Geometry with an Introduction to Cosmic Topology

Example 8.1.2. The 3-torus.

We may think of the 3-torus \(\mathbb{T}^3\) as a cubical room with the following face (or wall) identifications: if we propel ourselves (with a jet pack) up through the ceiling, we reappear through the floor directly below where we had been. For instance, the elliptical points in the following figure are identified. Each point on the top face is identified with the corresponding point on the bottom face. If we fly through the left wall we reappear through the right wall at the corresponding spot (see the triangles in the figure). If we fly through the front wall we reappear through the back wall at the corresponding spot (see the squares in the figure).

The 3-torus is the three-dimensional analogue of the torus video screen we saw on the first pages of this text. In fact, a slice of the 3-torus in a plane parallel to one of the faces is a torus. One checks that under the face identification, all eight corners come together in a single point. The angles at these corners are such that they form a perfect patch of three-dimensional space. We do not need to inflate the corner angles (in elliptic space) or shrink the angles (in hyperbolic space) in order for the corners to come together perfectly.

We can also view the 3-torus as an orbit space as discussed in Section 7.7. In particular, let \(T_x, T_y, T_z\) represent translations of \(\mathbb{R}^3\) by one unit in the \(x\)-direction, \(y\)-direction, and \(z\)-direction, respectively. That is, \(T_x(x,y,z) = (x+1,y,z)\text{,}\) \(T_y(x,y,z) = (x,y+1,z)\) and \(T_z(x,y,z) = (x,y,z+1)\text{.}\) These three transformations generate a group of transformations \(\Gamma\) of \(\mathbb{R}^3\text{,}\) and any transformation in the group has the form \(T(x,y,z) = (x + a, y + b, z + c)\) where \(a, b, c\) are integers. A fundamental domain of the quotient space \(\mathbb{R}^3/\Gamma\) is the unit cube

and all the images of this cube under the transformations in \(\Gamma\) tile \(\mathbb{R}^3\text{.}\) The 3-torus is a quotient of \(\mathbb{R}^3\) generated by Euclidean isometries, and it inherits Euclidean geometry.

Example 8.1.3. The 3-sphere.

Recall, the 3-sphere consists of all points in 4-dimensional space 1 unit from the origin:

One model of \(\mathbb{S}^3\) consists of two solid balls in \(\mathbb{R}^3\) whose boundary 2-spheres are identified point for point, as in the following figure. The figure shows a path from point \(a\) to point \(b\) in \(\mathbb{S}^3\) that goes through the point \(p\) on the boundary of the left-hand solid ball before entering the right-hand solid ball through the point on its boundary with which \(p\) has been identified (also called \(p\)).

The 3-sphere has interesting features, often accessible by analogy with the 2-sphere. If you slice the 2-sphere with a plane in \(\mathbb{R}^3\text{,}\) the intersection is a circle (or a single point if the plane is tangent to the 2-sphere). Moreover, the circle of intersection is a geodesic on the 2-sphere if it is a great circle: a circle of maximum radius drawn on the 2-sphere. Bumped up a dimension, if you “slice” the 3-sphere with \(\mathbb{R}^3\text{,}\) the intersection is a 2-sphere (or a single point if the slice is tangent to \(\mathbb{S}^3\)). Moreover, the 2-sphere of intersection is a great 2-sphere if it is a 2-sphere of maximum diameter drawn in \(\mathbb{S}^3\text{.}\) The boundary 2-sphere common to both solid balls in the figure is a great 2-sphere of \(\mathbb{S}^3\text{.}\)

One may also view \(\mathbb{S}^3\) by stereographically projecting it into the space obtained from \(\mathbb{R}^3\) by adding a point at \(\infty\text{,}\) much as we identified \(\mathbb{S}^2\) with the extended plane via stereographic projection in Section 3.3. Let \(\widehat{\mathbb{R}^3}\) denote real 3-space with a point at infinity attached to it. To construct the stereographic projection map, it is convenient to first express \(\mathbb{R}^3\) in terms of pure quaternions. A pure quaternion is a quaternion \(q\) whose scalar term is 0. That is, a pure quaternion has the form \(q = b\textbf{i} + c\textbf{j} + d\textbf{k}\text{.}\) The point \((x,y,z)\) in \(\mathbb{R}^3\) is identified with the pure quaternion \(q = x\textbf{i} + y\textbf{j} + z \textbf{k}.\)

Though we are working in four-dimensional space, Figure 3.3.4 serves as a guide in the construction of the stereographic projection map. Let \(N = (0,0,0,1)\) be the north pole on \(\mathbb{S}^3\text{.}\) If \(P = (a,b,c,d)\) is any point on \(\mathbb{S}^3\) other than \(N\text{,}\) construct the Euclidean line through \(N\) and \(P\text{.}\) This line has parametric form

We define the image of \(P\) under stereographic projection to be the intersection of this line with the three-dimensional subspace consisting of all points \((x,y,z,0)\) in \(\mathbb{R}^4\text{.}\) This intersection occurs when \(1 + t(d-1) = 0\text{,}\) or when

Then the stereographic projection map \(\phi:\mathbb{S}^3 \to \widehat{\mathbb{R}^3}\) is given by

In this way, we can transfer the geometry of the 3-sphere into real 3-space with a point at infinity attached to it. For instance, if \(P = (a,b,c,d)\) is on the 3-sphere, its antipodal point is \(-P = (-a,-b,-c,-d)\) and one can show (in the exercises) that these points get mapped by \(\phi\) to points in \(\mathbb{R}^3\) (pure quaternions) \(q = \phi(P)\) and \(u = \phi(-P)\) with the property that \(q \cdot u^* = -1\text{.}\) As a result, two points \(q\) and \(u\) in \(\widehat{\mathbb{R}^3}\) are called antipodal points if they satisfy the equation \(q \cdot u^* = -1\text{.}\)

Example 8.1.4. The Poincaré dodecahedral space.

The Poincaré dodecahedral space has been given consideration as a model for the shape of our universe (see [25]). Start with a dodecahedron living in \(\mathbb{R}^3\) (we're thinking of it as a solid now, not a surface) and identify opposite faces with a one-tenth clockwise twist (rotation by \(2\pi/10\) radians). It is cumbersome to indicate all face identifications, so the figure below shows the identification of the front face with the back face. For instance, the two corners labeled with a 1 get matched.Notice: no points in the interior of the dodecahedron get identified.

With this face identification the twenty corners of the dodecahedron come together in five groups of four (one group of 4 is marked with vertices in the figure), and the angles are too small for the corners to create a full open ball of three-dimensional space when they come together. Placing the dodecahedron in elliptic space inflates the corner angles, and we may make these angles fat enough to determine a perfect three-dimensional patch of space.

Extending the ideas of Section 7.7 to this case, one can show that the Poincaré dodecahedral space space is a quotient of the 3-sphere \(\mathbb{S}^3\) by a group of isometries. The dodecahedron pictured here is a fundamental domain of the space. Just as the polygons in Section 7.7 tiled the space in which they lived, and the cubical fundamental domain of the 3-torus tiles \(\mathbb{R}^3\text{,}\) the dodecahedron tiles \(\mathbb{S}^3\text{.}\) Although far from obvious, it turns out that \(\mathbb{S}^3\) is tiled by 120 copies of this dodecahedron!

Example 8.1.5. The Seifert-Weber space.

Identify opposite faces of the dodecahedron with a three-tenths clockwise turn. With this identification, all twenty corners of the dodecahedron come together to a single point, and no such identification is possible unless we drastically shrink the corner angles. We may do this by placing the dodecahedron in hyperbolic space. The resulting hyperbolic 3-manifold is called the Seifert-Weber space.

Example 8.1.6. Lens spaces.

The lens spaces \(L(p,q)\) form an infinite family of elliptic 3-manifolds. Assume \(p > q\) are positive integers whose largest common factor is 1. The lens spaces may be defined as follows. Consider the unit solid ball living in \(\mathbb{R}^3\) whose boundary is the unit 2-sphere. The ball consists of all points \((x,y,z)\) such that \(x^2 + y^2 + z^2 \leq 1\text{.}\) Any point on the boundary 2-sphere can be expressed in coordinates \((z,t)\) where \(z\) is a complex number, \(t\) is real, and \(|z|^2 + t^2 = 1\text{.}\) The lens space \(L(p,q)\) is obtained from the solid ball by identifying each point \((z,t)\) on the boundary 2-sphere with the point \((e^{(2\pi q/p)i} z,-t)\text{.}\) Note that the north pole of the boundary 2-sphere is \((0,1)\) and it is identified with the south pole \((0,-1)\text{.}\) Any other point \(u\) in the northern hemisphere is identified with a single point in the southern hemisphere. This point is found by reflecting \(u\) across the \(xy\)-plane, then rotating around the \(z\)-axis by \(2\pi q/p\) radians. Each point on the equator is identified with \(p-1\) other points.

The lens space \(L(p,q)\) can be obtained from a cell division of the unit solid ball. The cell division has \(p+2\) vertices: the north pole \(n\text{,}\) the south pole \(s\text{,}\) and \(p\) equally spaced vertices along the equator, label them \(v_0, v_1, \cdots, v_{p-1}\text{.}\) The following figure depicts this scene for \(p = 5\text{.}\) There is an edge connecting \(v_i\) to each of its neighbors on the equator, creating \(p\) edges around the equator. There is an edge connecting each \(v_i\) with \(n\) and an edge connecting each \(v_i\) with \(s\text{,}\) as pictured, for a total of \(3p\) edges. The vertices and edges create 2\(p\) triangular faces on the boundary sphere, half of them in the northern hemisphere. The 3-complex has a single 3-cell, corresponding to the interior of the solid ball.

Let \(N_i\) denote the face in the northern hemisphere whose vertices are \(v_i, v_{i+1}, n\text{.}\) Let \(S_i\) be the face in the southern hemisphere whose vertices are \(v_i, v_{i+1}\text{,}\) and \(s\text{.}\) The lens space \(L(p,q)\) is created by identifying face \(N_i\) with face \(S_{i+q}\) where the sum \(i + q\) is taken modulo \(p\text{.}\) For instance, in \(L(5,2)\text{,}\) the shaded face \(N_0\) in the figure is identified with the shaded face \(S_2\) (and the circle of points pictured in \(N_0\) gets sent to the circle of points in \(S_2\)). The faces \(N_1\) and \(S_3\) are identified, as are \(N_2\) with \(S_4\text{,}\) \(N_3\) with \(S_0\) and \(N_4\) with \(S_1\text{.}\)

Example 8.1.7. The Hantschze-Wendt manifold.

This interesting manifold can be constructed from two identically sized cubes that share a face, as in the following figure. The resulting box has 10 faces to be identified in pairs. A three-dimensional bug with a jet pack living in the manifold would experience the following face identifications. The top and bottom faces are identified just as they were in the 3-torus. Each point on the top back face (see the circle in the figure) is identified with a single point on the bottom back face. This point is the reflection of the given point across the vertical segment bisecting the face. The top front face and the bottom front face are identified the same way. The figure shows two square points on these faces that are identified. The top left face and bottom right face are identified with a \(180^\circ\) rotation (see the elliptical point and its image point), as are the bottom left face and the top right face (the two triangular points are identified). This 3-manifold also inherits Euclidean geometry.

Example 8.1.9. Four more Euclidean 3-manifolds.

Two orientable 3-manifolds arise from slight changes to the 3-torus construction of Example 8.1.2. The quarter turn manifold results when the front and back faces of a cube are identified with a \(90^\circ\) rotation, while the remaining 2 pairs of opposite faces are identified directly, as they were for the 3-torus. Figure 8.1.10(a) depicts this identification. The F on the front face is identified with an F on the back face by a one-quarter turn.

The half turn manifold results when the front and back faces of a cube are identified with a \(180^\circ\) rotation, while the remaining 2 pairs of opposite faces are identified directly, as they were for the 3-torus. See Figure Figure 8.1.10(b).

The remaining two connected, compact, orientable Euclidean 3-manifolds have a hexagonal prism as the fundamental domain. A hexagon can be used to tile the plane, and copies of a hexagonal prism as in Figure 8.1.10(c) can tile \(\mathbb{R}^3\text{.}\) Note that the prism has two hexagonal faces and the other six faces are parallelograms. The one-third turn manifold is obtained as follows: directly identify parallelograms that are opposite one another, and identify the two hexagonal faces with a rotation of \(120^\circ\text{,}\) as in Figure 8.1.10(c). The one-sixth turn manifold is obtained by identifying the parallelograms as before, but the two hexagonal faces by a rotation of just \(60^\circ\text{,}\) as in Figure 8.1.10(d). More detailed discussions of these and the non-orientable Euclidean 3-manifolds can be found in [12] and [16].