Pure singularities covered by directional singularities

Determine whether every directional singularity in the abstract boundary of a maximally extended pseudo-Riemannian manifold must cover a pure singularity in another envelopment, and, if so, determine whether that pure singularity and the pure point at infinity covered by the same directional singularity must be separate.

Background

A directional singularity is an essential singularity that covers a regular boundary point or a point at infinity. For maximally extended manifolds, the paper shows that a directional singularity must cover a pure point at infinity, but it does not establish whether a pure singularity must also occur lower in the covering relation.

The unresolved issue is connected with the existence of intertwined geodesics. The paper proves the desired conclusion in settings where an appropriate envelopment contains no intertwined geodesic pairs approaching the relevant boundary point, but leaves the general case open. A related question is whether the pure singularity and pure point at infinity covered by a directional singularity are necessarily separable.