  We generalise uniqueness theorems for non-extremal black holes with three
mutually independent Killing vector fields in five-dimensional minimal
supergravity in order to account for the existence of non-trivial 2-cycles in
the domain of outer communication. The black hole space-times we consider may
contain multiple disconnected horizons and be asymptotically flat or
asymptotically Kaluza-Klein. We show that in order to uniquely specify the
black hole space-time, besides providing its domain structure and a set of
asymptotic and local charges, it is necessary to measure the magnetic fluxes
that support the 2-cycles as well as fluxes in the two semi-infinite rotation
planes of the domain diagram.
