  We consider maximal slices of the Myers-Perry black hole, the doubly spinning
black ring, and the Black Saturn solution. These slices are complete,
asymptotically flat Riemannian manifolds with inner boundaries corresponding to
black hole horizons. Although these spaces are simply connected as a
consequence of topological censorship, they have non-trivial topology. In this
note we investigate the question of whether the topology of spatial sections of
the horizon uniquely determines the topology of the maximal slices. We show
that the horizon determines the homological invariants of the slice under
certain conditions. The homological analysis is extended to black holes for
which explicit geometries are not yet known. We believe that these results
could provide insights in the context of proving existence of deformations of
this initial data. For the topological slices of the doubly spinning black ring
and the Black Saturn we compute the homotopy groups up to dimension 3 and show
that their 4-dimensional homotopy group is not trivial.
