The classification of Seifert fibred 3-orbifolds.

*(English)*Zbl 0571.57011
Low dimensional topology, 3rd Topology Semin. Univ. Sussex 1982, Lond. Math. Soc. Lect. Note Ser. 95, 19-85 (1985).

[For the entire collection see Zbl 0561.00016.]

In a fundamental paper [Acta Math. 60, 147-238 (1933; Zbl 0006.08304)] H. Seifert studied a class of compact 3-manifolds, now called Seifert fiber spaces, which possess a nice decomposition into circles \(S^ 1\). Locally the decomposition looks like \((D^ 2\times S^ 1)/G\), with its natural decomposition into circles, where G is a finite cyclic group acting as rotations on \(D^ 2\) and \(S^ 1\). Seifert fiber spaces (together with hyperbolic manifolds) turned out to belong to the basic building blocks of compact 3-manifolds; for example, every spherical or Euclidean (flat) 3-manifold is a Seifert fiber space. A Seifert 3- orbifold O possesses a nice decomposition into circles and closed intervals; now, in the local model, G is a finite cyclic or dihedral group acting as rotations and/or reflections on \(D^ 2\) and \(S^ 1\). It is a quotient of \(S^ 3\), \({\mathbb{E}}^ 2\times {\mathbb{R}}\) or \({\mathbb{H}}^ 2\times {\mathbb{R}}\) by a properly discontinuous group \(\pi\) of fiber- preserving homeomorphisms (with \(\pi \cong \pi_ 1(O)\), the orbifold fundamental group). These homeomorphisms may have fixed points, so the quotient space is an orbifold in the sense of Thurston and not necessarily a manifold (the present paper contains a general discussion of orbifolds and orbifold fibrations).

It is not surprising that these spaces also occur at various places. For example many quotient spaces of the classical 3-dimensional crystallographic groups possess such a structure (184 out of 219); another example is the class of Montesinos-knots and links \(L\subset S^ 3\); here \(S^ 3\) occurs as a Seifert 3-orbifold, with L equal to the union of the endpoints of all interval-fibers (174 of the 250 ten- crossing knots belong to this class; for a description, see G. Burde and H. Zieschang [Knots, (1985; Zbl 0568.57001), Chapter 12]). The present paper presents the local and global classification of Seifert-3-orbifolds. The classification is similar in spirit to Seifert’s original classification of Seifert fiber spaces, by geometrically defined invariants. However the situation is much more complicated in the present situation: there are many more local types, and the dependence of the various invariants can be quite subtle. The results are too complicated to be stated here. Another more algebraic approach to the classification has been given by H. Zieschang and the reviewer [Math. Ann. 259, 29-51 (1982; Zbl 0466.57002)], in terms of an isomorphism classification of the orbifold fundamental groups \(\pi\). It should be noted that the general theory of P. E. Conner and F. Raymond on injective Seifert fibrations can be applied to these spaces [Bull. Am. Math. Soc. 83, 36-85 (1977; Zbl 0341.57003)]; see also the survey article by K.-B. Lee and F. Raymond [Contemp. Math. 36, 367-425 (1985; 564.57002)]. Analogously to the Seifert 3-manifolds, the Seifert 3- orbifolds possess geometric structures modelled on one of six 3- dimensional geometries [see, e.g., P. Scott, Bull. Lond. Math. Soc. 15, 401-487 (1983; Zbl 0564.57002)].

In a fundamental paper [Acta Math. 60, 147-238 (1933; Zbl 0006.08304)] H. Seifert studied a class of compact 3-manifolds, now called Seifert fiber spaces, which possess a nice decomposition into circles \(S^ 1\). Locally the decomposition looks like \((D^ 2\times S^ 1)/G\), with its natural decomposition into circles, where G is a finite cyclic group acting as rotations on \(D^ 2\) and \(S^ 1\). Seifert fiber spaces (together with hyperbolic manifolds) turned out to belong to the basic building blocks of compact 3-manifolds; for example, every spherical or Euclidean (flat) 3-manifold is a Seifert fiber space. A Seifert 3- orbifold O possesses a nice decomposition into circles and closed intervals; now, in the local model, G is a finite cyclic or dihedral group acting as rotations and/or reflections on \(D^ 2\) and \(S^ 1\). It is a quotient of \(S^ 3\), \({\mathbb{E}}^ 2\times {\mathbb{R}}\) or \({\mathbb{H}}^ 2\times {\mathbb{R}}\) by a properly discontinuous group \(\pi\) of fiber- preserving homeomorphisms (with \(\pi \cong \pi_ 1(O)\), the orbifold fundamental group). These homeomorphisms may have fixed points, so the quotient space is an orbifold in the sense of Thurston and not necessarily a manifold (the present paper contains a general discussion of orbifolds and orbifold fibrations).

It is not surprising that these spaces also occur at various places. For example many quotient spaces of the classical 3-dimensional crystallographic groups possess such a structure (184 out of 219); another example is the class of Montesinos-knots and links \(L\subset S^ 3\); here \(S^ 3\) occurs as a Seifert 3-orbifold, with L equal to the union of the endpoints of all interval-fibers (174 of the 250 ten- crossing knots belong to this class; for a description, see G. Burde and H. Zieschang [Knots, (1985; Zbl 0568.57001), Chapter 12]). The present paper presents the local and global classification of Seifert-3-orbifolds. The classification is similar in spirit to Seifert’s original classification of Seifert fiber spaces, by geometrically defined invariants. However the situation is much more complicated in the present situation: there are many more local types, and the dependence of the various invariants can be quite subtle. The results are too complicated to be stated here. Another more algebraic approach to the classification has been given by H. Zieschang and the reviewer [Math. Ann. 259, 29-51 (1982; Zbl 0466.57002)], in terms of an isomorphism classification of the orbifold fundamental groups \(\pi\). It should be noted that the general theory of P. E. Conner and F. Raymond on injective Seifert fibrations can be applied to these spaces [Bull. Am. Math. Soc. 83, 36-85 (1977; Zbl 0341.57003)]; see also the survey article by K.-B. Lee and F. Raymond [Contemp. Math. 36, 367-425 (1985; 564.57002)]. Analogously to the Seifert 3-manifolds, the Seifert 3- orbifolds possess geometric structures modelled on one of six 3- dimensional geometries [see, e.g., P. Scott, Bull. Lond. Math. Soc. 15, 401-487 (1983; Zbl 0564.57002)].

Reviewer: B.Zimmermann