  Nielsen realization problem for the mapping class group $\text{Mod}(S_g)$
asks whether the natural projection $p_g: \text{Homeo}_+(S_g)\to
\text{Mod}(S_g)$ has a section. While all the previous results use torsion
elements in an essential way, in this paper, we focus on the much more
difficult problem of realization of torsion-free subgroups of
$\text{Mod}(S_g)$. The main result of this paper is that the Torelli group has
no realization inside the area-preserving homeomorphisms.
