A note on causality conditions on covering spacetimesy

A number of techniques in Lorentzian geometry, such as those used in the proofs of singularity theorems, depend on certain smooth coverings retaining interesting global geometric properties, including causal ones. In this note we give explicit examples showing that, unlike some of the more commonly adopted rungs of the causal ladder such as strong causality or global hyperbolicity, less-utilized conditions such as causal continuity or causal simplicitydo notin general pass to coverings, as already speculated by one of the authors (EM). As a consequence, any result which relies on these causality conditions transferring to coverings must be revised accordingly. In particular, some amendments in the statement and proof of a version of the Gannon-Lee singularity theorem previously given by one of us (IPCS) are also presented here that address a gap in its original proof, simultaneously expanding its scope to spacetimes with lower causality.