The archetype of a large diffeomorphism is a Dehn twist of a torus, which may be described as the operation of cutting along a circumference to obtain a cylinder, twisting one end by , and regluing. Similar transformations exist for any closed surface , and in fact the Dehn twists around generators of generate [57, 56] . It is easy to see that the mapping class group of a spacetime acts on , and therefore on the holonomies of Section 2.2 . As diffeomorphisms, elements of the mapping class group also acts on the constant curvature metrics , and hence on the moduli of Section 2.4 .
Classically, geometries that differ by actions of are exactly equivalent, so the “true” space of vacuum solutions for a spacetime with the topology is really , where is the moduli space (11). Quantum mechanically, it is not clear whether one should impose mapping class group invariance on states or whether one should merely treat as a symmetry under which states may transform nontrivially (see, for instance, ). In 2+1 dimensions, though, there seems to be a strong argument in favor of treating the mapping class group as a genuine invariance, as follows. Using the Chern-Simons formalism, one can compute the quantum amplitude for the scattering of a point particle off another particle , a black hole , or a handle  . In each case, it is only when one imposes invariance under the mapping class group that one recovers the correct classical limit. It may still be that simple enough representations of lead to sensible physical results, but it is at least clear that the mapping class group cannot be ignored.
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