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The codimension 2 null cut locus with applications to spacetime topology. (English) Zbl 1509.83036

Summary: In this paper, we review and extend some results in the literature pertaining to spacetime topology while naturally utilizing properties of the codimension 2 null cut locus. Our results fall into two classes, depending on whether or not one assumes the presence of horizons. Included among the spacetimes we consider are those that apply to the asymptotically (locally) anti-de Sitter (AdS) setting.
©2022 American Institute of Physics

MSC:

83C75 Space-time singularities, cosmic censorship, etc.
83E15 Kaluza-Klein and other higher-dimensional theories
53Z05 Applications of differential geometry to physics
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[24] ψ is an immersion; hence, \( \overline{\psi}\) is an immersion. Then, that \(\overline{\psi}\) is an embedding follows from \(\overline{\psi}\) being an injective open map onto its image. Injectivity of \(\overline{\psi}\) follows from the homotopy lifting property applied to the commutative diagram along with the fact that ι_* is injective, which follows from V being a retract of M. Then, that \(\overline{\psi}\) is an open map onto its image follows from p and P being local diffeomorphisms along with ι being a topological embedding.
[25] We just observed that \(P(\mathcal{H}) \subset H\). Similarly, it is not hard to see that \(\mathcal{H} = P^{- 1}(H)\); this holds in Sec. IV C as well.
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