## 0-efficient triangulations of 3-manifolds.(English)Zbl 1068.57023

The paper under review studies triangulations on 3-manifolds from an algorithmic perspective. A surface in a triangulated 3-manifold is called normal if each component of the surface in each tetrahedron is a normal disk (basically, a triangle or a quadrilateral). Normal surfaces date back to Kneser in the 20’s and Haken in the 60’s and have proven to be very useful. However, normal surfaces of nonnegative Euler characteristic are often a problem, as they make Euler characteristic count difficult. Roughly, the goal of the authors is to produce a triangulation without normal spheres. The authors prove that given a triangulation $${\mathcal T}$$ of an irreducible 3-manifold other than $$S^3$$, $$S^2\times S^1$$ and the lens space $$L(3,1)$$, $${\mathcal T}$$ can be modified algorithmically to produce a triangulation with exactly one normal sphere, a vertex linking sphere. Note that for any vertex in a triangulation, the vertex link is a normal sphere; thus this result is best possible. The authors call such a triangulation $$0$$-efficient. The main tools in the proof are collapsing 3-balls with normal boundary and then modifying the resulting triangulation, and shrinking (normalizing) and barriers.
The notion of $$0$$-efficiency is extended to compact manifolds where the $$0$$-efficient triangulation has exactly one vertex on each boundary component. It is extended further to certain non-compact manifolds, namely, manifolds with cusps homeomorphic to a torus cross $$\mathbb{R}$$. For these manifolds the authors prove existence of $$0$$-efficient idea triangulations.
Several applications are given, including algorithms in 3-manifolds (for example for finding maximal collection of non-parallel 2-spheres) and construction of irreducible knots (originally due to Bing); in fact, it is shown that every edge of an efficient triangulation is an irreducible knot.
$$0$$-efficient triangulations are useful for the study of various structures on 3-manifolds, for example see [I. Agol and T. Li, Geom. Topol. 7, 287–309 (2003; Zbl 1037.57008)] for an algorithm to decide if a 3-manifold admits a Reebless essential lamination and [T. Li, Heegaard surfaces and measured laminations. I: The Waldhausen conjecture, arXiv:math.GT/0408198 and Heegaard surfaces and measured laminations, II: Non-Haken 3-manifolds, arXiv:math.GT/0408199] for finiteness of Heegaard splittings in non-Haken, a-toroidal 3-manifolds.

### MSC:

 57Q15 Triangulating manifolds 57N10 Topology of general $$3$$-manifolds (MSC2010)

### Keywords:

normal surfaces algorithms

Zbl 1037.57008
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