Topology change in ISO(2,1) Chern-Simons gravity

In 2+1 dimensional gravity, a dreibein and the compatible spin connection can represent a space-time containing a closed spacelike surface $\Sigma$ only if the associated SO(2,1) bundle restricted to $\Sigma$ has the same non-triviality (Euler class) as that of the tangent bundle of $\Sigma.$ We impose this bundle condition on each external state of Witten's topology-changing amplitude. The amplitude is non-vanishing only if the combination of the space topologies satisfies a certain selection rule. We construct a family of transition paths which reproduce all the allowed combinations of genus $g \ge 2$ spaces.