Fundamental Domain and Cellular Decomposition of Tetrahedral Spherical Space Forms

Cellularization for exceptional spherical space forms and the flag manifold of $SL_3(\mathbb{R})$

“…It turns out that all such groups were entirely classified by Milnor in [Mil57] and there is five families of them: the quaternionic groups, the metacyclic groups, the generalized tetrahedral groups, the binary octahedral and icosahedral groups. So far, explicit equivariant cellular structures were found for the quaternionic groups ( [MNdMS13]), the metacyclic groups ( [FGMNS13] and [CS17]) and the generalized tetrahedral groups ( [FGMNS16]). According to Milnor's classification, the only remaining cases are the binary octahedral group P 48 and the binary icosahedral group P 120 .…”

“…In Sections 3, 4 and in the Appendix A, we apply the orbit polytope techniques to the cases where G is O, I or T acting on S 3 ; the later having been treated previously in [FGMNS16]. In particular, we explicitly describe a fundamental domain for the boundary of the polytope, and we use it to determine a G-equivariant cellular decomposition of S 3 .…”

“…Even if the case of T has already been treated in [FGMNS16], we can recover it by applying the above methods to this case. Note that all the groups in the tetrahedral family are studied in [CS17], but there T is excluded since, while it is the simplest one of the family, it is somehow different from all the other ones.…”