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We investigate the space ${\cal M}$ of classical solutions to Witten'sformulation of 2+1 gravity on the manifold ${\bf R} \times T^2$. ${\cal M}$ isconnected, unlike the spaces of classical solutions in the cases where $T^2$ isreplaced by a higher genus surface. Although ${\cal M}$ is neither Hausdorffnor a manifold, removing from ${\cal M}$ a set of measure zero yields amanifold which is naturally viewed as the cotangent bundle over a non-Hausdorffbase space~${\cal B}$. We discuss the relation of the various parts of ${\calM}$ to spacetime metrics, and various possibilities of quantizing~${\cal M}$.There exist quantizations in which the exponentials of certain momentumoperators, when operating on states whose support is entirely on the part of${\cal B}$ corresponding to conventional spacetime metrics, give states whosesupport is entirely outside this part of~${\cal B}$. Similar results hold whenthe gauge group ${\rm SO}_0(2,1)$ is replaced by ${\rm SU}(1,1)$.Comment: 23 pages, REVTeX v3.0, SU-GP-93/7-6, CGPG-93/8-3. (Discussion on the  gauge equivalence of degenerate and nondegenerate metrics extended.

The solution space of 2 + 1 gravity on in Witten's connection formulation