  We investigate the space ${\cal M}$ of classical solutions to Witten's
formulation of 2+1 gravity on the manifold ${\bf R} \times T^2$. ${\cal M}$ is
connected, unlike the spaces of classical solutions in the cases where $T^2$ is
replaced by a higher genus surface. Although ${\cal M}$ is neither Hausdorff
nor a manifold, removing from ${\cal M}$ a set of measure zero yields a
manifold which is naturally viewed as the cotangent bundle over a non-Hausdorff
base space~${\cal B}$. We discuss the relation of the various parts of ${\cal
M}$ to spacetime metrics, and various possibilities of quantizing~${\cal M}$.
There exist quantizations in which the exponentials of certain momentum
operators, when operating on states whose support is entirely on the part of
${\cal B}$ corresponding to conventional spacetime metrics, give states whose
support is entirely outside this part of~${\cal B}$. Similar results hold when
the gauge group ${\rm SO}_0(2,1)$ is replaced by ${\rm SU}(1,1)$.
