MarkGrant: I voted to close because it is more suitable for MSE. To begin with, it is based on a false premise that non-epimorphism can be induced by a non-surjective map (think of finite covers of the torus to itself); however, this false premise does hold in the higher genus case (why?). Secondly, any continuous map $f: M^m\to N^n$ is homotopic to a surjective map (think of a Peano curve). $\endgroup$
    – Misha
    Commented Nov 8, 2013 at 8:31
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    $\begingroup$ @J.C.Wu, you might be interested in Scott's paper 'Subgroups of surface groups are almost geometric'. $\endgroup$
    – HJRW
    Commented Nov 8, 2013 at 10:38
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    $\begingroup$ @HJRW: Yes, I should have, but I was doing this on a tiny screen... $\endgroup$
    – Misha
    Commented Nov 8, 2013 at 12:47

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