Triangulations of \(3\)-manifolds with essential edges (Q318704)

The authors focus on one-vertex triangulations of closed manifolds and ideal triangulations of the interiors of compact manifolds with boundary. For one-vertex triangulations in the closed case, if no edge loop is null-homotopic (keeping the vertex fixed) then the triangulation is called essential, and if no two edge loops are homotopic (keeping the vertex fixed) then the triangulation is called strongly essential. Similarly, for ideal triangulations of the interiors of compact manifolds, the triangulation is called essential if no ideal edge can be homotoped into a vertex neighbourhood, keeping its ends in the vertex neighbourhood, and is called strongly essential if no two ideal edges are homotopic, keeping their ends in the respective vertex neighbourhoods.NEWLINENEWLINEThe authors give four different constructions (Heegaard splittings, hierarchies of Haken 3-manifolds, Epstein-Penner decompositions, and cut loci of Riemannian manifolds) of essential and strongly essential triangulations, each using a different condition on the underlying manifold. Each construction is equally important and contains some important examples.NEWLINENEWLINEThe authors also show that an ideal triangulation of a 3-manifold admitting a strict angle structure must be strongly essential and if a triangulation has a semi-angle structure, then it must be essential. They also show that there are algorithms to decide whether a given triangulation of a closed 3-manifold is essential or strongly essential, based on the fact that the word problem is uniformly solvable in the class of fundamental groups of compact, connected 3-manifolds.