On the relation between 2+1 Einstein gravity and Chern-Simons theory

A simple example is given to show that the gauge equivalence classes of physical states in Chern-Simons theory are not in one-to-one correspondence with those of Einstein gravity in three spacetime dimensions. The two theories are therefore not equivalent. It is shown that including singular metrics into general relativity has more, and in fact a quite counter-intuitive, impact on the theory than one naively expects.