4.5 KK topological censorship

Black-hole uniqueness in four-dimensions uses simple connectedness of the event horizon extensively. But the Schwarzschild metric multiplied by a flat torus shows that simple connectedness does not hold for general domains of outer communications of Kaluza–Klein black holes. Fortunately, simple connectedness of the orbit space n−2 ⟨⟨Mext ⟩⟩∕ (ℝ × 𝕋 ) suffices: for instance, to prove that the (n − 1)-dimensional orbit generated by the stationary and axial vectors is timelike in ⟨⟨Mext ⟩⟩ away from the axes (which in turn is essential to the construction of Weyl coordinates), to guarantee the existence of global twist potentials [65] and to exclude the existence of exceptional orbits of the toroidal action (see Section 4.4). The generalized topological censorship theorem of [73] shows that this property follows from the simple connectedness of the orbit space in the asymptotic end 𝒮ext ∕𝕋n−2 ≈ ℝN ∖ B.
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