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 S3, S2 × S1 and lens-space L (p, q) 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 [169Jump To The Next Citation Point]; for instance, for N = 4 and n ≥ 4, each connected component of the horizon has necessarily one of the following topologies: 3 n− 4 S × 𝕋, 2 n−3 S × T and n−4 L (p,q) × 𝕋.

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.

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