### 4.4 Orbit space structure

The structure theorem [65] applies to stationary toroidal Kaluza–Klein black holes and provides the
following product structure
with
the stationary vector being tangent to the factor and where , endowed with the induced metric, is
an -dimensional Riemannian manifold admitting a action by isometries. A careful analysis of the
topological properties of such toroidal actions [169], based on deep results from [260, 261], allows one to
show that the orbit space is homeomorphic to a half plane with boundary composed
of segments and corners; the segments being the projection of either a component of the event horizon or an
axis of rotation (the set of zeros of a linear combination of axial vectors), and the corners being the
projections of the intersections of two axes. Moreover, the interiors are in fact diffeomorphic. To
establish this last fundamental result it is necessary to exclude the existence of exceptional
orbits of the toroidal action; this was done by Hollands and Yazadjiev in [169] by extending
the results in [260] to the -black hole setting. In particular one obtains the following
decomposition
where is the union of all axes; we note that such product structure is necessary to the construction of
Weyl coordinates [76, 65] and, consequently, indispensable to perform the desired reduction of the vacuum
equations.
As already discussed, basic properties of black rings show that a classification of -black holes in
terms of mass, angular momenta and horizon topology is not possible. But, as argued by Hollands and
Yazadjiev [169], the angular momenta and the structure of the orbit space characterize such black holes if
one further assumes non-degeneracy of the event horizon. This orbit space structure is in turn determined
by the interval structure of the boundary of the quotient manifold, a concept related to Harmark’s rod
structure developed in [137] (see also [101, Section 5.2.2.1]). Note that the interval structure codifies the
horizon topology.