Plumbing Constructions and the Domain of Outer Communication for 5-Dimensional Stationary Black Holes

TL;DR

This paper classifies the topology of the domain of outer communication for 5D stationary black holes using plumbing constructions, establishing a correspondence with rod structures and describing possible topologies of these spacetimes.

Contribution

It introduces an algorithmic bijection between plumbing of disc bundles and rod structures, and characterizes all possible topologies of the domain of outer communication in 5D black holes.

Findings

Classifies topologies as $S^4$, connected sums of $S^2\times S^2$, or $\mathbb{CP}^2$

Establishes a bijective correspondence between plumbing and rod structures

Shows all topologies are realizable by vacuum solutions

Abstract

The topology of the domain of outer communication for 5-dimensional stationary bi-axisymmetric black holes is classified in terms of disc bundles over the 2-sphere and plumbing constructions. In particular we find an algorithmic bijective correspondence between the plumbing of disc bundles and the rod structure formalism for such spacetimes. Furthermore, we describe a canonical fill-in for the black hole region and cap for the asymptotic region. The resulting compactified domain of outer communication is then shown to be homeomorphic to $S^4$, a connected sum of $S^2\times S^2$'s, or a connected sum of complex projective planes $\mathbb{CP}^2$. Combined with recent existence results, it is shown that all such topological types are realized by vacuum solutions. In addition, our methods treat all possible types of asymptotic ends, including spacetimes which are asymptotically flat, asymptotically Kaluza-Klein, or asymptotically locally Euclidean.