THE SOLUTION SPACE OF 2-X-T2 IN WITTEN CONNECTION FORMULATION(1 GRAVITY ON R)

We investigate the space M of classical solutions of Witten's formulat ion of 2 + 1 gravity on the manifold R x T2. M is connected, unlike th e spaces of classical solutions in the cases where T2 is replaced by a higher genus surface. Although M is neither Hausdorff nor a manifold. removing from M a set of measure zero yields a manifold which is natu rally viewed as the cotangent bundle over a non-Hausdorff base space B . We discuss the relation of the various parts of M with spacetime met rics, and various possibilities of quantizing M. There exist quantizat ions in which the exponentials of certain momentum operators, when ope rating on states whose support is entirely on the part of B correspond ing to conventional spacetime metrics, give states whose support is en tirely outside this part of B. Similar results hold when the gauge gro up SO0(2, 1) is replaced by SU(1, 1).