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The Weber-Seifert dodecahedral space is non-Haken. (English) Zbl 1250.57033

The Seifert-Weber dodecahedral space \(WS\) is formed by identifying opposite faces of a solid dodecahedron with a \(3/10\)-twist. It is a closed orientable irreducible 3-manifold, and it was one of the first known examples of a hyperbolic manifold [C. Weber and H. Seifert, Math. Z. 37, 237–253 (1933; Zbl 0007.02806)]. It is different from the Poincaré dodecahedral space, which is also obtained by identifying opposite faces of a dodecahedron. W. Thurston conjectured that \(WS\) is non-Haken, that is, it does not contain an embedded 2-sided \(\pi_1\)-injective surface. The main result of the paper under review proves this conjecture. The proof is computational, and it is perhaps the first non-trivial implementation of the known algorithms for 3-manifolds.
W. Jaco and U. Oertel [Topology 23, 195–209 (1984; Zbl 0545.57003)] gave an algorithm to determine whether a given 3-manifold \(M\) is Haken. This is based in Haken’s normal surface theory. Let \(M\) be a 3-manifold with a fixed triangulation \(\mathcal T\). A normal surface is an embedded surface in \(M\) which intersects each tetrahedron of \(\mathcal T\) in triangles and squares. It is proved that if \(M\) contains an incompressible surface, then there is one incompressible surface among the normal vertex surfaces. Then the algorithm works, roughly speaking, as follows: First, enumerate all vertex surfaces, there are only finitely many of them. Now for each surface \(S\), check whether it is incompressible. To do that cut \(M\) along \(S\), retriangulate, enumerate the normal surfaces in the new manifold, and check whether any of these is a compression disk. So, it may seem that an application of this algorithm will suffice to prove that \(WS\) is non-Haken. However this algorithm is very slow in practice.
The present paper improves the Jaco-Oertel algorithm. It is shown that if a 3-manifold is Haken, then it contains a pair of incompressible compatible vertex surfaces, unless some special situation arises. Then a heuristic method to check compressibility is given, which is not conclusive, but useful in many cases. The proof of the main result goes as follows: A triangulation \(\mathcal T\) of \(WS\) is given which has 23 tetrahedra. Then an enumeration is done of all normal vertex surfaces, which uses some new enumeration algorithms [B. Burton, Math. Comput. 79, No. 269, 453–484 (2010; Zbl 1246.57038)], obtaining 1751 vertex surfaces. The heuristic algorithm shows that most of these surfaces are compressible, except 16 of them. For the remaining surfaces, it is shown that no two of them are compatible, completing the proof. All the computations were carried out using the open-source software package Regina [B. Burton, Exp. Math. 13, No. 3, 267–272 (2004; Zbl 1090.57003)].

MSC:

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

Software:

SnapPea; Regina
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References:

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