Minimal crystallizations of 3-manifolds
Summary: We introduce the weight of a group which has a presentation where the number of relations is at most the number of generators. We show 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 \(\mathbb{RP}^3\), \(L(3,1)\), \(L(5,2)\), \(S^1\times S^1 \times S^1\), \(S^2 \times S^1\), the twisted product of \(S^2\)~and \(S^1\) and \(S^3/Q_8\), where \(Q_8\) is the quaternion group. Moreover, there is a unique such facet minimal pseudotriangulation in each of these seven cases. We also construct contracted pseudotriangulations of \(L(kq-1,q)\) with \(4(q+k-1)\) facets for \(q \geq 3\), \(k \geq 2\) and \(L(kq+1,q)\) with \(4(q+k)\) facets for \(q\geq 4\), \(k\geq 1\). By a recent result of Swartz, our pseudotriangulations of \(L(kq+1, q)\) are facet minimal when \(kq+1\) is even. In [Boll. Unione Mat. Ital., V. Ser., A 16, 441--449 (1979; Zbl 0414.57004)], \textit{C. 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.
scientific article; zbMATH DE number 6340159
scientific article