Thurston's geometrization conjecture (now, a theorem of Grigory Perelman), which aims to answer the question: How could one describe the possible shapes of our universe? Here we will be assuming (although, some scientists think otherwise) that our space is 3-dimensional (3-D) [it has three directions: width, depth, and height]; closed (which means that you cannot travel infinitely far away from a given point in space, and that it does not have a boundary); connected (you can get from any point to any point); and orientable (if you travel along a path and return to the point of departure, your left and right hands will not get switched).
Constructing spaces from building blocks
To understand the answer to this question, we first look at the 2-dimensional (2-D) case for guidance. If our universe were a 2-D surface S, we could describe its shape by combining copies of three basic building blocks:
1. The disk D2 (given by the inequality x2 + y2 ≤ 1)
2. The annulus A2, also known as the cylinder (disk with one hole)
3. The pair of pants P2 (disk with two holes)
Each surface S is obtained by gluing these blocks along their boundary circles. For example, the 2-D sphere S2 (given by the equation x2 + y2 + z2 = 1) can be obtained by gluing together two disks (lower and upper hemispheres in S2). The 2-D torus T 2 (the surface of a doughnut) is obtained by identifying (gluing together) boundary circles of the cylinder (Fig. 1). [In our description of shapes we are not concerned with the measurements of distance. For example, instead of the disk of radius 1 we could have taken, say, an ellipse. Mathematicians define this field of study as topology.]
Every surface S can also be obtained as a connected sum of tori. This is done by taking a collection T1, …, Tg of 2-D tori and attaching them to the 2-sphere S2 by first removing a disk from each torus and g disks from S2 and then identifying the boundary circles. Both descriptions of surfaces will be useful when dealing with 3-D universes, which mathematicians call 3-D manifolds. See also: Manifold (mathematics)
How can these descriptions of 2-D surfaces be generalized to dimension 3? We can start by promoting the 2-D building blocks to 3-D ones by adding an extra spatial dimension (a circle) to each. For example, the solid torus, D2 × S1, is obtained by rotating the 2-D disk around an axis in 3-D space that is far enough from the disk. The result is the 3-D doughnut. Similarly, one obtains A2 × S1 (a doughnut with a thinner concentric doughnut removed) and P2 × S1 (a doughnut from which we have removed two thinner doughnuts). The boundary of each 3-D block we obtained consists of 1, 2, or 3 tori. Now, take a collection of these 3-D blocks and glue them together along their boundary tori. The resulting spaces are called graph manifolds. For example, we can take the 3-D sphere S3 given by Eq. (1)
in four-dimensional (4-D) space. One can describe S3 as a graph manifold by gluing together two solid tori: Take a small spherical cap on S3 around the north pole, (0, 0, 0, 1), then rotate this cap around the xy plane. The result of the rotation is a solid torus. Removing this solid torus from S3 leaves us with another solid torus. As an additional example, we can consider the 3-D torus T 3 = S1 × S1 × S1. It can be obtained by gluing together the two boundary tori of A2 × S1.
By analogy with the description of 2-D surfaces above, one could have expected that all 3-D manifolds are graph manifolds. It turns out that this is far from being the case. The missing manifolds are best described by means of hyperbolic geometry.
On a small scale (say, the scale of a single planet), the universe appears flat: The distances between points in space can be computed by the Pythagorean formula, or, as mathematicians would say, by means of the Euclidean (flat Riemannian) metric, which, in coordinates, is given by Eq. (2).
However, on the scale of a planetary system (and beyond), the universe is curved by gravity. The distance measurements are described by means of a Riemannian metric. To define such a metric, we will use from now on the coordinates x1, x2, x3; the notation x will be reserved for a point in a manifold. Then a Riemannian metric is given by a metric tensor, gij, through Eq. (3).
Hyperbolic 3-space, H3, can be described as the upper half-space (x3>0) in 3-D space, with the Riemannian metric given by Eq. (4).
In this description, hyperbolic space does not seem to have any preferred origin, but appears to have a preferred direction. To dispel this misperception, we can change the coordinates and describe the hyperbolic metric on the unit ball, x21 + x22 + x23 < 1, given by Eq. (5).
It is then clear that such a metric is homogeneous (it looks the same at each point) and isotropic (is the same in every direction). Mathematicians describe this property by saying that the hyperbolic metric has constant curvature. There are three classes of constant-curvature metrics: of positive curvature (the metric on the sphere), zero curvature (the metric of flat space), and negative curvature (the hyperbolic metric). To visualize what negative curvature means, we note that the volume enclosed by a sphere of (hyperbolic) radius r in hyperbolic space grows exponentially fast for large r. (In flat 3-D space, the volume grows only cubically.)
How can our universe possibly have negative curvature? At first glance, this would violate our assumption that it is closed (has finite size and is borderless). Let us reexamine the flat metric. We can start with a cube Q in the flat 3-D space, take opposite faces of Q, and identify them by parallel translations. The result has the shape of the 3-D torus T 3, which then inherits a flat Riemannian metric from Q, because the translations are isometries (that is, they preserve the metric).
How does this help to construct closed hyperbolic manifolds? Imagine that you have a finite polyhedral convex solid P in H3 and a collection of isometries of H3 identifying the faces of P. The result of identification (subject to certain conditions) is a closed hyperbolic 3-D manifold M3. Its Riemannian metric is the one inherited from P. One of the earliest examples of this construction is the Seifert-Weber dodecahedral space, obtained from a right-angled hyperbolic dodecahedron P (Fig. 2).
More generally, one can take a convex finite-volume solid P in H3 with finitely many faces (P could have infinite diameter; Fig. 3) to obtain a hyperbolic 3-D manifold M3 of finite volume. The resulting manifolds M3 are hyperbolic and every finite-volume hyperbolic manifold appears in this way. Even if M3 has infinite diameter, one can compactify it: There exists a manifold (with boundary) N3 of finite diameter (and a metric different from the one of M3), so that M3 is obtained by removing from N3 its boundary. The boundary of N3 is called the ideal boundary of M3, as it appears infinitely far from the points of the manifold M3. This boundary is a finite collection of 2-D tori T2. The manifold N3 (unlike M3) is compact, that is, one cannot travel indefinitely far from a given point; therefore, this procedure is said to compactify M3. To visualize the compactification, imagine that the solid P consists of a finite part (a house) and an infinite part, a rectangular chimney rising from the roof (this chimney has infinite height). Identification of the faces of P is such that the opposite vertical sides of the chimney are identified by horizontal translations. After this identification, at each finite height level of the chimney we see a 2-D torus. As the height increases, these tori move away from an observer staying in the house. At the infinite height, the tori reach a limit that is the ideal boundary torus of M3. The compactification process amounts to compressing an infinite chimney to a finite one, so that the ideal boundary torus lowers to a finite height (bringing it “from heaven to earth”). It is common to say that a 3-D manifold (with unspecified geometry) is hyperbolic if it admits a hyperbolic metric of finite volume. (Such a metric is known to be unique.) To simplify the terminology, one also refers to the compactifications N3 as hyperbolic manifolds. The boundary tori of a hyperbolic manifold satisfy an important technical property: They are incompressible. A boundary surface S of a manifold N3 is called incompressible if any circle on S that bounds a 2-D disk in N3 already bounds a disk in S.
One can show that hyperbolic manifolds are never graph manifolds, so we indeed have a new class of possible universes. More generally, a manifold is called geometric if it admits a Riemannian metric that is locally homogeneous (that is, it locally looks the same at all points). See also: Hyperbolic geometry
Thurston's geometrization conjecture
We are now ready to state Thurston's geometrization conjecture: Every (closed, oriented) 3-D manifold M3 can be obtained as follows: Start with a finite collection of graph manifolds (with incompressible boundary, if there is any) and hyperbolic manifolds. Then glue these manifolds along their respective boundary tori so that the result has no boundary. It is possible that the resulting manifold M′ is disconnected (you cannot reach one point from another). To remedy this, take a connected sum of the components of M′: Remove small balls from distinct components and glue together the exposed boundary 2-D spheres. The result is M3.
In other words, one can build M3 by using the three types of basic 3-D building blocks appearing in graph manifolds, together with hyperbolic manifolds. Equivalently, one can state the Thurston geometrization conjecture by saying that M3 can be built from a collection of geometric manifolds by gluing them along incompressible boundary tori and then taking a connected sum. (The conjecture derives its name from this formulation.)
William Thurston proved his conjecture in the 1970s under the technical assumption that M3 is a Haken manifold. This work covered a large class of 3-D manifolds, but far from all of them.
An important special case of the Thurston geometrization conjecture is the Poincaré conjecture, to which Thurston's technique did not apply: If M3 is a closed, simply connected 3-D manifold, then M3 is the 3-D sphere. (The manifold M3 is said to be simply connected if every circle in M3 can be contracted to a point.)
This conjecture was first formulated by Henri Poincaré in the early twentieth century and, despite numerous efforts, remained out of reach until Perelman's work. Its analogs were proven for higher-dimensional manifolds (dimensions equal to or greater than 5 by the 1960s, and dimension 4, in the topological form, by the 1980s) through the efforts of many mathematicians (most notably, Stephen Smale, John Stallings, and Michael Freedman).
Spherical space forms conjecture
Another special case of the Thurston geometrization conjecture is the spherical space forms conjecture: Suppose that M3 is a closed 3-D manifold with finite fundamental group. (In the context of 3-D manifolds, this property can be formulated as follows: If c is any circle in the manifold M3, then, for some n > 0, the circle cn, obtained by tracing the original circle n times, can be contracted to a point in M3.) Then M3 is a spherical space form; that is, it admits a metric of constant positive curvature. In particular, M3 is a graph manifold.
The Ricci flow was introduced by Richard Hamilton as a possible approach to the Thurston geometrization conjecture. Hamilton used it to prove this conjecture for manifolds of positive curvature in 1982. He also wrote a number of other papers establishing a program for proving the Thurston geometrization conjecture using the Ricci flow.
Roughly speaking, the idea is to define a flow on the space of Riemannian metrics that, starting with an arbitrary metric tensor g on a 3-D manifold, “homogenizes” g and in the limit separates the manifold into graph manifolds and homogeneous pieces. To define the Ricci flow, consider a metric tensor gij(x) on a 3-D manifold M. This metric can be encoded in a 3-by-3 array of numbers, indexed by i and j that depend on the point x in M. The curvature of g is described by means of the Riemannian curvature tensor Ripjq (given by a 4-D array of numbers). This tensor has an important simplification, the Ricci tensor, given by Eq. (6),
which is a symmetric (Ricij = Ricji) 2-D array of numbers. Then, the Ricci flow is given by the differential equation (7).
The Ricci flow can be regarded as an analog of the heat flow, describing the evolution of temperature in flat space. As the heat flow homogenizes the temperature distribution in space, the Ricci flow homogenizes Riemannian metrics.
By rescaling both space and time, one gets the normalized Ricci flow of metrics of constant volume, given by Eq. (8).
Here r = r(t) is some scalar function. Suppose that ĝ(t0) is a stationary point of the normalized Ricci flow, that is, ĝ′(t0) = 0. Then ĝ(t0) is an Einstein metric: Its Ricci tensor is a scalar multiple of ĝ(t0). In dimension 3, Einstein metrics all have constant curvature; hence, M3 is geometric. This suggests the strategy: Start with an arbitrary metric g(0) on M3 and let it evolve via the normalized Ricci flow, hoping that it will converge to a stationary metric (that will have to be homogeneous). Hamilton showed that this strategy actually works if g(0) has positive curvature. However, it has been known since the early 1980s that in many cases 2-D spheres inside of M3 can cause the normalized Ricci flow to blow up (the curvature becoming infinite) in a finite amount of time.
An example of this phenomenon is known as the “dumbbell”: Take two copies of the round 3-D sphere and connect them by a thin neck. The neck will get pinched (under both the Ricci flow and the normalized Ricci flow) in finite time as the 2-D sphere in its cross section gets pinched into a point (and has curvature blowing up to infinity in the process) as t → T1 < ∞ and our “universe” splits in two (Fig. 4).
In this example the Ricci flow “finds” the 2-D spheres along which M3 has to decompose as a connected sum (before becoming geometric). The example also suggests that one should look for these spheres in the part, M(t)+, of M3 where the curvature of g(t) is “high.” Hamilton was hoping that one could prove that the regions of high curvature on M3(t) [and the corresponding singularities at the first blow-up time] are “nearly standard,” similar to the dumbbell example.
Understanding singularities of g at the first blow-up time T1 was the first among major obstacles in the way of Hamilton's program by the mid-1990s. Working in nearly total isolation and secrecy from the mid-1990s to 2002, Perelman introduced several new important tools and ideas and modified Hamilton's program; his work culminated in a sequence of three preprints in 2002–2003 proving the entire Thurston geometrization conjecture. Instead of looking at the Ricci flow as just an evolution of a metric on a 3-D manifold M3, he analyzed the geometry of the 4-D manifold N4 obtained from M3 by adding the extra dimension (the time t). This 4-D geometry is specifically designed to reflect properties of the Ricci flow. Using this geometry (in combination with the Alexandrov geometry of singular metric spaces), Perelman established that singularities of the Ricci flow are, indeed, standard after appropriate rescaling and taking a limit. This allowed him to prove, without identifying the geometry of singularities precisely, that the topological type of high-curvature regions M(t)+ in M(t) is standard, the main examples being: a cup (a 3-D ball), a spherical neck (the product of the 2-D sphere and the interval), or a spherical space form (a 3-D manifold that admits a metric of positive curvature). On the other hand, Perelman managed to understand the geometry of M(t)+ well enough to be able to cut it off from the rest of M3 along some spheres and attach nearly round spherical caps along these spheres (Fig. 5).
As a result, Perelman defined (realizing an earlier idea of Hamilton) the Ricci flow with surgeries, which evolves both the geometry and the topology of M3. He proved that the surgery times Ti are well-separated and the Ricci flow with surgeries thus exists for all times t ≥ 0. Perelman also proved that (under certain assumptions) the entire manifold becomes extinct in a finite time, the curvature becoming infinitely high everywhere. (This was also independently established by Tobias Colding and William Minicozzi.) This means that the original manifold was the connected sum of spherical space forms. In particular, this happens if M3 is simply connected or, more generally, has a finite fundamental group; this yields the Poincaré conjecture and the spherical space forms conjecture. Lastly, extending prior work of Hamilton, Perelman proved that at t = ∞ the manifold M(t) splits along incompressible tori into the pieces Mthick, where the metric becomes hyperbolic, and the pieces Mthin, where the metric collapses to metrics of dimension 1 or 2.
This work was described by Perelman in three preprints that he posted on the Internet in 2002–2003: “Entropy formula for the Ricci flow and its geometric applications,” “Ricci flow with surgery on 3-manifolds,” and “Finite extinction time for the solutions to the Ricci flow on certain three-manifolds.” In order to establish the full Thurston geometrization conjecture, one also needs to show that Mthin is a graph manifold. A proof of this proposition (originally promised by Perelman) was soon provided by Takashi Shioya and Takao Yamaguchi, and later on by other mathematicians.
So far, Perelman's papers have not been published in a peer-reviewed journal (and he does not seem to be interested in this), but only posted on the Mathematical Archive, which is a common place for mathematicians to post preliminary versions of their work. Many arguments in Perelman's proofs are rather sketchy and some important details are missing. Soon after the appearance of Perelman's preprints, several teams of mathematicians started to work toward filling in the missing details. Three detailed accounts of Perelman's proof of the Thurston geometrization conjecture have emerged:
1. In the summer of 2003, Bruce Kleiner and John Lott started to post missing details of Perelman's proof on the Internet; their work culminated in an article covering the entire proof.
2. John Morgan and Gang Tian published a book with a self-contained treatment of the Poincaré conjecture (and spherical space forms conjecture) part of the Thurston geometrization conjecture.
3. An account of the entire proof was published by Huai-Dong Cao and Xi-Ping Zhu and caused a great deal of controversy.
The proof of the Thurston geometrization conjecture solved a number of important open problems (for example, the Poincaré conjecture). The Thurston geometrization conjecture also makes it possible to reduce many open problems about 3-D manifolds to questions about hyperbolic manifolds, which then can be attacked using geometric, analytic, or algebraic methods. For example, using the Thurston geometrization conjecture, one can define an algorithm that will determine if two 3-D manifolds have the same topological shape or not. Such an algorithm is known to be impossible for manifolds of dimension equal to or greater than 4.
Relevance outside of mathematics
The Thurston geometrization conjecture deals with possible shapes of our universe. Determining the actual shape of our space is an important problem in cosmology, which is best understood in terms of the Thurston geometrization conjecture. For example, astronomers, following a suggestion of Jean-Pierre Luminet, have attempted to determine, by observing the cosmic background radiation, whether our universe has the shape of a certain space form (Poincaré dodecahedral space). See also: Cosmic background radiation; Cosmology; The shape of the universe; Topology