Cellular decomposition and free resolution for split metacyclic spherical space forms

Abstract

Given a free isometric action of a split metacyclic group on odd dimensional sphere, we obtain an explicit finite cellular decomposition of the sphere equivariant with respect to the group action. A cell decomposition of the factor space and an explicit description of the associated cellular chain complex of modules over the integral group ring of the fundamental group follow. In particular, the construction provides a simple explicit 4-periodic free resolution for the split metacyclic groups.

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 .…”

Decomposição celular e torção de Reidemeister para formas espaciais esféricas tetraedrais

“…Em [16], foram estudados os grupos quatérnios generalizados, e em [11], foram estudados os grupos split metacíclicos. Em particular, seguimos a ideia introduzida por Cohen [7].…”