Higher-dimensional black-hole topology conjecture with maximal rotational symmetry

Let $(\mathbb{T}{M},g)$ be a spin spacetime that is asymptotically flat or Kaluza–Klein, whose Ricci tensor satisfies the null-convergence condition, and that admits an action of $U(1)^{D-3}$. Let $\Sigma$ be a Cauchy surface and let ${\rm BH}$ denote the black-hole region. The asymptotic region is determined by the precise boundary conditions; for example, in the standard Kaluza–Klein setup it is $\mathbb{R}^3\times\mathbb{T}^{D-4}$. Higher-dimensional black-hole topology conjecture. The Cauchy surface $\Sigma$ can be decomposed as

$$\Sigma \cong \left[\#_{k=2}^{D-3} m_k\cdot\left(S^k\times S^{D-1-k}\right)\ \#\ ({\rm asymptotic\ region})\right]\setminus {\rm BH}.$$

This conjecture proposes a classification of the topology of Cauchy surfaces under maximal $U(1)^{D-3}$ symmetry, extending the restrictions known in lower dimensions. The multiplicities $m_k$ and the precise asymptotic summand depend on the spacetime and boundary conditions.