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Cosmic Topology

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M. Lachieze-Rey (1), J.P.Luminet (2) ((1) CE-Saclay/Service d'Astrophysique, (2) Observatoire de Meudon/DARC)

General Relativity and Quantum Cosmology · Submitted 6 May 1996 · 165 references

Pith's verdict: A solid, honest 1996 review that is still the standard entry point to cosmic topology, with one overreaching paragraph in §11.7 where a completeness assumption gets dropped.

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Citation(2) Observatoire de Meudon/DARC), J.P.Luminet (2) ((1) CE-Saclay/Service d'Astrophysique, M. Lachieze-Rey (1). "Cosmic Topology." arXiv:gr-qc/9605010 (1996). https://pith.science/paper/2K65QTHQ
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The paper

The paper claims

General relativity fixes the shape of space but not its topology, so the universe may be finite and wrapped up in itself, and that wrapping would show in the sky as repeated images of the same galaxies.

The reading

Einstein's equations are local: they constrain the curvature of space at each point but say nothing about whether space closes back on itself. This review argues that the simply-connected Friedmann-Lemaitre models, in which space is the whole sphere, the whole Euclidean plane, or the whole hyperbolic space, are only the default choice, not a consequence of the theory, and that multi-connected alternatives deserve equal standing. The authors set out the mathematical toolkit for classifying such spaces, the fundamental polyhedron and its holonomy group, and then follow it into cosmology: if the repeating cell is smaller than the horizon, a single galaxy is seen several times along different light paths, at different redshifts and look-back times. They collect the observational searches for those repeated images, in galaxy and cluster catalogs, quasar pairs, gamma-ray bursts, and the microwave background, and report the limits obtained so far. The value of the programme is that a positive detection would simultaneously reveal the topology, the sign of the spatial curvature, and the size of the universe.

What carries the argument

The load-bearing structure is the triple of universal covering space, fundamental polyhedron, and holonomy group. The universal covering is the simply-connected space that locally looks identical to the real space, the fundamental polyhedron is the largest simply-connected cell around an observer, and the holonomy group is the discrete group of isometries, acting freely and without fixed points, that tiles the covering space with copies of that cell. Every ghost image of a source corresponds to one group element, so the observable pattern of replicas is a direct picture of the holonomy. The practical test the authors highlight is a histogram of comoving pair separations in a catalog: if space repeats at scale $a$, a peak must appear at that separation, independently of which topology is responsible.

Extended reading notes

Core claim

The paper's central claim is that multi-connectedness of space is a live and testable possibility rather than a mathematical curiosity. Because the metric of a Friedmann-Lemaitre model is compatible with many topologies, the same local expansion history and the same curvature can be realised in spaces of very different volume, including compact spaces with zero or negative curvature. The authors classify the admissible three-dimensional space forms of constant curvature, Euclidean, spherical and hyperbolic, and give their holonomy groups, then argue that the observable signature is the ghost image: a source inside a cell smaller than the horizon is reached by many null geodesics, producing replicas whose separations in comoving distance are set by the generators of the holonomy group. They review the resulting observational constraints, notably that the Abell catalog of clusters excludes an identification scale below roughly $600~h^{-1}$ Mpc whatever the topology, and they discuss the CMB as the other main channel, where a small universe would have been homogenised causally before recombination and would imprint a cutoff on large angular scales.

Load-bearing premise

The entire reduction of the topology question to the topology of a single spatial hypersurface rests on assuming the spacetime is globally hyperbolic and space-orientable, an assumption the authors themselves say is hard to justify on physical grounds beyond a belief in strong determinism.

Editorial extensions

If this is right

  • A detection of repeated images would fix the topology of space, and with it the sign of the spatial curvature and the finite volume of the universe.
  • Existing cluster catalogs already rule out any topology whose fundamental cell is smaller than a few hundred megaparsecs, a bound that applies whatever the holonomy group is.
  • A small universe would have become causally connected before recombination, offering an alternative to inflation for explaining the homogeneity of the microwave background.
  • Compact hyperbolic models with the smallest known volumes, such as the Weeks manifold, remain admissible against present data and are the most promising targets for ghost-image searches.
  • Quantum cosmology gives no reason to prefer the simply-connected case; creation probabilities favour small compact spaces, so topology should be treated as a parameter to be measured.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same pair-separation histogram could be run on forthcoming large photometric and spectroscopic surveys, where the number of catalogued objects is orders of magnitude larger, pushing the excluded cell size close to the horizon and turning a null result into a genuine statement that we live in the simply-connected case.
  • If a peak were found, the principal directions recovered from it would define preferred axes on the sky, which should also appear as anomalies in the microwave background at large angular scales; a joint detection in both channels would be far harder to explain away than either alone.
  • The authors treat orientability and global hyperbolicity as simplifying assumptions, but a compact non-orientable space would still be observationally distinguishable through a parity flip in ghost images, which no catalog search has yet been designed to look for.
  • Because ghost images arrive with different look-back times, a confirmed pair would give a direct measurement of the light travel time around the universe, and hence of the cell size, independent of any redshift-distance relation.
Original abstract

General relativity does not allow one to specify the topology of space, leaving the possibility that space is multi-- rather than simply--connected. We review the main mathematical properties of multi--connected spaces, and the different tools to classify them and to analyse their properties. Following the mathematical classification, we describe the different possible muticonnected spaces which may be used to construct universe models. We briefly discuss some implications of multi--connectedness for quantum cosmology, and its consequences concerning quantum field theory in the early universe. We consider in details the properties of the cosmological models where space is multi--connected, with emphasis towards observable effects. We then review the analyses of observational results obtained in this context, to search for a possible signature of multi--connectedness, or to constrain the models. They may concern the distribution of images of cosmic objects like galaxies, clusters, quasars,..., or more global effects, mainly those concerning the Cosmic Microwave Background, and the present limits resulting from them.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,548 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.02 · states limitation 0.11 · human participants 0.02 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.10 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

General relativity does not allow one to specify the topology of space, leaving the possibility that space is multi-- rather than simply--connected. We review the main mathematical properties of multi--connected spaces, and the different tools to classify them and to analyse their properties. Following the mathematical classification, we describe the different possible muticonnected spaces which may be used to construct universe models. We briefly discuss some implications of multi--connectedness for quantum cosmology, and its consequences concerning quantum field theory in the early universe. We consider in details the properties of the cosmological models where space is multi--connected, with emphasis towards observable effects. We then review the analyses of observational results obtained in this context, to search for a possible signature of multi--connectedness, or to constrain the models. They may concern the distribution of images of cosmic objects like galaxies, clusters, quasars,..., or more global effects, mainly those concerning the Cosmic Microwave Background, and the present limits resulting from them.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 General relativity does not allow one to specify the topology of space, leaving the possibility that space is multi-- rather than simply--connected. background 1.00 0.04 0.07 asserted 0.76
s1 We review the main mathematical properties of multi--connected spaces, and the different tools to classify them and to analyse their properties. contribution 0.86 0.24 0.04 asserted 0.67
s2 Following the mathematical classification, we describe the different possible muticonnected spaces which may be used to construct universe models. contribution 0.69 0.41 0.05 hedged possibility 0.56
s3 We briefly discuss some implications of multi--connectedness for quantum cosmology, and its consequences concerning quantum field theory in the early universe. contribution 0.78 0.15 0.04 unclear 0.59
s4 We consider in details the properties of the cosmological models where space is multi--connected, with emphasis towards observable effects. contribution 0.90 0.43 0.06 asserted 0.73
s5 We then review the analyses of observational results obtained in this context, to search for a possible signature of multi--connectedness, or to constrain the models. contribution 0.55 0.15 0.06 asserted 0.46
s6 They may concern the distribution of images of cosmic objects like galaxies, clusters, quasars,..., or more global effects, mainly those concerning the Cosmic Microwave Background, and the present limits resulting from them. method 0.44 0.08 0.07 hedged possibility 0.67

What kind of paper the classifier reports this is: review 1.00 · evidence from literature 0.96.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. Chronological future in spacetime 2.2. Causality The notion of causality, which intuitively requires that the cause precedes the effect, is a rule imposed by logics and common sense, not by the theory of relativity. Causality is implicit in special relativity, because the travel into the past is strictly equivalent to a motion along spacelike curves, which is forbidden for real particles. On the contrary, in general relativity, certain subtle distortions imprinted on curved spacetime by particular gravitational fields – for instance the one generated by a rotating black hole or a…
Figure 2. The non-oriented M¨obius band The following definition arises : any two-dimensional manifold lying in IR3 is orientable if and only if it is two-sided. This can be generalized to higher dimensions. At any point of the spacetime, one can define two classes of spacelike triads : the left–handed class and the right–handed class. The spacetime is space–orientable if every closed curve preserves the spatial parity. If time orientability can be justified on physical grounds such as the existence of an “arrow of time”, the requirement of spatial orientability is less stringent. In particle…
Figure 3. Classes of homeomorphic surfaces. The digits below the columns denote the number of holes, a topological invariant. 3. Basic topology for Riemannian manifolds 3.1. What is topology ? In simple words, topology is the mathematical framework within which to study continuity : the topological properties are those which remain insensitive to continuous transformations. Thus, size and distance are in some sense ignored in topology : stretching, squeezing or “kneading” a manifold change the metric but not the topology; cutting, tearing or making holes and handles change the latter. As a…
Figure 4. Construction of the flat torus I I J J 1 2 3 4
Figure 5. The turtle’s walk on a flat torus immersed in IR3 . It is only topologically speaking that these two tori are the same because there is an homeomorphism between them. As food for thought we provide a more precise, although elementary, statement of this. The usual torus T1 can be endowed with a natural riemannian metric gij by taking the Euclidean metric in IR3 and imposing the restriction that the points in IR3 lie on the torus. In polar coordinates we obtain for instance ds2 = (R + rcosφ) 2 (dθ2 + r 2 dφ2 ) (2) (for the torus obtained by rotation of a circle of radius r along
Figure 6. The two-torus as the connected sum of two simple tori. are the only possible compact oriented (two–sided) surfaces. g is called the genus of the surface. The non–oriented surfaces are similarly defined by their genus. The major triumph of topology in the nineteenth century was the complete classification of all compact surfaces in terms of two and only two items of data : the number of holes g and the orientability / non–orientability property. It may be useful here to recall the link between the genus and the Poincar´e-Euler characteristic (whose general definition was given in…
Figure 7. Classes of homotopy. Left : the Euclidean plane. Any loop can be shrunk to a point. The fundamental group reduces to identity. Center : a plane with a hole. γ is not homotopic to γ ′ . Every homotopy class hn is associated to an integer n : γ ∈ hn if it winds n times round the hole in clockwise direction (n > 0), n < 0 in the anticlockwise direction, n = 0 if it does not wind. Thus the fundamental group is the infinite cyclic group of integers ZZ. Right : a torus S 1 × S 1 . Loops can wind m times around the central hole and n times around the body of the torus. Thus the fundamental…
Figure 8. Development of a simply-connected domain of the cylinder. its development (figure 8). There is a one-to-one correspondence between the points of D and those of ∆, and all the distances remain unchanged. Inside D, all the properties of Euclidean geometry are valid : the sum of the angles of a triangle is 180 degrees; one and only one geodesic joins two any distinct points; and so on . . . Now consider the domain D′ bounded by two circular sections of the cylinder (figure 9). D′ is obviously multi–connected because between two arbitrary points P and P ′ can now pass an infinite number
Figure 9. Development of a multi-connected domain of the cylinder. Such a procedure can be generalized to any manifold. Start with a manifold M with metric g. Choose a base point x in M and consider the differents paths from x to an other point y. Each path belongs to a homotopy class γ of loops at x. We construct the universal covering space as the new manifold (Mf,g˜) such that each point y˜ of Mf is obtained as a pair (y, γ), y varying over the whole of M while x remains fixed . The metric g˜ is obtained by defining the interval from x˜ = (x, γ) to a nearby point x˜′ = (x ′ , γ) in Mf to be
Figure 10. Universal covering space of the torus. Equipped with such properties, the holonomy group is said to act freely and discontinuously on Mf. The holonomy group is isomorphic to the fundamental group π1(Mf) (see for instance [14]). 3.4.4. Fundamental polyhedron The geometrical properties of a manifold M within a simply– connected domain are the same as those of its development in the universal covering Mf. It may be asked what is the largest simply–connected domain containing a given point x of M, namely the set  y ∈ M, d(y, x) ≤ d(y, γ(x)), ∀γ ∈ Γ
Figure 11. Development of geodesics of the cylinder. a tesselation of Mf, each image γP being a cell of the tesselation. The F P presents two major interests: • The fundamental group of a given topological manifold M is isomorphic to the fundamental group of the F P. Since routine methods are available to determine the holonomy group of as a polyhedron, the problem is considerably simplified. • The F P allows one to represent any curve in M, since any portion of a curve lying outside the F P can be carried inside it by appropriate holonomies (figure 11). As a general conclusion of this…
Figure 12. The three kinds of geometries for Riemannian surfaces, with their universal covering space. 4. Classification of Riemannian surfaces In addition to pedagogical and illustrative interest, the classification of two–dimensional Riemannian surfaces plays an important role in physics for understanding (2+1)–dimensional gravity, a toy model to gain insight into the real world of (3+1)–dimensional quantum gravity [146, 74, 68, 28, 157, 64]. Also, from a mathematical point of view, three–dimensional spaces can be constructed from surfaces. As we have seen in section 3.3.3, any Riemannian…
Figure 13. The four types of multi-connected Euclidean surfaces. order† two. It is generated by the antipodal map of S 2 which identifies diametrically opposite points on the surface of the sphere. As a result there are only two spherical surface forms† : • the sphere S 2 iself : C, SC, OR • the projective plane IP2 ≡ S 2 /ZZ2 (also called the elliptic plane) : C, MC, NOR. Whereas the surface of the unit sphere is 4π, the surface of the unit projective plane is only 2π, and its diameter, i.e., the distance between the most widely separated points, only π/2. † The order of a group is the num
Figure 14. Three representations of IH2 : the upper-half space U, the Poincar´e model DP and the Klein model DK. Representative geodesics are shown. The model is also conformally flat, and the hyperbolic geodesics are mapped onto arcs of circle which meet the frontier of DP at right angles. • The Klein representation also represents IH2 in an unit open disc DK ≡ {(x ′′, y′′) ∈ IR2 , x2′′+y 2′′ < 1} with a mapping DP 7→ DK such that z ′′ ≡ x ′′ + iy′′ = 2z ′ 1+z ′z¯ ′ . The hyperbolic geodesics are mapped onto Euclidean lines, but this model is not conformally flat. 4.3.2. The holonomi
Figure 15. Tesselation of IH2 by octogons in the Poincar´e representation. The vertices have the coordinates z(Jk) = 0.841 exp (3−2k)πi 8 . 5. Three-dimensional homogeneous spaces We now consider a three-dimensional Riemannian manifold M admitting at least a 3–dimensional discrete isometry group Γ simply transitive on M (cf. section 3.3.3). Such a (locally) homogeneous manifold can be written as the quotient Mf/Γ, where Mf is the universal covering space of M. Let G be the full group of isometries of M (containing Γ as a discrete subgroup). In the terminology used in the theory of classificat
Figure 16. The quasi-hyperbolic model of Fagundes. The edges of the 4g-gon are identified by pairs, so that the two Mi are the same, and all the 4g points J are the same. Resolution of the triangle CJMi gives CMi = a cosh−1 (cotπ/4g), CJ = a cosh−1 (cot2π/4g). The volume of Σ is V = 8πgλ(g − 1)a 2 b. The anisotropy of Σ is manifest in the horizontal section : from the point of view of Mi , the points C and J have opposite images at distance | ai | /2, whereas from the point of view of C only the Mi provide opposite images at this distance. 5.5. Spaces of constant curvature We emphasize that
Figure 17. The six locally Euclidean, closed, oriented 3-spaces When Γ includes a glide reflection, the space forms IR3 /Γ are not orientable. We shall not describe them because of their lack of interest for cosmology (cf. section 2) 6.2. Closed models The compact models can be better visualised by identifying appropriate faces of fundamental polyhedra. Six of them are orientable (figure 17) The fundamental polyhedron can be a parallelepiped. The possible identifications are then : • E1 - opposite faces by translations. The hypertorus T 3 already mentioned in section 3.2.3, which is homeomorph
Figure 18. Top : Representation of S 3 by two balls in IR3 glued together. Bottom : Behaviour of coordinates within slices sinφ = 0. quite analogous to the 2–sphere which can be sliced by planes into circles). Another convenient way (genially guessed in the Middle Ages by Dante in his famous Divine Comedy, see [124]) is to consider S 3 as composed of two solid balls in Euclidean space IR3 , glued together along their boundaries (figure 18) : each point of the boundary of one ball is the same as the corresponding point in the other ball. The result has twice the volume of one of the balls. 7
Figure 19. The dihedral group D5. The pentagon J1 . . . J5 is invariant by rotations of angle 2π/5 about the line L. T O I
Figure 20. The polyhedral groups. Left: Tetrahedral group. Center: Octahedral group. Right: Icosahedral group. SAS−1 = cA−1 , where c = exp(2π ki/m) is a mth root of unity. In a more geometrical representation, Dm can be viewed as generated by the rotations in the plane by an angle 2π/m and a flip about a line through the origin. Such symmetries preserve a regular m-gon lying in the plane and centered on the origin (figure 19). • the polyhedral groups They are the symmetry groups of the regular polyhedra in IR3 (figure 20), namely : – the group T of the tetrahedron (4 vertices, 6 edges, 4 face
Figure 21. The spherical Poincar´e space and the closed hyperbolic Seifert-Weber space are respectively obtained by identifying opposite faces of a regular dodecahedron after rotation by π/5 and 3π/5
Figure 22. The FP for the L¨obell topology is made of 8 such 14–hedra pasted together. All the angles in the figure are right angles in IH3 . compact hyperbolic 3-spaces can be build by pasting together various numbers of these 14–hedra, and suitably identifying the unattached faces. The Best Spaces. Best [11] has constructed several compact hyperbolic manifolds whose FP is a regular icosahedron. One of them was studied in details by Fagundes [48, 49] in a cosmological context. Its outer structure is represented in figure (23). The corresponding generators of the holonomy group are expressible
Figure 23. The closed hyperbolic Best space. The pairwise identifications of faces are ADI ↔ BGC, ICA ↔ ABC, EF L ↔ DIK, DEL ↔ KLD, GF B ↔ HJK, F GJ ↔ GJH, ABE ↔ LJF, F EB ↔ CGH, AED ↔ KIH, JKL ↔ CHI (where the order of vertices in the faces in maintained in the identification). Appendix A of [49]. The manifold is avantageously described in the Klein coordinates (33). The Weeks Space The Weeks manifold [154] is a polyhedron with 26 vertices and 18 faces, among which 12 are pentagons and 6 are tetragons. Its outer structure is represented in figure (24). It has the peculiarity to be the…
Figure 24. The closed hyperbolic Weeks space. The pairwise identifications of faces are ABT CD ↔ HLDAO, SEF BA ↔ FMP JE, RHIJE ↔ IKQLH, FMXT B ↔ UKZV N, W NUY C ↔ IJP ZK, CY QLD ↔ MXW NV , CT XW ↔ HOGR, KQY U ↔ ERGS, MP ZV ↔ ASGO. 9.1. Simply and multi–connected models The metric of spacetime specifies its local geometry but not its topology. Such a metric, solution of the Einstein’s equations for a given form of the cosmic stress–energy tensor, may correspond to different models of the Universe with different spatial topologies. On the other hand, the local property of being a solution of…
Figure 25. Ghosts of the second kind in a spherical (simply–connected) Universe. spacetime geodesics link a spatial position to the observer (at present time).The nearest image is called “real” and the others are called “ghosts” (figure 26). Each image corresponds to a different null geodesic linking the source to us. To each of them is associated a set Sobs of observable quantities. To deal with this situation, it is convenient towork in the UC space : the one–to–one correspondence holds between Sobs and a spatial position in the UC, which can be called the “observer’s world”. But the…
Figure 26. Real image and first ghost in a multi–connected universe. between z, the observable distance dobs and the look–back time hold exactly as in the corresponding SCM : to each redshift z is associated an instant t of emission, by the same formula 1 + z = R0 R(t) than in a SCM (which involves the curvature of spacetime), where R0 = R(t0) is the present value of the scale factor. Most of the usual cosmological formulae may still be used, although they take their sense and validity in the UC, and not in the real space. 10.2. Comoving space and real space In multi–connected models like in S
Figure 27. Different ghosts of the same galaxy in a multi–connected universe. space at any other time, for instance at the present time t0, with the ratio R(t)/R0 = 1/1 + z. To any position P in space at time t corresponds, by the same homothety, a comoving position P0 in the present space. To avoid comparisons of spatial properties at different times, all spatial positions of cosmic objects are considered in the comoving space. This is allowed by the fact that all spatial structures and relations, as well as the topology, are preserved by the homothety. For instance, rather than using the…
Figure 28. Ghosts in spacetime and in the universal covering. all topological properties. The UC at time t is similarly imaged to the comoving UC. The relation between space and its universal covering is exactly identical to that between the comoving space and its comoving universal covering, and the comoving universal covering may be identified with the comoving space of the associated SCM. Thus the search for the topology of spacetime, already reduced to that of space, reduces further to that of the comoving space. Throughout this paper, if not specified otherwise, space will refer to…
Figure 29. The horizon length in curvature radius units. 10.3.3. Spatial scales associated to multi–connectedness In a MCM, additional spatial scales are associated with the topology, those of the fundamental polyhedron. The geometry suggests to compare them with R0 but it is often more convenient, for observations, to compare them to LHubble or Lh, or to evaluate them in Mpc or h −1 Mpc. Observable effects linked to the multi–connectedness will only occur if these scales are smaller than the size of the observable universe, i.e., the horizon length. We call α the smallest length associated…
Figure 30. Characteristic lengths of a toroidal universe. of the diagonal. If real space has the topology of such a 3–torus, immediate observations impose that α cannot be too small, for instance smaller than the size of the Milky Way. The observations detailed in the next sections allow to increase this lower limit. 10.4. Multi–connected universes with flat spatial sections For simply or multi–connected models, space is Euclidean if Ω + λ = 1. The universal covering is IR3 and space has no characteristic scale : the present value R0 of the scale factor may be chosen arbitrarily and thus…
Figure 31. Multiplication of images in the universal covering space of a toroidal universe. cell : R2 = β/2, where β = (α 2 x + α 2 y + α 2 z ) 1/2 is the diagonal. The real image of any celestial object is always closer than β/2 to the Earth. The torus considered by Gott [72] has all fundamental lengths equal to α = 28.5 h −1 Mpc, so that β = 50 h −1 Mpc (note that he calls RH the quantity β/2). In the more general case, the relation between β and α depends on the geometry. In a MCM, many null geodesics start from the position of an object (e.g., a light source) in real space, to reach
Figure 32. The histogram of pair separations (in comoving proper distances) in the galaxy distribution, for a computer generated toroidal Einstein–de Sitter universe. 50 galaxies are randomly distributed in the cubic fundamental polyhedron, of size 2 500 Mpc. The images are assumed to be visible up to a redshift z = 4. The peaks reveal the holonomies. Experiment) suggest a cosmological distribution. However Quashnock & Lamb [129] analyzed the angular distribution of detected bursts and concluded that they appear significantly clustered on angular scales ≈ 4 ◦ . From this they suggested that…

Figures reproduced from arXiv:gr-qc/9605010 by the authors.

References
How the literature uses it

43 arXiv papers cite this in 65 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 30 sentences without a confident label

Cites it as background 27 papers
  • We direct the reader to [ref] for the precise definitions of the topological notions involved in the definition of the topo-GR theory, involving in particular the Thurston classification of closed 3-manifolds.

    A natural model for curved inflation · Introduction · 0.86

  • It has generally been thought that if the spatial sections of the Universe are compact, then the topology of spatial sections of the Universe cannot have evolved at post-quantum epochs, because this would imply the existence of closed timelike curves or a discontinuity in the choice of the forward light cone, as a consequence of Geroch's Theorem 2 [ref] , [ref] , [ref] , and both are generally considered unphysical.

    Topological implications of inhomogeneity · Conclusion · 0.84

  • More precision on these notions are given in [ref] (Section IV.A.) or in [ref] (Section 2), see also [ref] .

    Non-relativistic regime and topology: topological term in the Einstein equation · What we mean by Euclidean and non-Euclidean topologies · 0.97

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