Global structure of Witten's 2+1 gravity on $\mathbf{R}\times T^2$

We investigate the space $\mathcal{M}$ of classical solutions to Witten’s formulation of 2+1 gravity on the manifold $\mathbf{R}\times T^2$. $\mathcal{M}$ is connected, but neither Hausdorff nor a manifold. However, removing from ${\cal M}$ a set of measure zero yields a connected manifold which is naturally viewed as the cotangent bundle over a non-Hausdorff base space. Avenues towards quantizing the theory are discussed in view of the relation between spacetime metrics and the various parts of~$\mathcal{M}$. (Contribution to the proceedings of the Lanczos Centenary Conference, Raleigh, NC, December 12–17, 1993.)