On the classification of Heegaard splittings. (English) Zbl 1403.57013
It is a long-standing problem for Heegaard splittings of 3-manifolds to exhibit (produce) for each closed 3-manifold a complete list, without duplications, of all its irreducible Heegaard splittings, up to isotopy. In the present paper, an algorithmic solution of this problem is given for the case of non-Haken hyperbolic 3-manifolds \(N\): there is an effectively constructible finite set of Heegaard surfaces of \(N\) such that every Heegaard surface of an irreducible Heegaard splitting is isotopic to exactly one of the surfaces of the list.
In previous work T. Li [Geom. Topol. 15, No. 2, 1029–1106 (2011; Zbl 1221.57034)] showed how to construct, for each closed non-Haken 3-manifold \(N\), a finite list of genus-\(g\) Heegaard surfaces which contains every genus-\(g\) Heegaard surface, up to isotopy. Moreover, by work of the first two authors [T. H. Colding and D. Gabai, Duke Math. J. 167, No. 15, 2793–2832 (2018; Zbl 1403.57012)], there esists an effectively computable constant \(C(N)\) such that every irreducible Heegaard splitting of \(N\) has genus at most \(C(N)\), and an effectively constructible set of Heegaard surfaces that contains every irreducible Heegaard surface. However this list may contain reducible splittings and duplications, and “the main goal of the present paper is to give an effective algorithm that weeds out the duplications and reducible splittings”.
“It is interesting to note that our algorithm is elementary and combinatorial, yet the proof that it works requires a 2-parameter sweep-out argument and a multiparameter min-max argument”. The authors close with three “fundamental problems” for Heegaard splittings of general 3-manifolds: for the case of Haken 3-manifolds and Seifert fiber spaces, find an algorithm to decide whether a Heegaard splitting is reducible, and whether two irreducible Heegaard splittings are isotopic; finally, given a closed non-Haken 3-manifold, construct the tree of Heegaard splittings (which is finite by another result of T. Li [J. Am. Math. Soc. 19, No. 3, 625–657 (2006; Zbl 1108.57015)]).
In previous work T. Li [Geom. Topol. 15, No. 2, 1029–1106 (2011; Zbl 1221.57034)] showed how to construct, for each closed non-Haken 3-manifold \(N\), a finite list of genus-\(g\) Heegaard surfaces which contains every genus-\(g\) Heegaard surface, up to isotopy. Moreover, by work of the first two authors [T. H. Colding and D. Gabai, Duke Math. J. 167, No. 15, 2793–2832 (2018; Zbl 1403.57012)], there esists an effectively computable constant \(C(N)\) such that every irreducible Heegaard splitting of \(N\) has genus at most \(C(N)\), and an effectively constructible set of Heegaard surfaces that contains every irreducible Heegaard surface. However this list may contain reducible splittings and duplications, and “the main goal of the present paper is to give an effective algorithm that weeds out the duplications and reducible splittings”.
“It is interesting to note that our algorithm is elementary and combinatorial, yet the proof that it works requires a 2-parameter sweep-out argument and a multiparameter min-max argument”. The authors close with three “fundamental problems” for Heegaard splittings of general 3-manifolds: for the case of Haken 3-manifolds and Seifert fiber spaces, find an algorithm to decide whether a Heegaard splitting is reducible, and whether two irreducible Heegaard splittings are isotopic; finally, given a closed non-Haken 3-manifold, construct the tree of Heegaard splittings (which is finite by another result of T. Li [J. Am. Math. Soc. 19, No. 3, 625–657 (2006; Zbl 1108.57015)]).
Reviewer: Bruno Zimmermann (Trieste)
MSC:
| 57M50 | General geometric structures on low-dimensional manifolds |
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