Cellular decomposition of quaternionic spherical space forms

Cellular decomposition and free resolution for split metacyclic spherical space forms

“…The proof of the following proposition is straightforward and will be omitted (compare with [8,Section 12] and [7,Lemma 2] …”

“…In order to deal with the fundamental domain, we will use the approach and the notation introduced in [7]. First, recall that if G is a finite group acting on a space S, then a fundamental domain of the action of π on S is a connected closed subset F of S such that S = g gF and gF ∩ g ′ F has empty interior for all g = g ′ ∈ G. The first important difference with respect to the case of the quaternion studied in [7] is that in the present case the fundamental domain can change when we change the representation.…”

“…First, recall that if G is a finite group acting on a space S, then a fundamental domain of the action of π on S is a connected closed subset F of S such that S = g gF and gF ∩ g ′ F has empty interior for all g = g ′ ∈ G. The first important difference with respect to the case of the quaternion studied in [7] is that in the present case the fundamental domain can change when we change the representation. We start by observing that the image of the cyclic subgroup generated by x, π k,l ( x ) is independent on k and l, while the image of the subgroup generated by y, π k,l ( y ) is independent on k but depends on l. This means that the fundamental domain cannot change when we change the parameter k, but, in principle, depends on l. Thus, in order to determine the fundamental domain, we can fix k = 1 in the representation.…”

“…Of course, the main problem in applying this approach is computational, and this is the reason why, after it was successfully exploited for the cyclic groups, it was somehow abandoned. We were back to this technique in [7], where we studied the generalized quaternionic groups. In particular, we followed the clever geometric setting introduced by F. Cohen in [3,Chap.…”

“…We refer to that book for all the basic details of the construction. In [7], the ideas of Cohen were used, and after some substantial improvement of his technique, a cellular decomposition of the sphere S n , equivariant with respect to the action of the generalized quaternionic groups, was obtained. We show in this work that this technique is, in fact, powerful enough in order to deal with another class of groups: the split metacyclic groups.…”