2020/06/25 by Birch Bryant, William Jaco, Bryant, Birch +3 · 1 citation
Mathematics · #57M99 #FOS: Mathematics #Geometric Topology (math.GT) #Primary 57N10 #Secondary 57M50 #math.GT #msc:57M50 #msc:57M99 #msc:57N10
paper · pdf · doi:10.48550/arxiv.2006.14701
21 pages, 6 figures; revised and improved version of an earlier paper arXiv:1108.2936, Annular efficient triangulations of 3-manifolds
arxiv created 2020/06/25 · arxiv updated 2020/06/29
For a compact, irreducible, ∂-irreducible, an-annular bounded 3-manifold M≠\mathbbB3, then any triangulation T of M can be modified to an ideal triangulation T^* of \stackrel∘M. We use the inverse relationship of crushing a triangulation along a normal surface and that of inflating an ideal triangulation to introduce and study boundary-efficient triangulations and end-efficient ideal triangulations. We prove that the topological conditions necessary for a compact 3-manifold M admitting an annular-efficient triangulation are sufficient to modify any triangulation of M to a boundary-efficient triangulation which is also annular-efficient. From the proof we have for any ideal triangulation T^* and any inflation TΛ, there is a bijective correspondence between the closed normal surfaces in T^* and the closed normal surfaces in TΛ with corresponding normal surfaces being homeomorphic. It follows that for an ideal triangulation T^* that is 0-efficient, 1-efficient, or end-efficient, then any inflation TΛ of T^* is 0-efficient, 1-efficient, or ∂-efficient, respectively. There are algorithms to decide if a given triangulation or ideal triangulation of a 3-manifold is one of these efficient triangulations. Finally, it is shown that for an annular-efficient triangulation, there are only a finite number of boundary slopes for normal surfaces of a bounded Euler characteristic; hence, in a compact, orientable, irreducible, ∂-irreducible, and an-annular 3-manifold, there are only finitely many boundary slopes for incompressible and ∂-incompressible surfaces of a bounded Euler characteristic.