Quantization of 2+1 Gravity on the Torus

We use the polygon representation of 2+1--dimensional gravity to explicitly carry out the canonical quantization of a universe with the topology of a torus. The mapping-class-invariant wave function for a quantum ''big bounce'', is reminiscent of the interference patterns of linear gratings. We consider the ``problem of time'' of quantum gravity: for one choice of internal time the universe recovers a semiclassical interpretation after the bounce, with a wave packet centered at a single geometry; for another choice of internal time, the quantum solutions involve interference between macroscopically distinct universes.