§ 2.5. Spherical 3-manifolds.

Spherical 3-manifolds have the 3-sphere $S^{3}$ as their simply connected universal cover.

In line with the reasoning given above, any spherical 3-manifold forms the prototile of a tiling of the 3-sphere. A particular family of prototiles arises from the five regular Platonic polyhedra: the tetrahedron, cube, octahedron, dodecahedron and icosahedron in their spherical form. These polyhedra are bounded by reflection hyperplanes passing through the origin of the 3-sphere embedded in the space $R^{4}$ . The so-called Weyl reflections are uniquely characterized as $W_{a}$ by a unit vector $a$ perpendicular to the reflecting hyperplane. Groups generated by Weyl reflections are Coxeter groups $\Gamma$ denoted by diagrams. In Table 15 we list sets of four Weyl vectors appropriate for the Coxeter groups, and corresponding Platonic spherical polyhedra given in Table 2. Sets of four Coxeter reflection planes bound what are called Coxeter simplices. The Platonic polyhedra consist of all simplices sharing a single vertex.

$\begin{array}[]{|l|l|l|l|l|}\hline\Gamma&a_{1}&a_{2}&a_{3}&a_{4}\\ \hline\circ-\circ-\circ-\circ&(0,0,0,1)&(0,0,\sqrt{\frac{3}{4}},\frac{1}{2})&(0,\sqrt{\frac{2}{3}},\sqrt{\frac{1}{3}},0)&(\sqrt{\frac{5}{8}},\sqrt{\frac{3}{8}},0,0)\\ \hline\circ\stackrel{4}{-}\circ-\circ-\circ&(0,0,0,1)&(0,0,-\sqrt{\frac{1}{2}},\sqrt{\frac{1}{2}})&(0,\sqrt{\frac{1}{2}},-\sqrt{\frac{1}{2}},0)&(-\sqrt{\frac{1}{2}},\sqrt{\frac{1}{2}},0,0)\\ \hline\circ-\circ\stackrel{4}{-}\circ-\circ&(0,\sqrt{\frac{1}{2}},-\sqrt{\frac{1}{2}},0)&(0,0,-\sqrt{\frac{1}{2}},\sqrt{\frac{1}{2}})&(0,0,0,1)&(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2})\\ \hline\circ-\circ-\circ\stackrel{5}{-}\circ&(0,0,1,0)&(0,-\frac{\sqrt{-\tau+3}}{2},\frac{\tau}{2},0)&(0,-\sqrt{\frac{\tau+2}{5}},0,-\sqrt{\frac{-\tau+3}{5}})&(\frac{\sqrt{2-\tau}}{2},0,0,-\frac{\sqrt{\tau+2}}{2})\\ \hline\end{array}$

(15)

Table 1. The four Weyl vectors $a_{s}$ for the four Coxeter groups $\Gamma$ in Table 2 with $\tau:=\frac{1+\sqrt{5}}{2}$ .

Each Platonic spherical polyhedron forms the prototile of a regular tiling of the 3-sphere. The Platonic tilings are well-known regular m-cells, known as convex regular 4-polytopes. The $8$ -cell is illustrated in Fig.21.

All possible homotopies of the Platonic polyhedra on the 3-sphere were derived by Everitt [11]. His algorithm gives the gluing prescriptions of polyhedral faces and edges. They describe how faces are rotated around their midpoints and then glued in pairs. A remarkable outcome of his analysis is that, for a fixed polyhedral shape of the regular polyhedron, there are distinct possible homotopies. We list them in Table 2 from [41]. An example is given in subsection 2.6.

Coxeter diagram $Gamma$	$\|\Gamma\|$	Polyhedron ${\cal M}$	$H={\rm deck}({\cal M})$	$\|H\|$	Reference
$\circ-\circ-\circ-\circ$	$120$	tetrahedron $N1$	$C_{5}$	$5$	[38]
$\circ\stackrel{4}{-}\circ-\circ-\circ$	$384$	cube $N2$	$C_{8}$	$8$	[39]
		cube $N3$	$Q$	$8$	[39]
$\circ-\circ\stackrel{4}{-}\circ-\circ$	$1152$	octahedron $N4$	$C_{3}\times Q$	$24$	[40]
		octahedron $N5$	$B$	$24$	[40]
		octahedron $N6$	${\cal T}_{2}$	$24$	[40]
$\circ-\circ-\circ\stackrel{5}{-}\circ$	$120\cdot 120$	dodecahedron $N1^{{\prime}}$	${\cal J}_{2}$	$120$	[36], [37]

Table 2. 4 Coxeter groups $\Gamma$ , 4 Platonic polyhedra ${\cal M}$ , 7 groups $H={\rm deck}({\cal M})$ of order $|H|$ . In the Table, $C_{n}$ denotes a cyclic, $Q$ the quaternion, ${\cal T}_{2}$ the binary tetrahedral, ${\cal J}_{2}$ the binary icosahedral group. The symbols $Ni$ are adapted from [11].

The first distinction between them is in the gluing of faces and edges [41]. For a fixed polyhedron, different homotopic boundary conditions are fulfilled.

In the following figures we show the tetrahedral, cubic and octahedral 3-manifolds and their face and edge enumeration.

Figure 14. Tetrahedron: The Weyl vectors $a_{1},a_{2},a_{3}$ of the Coxeter group $\Gamma=\circ-\circ-\circ-\circ$ , and the Coxeter simplex bounded by the Weyl reflection planes. $24$ Coxeter simplices share a vertex and form the tetrahedral manifold $N1$ . In the following figures we replace the Platonic spherical polyhedra by their Euclidean counterparts.

Figure 15. Tetrahedron: Enumeration of the four faces Fs and six directed edges ej of the tetrahedral spherical manifold from [11].

In the following Fig16 we illustrate the action of the cyclic generator of deck transformations for the tetrahedron $T$ .

Figure 16. Deck transformations of the tetrahedron: The action of the cyclic permutation $(5,4,3,2,1)=(5,4,3)(3,2,1)$ on the tetrahedron $T$ with vertices $\left\{ 1,2,3,4\right\}$ is a cyclic permutation from the symmetric group $\Gamma=S(5)$ . This cyclic permutation generates the deck transformations of the tetrahedron. Weyl reflections here act as transpositions. a: initial tetrahedron $T$ , b: $(3,2,1)T$ , c: $(4,3)(3,2,1)T$ , d: $(5,4)(4,3,2,1)T$ . The Weyl reflection plane for $(5,4)$ contains the vertices $\left\{ 1,2,3\right\}$ in c and d. $(5,4)$ in d reflects the tetrahedron shown in c from the dashed into the undashed position.

Figure 17. Cube: The unit vectors $1,2,3$ , the Weyl vectors $a_{1},a_{2},a_{3}$ of the Coxeter group $\Gamma=\circ\stackrel{4}{-}\circ-\circ-\circ$ , and the Coxeter simplex bounded by the Weyl reflection planes. $48$ Coxeter simplices share a vertex and form the cubic manifolds $N2,N3$ .

Figure 18. Cube: Enumeration of faces $F1,\ldots,F6$ and edges $e1,\ldots,e12$ for the cubic prototile according to Everitt [11] p. 260 Fig. 2.

Figure 19. Octahedron: The unit vectors $1,2,3$ , the Weyl vectors $a_{1},a_{2},a_{3}$ of the Coxeter group $\Gamma=\circ-\circ\stackrel{4}{-}\circ-\circ$ , and the Coxeter simplex bounded by the Weyl reflection planes. $48$ Coxeter simplices share a vertex and form the octahedral manifolds $N4,N5,N6$ .

Figure 20. Octahedron: The octahedron projected to the plane with faces $F1\ldots F8$ and directed edges $e1\ldots e12$ according to [11]. The products of Weyl reflections $(W_{1}W_{2})$ and $(W_{2}W_{3})$ generate right-handed $3$ fold and $4$ fold rotations respectively.

Figure 21. 8-cell, projected to the plane. Eight spherical cubes are replaced by deformed Euclidean cubes to illustrate the matching of faces. Any face bounds two, any edge three, any vertex four out of the eight cubes.

Figure 22. Cubic spherical manifolds $N2$ and $N3$ . The cubic prototile and three neighbour tiles sharing its faces $F1,F2,F3$ . The four cubes are replaced by their Euclidean counterparts and separated from one another. Visible faces are denoted by the numbers from Fig. 16. The actions transforming the prototile into its three neighbours generate the deck transformations and the 8-cell tiling of $S^{3}$ . In the tiling, homotopic face gluing takes the form of joined pairs of faces $N2:F3\cup F1,F4\cup F2,F6\cup F5$ and $N3:F1\cup F6,F2\cup F4,F3\cup F5$ . In the figure, glued pairs of faces are joined by heavy lines or arcs. The full deck transformation groups build up the $8$ -cell tiling of the 3-sphere by $8$ spherical cubes, illustrated in Fig. 21