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Cosmic Topology - Scholarpedia

# Cosmic Topology

Post-publication activity

Curator: Jean-Pierre Luminet

Cosmic Topology is the name given to the study of the overall shape of the universe, which involves both global topological features and more local geometrical properties such as curvature. Whether space is finite or infinite, simply-connected or multi-connected like a torus, smaller or greater than the portion of the universe that we can directly observe, are questions that refer to topology rather than curvature. A striking feature of some relativistic, multi-connected "small" universe models is to create multiples images of faraway cosmic sources. While the most recent cosmological data fit the simplest model of a zero-curvature, infinite space model, they are also consistent with compact topologies of the three homogeneous and isotropic geometries of constant curvature, such as, for instance, the spherical Poincaré Dodecahedral Space, the flat hypertorus or the hyperbolic Picard horn. After a "dark age" period, the field of Cosmic Topology has recently become one of the major concerns in cosmology, not only for theorists but also for observational astronomers, leaving open a number of unsolved issues.

## History

The notion that the universe might have a non-trivial topology and, if sufficiently small in extent, display multiple images of faraway sources, was first discussed in 1900 by Karl Schwarzschild (see Starkman, 1998 for reference and English translation). With the advent of Einstein's general relativity theory and the discoveries of non-static universe models by Friedmann and Lemaître in the decade 1922-1931, the face of cosmology definitively changed. While Einstein's cosmological model of 1917 described space as the simply-connected, positively curved hypersphere $\bf{S}^3$, de Sitter in 1917 and Lemaître in 1927 used the multi-connected projective sphere $\bf{P}^3$ (obtained by identifying opposite points of $\bf{S}^3$) for describing the spatial part of their universe models. On his side, Friedmann clearly pointed out that Einstein's equations are not sufficient for deciding if space is finite or infinite: Euclidean and hyperbolic spaces, which in their "trivial" (i.e. simply-connected) topology are infinite in extent, can become finite (although without an edge) if one identifies different points - an operation which renders the space multi-connected. Friedmann also foresaw how this possibility allowed for the existence of "phantom" sources, in the sense that at a single point of space an object coexists with its multiple images.

The whole problem of cosmic topology was thus posed, but as the cosmologists of the first half of XXth century had no experimental means at their disposal to measure the topology of the universe, the vast majority of them lost all interest in the question. However in 1971, George Ellis published an important article taking stock of recent mathematical developments concerning the classification of 3-D manifolds and their possible application to cosmology. A brief revival of interest in multi-connected cosmologies ensued, under the lead of theorists like Sokoloff, Starobinsky, Gott, and especially Fagundes, who investigated several kind of topologies (see Lachièze-Rey and Luminet, 1995 for exhaustive references). An observational program was even started up in the Soviet Union (Sokolov and Shvartsman, 1974), and the "phantom" sources of which Friedmann had spoken in 1924, meaning multiple images of the same galaxy, were sought. All these tests failed: no ghost image of the Milky Way or of a nearby galaxy cluster was recognized. This negative result allowed for fixing some constraints on the minimal size of a multi-connected space, but it hardly encouraged the researchers to pursue this type of investigation. The interest again subsided. Although the July 1984 Scientific American article by Thurston and Weeks on hyperbolic manifolds with compact topology was very cosmologically oriented, the idea of multi-connectedness for the real universe did not attract much support. Most cosmologists either remained completely ignorant of the possibility, or regarded it as unfounded.

Fortunately, a second revival occurred in the early 1990's. The new data on the Cosmic Microwave Background provided by the COBE telescope gave access to the largest possible volume of the observable universe, and the term "Cosmic Topology" itself was coined in 1995 in a Physics Reports issue discussing the underlying physics and mathematics, as well as many of the possible observational tests for topology. Since then, hundreds of articles have considerably enriched the field of theoretical and observational cosmology.

## Topology and Relativity

At very large scale our Universe seems to be correctly described by a Friedmann-Lemaître-Robertson-Walker model (hereafter FLRW). The FLRW models are homogeneous and isotropic solutions of Einstein's equations, of which the spatial sections have constant curvature. They fall into 3 general classes, according to the sign of curvature $k= -1, 0, +1$. The space-time manifold is described by the line element $ds^2 = c^2dt^2 - R^2(t)d\sigma^2$, where $d\sigma^2 = d\chi^2 + {S_k}^2(\chi) (d\theta^2+\sin^2{\theta}d\varphi^2)$ is the metric of a 3D homogeneous manifold, flat [$k=0$] or with curvature [$k \pm 1$]. The function $S_k(\chi)$ is defined as $\sinh(\chi)$ if $k= -1$, $\chi$ if $k=0$, $\sin(\chi)$ if $k=1$; $R(t)$ is the scale factor, which can be chosen to be the spatial curvature radius for non flat models.

Figure 1: Open or Closed? Contrary to a widespread opinion, the curvature of space dictates neither the time evolution of the Universe (unless the cosmological constant is zero), nor the finite or infinite extent of space (unless the topology is simply-connected). The table summarizes the various cases for FLRW "big bang" models. The first column shows the sign of the spatial curvature $k$, the second the finite or infinite character of space, the third the time behavior of the scale factor depending on the cosmological constant $\lambda$, the fourth the open or closed timelike character of the model.

In most studies, the spatial topology is assumed to be that of the corresponding simply-connected space: the hypersphere, Euclidean space or 3D-hyperboloid, the first being finite and the other two infinite. However, there is no particular reason for space to have a trivial topology. In any case, general relativity says nothing on this subject: the Einstein field equations are local partial differential equations which relate the metric and its derivatives at a point to the matter-energy contents of space at that point. Therefore, to a metric element solution of Einstein field equations there are several, if not an infinite number, of compatible topologies, which are also possible models for the physical universe. For example, the hypertorus $\bf{T}^3$ and the usual Euclidean space $\bf{E}^3$ are locally identical, and relativistic cosmological models describe them with the same FLRW equations, even though the former is finite in extent while the latter is infinite. Only the boundary conditions on the spatial coordinates are changed. The multi-connected cosmological models share exactly the same kinematics and dynamics as the corresponding simply-connected ones; in particular, the time evolutions of the scale factor $R(t)$ are identical.

In FLRW models, the curvature of physical space (averaged on a sufficiently large scale) depends on the way the total energy density of the universe may counterbalance the kinetic energy of the expanding space. The normalized density parameter $\Omega_0$, defined as the ratio of the actual energy density to the critical value that strictly Euclidean space would require, characterizes the present-day contents (matter, radiation and all forms of energy) of the universe. If $\Omega_0$ is greater than 1, then the space curvature is positive and the geometry is spherical; if $\Omega_0$ is smaller than 1, the curvature is negative and geometry is hyperbolic; otherwise $\Omega_0$ is strictly equal to 1, and space is locally Euclidean (currently said "flat", although the term can be misleading).

Figure 2: The Illusion of the Universal Covering Space. In the case of a 2D torus space, the fundamental domain, which represents real space, is the interior of the gray rectangle, whose opposite edges are identified. The observer O sees rays of light from the source S coming from several directions. He has the illusion of seeing distinct sources $S_1$, $S_2$, $S_3$, etc., distributed along a regular canvas which covers the UC space - an infinite plane.

The next question about the shape of the Universe is to know whether its topology is trivial or not. A subsidiary question - although one much discussed in the history of cosmology and philosophy - is whether space is finite or infinite in extent. Of course no physical measure can ever prove that space is infinite, but a sufficiently small, finite space model could be testable. Although the search for space topology does not necessarily solve the question of finiteness, it provides many multi-connected universe models of finite volume. The effect of a non-trivial topology on a cosmological model is equivalent to considering the observed space as a simply-connected 3D-slice of space-time (known as the "universal covering space", hereafter UC) being filled with repetitions of a given shape (the "fundamental domain") which is finite in some or all directions, for instance a convex polyhedron; by analogy with the two-dimensional case, we say that the fundamental domain tiles the UC space. For the flat and hyperbolic geometries, there are infinitely many copies of the fundamental domain; for the spherical geometry with a finite volume, there is a finite number of tiles. Physical fields repeat their configuration in every copy and thus can be viewed as defined on the UC space, but subject to periodic boundary conditions.

Figure 3: Six "Small" Orientable Euclidean Spaces. By properly identifying the opposing faces of a parallelepiped or a hexagonal prism, one obtains six finite and orientable Euclidean spaces. The unmarked pairs of walls are glued by simple translations. The others are glued with the orientations shown by the doors. (a) hypertorus (b) hypertorus with a quarter turn (c) hypertorus with a half-turn (d) hypertorus with a one-sixth turn (e) hypertorus with a one-third turn (f) Hantzsche-Wendt space. Courtesy Adam Weeks Marano.

For 3D-Euclidean spaces, the fundamental domains are either a finite or infinite parallelepiped, or a prism with a hexagonal base, corresponding to the two ways of tiling Euclidean space. The various combinations generate seventeen multi-connected Euclidean spaces (for an exhaustive study, see Riazuelo et al. 2004), the simplest of which being the hypertorus $\bf{T}^3$, whose fundamental domain is a parallelepiped of which opposite faces are identified by translations. Seven of these spaces, called slabs and chimneys, have an infinite volume. The ten other are of finite volume, six of them being orientable hypertori. All of them could correctly describe the spatial part of the flat universe models, as they are consistent with recent observational data which constrain the space curvature to be very close to zero. They are also consistent with current inflationary scenarios for the big bang, according to which the observable universe can appear to be arbitrarily close to be flat. Note also that calculations about the quantum creation of the universe from vacuum energy fluctuations favor the compact case, but it is by no means a cut and dried issue, given the lack of a satisfactory quantum gravity theory.

Figure 4: The Poincaré Dodecahedral Space. The PDS can be described as the interior of a spherical ball whose surface is tiled by 12 curved regular pentagons. When one leaves through a pentagonal face, one returns to the ball through the opposite face after having turned by $36^{\circ}$. As a consequence, the space is finite but without boundaries, therefore one can travel through it indefinitely. One has thus the impression of living in a UC space 120 times larger, paved with dodecahedra that multiply the images like a hall of mirrors. The return of light rays that cross the walls produces optical mirages: a single object has several images. This numerical simulation calculates the closest phantom images of the Earth which would be seen in the UC space. (Image courtesy of J. Weeks).

In spaces with non-zero curvature, the situation is notably different: the presence of a length scale - the curvature radius - precludes topological compactification at an arbitrary scale. The size of the space must now reflect its curvature, linking topological properties to the total energy density $\Omega_0$. All spaces of constant positive curvature are finite whatever be their topology. The reason is that the UC space - the simply-connected hypersphere $\bf{S}^3$ - is itself compact; thus, if one identifies the points of the hypersphere by transformations belonging to one of the cyclic groups of order p, the dihedral groups of order 2m or the three binary polyhedral groups which preserve the shapes of the regular polyhedra, the resulting spaces are spherical, multi-connected and compact (for a complete classification, see Gausmann et al. 2002). There is a countable infinity of these because of the integers p and m which parametrize the cyclic and dihedral groups; but there is only a finite set of "well-proportioned" topologies, i.e., those with roughly comparable sizes in all directions, which are of a particular interest for cosmology. As a now celebrated example, let us mention the Poincaré Dodecahedral Space (hereafter PDS), obtained by identifying the opposite pentagonal faces of a regular spherical dodecahedron after rotating by $36^{\circ}$ in the clockwise direction around the axis orthogonal to the face. Its volume is 120 times smaller than that of the hypersphere with the same curvature radius. After PDS was proposed, in 2003, as a specific candidate compatible with the Cosmic Microwave Background power spectrum WMAP data (Luminet et al. 2003), its mathematical properties were extensively studied. This provides an interesting example of how cosmological considerations may drive new developments in pure mathematics. However, as discussed below, the cosmological pertinence of such a model was immediately and vigorously disputed (Cornish et al. 2004).

Hyperbolic manifolds can be viewed as 3D generalizations of an infinitely extended saddle shape. According to the pioneering work of Thurston they represent the "generic" case for homogeneous three-dimensional geometries, since almost all 3-manifolds can be endowed with a hyperbolic structure. There is an infinite number of hyperbolic manifolds, with finite or infinite volumes, but their classification is not well understood. However, they have a remarkable property that links topology and geometry: the "rigidity theorem" implies that geometrical quantities such as the volume, the length of its shortest closed geodesics, etc., are topological invariants. This suggests the idea of using the volumes to classify the compact hyperbolic space forms. Such volumes are bounded below by V = 0.94271 (in units of the curvature radius cubed), which correspond to the so-called Weeks manifold. The computer program SnapPea (http://www.geometrygames.org/SnapPea/) is especially useful to unveil the rich structure of compact hyperbolic manifolds. Several millions of them with volumes less than 10, i.e. small enough to fit entirely within the observable universe, could be calculated. Quite recently, it was shown that the present-day observational constraints on the curvature of space, as well as large-scale anomalies observed in the CMB power spectrum, remain compatible with a marginally hyperbolic space. In particular, horned topologies such as the Picard space have been invoked to explain the suppression of the lower multipoles in the CMB anisotropy (Aurich et al. 2004), but these claims have been disputed by those arguing that reliable predictions of the amplitude of low-lying eigenmodes are not available.

## Probing Cosmic Topology

From an astronomical point of view, it is necessary to distinguish between the observable universe, which is the interior of a sphere centered on the observer and whose radius is that of the cosmological horizon (roughly the radius of the last scattering surface, presently estimated at 14.4 Gpc), and the whole universe, the topology of which is involved in Cosmic Topology. There are only three logical possibilities.

• First, the whole space is infinite - like for instance the simply-connected flat and hyperbolic spaces. In this case, the observable universe is an infinitesimal patch of the whole universe and, although it has long been the standard "mantra" of many theoretical cosmologists, this is not and will never be a testable hypothesis.
• Second, the whole universe is finite (e.g. a hypersphere or a closed multi-connected space), but greater than the observable universe. In that case, one easily figures out that if whole space widely encompasses the observable one, no signature of its finiteness will show in the experimental data. But if space is not too large, or if space is not globally homogeneous (as is permitted in many space models with multi-connected topology), and if the observer occupies a special position, some imprints of the space finiteness could be observable.
• Third, the whole space is smaller than the observable universe. Such an apparently odd possibility is due to the fact that space can be multi-connected, have a small volume and produce topological lensing. There are a lot of testable possibilities, whatever the curvature of space.

The present observational constraints on the $\Omega_0$ parameter favor a spatial geometry that is nearly flat with a 0.4% margin of error (Hinshaw et al 2013). However, even with the curvature so severely constrained by cosmological data, there are still possible multi-connected topologies that support positively curved, negatively curved, or flat metrics. Such small universe models generate multiple images of light sources, in such a way that the hypothesis of multi-connected topology can be tested by astronomical observations. The smaller the fundamental domain, the easier it is to observe the multiple images of luminous sources in the sky (generally not seen at the same age, except for the CMB). Note, however, the coincidence problem that occurs in order to get an observable non-trivial topology: for flat space, we need to have the topology scale length near the horizon scale, while for curved spaces, the curvature radius needs to be near the horizon scale.

How do the present observational data constrain the possible multi-connectedness of the universe and, more generally, what kinds of tests are conceivable? (See Luminet (2008) for a non-technical book about all aspects of topology and its applications to cosmology).

Figure 5: The Cosmic Crystallography Method in the Poincaré Dodecahedral Space. In this histogram for pairwise separations calculated in a PDS with $\Omega_0 = 1.1$, the presence of spikes signals a multi-connected topology, their positions reflect the size of the fundamental domain (a dodecahedron), and their relative heights characterizes the group of transformations that glue together the opposite faces of the dodecahedron. Although the main spike is situated at a spectral shift too large to be detectable, the first one is at $z \approx 1.5$, accessible to the deep observational surveys expected in this decade.

Different approaches have been proposed for extracting information about the topology of the universe from experimental data. One approach is to use the 3D distribution of astronomical objects such as galaxies, quasars and galaxy clusters: if the Universe is finite and small enough, we should be able to see "all around" it because the photons might have crossed it once or more times. In such a case, any observer might recognize multiple images of the same light source, although distributed in different directions of the sky and at various redshifts, or to detect specific statistical properties in the distribution of faraway sources. Various methods of cosmic crystallography, initially proposed by Lehoucq et al. 1996, have been widely developed by other groups (Fujii and Yoshii 2011 and references therein). However, for plausible small universe models, the first signs of topological lensing would appear only at pretty high redshift, say $z \approx 2$. The main limitation of cosmic crystallography is that the presently available catalogs of observed sources at high redshift are not complete enough to perform convincing tests for topology. But the large and deep surveys (up to redshift $z=6$), such as the LSST (Large Synoptic Survey Telescope) planned in the next decade, should make such methods applicable.

The other approach uses the 2D cosmic microwave background (CMB) maps (for a review, see Levin 2002). The last scattering surface from which the CMB is released represents the most distant source of photons in the Universe, and hence the largest scales with which we can probe the topology of the universe. The two most important CMB methods are the analysis of the angular power spectrum and the circles-in-the-sky tests.

The early Universe was crossed by acoustic waves generated soon after the Big Bang. Such vibrations left their imprints 380 000 years later as tiny density fluctuations in the primordial plasma. Hot and cold spots in the present-day 2.7 K CMB radiation reveal those density fluctuations and yield a wealth of information about the physical conditions that prevailed in the early Universe, as well as the present geometric properties like space curvature and topology. Density fluctuations may be expressed as combinations of the vibrational modes of space, whose shape can be "heard" in a unique way. The relative amplitudes of each spherical harmonics determine the angular power spectrum, which is a signature of the space geometry and of the physical conditions which prevailed at the time of CMB emission. The harmonics (precisely the eigenmodes of the Laplace-Beltrami operator) have been computed for all 18 Euclidean spaces, for most of the spherical spaces, and for some of compact hyperbolic manifolds.

The idea that a small universe model could lead to a suppression of power on large angular scales in the fluctuation spectrum of the CMB had been proposed by Fagundes as soon as 1983: in some way, space would be not big enough to sustain long wavelengths. After the release of COBE data in 1992, the idea was developed by several authors, such as Sokolov and Starobinskii, and used to constrain the models. Then, the CMB angular power spectrum derived from the first-year WMAP telescope release in 2003 exhibited unusually low values of the quadrupole and octupole moments (as compared to the standard $\Lambda$CDM cosmology), as well as a vanishing two-point angular correlation function above $60^{\circ}$. Various explanations were proposed, involving non-trivial topology, anisotropic Bianchi models or peculiar inflationary models. Regarding topological proposals, the best fits between theoretical power spectra computed for various topologies and the observed one were obtained with the positively curved Poincaré Dodecahedral Space and the flat hypertorus. In addition, it was shown that the low-order multipoles tended to be relatively weak in "well-proportioned" spaces (i.e. whose dimensions are approximately equal in all directions), so that several authors proposed various flat and non-flat small universe models to account for the "missing fluctuations". However, in subsequent WMAP data releases, the value of the octupole was found completely "normal", and since the remaining low value of the quadrupole could be merely explained by cosmic variance, the arguments favoring small universe models failed. In any case, to gain all the possible information from the correlations of CMB anisotropies, one has to consider the full covariance matrix rather than just the power spectrum.

Figure 6: The circle-in-the-sky method. The method is illustrated here in a 2D torus space. The fundamental polyhedron is a square (with a dotted outline), all of the red points are copies of the same observer. The two large circles (which are normally spheres in a three-dimensional space) represent the last scattering surfaces (lss) centered on two copies of the same observer. One is in position (0, 0), its copy is in position (3,1) in the universal covering space. The intersection of the circles is made up of the two points A and B (in three dimensions, this intersection is a circle). The observers (0,0) and (3, 1), who see the two points (A, B) from two opposite directions, are equivalent to a unique observer at (0, 0) who sees two identical pairs (A, B) and (A',B') in different directions. In three dimensions, the pairs of points (A, B) and (A',B') become a pair of identical circles, whose radius $r_{31}$ depends on the size of the fundamental domain and the topology.

A direct observational method of detecting topological signatures in CMB maps, called "circles-in-the-sky" (Cornish et al. 1998), uses pairs of circles with the same temperature fluctuation pattern, which may be intersections of the last-scattering surface and the observer's fundamental domain. If the Universe possesses a non-trivial topology, the UC space can be viewed as being tiled by copies of the fundamental domain, each one having a copy of the observer who sees the same CMB sky. If the observer and its nearest copy are not farther separated than the diameter of the last scattering surface, the two CMB spheres overlap in the UC, and their intersection will be a circle, seen by the observer and its copy in different directions. Since the observer and its copy are to be identified, two circles should exist on the CMB sky with identical temperature fluctuations seen in different directions and phases, but with the same radius. A non-trivial topology is thus betrayed by as many pairs of circles with the same temperature fluctuations as there are copies of the observer not farther away than the diameter of the last scattering surface. The PDS model, which predicts six pairs of antipodal circles with an angular radius comprised between $5^{\circ}$ and $55^{\circ}$ (sensitively depending on the cosmological parameters), became a particularly disputed candidate. In such a space, detection of the cosmic topology through the circles-in-the-sky method would give rise to very tight constraints on the density parameters (Rebouças et al. 2006). Several groups have searched for matched circles using various statistical indicators and massive computer calculations, and interpreted their results differently. Cornish et al. 2004 claimed that most of non-trivial topologies, including PDS and $\bf{T}^3$, were ruled out: they searched for antipodal or nearly antipodal pairs of circles in the WMAP data and found no such circles. However, their analysis could not be applied to more complex topologies, for which the matched circles deviate strongly from being antipodal. On the other hand, other groups claimed to have found hints of multi-connected topology, using different statistical indicators (Roukema et al. 2004 and 2008, Aurich et al. 2006).

Indeed, the circles-in-the-sky method has to take into account many effects that alter the CMB temperature fluctuations of the observer and its copy in a different way, so that temperature fluctuations on two circles are no longer strictly identical. The two most important CMB contributions in this respect are the Doppler contribution, whose magnitude depends on the velocity projection towards the observer, and the integrated Sachs-Wolfe contribution, which arises along the path from the last scattering surface to the observer or to its copy. These two paths are not identical and lead to different contributions to the total CMB signal. There are further degrading effects, like the finite thickness of the last scattering surface, and residuals left over by the subtraction of foreground sources, which have their own uncertainties. In addition, for generic multi-connected topologies (including the well-proportioned ones), matched circles are not antipodal, and the positions of matched circles in the sky depend on the observer’s position in the fundamental domain.

The corresponding larger number of degrees of freedom for the circles search in the CMB data generates a dramatic increase of the computer time. The search for matched circle pairs that are not back-to-back has nevertheless been carried out recently, with no obvious topological signal appearing in the seven-years WMAP data (Vaudrevange et al 2012). Other methods for experimental detection of non-trivial topologies have also been proposed and used to analyze the experimental data, such as the multipole vectors and the likelihood (Bayesian) method. The latter ameliorates some of the spoiling effects of the temperature correlations mentioned above (Kunz et al. 2006). The most up-to-date study used the 2013 and 2015 data from the Planck telescope (Planck Collaboration 2013). The circle-in-the-sky searches did not find any statistically significant correlation of antipodal circle pairs in any map. Thus they excluded, at the confidence level of 99 %, any topology that predicts matching pairs of back-to-back circles larger than 15° in angular radius, assuming that the relative orientation of the fundamental domain and mask allows its detection.

## Unresolved issues

After a difficult start, the overall topology of the universe has become an important concern in astronomy and cosmology. Even if particularly simple and elegant models such as the PDS and the hypertorus seem now to be ruled out at a subhorizon scale, many more complex models of multi-connected space cannot be eliminated as such. In addition, even if the size of a multi-connected space is larger (but not too much) than that of the observable universe, we could still discover an imprint in the fossil radiation, even while no pair of circles, much less ghost galaxy images, would remain. The topology of the universe could therefore provide information on what happens outside of the cosmological horizon (Fabre et al. 2013).

Whatever the observational constraints, a lot of unsolved theoretical questions remain. The most fundamental one is the expected link between the present-day topology of space and its quantum origin, since classical general relativity does not allow for topological changes during the course of cosmic evolution. Theories of quantum gravity should allow us to address the problem of a quantum origin of space topology. For instance, in quantum cosmology, the question of the topology of the universe is completely natural. Quantum cosmologists seek to understand the quantum mechanism whereby our universe (as well as other ones in the framework of multiverse theories) came into being, endowed with a given geometrical and topological structures. We do not yet have a correct quantum theory of gravity, but there is no sign that such a theory would a priori demand that the universe have a trivial topology. Wheeler first suggested that the topology of space-time might fluctuate at a quantum level, leading to the notion of a space-time foam. Also, some simplified solutions of the Wheeler-DeWitt equations for quantum cosmology show that the sum over all topologies involved in the calculation of the wavefunction of the universe is dominated by spaces with small volumes and multi-connected topologies. In the approach of brane worlds in string/M-theories, the extra-dimensions are often assumed to form a compact Calabi-Yau manifold; in such a case, it would be strange that only the ordinary dimensions of our 3-brane would not be compact like the extra ones. But still at an early stage of development, and quantum mechanics can only provide heuristic indications on the way multi-connected spaces would be favored.

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