On the role and determination of mapping-class groups in canonical quantum gravity

The diffeomorphism constraint in canonical quantum gravity eliminates a certain normal subgroup of the group of all spatial diffeomorphisms. The question remains as to how the ones left out are to be treated. They include, in particular, the ones that are not contained in the identity component of the diffeomorphism group, which may be identified with a certain mapping-class group of the underlying 3-manifold. We outline general aspects of their determination and discuss the problem of whether they should be viewed as gauge transformations or rather as proper symmetries. This highlights certain conceptual difficulties with the notion of "eliminating the constraints".