Localising Dehn’s lemma and the loop theorem in 3-manifolds. (English) Zbl 1067.57009

The authors introduce a “geometric” version of the hierarchies of Haken and Waldhausen (see for example [W. Haken, Acta Math. 105, 245–375 (1961; Zbl 0100.19402); Stud. in Math. 5 (Studies Modern Topol.), 39–98 (1968; Zbl 0194.24902); K. Johannson, Homotopy equivalences of 3-manifolds with boundaries (Lecture Notes in Mathematics 761, Berlin: Springer-Verlag) (1979; Zbl 0412.57007); F. Waldhausen, Ann. Math. (2) 87, 56–88 (1968; Zbl 0157.30603); Ann. Math. (2) 88, 272–280 (1968; Zbl 0167.52103)]. They work with a “very short hierarchy”, which is related to the short hierarchies discussed in [W. Jaco, Lectures on three-manifold topology (Regional Conference Series in Mathematics 43, Providence, R.I.: AMS) (1980; Zbl 0433.57001)], and prove that all orientable Haken 3-manifolds and non-orientable 3-manifolds without 2-sided Klein bottles contain such hierarchies. Then they give a new fine proof of the celebrated Dehn Lemma and Loop Theorem, which uses two basic tools of geometric topology of 3-manifolds, but no covering space theory. One technique is the method of normal surfaces, first developed by Haken (see the papers quoted above) after the earlier work of Kneser in [H. Kneser, Jahresbericht D. M. V. 38, 248–260 (1929; JFM 55.0311.03)]. The other method uses the concept of a simple hierarchy, introduced by Waldhausen in his second paper mentioned above for the solution of the word problem in the fundamental groups of Haken 3-manifolds. We recall here that the Dehn Lemma and the Loop Theorem are two fundamental results in the topology of 3-manifolds. It is well-known that Dehn’s Lemma was originally formulated by Dehn and the first correct proof was given by C. D. Papakyriakopoulos in his famous papers [Ann. Math. (2) 66, 1–26 (1957; Zbl 0078.16402) and Proc. Lond. Math. Soc., III. Ser. 7, 281–299 (1957; Zbl 0078.16305)]. As remarked above, the present authors give a new proof of the classical theorems of Papakyriakopoulos which does not use towers of coverings but their method uses the concept of boundary patterns of hierarchies as developed by Johannson.


57M35 Dehn’s lemma, sphere theorem, loop theorem, asphericity (MSC2010)
57N10 Topology of general \(3\)-manifolds (MSC2010)
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