A survey on Theorem Jean-Philippe Préaux

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Jean-Philippe Préaux. A survey on Seifert fiber space Theorem. International Scholarly Research Notices, Hindawi, 2014, 2014, 694106 (9 p.). ￿10.1155/2014/694106￿. ￿hal-01255703￿

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Distributed under a Creative Commons Attribution| 4.0 International License Hindawi Publishing Corporation ISRN Geometry Volume 2014, Article ID 694106, 9 pages http://dx.doi.org/10.1155/2014/694106

Review Article A Survey on Seifert Fiber Space Theorem

Jean-Philippe Préaux

Laboratoire d’Analyse Topologie et Probabilites,´ UMR CNRS 7353, Aix-Marseille Universite,´ 39 rue F.Joliot-Curie, 13453 Marseille Cedex 13, France Correspondence should be addressed to Jean-Philippe Preaux;´ [email protected]

Received 18 September 2013; Accepted 29 October 2013; Published 5 March 2014

Academic Editors: G. Martin and J. Porti

Copyright © 2014 Jean-Philippe Preaux.´ This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We review the history of the proof of the Seifert fiber space theorem, as well as its motivations in 3-manifold topology and its generalizations.

1. Introduction The first reference book on the subject originated in the annotated notes that the young student Seifert took The reader is supposed to be familiar with the topology of 3- during the courses of algebraic topology given by Threlfall. manifolds. Basic courses can be found in reference books (cf. In 1933, Seifert introduces a particular class of 3-manifolds, [1–3]). known as Seifert manifolds or Seifert fiber spaces.They In the topology of low-dimensional manifolds (dimen- have been since widely studied, well understood, and sion at most 3) the fundamental group, or 𝜋1,playsacentral have given a great impact on the modern understand- key role. On the one hand several of the main topological ing of 3-manifolds. They suit many nice properties, most properties of 2 and 3-manifolds can be rephrased in term of of them being already known since the deep work of properties of the fundamental group and on the other hand in Seifert. the generic cases the 𝜋1 fully determines their homeomorphic Nevertheless one of their main properties, the so- type. That the 𝜋1 generally determines their homotopy called Seifert fiber space theorem,hasbeenalongstand- type follows from the fact that they are generically 𝐾(𝜋, 1) ing conjecture before its proof was completed by a huge (i.e. their universal covering space is contractible); that the collective work involving Waldhausen, Gordon and Heil, homotopy type determines generically the homeomorphism Jaco and Shalen, Scott, Mess, Tukia, Casson and Jun- type appears as a rigidity property, or informally because greis, and Gabai, for about twenty years. It has become the lack of dimension prevents the existence of too many another example of the characteristic meaning of the 𝜋1 for manifolds, which contrasts with higher dimensions. 3-manifolds. This has been well known since a long time for surfaces; The Seifert fiber space conjecture characterizes the Seifert advancesinthestudyof3-manifoldshaveshownthatit fiber spaces with infinite 𝜋1 in the class of orientable remains globally true in dimension 3. For example, one irreducible 3-manifolds in terms of a property of their can think of the Poincareconjecture,theDehnandsphere´ fundamental groups: they contain an infinite cyclic normal theorems of Papakyriakopoulos, the torus theorem, the subgroup. It has now become a theorem of major importance rigidity theorem for Haken manifold, or Mostow’s rigidity in the understanding of (compact) 3-manifolds as we further theorem for hyperbolic 3-manifolds, and so forth. It provides explain. a somehow common paradigm for their study, much linked We review here the motivations and applications for to combinatorial and geometrical group theory, which has the understanding of 3-manifolds of the Seifert fiber space developed into an independent discipline: low-dimensional theorem, its generalizations for nonorientable 3-manifolds topology, among the more general topology of manifolds. and PD(3) groups, and the various steps in its proof. 2 ISRN Geometry

2. The Seifert Fiber Space Theorem and The exceptional fibers are the images of 𝑧×[0,1]for all Its Applications 𝑧 ∈ [−1, 1]; they are nonisolated and lie on the exceptional 1-sided annulus image of {𝑥 ∈ R;|𝑥|⩽1}×[0,1].Theregular 2.1. Reviews on Seifert Fiber Space. Seifert fibered spaces fibers are the images of all {𝑧, 𝑧} × [0, 1] for all 𝑧 ∉ [−1, 1]; originally appeared in a paper of Seifert [4]; they constitute they wrap 2 times around the exceptional fibers (cf. Figure 1). a large class of 3-manifolds and are totally classified by means Modern considerations have pointed out a need to of a finite set of invariants. They have since widely appeared enlarge the original definition of Seifert in order to englobe in the literature for playing a central key role in the topology this phenomenon for nonorientable 3-manifolds. In a more of compact 3-manifolds and being nowadays (and since the modern terminology, a compact 3-manifold is a Seifert fiber original paper of Seifert) very well known and understood. space whenever it admits a foliation by circles. We will Theyhaveallowedthedevelopmentofcentralconcepts talk of a Seifert bundle to emphasize the nuance. It is a in 3-manifold topology such as the JSJ-decomposition and consequence of Epstein’s result that such manifolds are those Thurston’s geometrization conjecture. thatdecomposeintodisjointunionsoffiberswithallfibers having a neighborhood fiber-preserving homeomorphic to a fiberedsolidtorusortoafiberedsolidKleinbottle,thelatter 2.1.1. Seifert Fiber Spaces. Let 𝑀 and 𝑁 be two 3-manifolds, appearingonlyfornonorientablemanifolds. each being a disjoint union of a collection of simple closed As above, given a Seifert fibration of 3-manifold, all fibers curves called fibers.A fiber-preserving homeomorphism from fall in two parts, depending only on the fibration: the regular 𝑀 to 𝑁 is a homeomorphism which sends each fiber of 𝑀 and the exceptional fibers; the latter can be either isolated onto a fiber of 𝑁;insuchcase𝑀 and 𝑁 are said to be fiber- or nonisolated. Let 𝑀 be a Seifert bundle; given a Seifert preserving homeomorphic. 2 fibration of 𝑀, the space of fibers, obtained by identifying Let D ={𝑧∈C;|𝑧| ⩽ 1} denotes the unit disk in the 2 each fiber to a point, is a surface 𝐵,calledthebase of the complex plane. A fibered solid torus is obtained from D × [0, 1] D2×0 D2 ×1 fibration. This surface is naturally endowed with a structure by identifying the bottom and top disks and of . Its singularities consists only of a finite number of via a homeomorphism (𝑧, 0) 󳨃→(𝑟(𝑧),1)for some rotation 𝑟:𝑧󳨃→𝑧× (𝑖𝜃) 𝜃 𝜋 cone points (which are the images of the isolated exceptional exp with angle commensurable with . fibers) and of a finite number of reflector circles and lines Therefore 𝑟 has a finite order, say 𝑛+1,sothattheimages 𝑛 𝑖 (whicharetheimagesofthenonisolatedexceptionalfibers). after identification of the union of intervals: ⋃𝑖=0 𝑟 (𝑧)×[0, 1] 2 This base orbifold is an invariant of the Seifert fibration. for 𝑧∈D , yield a collection of fibers. When 𝑟 has order >1 In a modern terminology involving , Seifert thecorefiber,imageof0×[0,1],issaidtobeexceptionaland bundles are those 3-manifolds in the category of 3-orbifolds otherwisetoberegular;allotherfibersaresaidtoberegular. which are circle bundles over a 2-dimensional orbifold In the original definition of Seifert (the English version without corner reflectors singularity. This yields a second of the original paper of H. Seifert introducing Seifert fibered invariant 𝑒,theEulernumberassociatedtothebundle(cf. spaces (translated from German by W. Heil), among other [7]). things he classifies them and computes their fundamental group [5]), a fibration by circles of a closed 3-manifold 𝑀 is a decomposition into a disjoint union of simple closed curves 2.1.2. Some Topological Properties of Seifert Fiber Spaces. A 𝑀 such that each fiber has a neighborhood fiber-preserving Seifert bundle isacirclebundleovera2-dimensional homeomorphic to a fibered solid torus. For a fiber, being orbifold, and it follows that the fibration lifts to the universal 𝑀̃ 𝑀 regular or exceptional does not depend neither on the covering space of into a fibration by circles or lines over an orbifold 𝐵̃ without proper covering. When the fibers are fiber-preserving homeomorphism nor on the neighborhood 2 involved. circles, it can only be 𝑆 with at most 2 cone points. In such The definition naturally extends to compact 3-manifolds case 𝑀 is obtained by gluing on their boundary two fibered solid tori, and therefore it is a lens space, so that its universal with nonempty boundary. It requires that the boundary be 3 2 a disjoint union of fibers; therefore all components in the covering space can only be 𝑆 or 𝑆 × R. When the fibers are ̃ 2 2 ̃ boundary are closed with 0 and can only lines 𝐵 can only be one of 𝑆 or R and 𝑀 is homeomorphic 2 3 consist of tori and Klein bottles. to 𝑆 × R or R (see [7] for details). A Seifert fibration is a foliation by circles; the converse istruefororientable3-manifolds:itisaresultofEpstein Proposition 1. The universal cover of a Seifert bundle has total 3 2 3 [6] that the orientable Seifert fiber spaces are characterized space homeomorphic to either R , 𝑆 × R or 𝑆 .Moreover,the among all compact orientable 3-manifolds as those that admit Seifert fibration lifts in the universal cover to a foliation by lines a foliation by circles. (in both two former cases) or circles (in the latter case). For nonorientable 3-manifolds the two notions do not agree because of the possibility for a foliation by circles to Recall that a 3-manifold 𝑀 is irreducible when every sphere embedded in 𝑀 bounds a ball. An irreducible 3- be locally fiber-preserving homeomorphic to a fibered solid 2 D2 × manifold which does not contain any embedded 2-sided P Klein bottle. A fibered solid Klein bottle is obtained from 2 3 [0, 1] D2×0 D2×1 is said to be P -irreducible. Now by Alexander’s theorems 𝑆 by identifying and via the homeomorphism 3 (𝑧, 0) 󳨃→(𝑧, 1). The manifold obtained is nonorientable, with and R areirreducible,anditfollowsthatany3-manifold 3 3 2 boundary a Klein bottle, and is naturally foliated by circles. covered by 𝑆 or R is P -irreducible. So that the only ISRN Geometry 3

Identify top and bottom after a rotation Identify top and bottom with angle 2𝜋/3 after a reflexion B A O A C B The exceptional fiber The annulus made of exceptional fibers

A regular fiber A regular fiber it wraps 3 times it wraps 2 times around the exceptional around any exceptional fiber core fiber

C B O B A A

(a) (b)

Figure 1: On the left, a fibered solid torus. On the right, a fibered solid Klein bottle.

2 2 non-P -irreducible Seifert bundles are covered by 𝑆 ×R,and those elements that act as the identity on 𝐵̃ or equivalently there are very few such manifolds (cf. [8]); one obtains the that preserve all the fibers of 𝑀̃. Therefore ker 𝜙 acts freely following. on each fiber. In case 𝑀̃ is fibered by lines, ker 𝜙 is infinite ̃ 3 3 3 2 1 2 1 cyclic. In case 𝑀 is S ,ker𝜙 is finite cyclic and 𝜋1(𝑀) is finite. Theorem 2. With the exceptions of P #P , 𝑆 ×𝑆 ,and𝑆 ×𝑆̃ , Finally, since the covering restricts onto the fibers of 𝑀̃ to a Seifert bundle is irreducible. The only irreducible and non- 𝑀 𝜙 P2 P2 ×𝑆1 covers of the fibers of ,ker is generated by any regular -irreducible Seifert fiber bundle is . fiber. 𝑀 𝑀 P2 A3-manifold is said to be Haken if is -irreducible Theorem 4. Let 𝑀 be a Seifert bundle with base orbifold 𝐵. 𝑀 𝐵3 𝑀 and either is or contains a 2-sided properly embed- Then one has the short exact sequence: ded 𝐹. Consider a two-sided simple curve in the base orbifold that consists only of nonsingular orb 1󳨀→𝑁󳨀→𝜋1 (𝑀) 󳨀→ 𝜋 1 (𝐵) 󳨀→ 1, (1) points and is closed or has its both extremities lying in the boundary. Whenever that curve does not bound on one of its where 𝑁 is the cyclic subgroup generated by any regular fiber. sides a disk with at most one cone point, its preimage yields a Moreover 𝑁 is infinite whenever 𝜋1(𝑀) is infinite. properly embedded two-sided incompressible surface. 𝜋orb(𝐵) 3 3 1 2 In particular from 1 and the homomorphism Theorem 3. With the exception of lens spaces, P #P , 𝑆 ×𝑆 , 𝜋orb(𝐵) → (𝑁) (𝑁) 1 2 2 1 1 Aut (note that Aut has order 1 or 2), one 𝑆 ×𝑆̃ ,andP ×𝑆 ,aSeifertbundleiseitherHakenorhasbase 2 deduces a finite presentation for any Seifert bundle, strongly 𝑆 and exactly 3 exceptional fibers; in this last case 𝑀 is Haken related to its Seifert fibration (see5 [ , 7, 9]). if and only if 𝐻1(𝑀, Z) is infinite.

2 Note that a P -irreducible 3-manifold 𝑀 whose first 2.2. The Seifert Fiber Space Theorem 𝐻 (𝑀, Z) homology group 1 is infinite does contain a 2-sided 2.2.1. Statement of the Seifert Fiber Space Theorem. As seen properly embedded incompressible surface and therefore is above an infinite fundamental group of a Seifert bundle =𝐵̸ 3 Haken; nevertheless, there exists Haken 3-manifolds contains a normal infinite cyclic subgroup. The Seifert fiber with finite first homology group (examples can be con- space theorem (or S.f.s.t.) uses this property of the 𝜋1 to structed by performing Dehn obturations on knots). characterizeSeifertfiberspacesintheclassoforientable irreducible 3-manifolds with infinite 𝜋1.Itcanbestatedas follows. 2.1.3. The Cyclic Normal Subgroup of the 𝜋1 of a Seifert Fiber Space. Existence of a Seifert fibration on a manifold Theorem 5 (Seifert Fiber Space Theorem, 1990s). Let 𝑀 be 𝑀 has a strong consequence on its fundamental group. We an orientable irreducible 3-manifold whose 𝜋1 is infinite and have seen that the fibration lifts to the universal cover space contains a nontrivial normal cyclic subgroup. Then 𝑀 is a 𝑀̃ into a foliation by circles or lines with basis 𝐵̃.The ̃ Seifert fiber space. action of 𝜋1(𝑀) onto 𝑀 preserves the circles or lines of the ̃ foliation, so that it induces an action of 𝜋1(𝑀) onto 𝐵 that It has become a theorem with the common work of a defines a homomorphism 𝜙 from 𝜋1(𝑀) onto the orbifold large number of topologists: for the Haken case: Waldhausen orb fundamental group 𝜋1 (𝐵) of 𝐵. The kernel ker 𝜙 consists of (Haken manifolds whose group has a nontrivial center 4 ISRN Geometry are Seifert fibered [10]), Gordon and Heil (S.f.s.t partially (ii) What about nonirreducible orientable 3-manifolds obtained in the Haken case: either 𝑀 is a Seifert fiber space with Z ⊲𝜋1(𝑀)? An orientable Seifert fiber space 1 2 or 𝑀 is obtained by gluing two copies of a twisted 𝐼-bundle is either irreducible or homeomorphic to 𝑆 ×𝑆 or 3 3 [11]),andJacoandShalen(theendoftheproofofS.f.s.t.in to P #P . As a consequence of the Kneser-Milnor the Haken case; but almost the famous Jaco-Shalen-Johansen theorem, an orientable nonirreducible 3-manifold 𝑀 theorem [12]); for the non-Haken case: the rigidity theorem whose 𝜋1 contains a nontrivial normal cyclic sub- 1 2 󸀠 󸀠 for Seifert fiber spaces Scott [13], Mess (reduces the proof group is either 𝑆 ×𝑆 ,or𝑀 #𝐶 with 𝑀 irreducible of the S.f.s.t. to prove that convergence groups of the circle and 𝐶 simply connected, or its 𝜋1 is the infinite arevirtuallysurfacegroups[14], unpublished), Tukia (solves dihedral group Z2 ∗Z2.WiththePoincareconjecture´ 1 2 partially the convergence groups conjecture [15]), Casson (now stated by Perelman) 𝑀 is obtained from 𝑆 ×𝑆 , 3 3 and Jungreis (provide an alternate solution independently P #P or from an irreducible 3-manifold by removing by giving solutions to the case remaining in Tukia; they a finite number of balls. Therefore the result becomes deduce the S.f.s.t. [16]), and Gabai (proves that convergence as follows. groupsofthecirclearefuchsiangroups;hededucestheS.f.s.t., torus theorem and geometrization in that case [17]); for the Theorem 7. Let 𝑀 be an orientable nonirreducible 3-manifold orientable non-Haken case, one can also cite Maillot (showed whose 𝜋1 contains an infinite cyclic normal subgroup. Then the that groups quasi-isometric to a simply connected complete manifold obtained by filling all spheres in 𝜕𝑀 with balls is a Riemannian surface are virtually surface groups [18]and Seifert fiber space. extends the strategy of Mess et al. to recover the S.f.s.t. in case of 3-orbifolds, and in corollary S.f.s.t. for 3-manifolds [19]) 2 and Bowditch (gives an independent proof of the S.f.s.t.; it (i) What about irreducible and non-P -irreducible 3- generalizes to PD(3) groups [20])whogiveanalternateproof manifolds with Z ⊲𝜋1(𝑀)? A nonorientable Seifert 2 1 2 2 including the unpublished key result of Mess. fiber space is either P -irreducible or 𝑆 ×𝑆̃ or P × 1 2 It has been generalized to the nonorientable case by 𝑆 .Alargeclassofirreducible,non-P -irreducible 3- Whitten. manifolds does not admit a Seifert fibration whereas their 𝜋1 contains Z as a normal subgroup. We will Theorem 6 (Seifert Bundle Theorem, (S.f.s.t. generalized to see that nevertheless the result generalizes also in that the nonorientable case) [21]). Let 𝑀 be a P2-irreducible 3- case by considering what Heil and Whitten call a manifold whom 𝜋1 is infinite and contains a nontrivial cyclic Seifert bundle mod P. normal subgroup. Then 𝑀 is a Seifert bundle. (ii) What about 3-manifolds with finite 𝜋1? According to Further Heil and Whitten have generalized the theorem the Kneser-Milnor decomposition, such manifolds, to the nonorientable irreducible 3-manifolds (cf. the case after eventually filling all spheres in the boundary 2 nonorientable irreducible non-P -irreducible of the S.f.s.t. with balls and replacing all homotopy balls with balls, They deduce the torus theorem and geometrization in that become irreducible. Note that as a consequence of 1 case (modulo fake P2 ×𝑆, that is, modulo the Poincare´ the Dehn’s lemma, their boundary can only consist 2 2 conjecture.) [22]); see Theorem 9. of S and P . The only nonorientable irreducible 2 We can also note that those two theorems can be 3-manifold with finite 𝜋1 is P ×𝐼(a result of D. generalized in several ways: for open 3-manifolds and 3- Epstein, cf. [3] Theorems 9.5 and 9.6). All known orbifolds [19], and for PD(3) groups [20], or by weakening orientable irreducible 3-manifolds with finite 𝜋1 turn the condition of existence of a normal Z by the existence to be elliptic manifolds, which are all Seifert fibered, of a nontrivial finite conjugacy class (the condition in S.f.s.t. and the orthogonalization conjecture (cf. section that the group contains a normal Z can be weakened by the 2.3.3 proved by the work of Perelman) has made a condition that it contains a nontrivial finite conjugacy class, statementofthat,oneofitsconsequencesbeingthe or equivalently that its Von-Neumann algebra is not a factor Poincareconjecturethatallhomotopyballsarereal´ of type 𝐼𝐼−1.ItappliesalsotoPD(3)groups[23]). balls. Thus the long standing conjecture has become a theorem. 2.2.2. Explanations of the Hypotheses Involved. Let us now discuss the hypotheses of the theorems. Theorem 8 (Consequence of the Orthogonalization Conjec- ture, 2000s). A3-manifoldwithfinite𝜋1 containing no sphere (i) Using the sphere theorem together with classical in its boundary is a closed Seifert fibered space when orientable arguments of algebraic topology one sees that a P2- 2 and P ×𝐼when nonorientable. irreducible manifold with infinite 𝜋1 has a torsion free fundamental group. So that in both conjectures one can replace: 2.3. Motivations. Three important questions in 3- dimensional topology have motivated the birth of the ... 𝜋 “ whose 1 is infinite and contains a nontrivial Seifert fiber space conjecture (S.f.s.c. for short). At first normal cyclic subgroup.” by: was the center conjecture (1960s), then the torus theorem “...whose 𝜋1 contains a normal infinite cyclic conjecture (1978), and finally Thurston’s geometrization subgroup.” conjecture (1980s). ISRN Geometry 5

2.3.1. The Center Conjecture. It is the problem 3.5 in Kirby’s (cf. [2]) has consequence that for any closed 3-manifold the list of problems in 3-dimensional topology [24]thathasbeen Euler characteristic vanishes to 0; the techniques used in attributed to Thurston. dimension 2 cannot be applied in dimension 3. Topological surfaces inherit more structure from Rie- Center Conjecture.Let𝑀 be an orientable irreducible 3- mannian geometry. The uniformization theorem of Riemann manifold with infinite 𝜋1 having a nontrivial center. Then 𝑀 shows that they can be endowed with a complete Riemannian is a Seifert fiber space. metric of constant curvature, so that they all arise as a The conjecture is an immediate corollary of the Seifert quotient space of one of the three 2-dimensional geometrical 2 2 2 fiber space theorem since any cyclic subgroup in the center spaces: euclidian E ,ellipticS , and hyperbolic H ,bya 𝐺 𝐺 of a group is normal in . discrete subgroup of isometry that acts freely. It has first been observed and proved for comple- Such property cannot generalize to dimension 3, for there ments: Murasugi [25] and Neuwirth (proof of the center are known obstructions: for example, a connected sum of conjecture for alternated knots [26]) in 1961 for alternated nonsimply connected 3-manifolds cannot be modeled on 3 3 3 knots and then Burde and Zieschang (proof of the cen- one of the geometrical spaces E , S ,andH ;otherwisean 𝜋 ter conjecture for knot complements: those whose 1 has essential sphere would lift to the universal cover into an =1̸ 3 3 3 center are the toric knot complements [27]) in 1966 essential sphere in E or H (S cannot arise since the 𝜋1 is 3 3 3 for all knots. In 1967 Waldhausen [10]hasprovedthe infinite), but E and H are both homeomorphic to R and more general case of Haken manifolds. The proof has been do not contain any essential sphere by Alexander’s theorem. achieved in the nineties with that of the Seifert fiber space Nevertheless, this fact generalizes to dimension 3 help theorem. to the topological decomposition of 3-manifolds. Any ori- From a presentation of a Seifert fiber space one shows that entable 3-manifold can be cut canonically along essential 𝜋 an infinite cyclic normal subgroup in the 1 of an orientable surfaces: spheres (Kneser-Milnor), discs (Dehn’s lemma), and Seifertfiberspaceiscentralifandonlyifthebaseisorientable. tori (Jaco-Shalen-Johanson) so that the pieces obtained are 𝜋 A Seifert fiber space with infinite 1 and with a nonorientable all irreducible with incompressible boundary and are either 𝜋 base has a centerless 1 containing a nontrivial normal cyclic Seifert fiber spaces or do not contain any incompressible subgroup. The S.f.s.c. generalizes the center conjecture in that torus. sense. Thurston’s geometrization conjecture asserts that all the pieces obtained in the topological decomposition of an 2.3.2. The Torus Theorem Conjecture. The long-standing so- orientable 3-manifold can be endowed in their interior called “torus theorem conjecture” asserts the following. with a complete locally homogeneous Riemannian metric or equivalently that their interior are quotients of an homoge- 𝑀 Torus Theorem Conjecture.Let be an orientable irreducible neous Riemannian simply connected 3-manifold by a discrete 𝜋 (𝑀) Z⊕Z 3-manifold with 1 containing as a subgroup. Then subgroup of isometry acting freely. 𝑀 𝑀 either contains an incompressible torus or is a (small) Thurston has proved that there are eight homogeneous Seifert fiber space. (nonnecessarily isotropic) geometries in dimension 3: the 3 3 InhasbeenproventobetrueforHakenmanifoldsby three isotropic geometries, elliptic S ,euclidianE ,and 3 2 2 Waldhausen in 1968 (announced in [28] (announcement hyperbolic H , the two product geometries, S ×R and H ×R, of the torus theorem for Haken manifolds.)), written and and the three twisted geometries, Nil, Sol,andtheuniversal published by Feustel in [29, 30]).Notethattogetherwiththe ̃ cover 𝑆𝐿 2R of 𝑆𝐿 2R (cf. [7]). theory of sufficiently large 3-manifolds developed by Haken It turns out that the orientable 3-manifolds modeled on during the 60s, this partial result has finally led to the Jaco- 3 3 2 2 one of the six geometries S , E , S × R, H × R, Nil,and Shalen-Johansen decomposition of Haken 3-manifolds (1979, 𝑆𝐿̃R cf. [12]). 2 are precisely the Seifert fiber spaces (and the geometry The proof in full generality of the conjecture reduces to involved only depends on the Euler characteristic of the base that of the S.f.s.c. help to the “strong torus theorem” proved and on the Euler number e of the bundle, see Theorem 5.3.ii by Scott in 1978 (the “strong torus theorem” [31]). in [7]). It follows from that all discrete subgroups of their isometry group preserve a foliation by lines or circles of the 3 2 3 Strong Torus Theorem.Let𝑀 be an orientable irreducible 3- underlying topological spaces 𝑆 , 𝑆 × R or R . manifold, with 𝜋1(𝑀) containing Z ⊕ Z as a subgroup. Then The strategy for proving the geometrization conjecture either 𝑀 contains an incompressible torus, or 𝜋1(𝑀) contains has naturally felt into three conjectures. a nontrivial normal cyclic subgroup. With the strong torus theorem, the torus theorem con- Conjectures.Let𝑀 be an orientable irreducible 3-manifolds. jecture follows as a corollary from the Seifert fiber space theorem. (1) (Orthogonalization conjecture). If 𝜋1(𝑀) is finite, then 𝑀 is elliptic. 2.3.3. The Geometrization Conjecture. Amajorproblemin 3-dimensional topology is the classification of 3-manifolds. (2) (S.f.s.c.). If 𝜋1(𝑀) is infinite and contains a nontrivial Surfaces have been classified at the beginning of the nineteen’s normal cyclic subgroup then 𝑀 is a Seifert fiber space century help to the Euler characteristic. The Poincare’sduality´ (S.f.s.c.) 6 ISRN Geometry

(3) If 𝜋1(𝑀) is infinite and contains no nontrivial normal 3.2. The Non-Haken Orientable Case: 1979–1994. Using the cyclic subgroup then 𝑀 is geometrizable. Haken case already proved, it suffices to restrict oneself to the closed 3-manifolds. The proof has proceeded in several main With the strong torus theorem [31]thethirdconjecture steps. reducestoconjecture2andalsotoconsiderthetwocases In 1983,Scott[13], by generalizing a result of Waldhausen when 𝜋1(𝑀) does not contain Z⊕Z and when 𝑀 contains an in the Haken case, shows a strong result of rigidity for Seifert incompressible torus. In the latter case, the JSJ decomposition fiber spaces. theorem applies. A manifold modeled on one of the seven Let 𝑀 and 𝑁 be two closed orientable irreducible 3- nonhyperbolic geometries has, with a few exceptions well manifolds, with 𝑁 a Seifert fiber space with infinite 𝜋1. 𝜋 Z ⊕ Z understood, a 1 containing .Finally,theconjecture If 𝜋1(𝑀) and 𝜋1(𝑁) are isomorphic, then 𝑀 and 𝑁 are 3 reduces to the conjecture 2 and to the following conjecture. homeomorphic. Note that with the sphere theorem and classical argu- 𝜋 (𝑀) (4) (Hyperbolization conjecture). If 1 is infinite and ments of algebraic topology, 𝑀 and 𝑁 are 𝐾(𝜋, 1) and the Z ⊕ Z 𝑀 contains no then is hyperbolic. conditions “𝜋1(𝑀) and 𝜋1(𝑁) which are isomorphic” can be replaced by “𝑀 and 𝑁 which have the same homotopy type.” Thurston has proved the conjecture to be true for Haken That reduces the proof of the Seifert fiber space theorem manifolds by proving the following. to a similar statement involving the fundamental group. The Thurston’s Hyperbolization Theorem.AnyHakenmanifold conjecture becomes as follows. 𝜋 (S.f.s.c.) If 𝑀 is a closed orientable 3-manifold and Γ= with infinite 1 that contains no incompressible torus is 𝜋 (𝑀) Γ hyperbolic. 1 contains an infinite cyclic normal subgroup, then is the 𝜋1 of a closed orientable Seifert fiber space. Indeed a Haken 3-manifold with finite 𝜋1 is a ball and Scott remarks also that Γ is the group of a closed it follows from Thurston’s hyperbolization theorem together Γ/Z with the Seifert fiber space theorem for Haken manifolds that orientable Seifert fiber space if and only if is the funda- the geometrization conjecture is true for Haken 3-manifolds. mental group of a 2-dimensional closed orbifold (eventually To prove the geometrization conjecture in all cases it nonorientable); an alternate proof is given in the Lemma 15.3 suffices to prove the orthogonalization conjecture, the S.f.s.c., of [20]. So finally, he reduces the proof of the Seifert fiber and the hyperbolization conjecture for non-Haken orientable space theorem to the proof of the following conjecture. (S.f.s.c.) If 𝑀 is a closed orientable 3-manifold and Γ= 3-manifolds. 𝜋 (𝑀) Γ/Z So that the S.f.s.c. appears as one of three parts of 1 contains an infinite cyclic normal subgroup, then the geometrization conjecture that the conjecture is true is the fundamental group of a (closed) 2-orbifold. Late 1980s,Mess[14]provesinanunpublishedpaper. for orientable irreducible 3-manifolds whose 𝜋1 contains Let 𝑀 be closed orientable and irreducible with Γ= aninfinitecyclicnormalsubgroup.Itistheoneusually 𝜋 (𝑀) 𝐶 considered as the easiest one and has been the first to be 1 containing an infinite cyclic normal subgroup .Then proved. The two other parts have been proved by the work (i) the covering of 𝑀 associated to 𝐶 is homeomorphic 2 1 of Perelman et al. following the Hamilton program making totheopenmanifoldD ×𝑆 ; 2 1 use of the Ricci flow (cf. [32]). (ii) the covering action of Γ/𝐶 on D ×𝑆 gives rise to an 2 almost everywhere defined action on D ×1:following 3. History of the Proof of the S.f.s. Theorems its terminology Γ/𝐶 is coarse quasi-isometric to the Euclidean or hyperbolic plane; The progress in proving the conjecture has been made as (iii) in the case where Γ/𝐶 is coarse quasi-isometric to the follows. Euclidean plane, then it is the group of a 2-orbifold; hence, (with the result of Scott) 𝑀 is a Seifert fiber 3.1. The Haken Orientable Case: 1967–1979. In 1967,Wald- space; hausen and Feustel [10, 29, 30]showthataHaken3-manifold Γ/𝐶 𝜋 (iv) in the remaining case where is coarse quasi- has a 1 with nontrivial center if and only if it is a Seifert isometric to the hyperbolic plane, Γ/𝐶 induces an fiberspacewithanorientablebase.ItmotivatestheS.f.s.c. action on the circle at infinity, which makes Γ/𝐶 a and solves the Haken case when the cyclic normal subgroup convergence group. is central. The result is announced by Waldhausen in [10]and a proof appears in two papers of Feustel [29, 30]. A convergence group is a group 𝐺 acting by orientation In 1975,GordonandHeil[11] show partially the S.f.s.c. preserving homeomorphism on the circle, in such a way that 𝑇 (𝑥,𝑦,𝑧)1 ∈𝑆 ×𝑆1 × in the Haken case: a Haken manifold 𝑀 whose 𝜋1 contains if denotes the set of ordered triples: 1 1 an infinite cyclic normal subgroup is either a Seifert space or 𝑆 , 𝑥 =𝑦̸ =𝑧̸ ,with𝑥, 𝑦, 𝑧 appearing in that order on 𝑆 in the is obtained by gluing two copies of a twisted 𝐼-bundle along direct sense, the action induced by 𝐺 on 𝑇 is free and properly a nonorientable surface. So that they reduce the remaining discontinuous. Haken cases to this case of 3-manifolds. With the work of Mess the proof of the S.f.s.c. reduces to In 1979, Jaco and Shalen [12] and independently that of: MacLachlan (unpublished) achieve the proof for the (Convergence groups conjecture). Convergence groups remaining Haken 3-manifolds. are fundamental groups of 2-orbifolds. ISRN Geometry 7

In 1988,Tukiashows(see[15]) that some of the conver- D C gence groups (when 𝑇/Γ is noncompact and Γ has no torsion elements with order >3) are Fuchsian groups and in particular P2 ×I orb are 𝜋1 of 2-orbifolds. In 1992,Gabaishows(see[17]) independently from other works and in full generality that convergence groups 2 1 C D A B D ×I areFuchsiangroups.Theyacton𝑆 , up to conjugacy in 1 1 2 Homeo(𝑆 ),astherestrictionson𝑆 =𝜕H of their natural 2 H action on . That proves the S.f.s.t. P2 ×I In 1994, At the same time Casson and Jungreis show B A the cases left remained by Tukia (see [16]). That proves independently from Gabai the S.f.s.t. The Seifert fiber space theorem is proved and then follow Figure 2: The nonorientable 3-manifold P and a fibered annulus in the center conjecture, the torus theorem, and one of three of the Klein bottle component of 𝜕P. the geometrization conjectures. In 1999, later, Bowditch obtains a different proof of the S.f.s.c. (see [20]). Its proof generalizes to other kinds of group as PD(3) groups (groups of type FP with cohomological Z, so that it can be fibered with fibers that generate the cyclic dimension ⩽3whose cohomology groups satisfy a relation infinite normal subgroup. somewhat analogous to Poincare´ duality for 3-manifolds; A Seifert bundle mod P is either P or is obtained from 𝜋 P2 it is conjectured that they coincide with 1 of closed - a Seifert bundle by gluing on its boundary a finite number 𝜋 irreducible 3-manifolds with infinite 1;see[20]). of copies of the manifold P along fibered annuli and such In 2000, in his thesis Maillot extends the techniques of that the fibers map onto fibers. Therefore, the 𝜋1 contains Mess to establish a proof of the S.f.s.c. in the more general an infinite cyclic normal subgroup. From the irreducibility 3 cases of open 3-manifolds and of -orbifolds. This result of P and the incompressibility of the fibered annuli follows already known as a consequence of the S.f.s.c. (as proved that Seifert bundles mod P, with the same exceptions as in by Mess et al.) and of the Thurston orbifolds theorem, has Theorem 2, are all irreducible. the merit to be proved by using none of these results and to state as a corollary a complete proof of the S.f.s.c. and of the Theorem 9. Let 𝑀 be a nonorientable irreducible 3-manifold techniques of Mess. which does not contain a fake P2 ×𝐼and such that 𝜋1(𝑀) In 2003, the argument of Mess is pursued by Maillot contains a nontrivial cyclic normal subgroup. Then 𝑀 is either in [19] where he reduces to groups quasi-isometric to a P2 ×𝐼or a Seifert bundle mod P. Riemannian plane, and he shows in [18]thattheyarevirtually surface groups (and hence Fuchsian groups). They deduce in the nonorientable case the torus theo- rem: “if 𝑀 is an irreducible nonorientable 3-manifold with 𝜋1(𝑀) ⊃ Z ⊕ Z,then𝑀 contains an incompressible torus or 3.3. The Nonorientable Case: 1992–94. The solution to the Klein bottle.” S.f.s.c. in the nonorientable case goes back to Whitten, The orientation covering space 𝑁 of a Seifert bundle improved by Whitten and Heil. mod P, 𝑀, becomes a Seifert bundle 𝑁 once all spheres in In 1992, Whitten proves in [21] the S.f.s.t. for nonori- its boundary have been filled up with balls. The covering entable and irreducible 3-manifolds, which are not a fake 1 involution of 𝑁 extends to 𝑁 with isolated fixed points and P2×𝑆 ,andwhose𝜋1 does not contain any Z2∗Z2.Heobtains quotient an orbifold 𝑀,obtainedfrom𝑀 by filling up all in particular the S.f.s.t. in the P2-irreducible case. 2 projective planes in its boundary with cones over P .The In1994,HeilandWhittenin[22]characterisethose 𝑀 nonorientable irreducible 3-manifolds which do not contain orbifold ismodeledinthesenseoforbifoldsontooneof the 6 Seifertic geometries, and finally Seifert bundles mod P afakeP2 ×𝐼and whose 𝜋1 contains a normal Z subgroup and are precisely those nonorientable 3-manifolds that are said to possibly Z2 ∗ Z2:theycallthem Seifert bundles mod P;they be modeled (with a noncomplete metric) onto one of the 6 constitute a larger class than Seifert bundles. Consider the 3 3 2 2 2 S E S × R H × R 𝑁𝑖𝑙 manifold P obtained from two copies of P ×𝐼by a connected geometries: , , , , ,andtheuniversal 2 2 2 𝑆𝐿̃R disksummand,thatis,bygluingaD ×𝐼 along D ×0 and D ×1 cover of 2 . 2 to one boundary components of each copies of P ×𝐼(see Figure 2). It is a nonorientable 3-manifold that is irreducible 3.4. With the PoincareConjecture.´ In 2003–06,theworkof 2 with a boundary made of two P and one Klein bottle. Its Perelman (2003 et al.) proves the Poincareconjectureasa´ fundamental group is the infinite dihedral group Z2 ∗ Z2, corollary of the orthogonalization conjecture. Hence, each which contains a unique infinite cyclic normal subgroup. The simply connected closed 3-manifold is a 3-sphere and there Kleinbottleintheboundarycontainsa“fibered”annulusthat exists no fake ball or fake P2×𝐼. As already said one can simply is made of two rectangular bands with vertices 𝐴, 𝐵,,and 𝐶 𝐷 avoid in the S.f.s. theorem the hypothesis of irreducibility. The (as in Figure 2)whose𝜋1 embeds in 𝜋1(P) onto the normal more general statement of the theorem becomes as follows. 8 ISRN Geometry

Theorem 10. Let 𝑀 be a 3-manifold whose 𝜋1 is infinite and [11] C. M. Gordon and W. Heil, “Cyclic normal subgroups of contains a nontrivial cyclic normal subgroup. After filling up fundamental groups of 3-manifolds,” Topology,vol.14,no.4,pp. all spheres in 𝜕𝑀 with balls one obtains either a connected sum 305–309, 1975. P ×𝐼 P P of 2 with itself or with 3,oraSeifertbundlemod . [12] W. Jaco and P. B. Shalen, Seifert Fibered Spaces in 3-Manifolds, Moreover, 𝑀 is a Seifert bundle exactly when 𝜋1(𝑀) does not vol. 21 of Memoirs of the AMS, no. 220, 1979. Z 𝜕𝑀 contain any 2 or equivalently when does not contain any 𝜋 P [13] P. Scott, “There are no fake Seifert fibre spaces with infinite 1,” 2. Annals of Mathematics,vol.117,no.1,pp.35–70,1983. [14] G. Mess, “Centers of 3-manifold groups, and groups which are 3.5. 3-Manifold Groups with Infinite Conjugacy Classes. 2005. coarse quasi-isometric to planes,” 1988, Unpublished. Itisshownin[4] that in both cases of 3-manifold groups and of PD(3)-groups the hypothesis in the S.f.s. and Seifert [15] P. Tukia, “Homeomorphic conjugates of Fuchsian groups,” bundles theorems: “contains a nontrivial cyclic subgroup”can Journal fur¨ die Reine und Angewandte Mathematik,vol.391,pp. be weakened into “contains a nontrivial finite conjugacy class” 1–54, 1988. or equivalently into “has a Von Neumann algebra which is not [16] A. Casson and D. Jungreis, “Convergence groups and Seifert a factor of type II-1”. Th e r e s u l t s b e c o m e: fibered 3-manifolds,” Inventiones Mathematicae,vol.118,no.3, pp.441–456,1994. 𝐺 𝜋 P2 (1) Let be either an infinite 1 of a -irreducible 3- [17] D. Gabai, “Convergence groups are Fuchsian groups,” Annals of manifold, or a PD(3)-group, that contains a nontrivial Mathematics,vol.136,no.3,pp.447–510,1992. finite conjugacy class. Then 𝐺 is the 𝜋1 of a Seifert [18] S. Maillot, “Quasi-isometries of groups, graphs and surfaces,” bundle. Commentarii Mathematici Helvetici,vol.76,no.1,pp.29–60, (2) Let 𝐺 be an infinite 𝜋1 of a irreducible 3-manifold. If 2001. 𝐺 𝐺 contains a nontrivial finite conjugacy class, then [19] S. Maillot, “Open 3-manifolds whose fundamental groups have 𝜋 P is the 1 of a Seifert bundle mod . infinite center, and a torus theorem for 3-orbifolds,” Transac- tions of the American Mathematical Society,vol.355,no.11,pp. The proof of these results makes use of the S.f.s. theorems 4595–4638, 2003. for 3-manifolds as well as for PD(3) groups. [20] B. H. Bowditch, “Planar groups and the Seifert conjecture,” Journal fur¨ die Reine und Angewandte Mathematik,vol.576,pp. Conflict of Interests 11–62, 2004. The author declares that there is no conflict of interests [21] W. Whitten, “Recognizing nonorientable Seifert bundles,” Jour- nal of Knot Theory and Its Ramifications,vol.1,no.4,pp.471– regarding the publication of this paper. 475, 1992. [22] W. Heil and W. Whitten, “The Seifert fiber space conjecture References and torus theorem for nonorientable 3-manifolds,” Canadian Mathematical Bulletin,vol.37,no.4,pp.482–489,1994. [1] H. Seifert and W. Threlfall, Lehrbuch der Topologie, 89 Teubner, 1934. [ 2 3 ] P. d e l a Har p e an d J. P. Pr eaux,´ “Groupes fondamentaux des [2]E.H.Spanier,Algebraic Topology, McGraw-Hill, New York, NY, variet´ es´ de dimension 3 et algebres` d’operateurs,”´ Annales de la USA, 1966. Faculte´ des Sciences de Toulouse,vol.16,no.3,pp.561–589,2007. [3] J. Hempel, 3-Manifolds, Princeton University Press, Princeton, [24] R. Kirby, “Problems in low dimensional topology,” 1995, NJ, USA, 1976. http://math.berkeley.edu/%7Ekirby/. [4] H. Seifert, “Topologie dreidimensionaler gefaserter Raume,”¨ [25] K. Murasugi, “Remarks on torus knots,” Proceedings of the Japan Acta Mathematica,vol.60,no.1,pp.147–288,1933,Anenglish Academy,vol.37,p.222,1961. translationbyW.HeilinH.SeifertandW.Threlfall,ATextbook [26] L. Neuwirth, “A note on torus knots and links determined by of Topology, Pure and Applied Mathematics no. 89, Academic their groups,” Duke Mathematical Journal,vol.28,pp.545–551, Press, 1980. 1961. [5] H. Seifert, “Topology of 3-dimensional fibered spaces,” Acta Mathematica, vol. 60, no. 1, pp. 147–288, 1933. [27] G. Burde and H. Zieschang, “Eine Kennzeichnung der Torus- knoten,” Mathematische Annalen, vol. 167, pp. 169–176, 1966. [6] D. B. A. Epstein, “Periodic flows on three-manifolds,” Annals of Mathematics,vol.95,pp.66–82,1972. [28] F. Waldhausen, “On the determination of some bounded 3- [7] P. Scott, “The geometries of 3-manifolds,” The Bulletin of the manifolds by their fundamental groups alone,”in Proceedings of London Mathematical Society,vol.15,no.5,pp.401–487,1983. the International Symposium on Topology and Its Applications, 2 1 pp. 331–332, Herceg Novi, Montenegro, 1968. [8] J. L. Tollefson, “The compact 3-manifolds covered by 𝑆 × R ,” Proceedings of the American Mathematical Society,vol.45,pp. [29] C. D. Feustel, “On the torus theorem and its applications,” 461–462, 1974. Transactions of the American Mathematical Society,vol.217,pp. [9] P. Orlik, Seifert Manifolds,vol.291ofLecture Notes in Mathe- 1–43, 1976. matics, Springer, Berlin, Germany, 1972. [30] C. D. Feustel, “On the torus theorem for closed 3-manifolds,” [10] F. Waldhausen, “Gruppen mit Zentrum und 3-dimensionale Transactions of the American Mathematical Society,vol.217,pp. Mannigfaltigkeiten,” Topology,vol.6,pp.505–517,1967. 45–57, 1976. ISRN Geometry 9

[31] P. Scott, “A new proof of the annulus and torus theorems,” The American Journal of Mathematics,vol.102,no.2,pp.241–277, 1980. [32] L. Bessieres,` G. Besson, M. Boileau, S. Maillot, and J. Porti, “Geometrisation of 3-manifolds,” EMS Tracts in Mathematics, vol. 13, 2010. Advances in Advances in Journal of Journal of Operations Research Decision Sciences Applied Mathematics Algebra Probability and Statistics Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014

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