Quantum mechanics and field theory on multiply connected and on homogeneous spaces

The basic framework for discussing quantum mechanics on multiply connected spaces is presented using the covering space concept. The theorem of Laidlaw and DeWitt is rederived and extended to the case of field theory. It is pointed out that chiral dynamics is similar to Skyrme's nonlinear theory and forms another example of Finkelstein's kink idea. The possible existence of ' pi geons' is raised, and the fact that the pion manifold may be any one of the Clifford-Klein constant curvature space-forms, rather than just the whole three-sphere, is suggested. The related formalism for quantum mechanics on homogeneous spaces is given in general terms.