WITTEN 2+1-GRAVITY ON RX(KLEIN-BOTTLE)

Witten's formulation of 2+1 gravity is investigated on the non-orienta ble 3-manifold R x (Klein bottle). The gauge group is taken to consist of all four components of the 2 + 1 Poincare group. We analyse in det ail several components of the classical solution space, and we show th at from four of the components one can recover non-degenerate spacetim e metrics. In particular, from one component we recover metrics for wh ich the Klein bottles are spacelike. An action principle is formulated for bundles satisfying a certain orientation compatibility property, and the corresponding components of the classical solution space are p romoted into a phase space. Avenues towards quantization are briefly d iscussed.