A new combinatorial class of \(3\)-manifold triangulations (Q1688495)

Within the study of 3-manifold triangulations (including semi-simplicial and singular ones), it is important to establish minimality conditions, in order to generate and analyze suitable 3-manifold censuses; see in particular \textit{W. Jaco} and \textit{J. H. Rubinstein} [J. Differ. Geom. 65, No. 1, 61--168 (2003; Zbl 1068.57023)], which involves the so called \textit{0-efficient triangulations}. The present paper introduces a new class of 3-manifold triangulations, satisfying a weak version of 0-efficiency and a weak version of minimality: they are \textit{face-generic} (i.e. all faces are triangles or Möbius bands and each tetrahedron has at most two Möbius faces) and \textit{face-pair-reduced} (i.e. certain simplification moves, which can be determined from the 2-skeleton, are not possible). The authors prove the existence of an algorithm which takes as input any triangulation \(K\) of a closed, orientable, irreducible 3-manifold \(M\), and either outputs a face-pair-reduced and face-generic triangulation of \(M\) having at most the same number of tetrahedra as \(K\) and precisely one vertex, or concludes that \(M\) is one of \(S^3\), \(\mathbb{R}P^3\), \(L(3, 1)\), \(L(4, 1)\), \(L(5, 1)\) or \(L(5, 2)\). In order to classify the possible combinatorial types of the 3-simplices in a face-pair-reduced and face-generic triangulation, the authors make use of the notion of \textit{twisted square}, which is a properly embedded disc in a tetrahedron \(\sigma\) of the triangulation, such that the boundary of the disc is the union of two pairs of opposite edges of \(\sigma.\) Moreover, as an application of the above tools, they obtain strong restrictions on the topology of a 3-manifold from the existence of non-smooth maxima of the volume function on the space of circle-valued angle structures. This improves results obtained by the first author in [J. Differ. Geom. 93, No. 2, 299--326 (2013; Zbl 1292.57018)]. Finally, it is worthwhile to note -- as the referee pointed out, and the authors themselves recall in the introduction -- ``that the main application of this paper can be viewed as a step towards realising a variant of Casson's program to prove the Geometrisation Theorem for 3-manifolds''; see also [the first author, Contemp. Math. 541, 183--204 (2011; Zbl 1236.57004)].