Let $\mathrm{Out}(F_g)$ denote the automorphism group of a free group, and $\mathrm{Mod}_g$ the mapping class group of a closed oriented genus $g$ surface. Is there a map, as indicated with the dashed arrow below, making the following diagram commute?

$$ \begin{array}{ccc} \mathrm{Out}(F_g) & \dashrightarrow & \mathrm{Mod}_g \\ \downarrow & & \downarrow \\ \mathrm{GL}(g,\mathbf Z) & \to & \mathrm{Sp}(2g,\mathbf Z) \\ \end{array} $$ The lower horizontal map is induced by the functor $V \mapsto V \oplus V^\ast$ taking a vector space of dimension $g$ to a $2g$-dimensional symplectic vector space.

I do have a reason for asking but it's too long of a story to write here.

3more comments