Aspects of 3-manifold theory in classical and quantum general relativity

Einstein's field equation of General Relativity can be cast into the form of evolution equations with well posed Cauchy problem. The object that undergoes evolution is then a Riemannian 3-manifold the instantaneous dynamical configuration of which is either described by a Teichmuller (Riemannian metrics modulo diffeomorphisms isotopic to the identity) or Riemannian moduli space (Riemannian metrics modulo all diffeomorphisms); the former being the universal cover of the latter. The two are related by the action of the mapping-class group of the underlying 3-manifold which may act as group of residual dynamical symmetries. In this way topological information regarding the Cauchy surface enters the dynamical description in an interesting way that has been speculated to be potentially significant in canonical quantum-gravity. In this contribution I will try to review these developments and also convey a flavour of the mathematical ideas involved.