  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.
