Layered-triangulations of 3-manifolds

TL;DR

This paper introduces layered-triangulations for 3-manifolds, providing detailed classifications, new proofs of known results, and constructing canonical triangulations for Dehn fillings, advancing understanding of 3-manifold topology.

Contribution

It develops the theory of layered-triangulations for 3-manifolds, including classifications, new proofs, and the construction of canonical triangulations, linking triangulations to Heegaard splittings and Dehn fillings.

Findings

Classified layered-triangulations for solid tori and lens spaces.

Provided new proofs for lens space classifications and non-orientable surfaces.

Constructed canonical triangulations for Dehn fillings.

Abstract

A family of one-vertex triangulations of 3-manifolds, layered-triangulations, is defined. Layered-triangulations are first described for handlebodies and then extended to all 3-manifolds via Heegaard splittings. A complete and detailed analysis of layered-triangulations is given in the cases of the solid torus and lens spaces, including the classification of all normal and almost normal surfaces in these triangulations. Minimal layered-triangulations of lens spaces 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. Canonical triangulations of Dehn fillings, triangulated Dehn fillings, are constructed and applied to the study of Heegaard splittings and efficient triangulations of Dehn fillings. A new presentation of 3-manifolds as being obtained from special layered-triangulations of handlebodies with one-vertex, 2-symmetric triangulations on their boundaries, called triangulated Heegaard splittings, is defined and explored. The 1-skeleton (L-graph) of the complex determined as the quotient of the flip-complex by considering those homemorphisms of the genus g surface that extend to a homeomorphism of the genus g handlebody is used to organize much of the work. Numerous questions remain open, particularly in relation to the L-graph, 2-symmetric triangulations of a closed orientable surface, minimal layered-triangulations of genus-g-handlebodies, and the relationship of layered-triangulations to foliations.