Geometry with an Introduction to Cosmic Topology

Subsection Orbit Spaces

We may construct a natural quotient set from a geometry \((X,G)\text{.}\)

The group structure of \(G\) defines an equivalence relation \(\sim_G\) on \(X\) as follows: For \(x, y \in X\text{,}\) let

Indeed, for each \(x \in X\text{,}\) \(x \sim_G x\) because the group \(G\) must contain the identity transformation, so the relation is reflexive. Next, if \(x \sim_G y\) then \(T(x) = y\) for some \(T\) in \(G\text{.}\) But the group contains inverses, so \(T^{-1}\) is in \(G\) and \(T^{-1}(y) = x\text{.}\) Thus \(y \sim_G x\text{,}\) and so \(\sim_G\) is symmetric. Third, transitivity of the relation follows from the fact that the composition of two maps in \(G\) is again in \(G\text{.}\)

Given geometry \((X,G)\) we let \(X/G\) denote the quotient set determined by the equivalence relation \(\sim_G\text{.}\) In this setting we call the equivalence class of a point \(x\) in \(X\text{,}\) the orbit of \(x\text{.}\) So, the orbit of \(x\) consists of all points in the space \(X\) to which \(x\) can be mapped under transformations of the group \(G\text{:}\)

Put another way, the orbit of \(x\) is the set of points in \(X\) congruent to \(x\) in the geometry \((X,G)\text{.}\)

Note that if the geometry \(G\) is homogeneous, then any two points in \(X\) are congruent and, for any \(x \in X\text{,}\) the orbit of \(x\) is all of \(X\text{.}\) In this case the quotient set \(X/G\) consists of a single point, which is not so interesting. We typically want to consider orbit spaces \(X/G\) in which \(G\) is a “small” group of transformations.

We say that a group of transformations \(G\) of \(X\) is a group of homeomorphisms of \(X\) if each transformation in \(G\) is continuous. In this case, we call \(X/G\) an orbit space. If \(X\) has a metric, we say that a group of transformations of \(X\) is a group of isometries if each transformation of the group preserves distance between points.

Example 7.7.6. Building a topological cylinder.

Consider the horizontal translation \(T_1(z) = z + 1\) of \(\mathbb{C}\text{.}\) This transformation is a (Euclidean) isometry of \(\mathbb{C}\) and it generates a group of isometries of \(\mathbb{C}\) as follows. Put \(T_1\) and \(T_1^{-1}\) in the group, along with any number of compositions of these transformations. Fortunately, any number of compositions of these two maps results in an isometry that is easy to write down. Any finite composition of copies of \(T_1\) and \(T_1^{-1}\) indicates a series of instructions for a point \(z\text{:}\) at each step in the long composition \(z\) moves either one unit to the left if we apply \(T_1^{-1}\) or one unit to the right if we apply \(T_1\text{.}\) In the end, \(z\) has moved horizontally by some integer amount. That is, any such composition can be written as \(T_n(z) = z + n\) for some integer \(n\text{.}\) We let \(\langle T_1 \rangle\) denote the group generated by \(T_1\text{,}\) and we have

\begin{equation*} \langle T_1 \rangle = \{T_n(z) = z + n ~|~ n \in \mathbb{Z}\}\text{.} \end{equation*}

The orbit of a point \(p\) under this group of isometries is

\begin{equation*} [p] = \{p + n ~|~ n \in \mathbb{Z}\}\text{.} \end{equation*}

A fundamental domain for the orbit space \(\mathbb{C}/\langle T_1 \rangle\) is the vertical strip consisting of all points \(z\) with \(0 \leq \text{Re}(z) \leq 1\text{,}\) as in the following figure. Every point in \(\mathbb{C}\) is related to a point in this shaded vertical strip. Furthermore, no two points in the interior of the strip are related. By passing to the quotient, we are essentially “rolling” up the plane in to an infinitely tall cylinder. The rolling up process is described by the map \(p:\mathbb{C} \to \mathbb{C}/\langle T_1 \rangle \) given by \(p(z) = [z]\text{.}\)

It turns out that every surface can be viewed as a quotient space of the form \(M/G\text{,}\) where \(M\) is either the Euclidean plane \(\mathbb{C}\text{,}\) the hyperbolic plane \(\mathbb{D}\text{,}\) or the sphere \(\mathbb{S}^2\text{,}\) and \(G\) is a group of isometries in Euclidean geometry, hyperbolic geometry, or elliptic geometry, respectively. In topology terminology, the space \(M\) is called a universal covering space of the orbit space \(M/G\text{.}\)

Example 7.7.8. \(H_1\) as quotient of \(\mathbb{C}\).

Suppose \(a\) and \(b\) are positive real numbers. Let \(\langle T_a, T_{bi}\rangle\) be the group of homeomorphisms generated by the horizontal translation \(T_a(z) = z + a\) and the vertical translation \(T_{bi}(z) = z + bi\text{.}\)

This group contains all possible compositions of these two transformations and their inverses. Thus, the orbit of a point \(z\) consists of all complex numbers to which \(z\) can be sent by moving \(z\) horizontally by some integer multiple of \(a\) units, and vertically by some integer multiple of \(b\) units. An arbitrary transformation in \(\Gamma = \langle T_a, T_{bi}\rangle\) has the form

\begin{equation*} T(z) = z + (ma + nbi) \end{equation*}

where \(m\) and \(n\) are integers. A fundamental domain for the orbit space consists of the rectangle with corners \(0, a, a + bi, bi\text{.}\) The resulting quotient space is homeomorphic to the torus. Notice that points on the boundary of this rectangle are identified in pairs. In fact, the fundamental domain, with its boundary point redundancies, corresponds precisely to our polygonal surface representation of the torus.

If the space \(M\) has a metric and our group of homeomorphisms is sufficiently nice, then the resulting orbit space inherits a metric from the universal covering space \(M\text{.}\) To be sufficiently nice, we first need our homeomorphisms to be isometries. The group of isometries must also be fixed-point free and properly discontinuous. The group \(G\) is fixed-point free if each isometry in \(G\) (other than the identity map) has no fixed points. The group \(G\) is properly discontinuous if every \(x\) in \(X\) has an open 2-ball \(U_x\) about it whose images under all isometries in \(G\) are pairwise disjoint. The interested reader is encouraged to see [10] or [9] for more detail. If our group \(G\) is a fixed-point free, properly discontinuous group of isometries, then the resulting orbit space inherits a metric from \(M\text{.}\)

Consider the quotient space in Example 7.7.7. The group here is a group of isometries, since rotations preserve Euclidean distance, but it is not fixed-point free. All maps in \(G\) have fixed points (rotation about the origin fixes 0). This prevents the quotient space from inheriting the geometry of its mother space. Indeed, a circle centered at [0] with radius \(r\) would have circumference \(\frac{2\pi r}{4}\text{,}\) which doesn't correspond to Euclidean geometry.

The group of isometries in the torus example is fixed-point free and properly discontinuous, so the following formula for the distance between two points \([u]\) and \([v]\) in the orbit space \(\mathbb{C}/\langle T_a, T_{bi}\rangle\) is well-defined:

Figure 7.7.9 depicts two points in the shaded fundamental domain, \([u]\) and \([v]\text{.}\) The distance between them equals the Euclidean distance in \(\mathbb{C}\) of the shortest path between any point in equivalence class \([u]\) and any point in equivalence class \([v]\text{.}\) There are many such nearest pairs, and one such pair is labeled in Figure 7.7.9 where \(z\) is in \([u]\) and \(w\) is in \([v]\text{.}\) Also drawn in the figure is a solid line (in two parts) that corresponds to the shortest path one would take within the fundamental domain to proceed from \([u]\) to \([v]\text{.}\) This path marks the shortest route a ship in the video game from Chapter 1 could take to get from \([u]\) to \([v]\text{.}\)

Example 7.7.11. \(H_2\) as quotient of \(\mathbb{D}^2\).

We may build a regular octagon in the hyperbolic plane whose interior angles equal \(\pi/4\) radians. We may also find a hyperbolic transformation that takes an edge of this octagon to another edge. Labelling the edges as in the following diagram, let \(T_{a}\) be the hyperbolic isometry taking one \(a\) edge to the other, being careful to respect the edge orientations. We construct such a map by composing two hyperbolic reflections about hyperbolic lines: the hyperbolic line containing the first \(a\) edge, and the hyperbolic line \(m\) that bisects the \(b\) edge between the \(a\) edges. Since the two lines of reflection do not intersect, the resulting map in \(\cal H\) is a translation in \({\cal H}\) and has no fixed points in \(\mathbb{D}\) (the fixed points are ideal points).

Define \(T_{b},T_{c}\text{,}\) and \(T_{d}\) similarly and consider the group of isometries of \(\mathbb{D}\) generated by these four maps. This group is a fixed-point free, properly discontinuous group of isometries of \(\mathbb{D}\text{,}\) so the resulting quotient space inherits hyperbolic geometry.

The distance between two points \([u]\) and \([v]\) in the quotient space is given by

\begin{equation*} d_H([u],[v]) = ~\text{min}\{d_H(z,w) ~|~ z \in [u], w \in [v]\}\text{.} \end{equation*}

Geodesics in the quotient space are determined by geodesics in the hyperbolic plane \(\mathbb{D}\text{.}\)

Topologically, the quotient space is homeomorphic to \(H_2\text{,}\) and the octagon pictured above serves as a fundamental domain of the quotient space. Moving copies of this octagon by isometries in the group produces a tiling of \(\mathbb{D}\) by this octagon. Each copy of the octagon would serve equally well as a fundamental domain for the quotient space. Figure 7.7.12 displays a portion of this tiling, including a geodesic triangle in the fundamental domain, and images of it in neighboring octagons.

All surfaces \(H_g\) for \(g \geq 2\) and \(C_g\) for \(g \geq 3\) can be viewed as quotients of \(\mathbb{D}\) by following the procedure in the previous example.

Start with a perfectly sized polygon in \(\mathbb{D}\text{.}\) The polygon must have corner angle sum equal to \(2\pi\) radians, and the edges that get identified must have equal length so that an isometry can take one to the other. (In every example so far, we have used regular polygons in which all sides have the same length, but asymmetric polygons will also work.) Next, for each pair of oriented edges to be identified, find a hyperbolic isometry that maps one onto the other (respecting the orientation of the edges). The group generated by these isometries creates a quotient space homeomorphic to the space represented by the polygon, and it inherits hyperbolic geometry. Note also that the initial polygon can be moved by the isometries in the group to tile all of \(\mathbb{D}\) without gaps or overlaps.

Subsection Dirichlet Domain

We end this section with a discussion of the Dirichlet domain, which is an important tool in the investigation of the shape of the universe. Suppose we live in a surface described as a quotient \(M/G\) where \(M\) is either \(\mathbb{C}\text{,}\) \(\mathbb{D}\text{,}\) or \(\mathbb{S}^2\text{,}\) and \(G\) is a fixed-point free and properly discontinuous group of isometries of the space. For each point \(x\) in \(M\) define the Dirichlet domain with basepoint \(x\) to consist of all points \(y\) in \(M\) such that

for all \(T\) in \(G\text{,}\) where it is understood that \(d(x,y)\) is Euclidean distance, hyperbolic distance, or elliptic distance, depending on whether \(M\) is \(\mathbb{C}\text{,}\) \(\mathbb{D}\text{,}\) or \(\mathbb{S}^2\text{,}\) respectively.

At each basepoint \(x\) in \(M\text{,}\) the Dirichlet domain is itself a fundamental domain for the surface \(M/G\text{,}\) and it represents the fundamental domain that a two-dimensional inhabitant might build from his or her local perspective. It is a polygon in \(M\) (whose edges are lines in the local geometry) consisting of all points \(y\) that are as close to \(x\) or closer to \(x\) than any of its image points \(T(y)\) under transformations in \(G\text{.}\)

We may visualize a Dirichlet domain with basepoint \(x\) as follows. Consider a small circle in \(M\) centered at \(x\text{.}\) Construct a circle of equal radius about all points in the orbit of \(x\text{.}\) Then, begin inflating the circle (and all of its images). Eventually the circles will touch one another, and as the circles continue to expand let them press into each other so that they form a geodesic boundary edge. When the circle has filled the entire surface, it will have formed a polygon with edges identified in pairs. This polygon is the Dirichlet domain.

At any basepoint in the torus of Example 7.7.8 the Dirichlet domain will be a rectangle identical in proportions to the fundamental domain. In general, however, the shape of a Dirichlet domain may be different than the polygon on which the surface was built, and the shape of the Dirichlet domain may vary from point to point, which is rather cool.

Example 7.7.14. Dirichlet domains in a Klein bottle.

Consider the surface constructed from the hexagon in Figure 7.7.13, which appeared in Levin's paper on cosmic topology [23]. Assume the hexagon is placed in \(\mathbb{C}\) with its six corners at the points 0, 1, 2, \(2 + i\text{,}\) \(1+i\) and \(i\text{.}\)

This polygonal surface represents a cell division of a surface with three edges, two vertices, and one face. The Euler characteristic is thus 0, so the surface is either the torus or Klein bottle. In fact, it is a Klein bottle because it contains a Möbius strip. We may tile the Euclidean plane with copies of this hexagon using the transformations \(T(z) = z + 2i\) (vertical translation) and \(r(z) = \overline{z}+(1+2i)\) (a transformation that reflects a point about the horizontal axis \(y = 1\) and then translates to the right by one unit). The following figure shows the shaded fundamental domain \(A\) and its images under various combinations of \(T\) and \(r\text{.}\)

If \(\Gamma\) is the group of transformations generated by \(T\) and \(r\text{,}\) the quotient space \(\mathbb{C}/\Gamma\) is the Klein bottle, and its geometry is Euclidean, inherited from the Euclidean plane \(\mathbb{C}\text{.}\)

It turns out that the Dirichlet domain at a basepoint in this space can vary in shape from point to point. Exercise 7.7.4 investigates the shape of the Dirichlet domain at different points.