Compact three-dimensional black hole: topology change and closed time-like curve

We present a compactified version of the three-dimensional black hole recently found by considering extra identifications and determine the analytical continuation of the solution beyond its coordinate singularity by extending the identifications to the extended region of the spacetime. In the extended region of the spacetime, we find a topology change and non-trivial closed time-like curves both in the ordinary three-dimensional black hole and in the compactified one. Especially, in the case of the compactified three-dimensional black hole, we show an example of topology change from one double torus to eight spheres with three punctures.