§2 Topology and harmonic analysis.

§ 2.7. Deck transformation groups for spherical 3-manifolds.

By use of the theorem by Seifert and Threlfall [54], we construct in [38], [39], [40], [41] from Everitt's homotopic glue generators of Platonic 3-manifolds isomorphic deck operations. These generators are those deck transformations which map the prototile to any one of its face neighbours.

Once we have found the relative position of a neighbour tile, we employ rotations from the Coxeter group to describe algebraically the corresponding deck transformation. From them we generate the full groups of deck transformations, These groups for different homotopies from the same polyhedra are distinct but share the order m equal to the number of tiles in the Platonic tiling of the 3-sphere.