# § 2.5. Spherical 3-manifolds.

Spherical 3-manifolds have the 3-sphere 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 . The so-called Weyl reflections are uniquely characterized as by a unit vector perpendicular to the reflecting hyperplane. Groups generated by Weyl reflections are Coxeter groups 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.

 (15)
Table 1. The four Weyl vectors for the four Coxeter groups in Table 2 with .

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 -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 Polyhedron Reference tetrahedron [38] cube [39] cube [39] octahedron [40] octahedron [40] octahedron [40] dodecahedron [36], [37]
Table 2. 4 Coxeter groups , 4 Platonic polyhedra , 7 groups of order . In the Table, denotes a cyclic, the quaternion, the binary tetrahedral, the binary icosahedral group. The symbols 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.

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