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The topology of the event horizon (TOEH) is usually believed to be a sphere.Nevertheless, some numerical simulations of gravitational collapse with atoroidal event horizon or the collision of event horizons are reported.Considering the indifferentiability of the event horizon (EH), we see that suchnon-trivial TOEHs are caused by the set of endpoints (the crease set) of theEH. The two-dimensional (one-dimensional) crease set is related to the toroidalEH (the coalescence of the EH). Furthermore, examining the stability of thestructure of the endpoints, it becomes clear that the spherical TOEH isunstable under linear perturbation. On the other hand, a discussion based oncatastrophe theory reveals that the TOEH with handles is stable and generic.Also, the relation between the TOEH and the hoop conjecture is discussed. It isshown that the Kastor-Traschen solution is regarded as a good example of thehoop conjecture by the discussion of its TOEH. We further conjecture that anon-trivial TOEH can be smoothed out by rough observation in its mass scale.Comment: 53 pages, revtex, Published in Prog. Theo. Phys. vol.99, 13 figure

Topological Appearance of Event Horizon: What Is the Topology of the Event Horizon That We Can See?