It has been known for some time that in more than 4 spacetime dimensions, there is a considerably larger variety of black "hole"
solutions, having e.g. different horizon topology. In particular, they are no longer fully characterized by their asymptotic charges (mass, angular momenta) alone. We give a partial classification theorem for higher dimensions, for solutions with sufficiently many axial Killing fields. We show that higher dimensional black holes may be fully characterized by their asymptotic charges, together with certain "moduli" and "winding numbers" that are analogous to those of Seiffert fibrations. In particular, we find constraints on the possible horizon topologies. In 5 dimensions, they may be either a black "hole" (sphere), black "ring", or a black "lens".
