  We introduce the domain structure for stationary black hole space-times.
Given a set of commuting Killing vector fields of the space-time the domain
structure lives on the submanifold where at least one of the Killing vector
fields have zero norm. Depending on which Killing vector field has zero norm
the submanifold is naturally divided into domains. A domain corresponds either
to a set of fixed points of a spatial symmetry or to a Killing horizon,
depending on whether the characterizing Killing vector field is space-like or
time-like near the domain. The domain structure provides invariants of the
space-time, both topological and geometrical. It is defined for any space-time
dimension and any number of commuting Killing vector fields. We examine the
domain structure for asymptotically flat space-times and find a canonical form
for the metric of such space-times. The domain structure generalizes the rod
structure introduced for space-times with D-2 commuting Killing vector fields.
We analyze in detail the domain structure for Minkowski space, the
Schwarzschild-Tangherlini black hole and the Myers-Perry black hole in six and
seven dimensions. Finally we consider the possible domain structures for
asymptotically flat black holes in six and seven dimensions.
