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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