Chapter 20

3-Manifolds — Shape of Space

Now we come to where we live. We live in a physical 3-dimensional space, that is (at least) locally like Euclidean 3-space. The fundamental question which we will investigate in this chapter is: How can we tell what is the shape of our universe? This is a very difficult question for which there is currently (as I write this) no clear answer. However, there are several things we can say:

Space as an Oriented Geometric 3-Manifold

Presumably our physical universe is globally a geometric 3-manifold. [A geometric 3-manifold is a space in which each point in the universe has a neighborhood which is isometric with a neighborhood of either Euclidean 3-space, a 3-sphere, or a hyperbolic 3-space.] I say "globally is a geometric 3-manifold" in the same sense in which we say that globally Earth is a sphere (and spherical geometry is the appropriate geometry for intercontinental airplane flights) even though it is clear almost anywhere on the earth that locally there are many hills and valleys that make Earth not locally isometric to a sphere. However, the highest point on earth (Mount Everest) is 8.85 km above sea level and the lowest point on the floor of the ocean (the Mariana Trench) is 10.99 km below sea level the difference is about 0.3% of the 6368 km radius of Earth (variations in the radius are of the same magnitude).

It is known that locally our physical universe is definitely NOT a geometric 3-manifold. It was predicted by Einstein's general theory of relativity that the local curvature of our physical universe is affected by any mass (especially large masses like our sun and other stars). [Albert Einstein (1879-1955) was born in Germany, educated in Switzerland, and worked first in Zurich and then in Princeton, NJ, USA.] This effect is fairly accurate illustrated by imagining a 2-dimensional universe which is the surface of a flat rubber sheet. If you place steel balls on this rubber sheet the balls will locally make dents or dimples in the sheet and thus will locally distort the flat Euclidean geometry. Einstein's prediction has been confirmed in two ways:

1. The orbits of the planets Mercury, Jupiter, and Saturn are (quite accurately) an ellipse and the major axes of these elliptical orbits change directions (precesses). Classical Newtonian mechanics (based on Euclidean geometry) predicts that in a century the precession will be:

Mercury - 1.48°, Jupiter - 1.20°, Saturn - 0.77°,

measured in degrees of an arc.
Astronomers noticed that the observed amount of precession agreed accurately with these values for Saturn and Jupiter; however, for Mercury (the closest planet to the Sun) the observed precession is 1.60°, which is 0.12° more than is predicted by Newtonian/Euclidean methods. But, if one does the computations based on the curvature of space near the sun that is predicted by Einstein, then the calculations agree accurately with the observed precession. For a mathematician's description of this calculation (which uses formulas from differential geometry), see Frank Morgan's Riemannian Geometry: A Beginner's Guide [DG: Morgan], Chapter 7.

2. In 1919 British astronomers led by Arthur Eddington measured the angle subtended by two stars from Earth: once when the Sun was not near the path of the light from the stars to Earth and once when the path of light from one of the stars went very close to the Sun. See Figure 20.1. In order to be able to see the star when its light passes close to the Sun they had to make the second observation during a total eclipse of the Sun. They observed that the angle (a) measured with the Sun near was smaller than the angle (b) measured when the Sun was not near. The difference between the two measured angles was exactly what Einstein's theory predicted. Some accounts of this experiment talk about the Sun "bending" the light rays, but it is more accurate to say that light follows intrinsically straight paths (geodesics) and that the Sun distorts the geometry (curvature) of the space nearby. For more details of this experiment and other experiments that verified Einstein's theory, see http://www.ncsa.uiuc.edu/Cyberia/NumRel/EinsteinTest.html.

Figure 20.1. Observing Local Non-Euclidean Geometry in the Universe.

But, most of our physical universe is empty space with only scattered planets, stars, galaxies and the "dimples" that the stars make in space is only a very small local effect near the star. The vast empty space appears to locally (on a medium scale much larger than the scale of the distortions near stars) be a geometry the local symmetries of Euclidean 3-space. (In particular, our physical space is observed to be locally (on a medium scale) to be the same in all directions with isometric rotations, reflections, and translations as in Euclidean 3-space.

Theorem 20.0. Euclidean 3-space, 3-spheres, and hyperbolic 3-spaces are the only simply-connected (every loop can be continuously shrunk to a point in the space) 3-dimensional geometries that locally have the same symmetries as Euclidean 3-space.

The condition of simply connected is to rule out general geometric manifolds modeled on Euclidean 3-space, 3-spheres, or hyperbolic 3-spaces. There is a discussion of the proof (and a more precise statement) of this theorem in [DG: Thurston], Section 3.8, where Thurston discusses eight possible 3-dimensional simply connected geometries, but only three have the same symmetries as Euclidean 3-space. A more elementary discussion without proof can be found in [DG: Weeks], Chapter 18.

Unfortunately (or maybe fortunately!), geometric 3-manifolds are not fully understood. At this point no one knows what all of the geometric 3-manifolds or how to distinguish one from the other. The theory of 3-manifolds is an area of current active research. For a very accessible discussion of this research see [DG: Weeks], Part III. For detailed discussions of this research see Thurston's Three-Dimensional Geometry and Topology, Volume 1, [DG: Thurston] and the second (and further ?) volumes as soon as they appear.

Problem 20.1. Is Our Universe Non-Euclidean?

The nineteenth century mathematician Carl Friedrich Gauss (1777-1855) is said to have tried to measure the angles of a triangle whose vertices were three mountain peaks in Germany. Gauss was born in Brunswick (Germany) and was educated and later became a professor at the University in Göttingen. If the sum of the angles had turned out to be other than 180°, then he would have surmised that the universe is not Euclidean (or that light does not travel in straight lines). However, his measurements were inconclusive because he measured the angles at 180° within the accuracy of his measuring instruments.

a. Could we now show that the universe is non-Euclidean by measuring the angles of a large triangle in our solar system? How accurately would we have to measure the angles?

[Hint: Note that, if the universe is a 3-sphere or a hyperbolic 3-space, the radius R of the universe would have to be at least as large as the diameter of our galaxy, which is about 1018 km. In the foreseeable future, the largest triangle whose angles we could measure has area less than the area of our solar system, which is about 8 ´ 1019 km2. Use the formulas you found in Problems 7.1 and 7.2 and extended in Problem 12.6.]

So, in order to determine the geometry of space we have to look further than our solar system.

b. If the stars were distributed uniformly in space (it is not clear to what extent this is actually true), how could you tell by looking at stars at different distances whether space was locally Euclidean, spherical, or hyperbolic?

[Hint: If you have trouble envisioning this then start with the analogous problem for a 2-dimensional bug on a plane, sphere, or hyperbolic plane. What would this bug observe? Assume that you can tell how far away each star is — this is something that astronomers know how to do.]

c. Suppose you know that certain types of stars (or galaxies) have a fixed know amount of brightness (astronomers call such stars or galaxies, standard candles), and you can see several of these standard candles at various distances from Earth. How could you tell whether the universe is Euclidean, spherical, or hyperbolic?

[Hint: The apparent brightness of a shining object in Euclidean space is inversely proportional to the square of the distance to the object.]

d. It is impractical to measure the excess of triangles in our solar system by only taking measurements of angles within our solar system. What observations would tell us that the universe is not Euclidean 3-space by also observing distant stars and galaxies?

[Hint: If you have trouble conceptualizing a 3-sphere or hyperbolic 3-space, then you can do this problem, first, for a very small bug on a 2-sphere or hyperbolic plane who can see distant points (stars), but who is restricted to staying inside its "solar system" which is so small that any triangle in it has excess (or defect) too small to measure. Always think intrinsically! You can assume, generally, that light will travel along geodesics, so think about looking at various objects and the relationships you would expect to find. For example, if the universe were a 3-sphere and you could see all the way around the universe (the distance of a great circle), how would you know that the universe is spherical. Why? What if we could see half way around the universe? Or a quarter of the way around? Think of looking at stars at these distances.]

All the ways discussed above have been tried by astronomers, but, up to now (1999), none of the observations have been accurate enough to make a definite determination. However, there is a satellite scheduled for a year 2000 launch (and another more accurate one scheduled for a launch in 2007) that are designed to make an accurate map of the microwave background radiation. An analysis of this map may provide the clues we need to definitely determine the global geometry of space. To describe how this will work we must first investigate geometric 3-manifolds.

Problem 20.2. Euclidean 3-Manifolds

We now consider the 3-dimensional analogue of the flat torus. Consider a cube in Euclidean 3-space with the opposite faces glued through a reflection in the plane that is midway between the opposite faces. See Figure 20.2. In this figure, I have drawn a closed straight path which starts from A on the bottom right edge and then hits the middle of the front face at B. It continues from the middle of the back face and finishes at the middle of the top left edge at a point which is glued to A.

Figure 20.2. A closed geodesic path on the 3-torus.

a. Show that the cube with opposite faces glued by a reflection through the plane midway between is a Euclidean 3-manifold. That is, check that a neighborhood of each point is isometric to a neighborhood in Euclidean 3-space. This Euclidean 3-manifold is call the 3-torus.

[Hint: Look separately at points (such as A) that are in the middle of edges, points (such as B) which are the middle of faces, and points which are vertices (such as C).]

Next glue the vertical faces of the cube the same way but glue the top and bottom faces by a quarter turn. In this case we get the closed geodesic depicted in Figure 20.3.

Figure 20.3. A closed geodesic in the quarter-turn manifold.

b. Show that you obtain a Euclidean 3-manifold from the cube with vertically opposite faces glued by a reflection through the plane midway between and the top and bottom faces glued by a quarter turn rotation. This Euclidean 3-manifold is called the quarter turn manifold.

c. Draw a picture similar to Figures 20.2 and 20.3, for the half turn manifold, which is the same as the quarter turn manifold except that it is obtained by gluing the top and bottom faces with a half turn.

In Problem 11.2 we represented the flat torus in 2 different ways — one starting with a rectangle or square and the other starting from a hexagon. The above discussion of the 3-torus corresponds to the construction of the flat torus from a square. Now we want to look at what happens if we use an analogue of the hexagon construction.

Consider a hexagonal prism as in Figure 20.4. We will make gluings on the vertical sides by gluing each vertical face with its opposite in such a way that each horizontal cross-section (which are all hexagons) has the same gluings as the hexagonal flat torus (see Problem 11.2b and Figure 11.8). The top and bottom face we glue in one of three ways. If we glue the top and bottom face through a reflection in the halfway plane then we obtain the hexagonal 3-torus. If we glue the top and bottom faces with a one-sixth rotation, then we obtain the one-sixth turn manifold. If we glue the top and bottom with a one-third rotation then we will get the one-third turn manifold.

Figure 20.4. Hexagonal 3-manifolds.

d. Show that the hexagonal 3-torus, the one-sixth turn manifold, and the one-third turn manifold are Euclidean 3-manifolds and that the hexagonal 3-torus is homeomorphic to the 3-torus. What happens if we consider the two-thirds turn manifold and the three-sixths turn manifold and the five-sixth turn manifold?

It can be shown that:

Theorem 20.2. There are exactly 10 Euclidean 3-manifolds up to homeomorphism. Of these 4 are non-orientable and 6 are orientable. Five of the six orientable Euclidean manifolds are the 3-torus, the quarter turn manifold, the half turn manifold, the one-sixth turn manifold, and the one-third turn manifold.

See [DG: Weeks], page 252, for a discussion of this theorem. For more detail and a proof see [DG: Thurston], Section 4.3.

Problem 20.3. Dodecahedral 3-Manifolds

Spherical and hyperbolic 3-manifolds are more complicated than Euclidean 3-manifolds. In fact, no one knows what all the hyperbolic 3-manifolds are. We will only look at a few examples in order to get an idea of how to construct spherical and hyperbolic 3-manifolds in this problem and the next.

There are two examples can be obtained by making gluings of the faces of a dodecahedron (see Problem 19.5). It will be best for this problem for you to have a model of the dodecahedron that you can look at and touch. We want to glue the opposite faces of the dodecahedron. Looking at your dodecahedron (or Figure 20.5) you should see that the opposite faces are not lined up but rather are rotated one-tenth of a full turn from each other. Thus there are three possibilities for gluings: We can glue with a one-tenth rotation, or a three-tenths rotation, or a five-tenths (= one-half) rotation. When making the rotations it is important to always rotate in the same direction (say clockwise as you are facing the face from the outside). You should check with your model that this is the same as rotating clockwise while facing the opposite face.

Figure 20.5. Dodecahedron.

a. When you glue the opposite faces of a dodecahedron with a one-tenth clockwise rotation, how many edges are glued together? What if you use a three-tenths rotation? Or a one-half rotation?

[Hint: I find the best way to do this counting is to take my model and mark one edge with tape. Then for each of the two pentagon faces on which the edges lies the gluing glues the edge to another edge on the other side of the dodecahedron — mark those edges also. Continue with those marked edges until you have marked all the edges that are glued together.]

Your answers to Part a should be 2, 3, 5 (NOT in that order!). Thus in order for these manifolds to be geometric manifolds we must use dodecahedrons with dihedral angles of 180°, 120°, and 72° in either Euclidean space, a sphere, or a hyperbolic 3-space. But before we go further we must figure out what is the size of the dihedral angles of the dodecahedron in Euclidean space, which from our model seems to be close to (if not equal to) 120°.

b. Calculate the size f of the dihedral angle of the (regular) dodecahedron in Euclidean 3-space.

[Hint: Imagine a small sphere with center at one of the vertices of the dodecahedron. This sphere will intersect the dodecahedron in a spherical equilateral triangle. This triangle is called the link of the vertex in the dodecahedron. Determine the lengths of the sides of this triangle and then use the Law of Cosines (Problem 17.2).]

Now imagine a very small dodecahedron in 3-sphere. Its dihedral angles will be very close to the Euclidean angle f (Why is this the case?). If you now imagine the dodecahedron growing in the 3-sphere, its dihedral angles will grow from f. If you start with a very small dodecahedron in a hyperbolic space then its dihedral angles will start very close to f and then decrease as the dodecahedron grows.

c. Show that the manifold from Part a with three edges being glued together is a spherical 3-manifold if f < 120°, or a Euclidean 3-manifold if f = 120°, or a hyperbolic 3-manifold if f > 120°. This geometric 3-manifold is called the Poincaré dodecahedral space in honor of Henri Poincaré (1854-1912, French) who first described (not using the dodecahedron) a space homeomorphic to this geometric 3-manifold.

[Hint: Show that each vertex of the dodecahedron is glued to three other vertices and that the four solid angles fit together to form a complete solid angle in the model (either Euclidean 3-space, 3-sphere, or hyperbolic 3-space).]

d. Can the dihedral angles of a dodecahedron in a 3-sphere grow enough to be 180°? What does such a dodecahedron look like? Is the manifold with two edges being glued together a spherical 3-manifold? This spherical 3-manifold is called the projective 3-space or RP3.

e. Can the dihedral angles of a dodecahedron in a hyperbolic 3-space shrink enough to be equal to 72°? If so, the dodecahedral manifold with five edges being glued together is a hyperbolic 3-manifold. This hyperbolic 3-manifold is called the Seifert-Weber dodecahedral space, after H. Seifert and C. Weber who first described both dodecahedral spaces in a 1933 article1.

[Hint: Imagine that the dodecahedron grows until its vertices are at infinity (thus on the bounding plane in the upper half space model). Use the fact that angles are preserved in the upper half space model and look at the three great hemispheres that are determined by the three faces coming together at a vertex. Remember to also check that the solid angles at the vertices of the dodecahedron fit together to form a complete solid angle.]

Problem 20.4. Some Other Geometric 3-Manifolds

We now look at three more examples of geometric 3-manifolds.

a. Start with a tetrahedron and glue the faces as indicated in Figure 20.6. Does this gluing produce a manifold? Can the tetrahedron be put in a 3-sphere or hyperbolic 3-space so that the gluings produce a geometric 3-manifold? We call this the tetrahedral space

[Hint: Investigate how many edges are glued together and what happens near the vertices.]

Figure 20.6. Tetrahedral space.

b. Start with a cube and glue each face to the opposite face with a one-quarter turn rotation. Does this gluing produce a manifold? Can the cube be put in a 3-sphere or hyperbolic 3-space so that the gluings produce a geometric 3-manifold? This is called the quaternionic manifold because its symmetries can be expressed in the quaternions (a four dimensional version of the complex numbers with three imaginary axes and one real axis).

[Hint: Again, investigate how many edges are glued together and what happens near the vertices.]

c. Start with an octahedron and glue each face to the opposite face with a one-sixth turn rotation. Does this gluing produce a manifold? Can the octahedron be put in a 3-sphere or hyperbolic 3-space so that the gluings produce a geometric 3-manifold? This is called the octahedral space.

[Hint: Again, investigate how many edges are glued together and what happens near the vertices and use your knowledge of solid angles from Chapter 19.]

Astronomers from earth bound observatories have noticed a radiation that is remarkably uniform coming to the Earth from all directions of space. In 1991, the USA's Cosmic Background Explorer (COBE) mapped large portions of this radiation to a resolution of about 10 degrees of arc. COBE determined that the radiation is uniform to nearly one part in 100,000, but there are slight variations (or texture) observed. It is this texture that gives us the possibility to determine the global shape of the universe. To understand how this determination may be possible we must first understand from (and when!) this background radiation came.

The generally accepted explanation of the cosmic background radiation is that in the early stages of the universe matter was so dense that no radiation could escape (the space was filled with matter so dense that all radiation was scattered). But at a certain point in time (about 300,000 years after the big bang and about 13 billion years ago) matter in the universe started coalescing and the density of matter decreased enough that radiation could start traveling though the universe in all directions but still dense enough that the radiating matter (radiating because it was hot) was fairly homogeneously distributed throughout the universe. It is this first escaped radiation that we see when we look at the cosmic background radiation.

Remember that all radiation (light and others) travels at the speed of light (~1013 kilometers per year) and thus the cosmic radiation that reaches us today has been traveling for about 13 billion years and has traveled about 1.3 ´ 1022 km. Thus the cosmic background radiation gives us a picture of a sphere that was a slice of the early universe roughly 13 billion years ago, the cosmologists call this the last scattering surface. It appears to us that the Earth is at the center of this sphere, but this is talking in space-time. The physical center of this sphere is the point in the early universe where 13 billion years later the milky way galaxy, solar system, and the Earth would form.

The cosmologists say that the above discussion does not involve any assumptions about the density of matter or the presence of a "cosmological constant", both of which are hotly debated subjects among cosmologists. In addition, the only assumption made about the geometry of our physical universe is the assumption that our physical universe is a geometric manifold and that, even though it is expanding, the topology is constant (or at least has been constant since the universe was big enough for the radiation to escape). In Theorem 20.5, below, we see that in fact if we know the topology we also know the geometry

Before going on with the 3-dimensional discussion let us look at an analogous situation in 2-dimensions. Imagine that the 2-dimensional bug's universe is a flat torus obtained from a square with opposite side glued. We may assume that the bug is at the center of the square looking out in all directions at a textured circle from the center of that circle, the last scattering circle for the bug. If this circle has diameter larger than the side of the square then the circle will intersect itself as indicated in Figure 20.7.

Figure 20.7. Seeing the 2-dimensional scattering circle.

Note, in Figure 20.7, that when the bug in the center looks toward the point A the bug will not see A but rather will see C. (The light from A will have reached the center of the square earlier!) Likewise, the bug will see the point D on the circle but not the point B. The important points to focus on are the points, E, F, G, H, where the circle intersects itself. The bug will see these points in two different directions. See Figure 20.7 where the point E is seen from both sides. If the texture is unique enough the bug should be able to tell that it is looking at the same point in two different directions. The pattern of these identical point pairs will indicate to the bug that its universe is the flat torus. (See Problem 20.5a.)

The three-dimensional situation is similar. Consider that our physical universe is a 3-torus which is the result of gluings on a cube as in Figure 20.2. If the earth is considered to be in the center of this cube and if the sphere of last scatter has reached the faces of the cube then the intersection of the sphere of last scatter with the faces of the cube will be circles and (because of the gluings) the circle on one face will be identified (by a reflection) with the circle on the opposite face. See Figure 20.8. Note that the pattern of circles shows the underlying cube and the gluings. In Problem 20.5b you will explore what the pattern of circles will look like in the other geometric 3-manifolds which we have discussed.

Figure 20.8. Self-intersections of the sphere of last scatter in the 3-torus.

In our physical universe, there are two space probes that are scheduled to be launched to study the texture of the microwave background radiation. In 2000 (about the time that this book is published) the USA's National Aeronautic and Space Administration plans to launch the Microwave Anisotropy Probe (MAP). See

http://map.gsfc.nasa.gov/

for more information. In 2007, the European Space Agency is scheduled to launch the Planck satellite. See

http://astro.estec.esa.nl/SA-general/Projects/Planck/planck.html

for more information.) These probes will produce detailed maps of the texture of the microwave radiation. MAP should produce 0.3° resolution and the Planck satellite is hoped to provide about 0.1° resolution. Remember that the current maps have only 10° resolution.

For further discussion of these measurements see Luminet, Starkman, and Weeks, "Is Space Finite?", Scientific American, April 1999, page 90; or Cornish & Weeks, "Measuring the Shape of the Universe", Notices of the American Mathematical Society, 1998, or the look for more recent articles that have been published since this book was written (May, 1999).

Problem 20.5. Circle Patterns Show the Shape of Space

*a. For each of the geometric 2-manifolds in Problems 11.1, 11.2, 11.4, 11.5, what would be the patterns of matching point pairs look like if the bug's last scattering circle was large enough to intersect itself?

[Hint: Draw pictures analogous to Figure 20.7.]

b. For each of the geometric 3-manifolds in Problems 20.2, 20.3, and 20.4, what would the pattern of matching circles look like if our physical universe were the shape of that manifold and if the sphere of first scatter reaches far enough around the universe for it to intersect itself.

[Hint: Draw and examine, as best you can, pictures analogous to Figure 20.8.]

We saw in Problem 11.6 that the area of a spherical or hyperbolic 2-manifold is determined by its topology and the radius (or curvature) of the model. Similarly, (but much more complicatedly) we have for spherical and hyperbolic 3-manifolds:

Theorem 20.5. If two orientable spherical or hyperbolic 3-manifold are homeomorphic then they are geometrically similar. That is, two such homeomorphic manifolds are isometric if the model spherical or hyperbolic space have the same radius (curvature).

Thus, if we are successful in finding a pattern of circles that determines the universe is a spherical or hyperbolic 3-manifold, then we will know what is the volume of our physical universe.

Watch the news media and science and mathematics journals for news as analysis of the data from Microwave Anisotropy Probe and Planck satellite start coming in (probably in late 2001). As they come in and I learn about them, I will post updates of these probes and the analyses of the data that are relevant to this text at the URL:

http://math.cornell.edu/~dwh/books/eg99/title.html

1 "Die beiden Dodekaederräume," Mathematische Zeitschrift, Vol. 37 (1933), no. 2, p. 237.