Minimal Crystallizations of 3-Manifolds

We have introduced the weight of a group which has a presentation with number of relations is at most the number of generators. We have shown that the number of facets of any contracted pseudotriangulation of a connected closed 3-manifold M is at least the weight of the fundamental group of M . This lower bound is sharp for the 3-manifolds RP, L(3, 1), L(5, 2), S1 × S1 × S1, S2 × S1, S×− S1 and S/Q8, where Q8 is the quaternion group. Moreover, there is a unique such facet minimal pseudotriangulation in each of these seven cases. We have also constructed contracted pseudotriangulations of L(kq − 1, q) with 4(q + k − 1) facets for q > 3, k > 2 and L(kq + 1, q) with 4(q + k) facets for q > 4, k > 1. By a recent result of Swartz, our pseudotriangulations of L(kq + 1, q) are facet minimal when kq + 1 are even. In 1979, Gagliardi found presentations of the fundamental group of a manifold M in terms of a contracted pseudotriangulation of M . Our construction is the converse of this, namely, given a presentation of the fundamental group of a 3-manifold M , we construct a contracted pseudotriangulation of M . So, our construction of a contracted pseudotriangulation of a 3-manifold M is based on a presentation of the fundamental group of M and it is computer-free.