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**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.

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.

Reviewer: Yo’av Rieck (Fayetteville)

### MSC:

57Q15 | Triangulating manifolds |

57N10 | Topology of general \(3\)-manifolds (MSC2010) |