Stable topologies of the event horizon

In our previous work, it was shown that the topology of the event horizon (TOEH) is determined by the past end points of the event horizon (EH). A toroidal EH (coalescing EH) is related to a two-dimensional (one-dimensional) set of end points. Therefore, we can see the stability of the TOEH by examining the stability of the end points (caustics). In the present article, we examine the stability of the TOEH by the discussion of linear perturbation and catastrophe theory. We see that a simple case of a single spherical EH is unstable under the linear perturbation. Remarkably, it is newly concluded that an EH with handles (torus, double torus, etc.) is more probable than coalescencing EHs by the analysis of catastrophe theory.