Notes on maximal slices of five-dimensional black holes

TL;DR

This paper investigates the topology of maximal slices in five-dimensional black hole solutions, analyzing how horizon topology influences the slices' topological invariants and homotopy groups, with implications for initial data deformations.

Contribution

It demonstrates that horizon topology determines the homological invariants of maximal slices and extends homological analysis to unknown geometries, providing new insights into black hole initial data.

Findings

Horizon topology determines homological invariants of slices.

Homotopy groups up to dimension 3 are computed for specific black hole slices.

The 4-dimensional homotopy group of certain slices is non-trivial.

Abstract

We consider maximal slices of the Myers-Perry black hole, the doubly spinning black ring, and the Black Saturn solution. These slices are complete, asymptotically flat Riemannian manifolds with inner boundaries corresponding to black hole horizons. Although these spaces are simply connected as a consequence of topological censorship, they have non-trivial topology. In this note we investigate the question of whether the topology of spatial sections of the horizon uniquely determines the topology of the maximal slices. We show that the horizon determines the homological invariants of the slice under certain conditions. The homological analysis is extended to black holes for which explicit geometries are not yet known. We believe that these results could provide insights in the context of proving existence of deformations of this initial data. For the topological slices of the doubly spinning black ring and the Black Saturn we compute the homotopy groups up to dimension 3 and show that their 4-dimensional homotopy group is not trivial.