  We propose a method to determine the topological structure of an event
horizon far in the future of a spacetime from the geometrical information of
its future null infinity. In the present article, we mainly consider spacetimes
with two black holes. Although, in most of cases, the black holes coalesce and
their event horizon is topologically a single sphere far in the future, there
are several possibilities that the black holes do not coalesce eternally and
such exact solutions. In our formulation, the geometrical structure of future
null infinity is related to the topological structure of the upper end of the
future null infinity through the Poincare\'-Hopf's theorem. Since the upper end
of the future null infinity determines the event horizon far in the future
under the conformal embedding, the topology of event horizon far in the future
will be affected by the geometrical structure of the future null infinity. Our
method is not only for the case of black hole coalescence. Also we can consider
more than two black holes or a black hole with non-trivial topology.
