Topology change in (2+1)-dimensional gravity with non-Abelian Higgs field

We study topology change in (2+1)D gravity coupling with non-Abelian SO(2,1) Higgs field from the point of view of Morse theory. It is shown that the Higgs potential can be identified as a Morse function. The critical points of the latter (i.e. loci of change of the spacetime topology) coincide with zeros of the Higgs field. In these critical points two-dimensional metric becomes degenerate, but the curvature remains bounded.