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An algorithm to recognise small Seifert fiber spaces. (English) Zbl 1061.57023

The homeomorphism problem asks: is there for any two compact \(n\)-manifolds an algorithm to decide whether they are homeomorphic or not? The homeomorphism problem has been solved for many important classes of 3-manifolds; in particular, the manifolds having embedded 2-sided incompressible surfaces, called Haken manifolds (see for example [W. Haken, Math. Z. 80, 89–120 (1962; Zbl 0106.16605); G. Hemion, Acta Math. 142, 123–155 (1979; Zbl 0402.57027) and The classification of knots and 3-dimensional spaces (Oxford Science Publications, Oxford: Oxford University Press) (1992; Zbl 0771.57001)]). It is also well-known that the homeomorphism problem is easily solvable for two 3-manifolds which admit geometries in the sense of Thurston (see for example [P. Scott, Bull. Lond. Math. Soc. 15, 401–487 (1983; Zbl 0561.57001); W. Thurston, The geometry and topology of 3-manifolds, Lect. Notes Princeton Univ. (1978)]). Six of these geometries have all examples which admit Seifert fibre structures, namely \(\mathbb{S}^3,\mathbb{S}^2\times\mathbb{R}\), \(\mathbb{H}^2\times\mathbb{R}\), \(\text{PSL}(2,\mathbb{R})\), \(\mathbb{R}^3\), and Solv, while all examples admitting Nil geometric structures are Haken manifolds. The methods of Haken, as extended by W. Jaco and U. Oertel (see for example [Topology 23, 195–209 (1984; Zbl 0545.57003); W. Jaco and J. L. Tollefson, Ill. J. Math. 39, No. 3, 358–406 (1995; Zbl 0858.57018)]) give an algorithm to recognise any 3-manifold in these seven geometries, which is Haken or reducible, that is, contains a 2-sided embedded incompressible surface or an embedded 2-sphere which does not bound a 3-cell. The aim of the author is to complete the recognition problem for all 3-manifolds which admit one of the first six geometries and are non-Haken and irreducible. Such examples occur only for \(\mathbb{S}^3\) and \(\text{PSL}(2,\mathbb{R})\) geometries. The author gives an algorithm to recognise the class of small Seifert fibered spaces, which either have finite fundamental group or have fundamental groups which are extensions of \(\mathbb{Z}\) by a triangle group and have finite abelianisation. A completely different solution has been announced recently by [Lao Li, “An algorithm to recognise vertical tori in small Seifert fibred spaces”, preprint (2003)]. Also Perelman’s announcement of a solution of the geometrisation conjecture would enable a complete solution of the homeomorphism problem by identifying which geometric structure a given manifold admits. However it is worth noting that practical algorithms for the homeomorphism and recognition problems, which can be implemented via software, are very useful for experimentation in 3-manifold topology (see for example [B. Burton, Minimal triangulations and normal surfaces, PhD thesis, University of Melbourne (2003); J. Weeks, Snap Pea (Hyperbolic 3-manifold software), http://www.northnet.org/weeks/index/SnapPea.html, (1991–2000)]).

MSC:

57N10 Topology of general \(3\)-manifolds (MSC2010)
57M50 General geometric structures on low-dimensional manifolds
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