Layered - Triangulations of 3 – Manifolds

A family of one-vertex triangulations of the genus-g-handlebody, called layered-triangulations, is defined. These triangulations induce a one-vertex triangulation on the boundary of the handlebody, a genus g surface. Conversely, any one-vertex triangulation of a genus g surface can be placed on the boundary of the genus-g-handlebody in infinitely many distinct ways; it is shown that any of these can be extended to a layered-triangulation of the handlebody. To organize this study, a graph is constructed, for each genus g ≥ 1, called the Lg graph; its 0–cells are in one-one correspondence with equivalence classes (up to homeomorphism of the handlebody) of one-vertex triangulations of the genus g surface on the boundary of the handlebody and its 1–cells correspond to the operation of a diagonal flip (or 2 ↔ 2 Pachner move) on a one-vertex triangulation of a surface. A complete and detailed analysis of layered-triangulations is given in the case of the solid torus (g = 1), including the classification of all normal and almost normal surfaces in these triangulations. An initial investigation of normal and almost normal surfaces in layered-triangulations of higher genera handlebodies is discussed. Using Heegaard splittings, layered-triangulations of handlebodies can be used to construct special one-vertex triangulations of 3-manifolds, also called layered-triangulations. Minimal layered-triangulations of lens spaces (genus one man-ifolds) provide a common setting for new proofs of the classification of lens spaces admitting an embedded non orientable surface and the classification of embedded non orientable surfaces in each such lens space, as well as a new proof of the uniqueness of Heegaard splittings of lens spaces, including S 3 and S 2 × S 1. Canonical triangulations of Dehn fillings called triangulated Dehn fillings are constructed and applied to the study of Heegaard splittings and efficient triangulations of Dehn fillings. It is shown that all closed 3–manifolds can be presented in a new way, and with very nice triangulations, using layered-triangulations of handlebodies that have special one-vertex triangulations of a closed surface on their boundaries, called 2–symmetric triangulations. We provide a quick introduction to a connection between layered-triangulations and foliations. Numerous questions remain unanswered, particularly in relation to the Lg-graph, 2–symmetric triangulations of a closed orientable surface, minimal layered-triangulations of genus-g-handlebodies, g ≥ 1 and the relationship of layered-triangulations to foliations.