### 4.3 Topology of the event horizon

A theorem of Galloway and Schoen [111] shows that compact cross-sections of the horizon must be of
positive Yamabe type, i.e., admit metrics of positive scalar curvature. In spacetime dimension
five, the positive Yamabe property restricts the horizon to be a finite connected sum of spaces
with , and lens-space topologies. Such results require no symmetry
assumptions but by further assuming stationarity and the existence of one axial Killing vector more
topological restrictions, concerning the allowed factors in the connected sum, appear in five
dimensions [162].
In the toroidal Kaluza–Klein case, the existence of a toroidal action leads to further restrictions [169];
for instance, for and , each connected component of the horizon has necessarily one of the
following topologies: , and .

It should be noted that no asymptotically-flat or Kaluza–Klein black holes with lens-space topology of
the horizon are known. Constructing a black lens, or establishing non-existence, appears to be a challenging
problem.