Internal symmetry groups of quantum geons

In canonical quantum gravity asymptotically trivial diffeomorphisms not deformable to the identity can act nontrivially on the quantum state space. We show that for many 3-manifolds, the inequivalent diffeomorphisms comprise coverings in SU(2) of crystallographic groups. When the diffeomorphism R_2π associated with 2π-rotation is nontrivial, state vectors can have half-integral angular momentum; we list all 3-manifolds with R_2π trivial.