Consider the surface group $S\_g=\langle a\_1,b\_1,a\_2,b\_2,\dots,a\_g,b\_g \mid [a\_1,b\_1][a\_2,b\_2]\cdots[a\_{g},b\_{g}]=1\rangle$, which is the fundamental group of the closed orientable genus-$g$ surface.


Suppose $2\leq m<n$ and let $p:S\_n\to S\_m$ be the canonical projection, which is the identity on generators $a\_i,b\_i$, $1\leq i\leq m$ and which trivializes $a\_i,b\_i$, $m+1\leq i\leq n$. Let $F\_r$ denote the free group with rank $r$.


**Question:** Is it possible to have finite rank free group $F\_r$ and epimorphisms $f:S\_n\to F\_r$ and $g:F\_r\to S\_m$ such that $g\circ f=p$?


There is apparently a kind of classification of epimorphisms of surface groups onto free groups that seems relevant (in the following reference). However, I have not been able to see exactly how it helps answer this question.



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> R. I. Grigorchuk, P. F. Kurchanov, and H. Zieschang, “Equivalence of homomorphisms of surface groups to free groups and some properties of 3-dimensional handlebodies,” Preprint, Ruhr-Universität Bochum, 1990.
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The classification: If $r>n$, then no epimorphism $S\_n\to F\_r$ can exist. Apparently, when $r\leq n$, then there is only one epimorphism $S\_n\to F\_r$ "up to automorphism." The representative is this: choose $F:S\_n\to F\_r$ to send $a\_i$, $1\leq i\leq r$ to the free generators and trivialize all other generators. This is unique in the sense that for any other epimorphism $f:S\_n\to F\_r$, we must have $F=f\circ \sigma$ for some $\sigma\in Aut(S\_n)$.


Considering this classification and the structure of $p$, my feeling is that the question has a negative answer but that I am perhaps overlooking something simple.



Factoring a projection of surface groups through a free group
The homomorphism $p$ is non-trivial on second (co)homology, since it is induced by a degree one map of surfaces ("collapse $n-m$ holes to a point"). This can also be seen using the structure of the cohomology rings of orientable surfaces.


Hence no such factorisation exists, as free groups have zero (co)homology above dimension one.


