I am given two handlebodies $H_g$ and $H_h$ and a continuous map $f : \partial H_g \to \partial H_h$ - under what conditions is it possible to extend $f$ to a map $F : H_g \to H_h$? Is it sufficient for null simple closed curves in $\partial H_g$ that bound disks in $H_g$ to be mapped by $f$ to simple closed curves in $\partial H_h$ that bound disks in $H_h$?
Thanks!