Spacetime Topology

In General > s.a. cosmic geometry; spacetime boundaries [including compactification]; types of geometries [low-regularity].
* Manifold topology: The topology \(\cal M\) spacetime inherits from the manifold structure; Its global structure can be studied using topological invariants, notably the Euler class and the Pontryagin class.
* Restrictions: An even-dimensional compact manifold without boundary with a Lorentz metric must have χ(M) = 0; In 4D, this implies that the manifold is not simply connected.
> Online resources: see Wikipedia page.

Path or Zeeman Topology
$ Def: The topology \(\cal P\) in which E ⊂ M is open iff for every timelike curve c there is an O ∈ \(\cal M\) such that E ∩ c = O ∩ c; The finest which induces the same topology as \(\cal M\) does on timelike curves.
* Properties: Strictly finer than the manifold topology, and therefore it is Hausdorff; Separable, but not locally compact, not Lindelöf, not normal, and not first countable; There are sequences of points in M which converge in the manifold topology (e.g., a sequence of distinct points on the light cone of p which converges to p) but does not converge in the path topology.
* Base: Sets of the form I ⁺(p, U) ∪ I ⁻(p, U) ∪ {p}, for convex normal neighborhoods U.
* For Minkowski space: The open (closed) sets in \(\cal P\) are the subsets of M whose intersections with all timelike geodesics and spacelike hyperplanes are open (closed) in the natural topology on those subsets; Any null geodesic is discrete (its points are isolated); The homeomorphism group is generated by the Poincaré group and dilatations; The space is path connected, but not simply connected.
* For curved spacetimes: The homeomorphism group is the group of all homothetic transformations.
* Variation: Fullwood's topology, defined only in terms of causal structure, equivalent to \(\cal P\) iff the distinguishing condition holds.
@ For Minkowski space: Zeeman JMP(64), Top(67); Whiston IJTP(72); Dossena a0704-laurea, JMP(07) [properties]; Sainz a0803-wd, a1003-wd [criticism]; Dossena a1103 [constructive response to criticism]; Papadopoulos a1811 [30 topologies].
@ For curved spacetimes: Göbel CMP(76), JMP(76); Hawking et al JMP(76); Fullwood JMP(92); Struchiner & Rosa mp/05 [for Kaluza-Klein and gauge theories]; Kim JMP(06); Papadopoulos & Papadopoulos MMAS(18)-a1706 [two distinct Zeeman topologies], G&C(19)-a1712 [more Zeeman topologies, and the Limit Curve Theorem]; Papadopoulos et al IJGMP(18)-a1710; > s.a. spacetime boundaries.

Alexandrov (or Interval) Topology > s.a. causality conditions [global hyperbolicity].
$ Def: The coarsest topology on M in which I ⁺(E) is open for all E ⊂ M.
* Base: In a full chronological space, one is given by the Alexandrov neighborhoods {[x, y]} [@ Lerner in(72)].
* Special cases: It coincides with the manifold topology iff (M, g) is strongly causal (in which case it is Hausdorff), but in general it is coarser; In the discrete case it is often trivial, in the sense that it gives the discrete topology.
@ General references: Papadopoulos & Kurt a2010 [completeness, remarks].
@ And strong causality: McWilliams IJTP(81); Martin & Panangaden gq/04 [globally hyperbolic case]; > s.a. causality.

Topology of Space > s.a. initial-value formulation of general relativity; topology at cosmological scales; topology change.
* Restrictions: There are none on the spatial topology for an asymptotically flat vacuum spacetime, although in most cases singularities will develop.
@ References: Isenberg et al AHP(03)gq/02 [vacuum].

References > s.a. boundaries in field theory; cosmological models in general relativity; initial-value formulation; particle models.
@ General: Alonso & Ynduráin CMP(67); Cel'nik SMD(68); Whiston IJTP(73), IJTP(74), IJTP(75); Briginshaw IJTP(80); Lee GRG(83); Heathcote BJPS(88); in Naber 88 (pr ch1); Lester JMP(89), Kirillov gq/94 [phenomenological description]; Acherjee et al a1710 [Euclidean topology, order topology from horismos, and global topological properties of spacetime manifolds]; Papadopoulos & Scardigli a1804-ch [critical review].
@ Dimension: Barrow PTRS(83); Mirman LNC(84); Zeilinger & Svozil PRL(85); Svozil & Zeilinger IJMPA(86); Müller & Schäfer PRL(86); Mirman IJTP(88); Hochberg & Wheeler PRD(91); NCA(91)469 [from wormholes]; Tegmark CQG(97)gq [from strings, anthropic]; Callender SHPMP(05) ["proofs" of 3-dimensionality]; Gersten FP(05) [proposed test of 4-dimensionality]; in Petkov 09; > s.a. fractals in physics.
@ Fundamental group: Smith AJM(60), PNAS(60); Lee GRG(75).
@ Special cases: Lee CMB(75), Yurtsever JMP(90) [compact]; Chamblin gq/95-conf [singular, and causality]; > types of spacetimes.
@ Phenomenology: Cassa PAMS(93) [and geodesics]; Parfionov & Zapatrin gq/97, Breslav et al HJ(99)qp [measurement, histories appproach]; > s.a. Detectors in Quantum Theory.
@ Related topics: in Steenrod 51, p207 [restrictions]; in Hawking & Ellis 73, 181-182 [orientability]; Kovár & Chernikava a1311 [causal sites, weakly causal topologies and their de Groot duals]; Sorkin et al CQG-sa1811 [manifold topology from \(K^+\)]; > s.a. stiefel-whitney classes.
> Related topics: see causal structures [recovery of spacetime structure]; diffeomorphisms; lines [topology on the space of causal lines/geodesics]; lorentzian geometry [topology on the space of lorentz metrics on a manifold].

Quantum Aspects > s.a. observables in gravity; particle statistics; quantum cosmology [sum over topologies]; quantum spacetime.
@ Topology at Planck length: Yetter ed-94; Madore & Saeger CQG(98)gq/97.
@ Scale-dependent topology: Seriu PLB(93), ViA(93); > s.a. Coarse Structures in Geometry.
@ Related topics: Friedman in(91) [and quantum gravity]; Jonsson PLB(98)ht [2D, handle width]; Raptis et al IJTP(06)gq/05, IJTP(06)gq/05 [tomographic histories approach]; Atyabi IJGMP(15)-a1412 [topology fluctuations and non-commutative spectral geometry, effect of matter].

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