  The domain of outer communication of five-dimensional asymptotically flat
stationary spacetimes may possess non-trivial 2-cycles. We discuss how this may
lead to a gross violation of black hole uniqueness, beyond the existence of
black rings, even for solutions with two commuting rotational symmetries. We
illustrate this with a simple example in minimal supergravity; a four parameter
family of supersymmetric black hole solutions, with spherical horizon topology
and a 2-cycle in the exterior. We show there are black holes in this family
with identical conserved changes to the BMPV black hole, thereby demonstrating
black hole non-uniqueness in this context. We find a decoupling limit of this
family of black holes that yields spacetimes asymptotic to the near-horizon
geometry of a BMPV black hole which contain a black hole and an exterior
2-cycle.
