a survey on seifert fiber space theorem

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A survey on Seifert fiber space TheoremJean-Philippe Preacuteaux

To cite this versionJean-Philippe Preacuteaux A survey on Seifert fiber space Theorem International Scholarly ResearchNotices Hindawi 2014 2014 694106 (9 p) 1011552014694106 hal-01255703

Review ArticleA Survey on Seifert Fiber Space Theorem

Jean-Philippe Preacuteaux

Laboratoire drsquoAnalyse Topologie et Probabilites UMR CNRS 7353 Aix-Marseille Universite 39 rue FJoliot-Curie13453 Marseille Cedex 13 France

Correspondence should be addressed to Jean-Philippe Preaux preauxcmiuniv-mrsfr

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

Academic Editors G Martin and J Porti

Copyright copy 2014 Jean-Philippe PreauxThis is an open access article distributed under theCreativeCommonsAttribution Licensewhich 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 itsgeneralizations

1 Introduction

The reader is supposed to be familiar with the topology of 3-manifolds Basic courses can be found in reference books (cf[1ndash3])

In the topology of low-dimensional manifolds (dimen-sion at most 3) the fundamental group or 120587

1 plays a central

key role On the one hand several of the main topologicalproperties of 2 and 3-manifolds can be rephrased in term ofproperties of the fundamental group and on the other hand inthe generic cases the120587

1fully determines their homeomorphic

type That the 1205871generally determines their homotopy

type follows from the fact that they are generically 119870(120587 1)

(ie their universal covering space is contractible) that thehomotopy type determines generically the homeomorphismtype appears as a rigidity property or informally becausethe lack of dimension prevents the existence of too manymanifolds which contrasts with higher dimensions

This has been well known since a long time for surfacesadvances in the study of 3-manifolds have shown that itremains globally true in dimension 3 For example onecan think of the Poincare conjecture the Dehn and spheretheorems of Papakyriakopoulos the torus theorem therigidity theorem for Haken manifold or Mostowrsquos rigiditytheorem for hyperbolic 3-manifolds and so forth It providesa somehow common paradigm for their study much linkedto combinatorial and geometrical group theory which hasdeveloped into an independent discipline low-dimensionaltopology among the more general topology of manifolds

The first reference book on the subject originated inthe annotated notes that the young student Seifert tookduring the courses of algebraic topology given by ThrelfallIn 1933 Seifert introduces a particular class of 3-manifoldsknown as Seifert manifolds or Seifert fiber spaces Theyhave been since widely studied well understood andhave given a great impact on the modern understand-ing of 3-manifolds They suit many nice properties mostof them being already known since the deep work ofSeifert

Nevertheless one of their main properties the so-called Seifert fiber space theorem has been a long stand-ing conjecture before its proof was completed by a hugecollective work involving Waldhausen Gordon and HeilJaco and Shalen Scott Mess Tukia Casson and Jun-greis and Gabai for about twenty years It has becomeanother example of the characteristic meaning of the 120587

1for

3-manifoldsThe Seifert fiber space conjecture characterizes the Seifert

fiber spaces with infinite 1205871in the class of orientable

irreducible 3-manifolds in terms of a property of theirfundamental groups they contain an infinite cyclic normalsubgroup It has now become a theorem of major importancein the understanding of (compact) 3-manifolds as we furtherexplain

We review here the motivations and applications forthe understanding of 3-manifolds of the Seifert fiber spacetheorem its generalizations for nonorientable 3-manifoldsand PD(3) groups and the various steps in its proof

Hindawi Publishing CorporationISRN GeometryVolume 2014 Article ID 694106 9 pageshttpdxdoiorg1011552014694106

2 ISRN Geometry

2 The Seifert Fiber Space Theorem andIts Applications

21 Reviews on Seifert Fiber Space Seifert fibered spacesoriginally appeared in a paper of Seifert [4] they constitutea large class of 3-manifolds and are totally classified bymeansof a finite set of invariants They have since widely appearedin the literature for playing a central key role in the topologyof compact 3-manifolds and being nowadays (and since theoriginal paper of Seifert) very well known and understoodThey have allowed the development of central conceptsin 3-manifold topology such as the JSJ-decomposition andThurstonrsquos geometrization conjecture

211 Seifert Fiber Spaces Let 119872 and 119873 be two 3-manifoldseach being a disjoint union of a collection of simple closedcurves called fibers A fiber-preserving homeomorphism from119872 to 119873 is a homeomorphism which sends each fiber of 119872onto a fiber of119873 in such case119872 and119873 are said to be fiber-preserving homeomorphic

Let D2 = 119911 isin C |119911| ⩽ 1 denotes the unit disk in thecomplex plane A fibered solid torus is obtained from D2 times

[0 1] by identifying the bottom and top disksD2times0 andD2times1via a homeomorphism (119911 0) 997891rarr (119903(119911) 1) for some rotation119903 119911 997891rarr 119911 times exp(119894120579) with angle 120579 commensurable with 120587Therefore 119903 has a finite order say 119899 + 1 so that the imagesafter identification of the union of intervals⋃119899

119894=0119903119894(119911)times[0 1]

for 119911 isin D2 yield a collection of fibers When 119903 has order gt 1

the core fiber image of 0times [0 1] is said to be exceptional andotherwise to be regular all other fibers are said to be regular

In the original definition of Seifert (the English versionof the original paper of H Seifert introducing Seifert fiberedspaces (translated from German by W Heil) among otherthings he classifies them and computes their fundamentalgroup [5]) a fibration by circles of a closed 3-manifold119872 is adecomposition into a disjoint union of simple closed curvessuch that each fiber has a neighborhood fiber-preservinghomeomorphic to a fibered solid torus For a fiber beingregular or exceptional does not depend neither on thefiber-preserving homeomorphism nor on the neighborhoodinvolved

The definition naturally extends to compact 3-manifoldswith nonempty boundary It requires that the boundary bea disjoint union of fibers therefore all components in theboundary are closed with Euler characteristic 0 and can onlyconsist of tori and Klein bottles

A Seifert fibration is a foliation by circles the converseis true for orientable 3-manifolds it is a result of Epstein[6] that the orientable Seifert fiber spaces are characterizedamong all compact orientable 3-manifolds as those that admita foliation by circles

For nonorientable 3-manifolds the two notions do notagree because of the possibility for a foliation by circles tobe locally fiber-preserving homeomorphic to a fibered solidKlein bottle A fibered solid Klein bottle is obtained fromD2 times

[0 1] by identifyingD2times0 andD2times1 via the homeomorphism(119911 0) 997891rarr (119911 1) The manifold obtained is nonorientable withboundary a Klein bottle and is naturally foliated by circles

The exceptional fibers are the images of 119911 times [0 1] for all119911 isin [minus1 1] they are nonisolated and lie on the exceptional1-sided annulus image of 119909 isin R |119909| ⩽ 1 times [0 1] The regularfibers are the images of all 119911 119911 times [0 1] for all 119911 notin [minus1 1]they wrap 2 times around the exceptional fibers (cf Figure 1)

Modern considerations have pointed out a need toenlarge the original definition of Seifert in order to englobethis phenomenon for nonorientable 3-manifolds In a moremodern terminology a compact 3-manifold is a Seifert fiberspace whenever it admits a foliation by circles We willtalk of a Seifert bundle to emphasize the nuance It is aconsequence of Epsteinrsquos result that such manifolds are thosethat decompose into disjoint unions of fibers with all fibershaving a neighborhood fiber-preserving homeomorphic to afibered solid torus or to a fibered solid Klein bottle the latterappearing only for nonorientable manifolds

As above given a Seifert fibration of 3-manifold all fibersfall in two parts depending only on the fibration the regularand the exceptional fibers the latter can be either isolatedor nonisolated Let 119872 be a Seifert bundle given a Seifertfibration of 119872 the space of fibers obtained by identifyingeach fiber to a point is a surface 119861 called the base of thefibration This surface is naturally endowed with a structureof orbifold Its singularities consists only of a finite number ofcone points (which are the images of the isolated exceptionalfibers) and of a finite number of reflector circles and lines(which are the images of the nonisolated exceptional fibers)This base orbifold is an invariant of the Seifert fibration

In a modern terminology involving orbifolds Seifertbundles are those 3-manifolds in the category of 3-orbifoldswhich are circle bundles over a 2-dimensional orbifoldwithout corner reflectors singularity This yields a secondinvariant 119890 the Euler number associated to the bundle (cf[7])

212 Some Topological Properties of Seifert Fiber Spaces ASeifert bundle 119872 is a circle bundle over a 2-dimensionalorbifold and it follows that the fibration lifts to the universalcovering space of119872 into a fibration by circles or lines overan orbifold 119861 without proper covering When the fibers arecircles it can only be 1198782 with at most 2 cone points In suchcase 119872 is obtained by gluing on their boundary two fiberedsolid tori and therefore it is a lens space so that its universalcovering space can only be 1198783 or 1198782 times R When the fibers arelines 119861 can only be one of 1198782 or R2 and is homeomorphicto 1198782 timesR or R3 (see [7] for details)

Proposition 1 Theuniversal cover of a Seifert bundle has totalspace homeomorphic to either R3 1198782 times R or 1198783 Moreover theSeifert fibration lifts in the universal cover to a foliation by lines(in both two former cases) or circles (in the latter case)

Recall that a 3-manifold 119872 is irreducible when everysphere embedded in 119872 bounds a ball An irreducible 3-manifold which does not contain any embedded 2-sided P2

is said to be P2-irreducible Now by Alexanderrsquos theorems 1198783and R3 are irreducible and it follows that any 3-manifoldcovered by 119878

3 or R3 is P2-irreducible So that the only

ISRN Geometry 3

core fiberaround the exceptional

A regular fiberit wraps 3 times

Identify top and bottomafter a rotation

The exceptional fiber

with angle 21205873

A

B

C

O

A

B

C

O

(a)

after a reflexionIdentify top and bottom

A regular fiberit wraps 2 timesaround any exceptional fiber

The annulus made ofexceptional fibers

A

B

A

B

(b)

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

non-P2-irreducible Seifert bundles are covered by 1198782timesR andthere are very few such manifolds (cf [8]) one obtains thefollowing

Theorem 2 With the exceptions ofP3P3 1198782times1198781 and 1198782times1198781a Seifert bundle is irreducible The only irreducible and non-P2-irreducible Seifert fiber bundle is P2 times 119878

1

A 3-manifold119872 is said to beHaken if119872 isP2-irreducibleand either119872 is 1198613 or119872 contains a 2-sided properly embed-ded incompressible surface 119865 Consider a two-sided simplecurve in the base orbifold that consists only of nonsingularpoints and is closed or has its both extremities lying in theboundary Whenever that curve does not bound on one of itssides a disk with at most one cone point its preimage yields aproperly embedded two-sided incompressible surface

Theorem 3 With the exception of lens spaces P3P3 1198781 times 1198782

1198781times1198782 andP2times1198781 a Seifert bundle is either Haken or has base

1198782 and exactly 3 exceptional fibers in this last case119872 is Hakenif and only if119867

1(119872Z) is infinite

Note that a P2-irreducible 3-manifold 119872 whose firsthomology group119867

1(119872Z) is infinite does contain a 2-sided

properly embedded incompressible surface and therefore isHaken nevertheless there exists Haken 3-manifolds = 119861

3

with finite first homology group (examples can be con-structed by performing Dehn obturations on knots)

213 The Cyclic Normal Subgroup of the 1205871of a Seifert

Fiber Space Existence of a Seifert fibration on a manifold119872 has a strong consequence on its fundamental group Wehave seen that the fibration lifts to the universal cover space into a foliation by circles or lines with basis 119861 Theaction of 120587

1(119872) onto preserves the circles or lines of the

foliation so that it induces an action of 1205871(119872) onto 119861 that

defines a homomorphism 120601 from 1205871(119872) onto the orbifold

fundamental group 120587orb1

(119861) of 119861 The kernel ker120601 consists of

those elements that act as the identity on 119861 or equivalentlythat preserve all the fibers of Therefore ker120601 acts freelyon each fiber In case is fibered by lines ker120601 is infinitecyclic In case isS3 ker120601 is finite cyclic and120587

1(119872) is finite

Finally since the covering restricts onto the fibers of tocovers of the fibers of 119872 ker120601 is generated by any regularfiber

Theorem 4 Let 119872 be a Seifert bundle with base orbifold 119861Then one has the short exact sequence

1 997888rarr 119873 997888rarr 1205871(119872) 997888rarr 120587

orb1

(119861) 997888rarr 1 (1)

where 119873 is the cyclic subgroup generated by any regular fiberMoreover119873 is infinite whenever 120587

1(119872) is infinite

In particular from 120587orb1

(119861) and the homomorphism120587orb1

(119861) rarr Aut(119873) (note that Aut(119873) has order 1 or 2) onededuces a finite presentation for any Seifert bundle stronglyrelated to its Seifert fibration (see [5 7 9])

22 The Seifert Fiber Space Theorem

221 Statement of the Seifert Fiber Space Theorem As seenabove an infinite fundamental group of a Seifert bundlecontains a normal infinite cyclic subgroup The Seifert fiberspace theorem (or Sfst) uses this property of the 120587

1to

characterize Seifert fiber spaces in the class of orientableirreducible 3-manifolds with infinite 120587

1 It can be stated as

follows

Theorem 5 (Seifert Fiber Space Theorem 1990s) Let 119872 bean orientable irreducible 3-manifold whose 120587

1is infinite and

contains a nontrivial normal cyclic subgroup Then 119872 is aSeifert fiber space

It has become a theorem with the common work of alarge number of topologists for the Haken case Waldhausen(Haken manifolds whose group has a nontrivial center

4 ISRN Geometry

are Seifert fibered [10]) Gordon and Heil (Sfst partiallyobtained in the Haken case either 119872 is a Seifert fiber spaceor 119872 is obtained by gluing two copies of a twisted 119868-bundle[11]) and Jaco and Shalen (the end of the proof of Sfst inthe Haken case but almost the famous Jaco-Shalen-Johansentheorem [12]) for the non-Haken case the rigidity theoremfor Seifert fiber spaces Scott [13] Mess (reduces the proofof the Sfst to prove that convergence groups of the circleare virtually surface groups [14] unpublished) Tukia (solvespartially the convergence groups conjecture [15]) Cassonand Jungreis (provide an alternate solution independentlyby giving solutions to the case remaining in Tukia theydeduce the Sfst [16]) and Gabai (proves that convergencegroups of the circle are fuchsian groups he deduces the Sfsttorus theorem and geometrization in that case [17]) for theorientable non-Haken case one can also citeMaillot (showedthat groups quasi-isometric to a simply connected completeRiemannian surface are virtually surface groups [18] andextends the strategy of Mess et al to recover the Sfst in caseof 3-orbifolds and in corollary Sfst for 3-manifolds [19])and Bowditch (gives an independent proof of the Sfst itgeneralizes to PD(3) groups [20]) who give an alternate proofincluding the unpublished key result of Mess

It has been generalized to the nonorientable case byWhitten

Theorem 6 (Seifert Bundle Theorem (Sfst generalized tothe nonorientable case) [21]) Let 119872 be a P

2-irreducible 3-

manifold whom 1205871is infinite and contains a nontrivial cyclic

normal subgroup Then119872 is a Seifert bundle

Further Heil and Whitten have generalized the theoremto the nonorientable irreducible 3-manifolds (cf the casenonorientable irreducible non-P2-irreducible of the SfstThey deduce the torus theorem and geometrization in thatcase (modulo fake P

2times 1198781 that is modulo the Poincare

conjecture) [22]) see Theorem 9We can also note that those two theorems can be

generalized in several ways for open 3-manifolds and 3-orbifolds [19] and for PD(3) groups [20] or by weakeningthe condition of existence of a normal Z by the existenceof a nontrivial finite conjugacy class (the condition in Sfstthat the group contains a normal Z can be weakened by thecondition that it contains a nontrivial finite conjugacy classor equivalently that its Von-Neumann algebra is not a factorof type 119868119868 minus 1 It applies also to PD(3) groups [23])

222 Explanations of the Hypotheses Involved Let us nowdiscuss the hypotheses of the theorems

(i) Using the sphere theorem together with classicalarguments of algebraic topology one sees that a P

2-

irreduciblemanifold with infinite1205871has a torsion free

fundamental group So that in both conjectures onecan replace

ldquo whose1205871is infinite and contains a nontrivial

normal cyclic subgrouprdquo byldquo whose 120587

1contains a normal infinite cyclic

subgrouprdquo

(ii) What about nonirreducible orientable 3-manifoldswith Z ⊲ 120587

1(119872) An orientable Seifert fiber space

is either irreducible or homeomorphic to 1198781times 1198782 or

to P3P3 As a consequence of the Kneser-Milnortheorem an orientable nonirreducible 3-manifold119872

whose 1205871contains a nontrivial normal cyclic sub-

group is either 1198781 times 1198782 or 1198721015840119862 with 119872

1015840 irreducibleand 119862 simply connected or its 120587

1is the infinite

dihedral groupZ2lowastZ2 With the Poincare conjecture

(now stated by Perelman)119872 is obtained from 1198781times1198782

P3P3 or froman irreducible 3-manifold by removinga finite number of balls Therefore the result becomesas follows

Theorem7 Let119872 be an orientable nonirreducible 3-manifoldwhose 120587

1contains an infinite cyclic normal subgroupThen the

manifold obtained by filling all spheres in 120597119872 with balls is aSeifert fiber space

(i) What about irreducible and non-P2-irreducible 3-manifolds with Z ⊲ 120587

1(119872) A nonorientable Seifert

fiber space is either P2-irreducible or 1198781times1198782 or P2 times1198781 A large class of irreducible non-P2-irreducible 3-manifolds does not admit a Seifert fibration whereastheir 120587

1contains Z as a normal subgroup We will

see that nevertheless the result generalizes also in thatcase by considering what Heil and Whitten call aSeifert bundle mod P

(ii) What about 3-manifolds with finite 1205871 According to

the Kneser-Milnor decomposition such manifoldsafter eventually filling all spheres in the boundarywith balls and replacing all homotopy balls with ballsbecome irreducible Note that as a consequence ofthe Dehnrsquos lemma their boundary can only consistof S2 and P2 The only nonorientable irreducible3-manifold with finite 120587

1is P2 times 119868 (a result of D

Epstein cf [3] Theorems 95 and 96) All knownorientable irreducible 3-manifolds with finite 120587

1turn

to be elliptic manifolds which are all Seifert fiberedand the orthogonalization conjecture (cf section233 proved by the work of Perelman) has made astatement of that one of its consequences being thePoincare conjecture that all homotopy balls are realballsThus the long standing conjecture has become atheorem

Theorem 8 (Consequence of the Orthogonalization Conjec-ture 2000s) A 3-manifold with finite 120587

1containing no sphere

in its boundary is a closed Seifert fibered space when orientableand P2 times 119868 when nonorientable

23 Motivations Three important questions in 3-dimensional topology have motivated the birth of theSeifert fiber space conjecture (Sfsc for short) At firstwas the center conjecture (1960s) then the torus theoremconjecture (1978) and finally Thurstonrsquos geometrizationconjecture (1980s)

ISRN Geometry 5

231 The Center Conjecture It is the problem 35 in Kirbyrsquoslist of problems in 3-dimensional topology [24] that has beenattributed toThurston

Center Conjecture Let 119872 be an orientable irreducible 3-manifold with infinite 120587

1having a nontrivial center Then119872

is a Seifert fiber spaceThe conjecture is an immediate corollary of the Seifert

fiber space theorem since any cyclic subgroup in the centerof a group 119866 is normal in 119866

It has first been observed and proved for knot comple-ments Murasugi [25] and Neuwirth (proof of the centerconjecture for alternated knots [26]) in 1961 for alternatedknots and then Burde and Zieschang (proof of the cen-ter conjecture for knot complements those whose 120587

1has

center = 1 are the toric knot complements [27]) in 1966for all knots In 1967 Waldhausen [10] has proved themore general case of Haken manifolds The proof has beenachieved in the nineties with that of the Seifert fiber spacetheorem

From a presentation of a Seifert fiber space one shows thatan infinite cyclic normal subgroup in the 120587

1of an orientable

Seifert fiber space is central if and only if the base is orientableA Seifert fiber space with infinite 120587

1and with a nonorientable

base has a centerless 1205871containing a nontrivial normal cyclic

subgroupThe Sfsc generalizes the center conjecture in thatsense

232 The Torus Theorem Conjecture The long-standing so-called ldquotorus theorem conjecturerdquo asserts the following

TorusTheorem Conjecture Let119872 be an orientable irreducible3-manifold with 120587

1(119872) containingZoplusZ as a subgroupThen

either 119872 contains an incompressible torus or 119872 is a (small)Seifert fiber space

In has been proven to be true for Haken manifolds byWaldhausen in 1968 (announced in [28] (announcementof the torus theorem for Haken manifolds)) written andpublished by Feustel in [29 30]) Note that together with thetheory of sufficiently large 3-manifolds developed by Hakenduring the 60s this partial result has finally led to the Jaco-Shalen-Johansen decomposition of Haken 3-manifolds (1979cf [12])

The proof in full generality of the conjecture reduces tothat of the Sfsc help to the ldquostrong torus theoremrdquo provedby Scott in 1978 (the ldquostrong torus theoremrdquo [31])

Strong Torus Theorem Let 119872 be an orientable irreducible 3-manifold with 120587

1(119872) containing Z oplus Z as a subgroup Then

either119872 contains an incompressible torus or1205871(119872) contains

a nontrivial normal cyclic subgroupWith the strong torus theorem the torus theorem con-

jecture follows as a corollary from the Seifert fiber spacetheorem

233 The Geometrization Conjecture A major problem in3-dimensional topology is the classification of 3-manifoldsSurfaces have been classified at the beginning of the nineteenrsquoscentury help to the Euler characteristicThePoincarersquos duality

(cf [2]) has consequence that for any closed 3-manifold theEuler characteristic vanishes to 0 the techniques used indimension 2 cannot be applied in dimension 3

Topological surfaces inherit more structure from Rie-mannian geometryThe uniformization theorem of Riemannshows that they can be endowedwith a complete Riemannianmetric of constant curvature so that they all arise as aquotient space of one of the three 2-dimensional geometricalspaces euclidian E2 elliptic S2 and hyperbolic H2 by adiscrete subgroup of isometry that acts freely

Such property cannot generalize to dimension 3 for thereare known obstructions for example a connected sum ofnonsimply connected 3-manifolds cannot be modeled onone of the geometrical spaces E3 S3 and H3 otherwise anessential sphere would lift to the universal cover into anessential sphere in E3 or H3 (S3 cannot arise since the 120587

1is

infinite) but E3 and H3 are both homeomorphic to R3 anddo not contain any essential sphere by Alexanderrsquos theorem

Nevertheless this fact generalizes to dimension 3 helpto the topological decomposition of 3-manifolds Any ori-entable 3-manifold can be cut canonically along essentialsurfaces spheres (Kneser-Milnor) discs (Dehnrsquos lemma) andtori (Jaco-Shalen-Johanson) so that the pieces obtained areall irreducible with incompressible boundary and are eitherSeifert fiber spaces or do not contain any incompressibletorus

Thurstonrsquos geometrization conjecture asserts that all thepieces obtained in the topological decomposition of anorientable 3-manifold can be endowed in their interiorwith a complete locally homogeneous Riemannian metric orequivalently that their interior are quotients of an homoge-neous Riemannian simply connected 3-manifold by a discretesubgroup of isometry acting freely

Thurston has proved that there are eight homogeneous(nonnecessarily isotropic) geometries in dimension 3 thethree isotropic geometries elliptic S3 euclidian E3 andhyperbolicH3 the two product geometriesS2timesR andH2timesRand the three twisted geometries Nil Sol and the universalcover 119878119871

2R of 119878119871

2R (cf [7])

It turns out that the orientable 3-manifolds modeled onone of the six geometries S3 E3 S2 times R H2 times R Nil and1198781198712R are precisely the Seifert fiber spaces (and the geometry

involved only depends on the Euler characteristic of the baseand on the Euler number e of the bundle see Theorem 53iiin [7]) It follows from that all discrete subgroups of theirisometry group preserve a foliation by lines or circles of theunderlying topological spaces 1198783 1198782 timesR or R3

The strategy for proving the geometrization conjecturehas naturally felt into three conjectures

Conjectures Let119872 be an orientable irreducible 3-manifolds

(1) (Orthogonalization conjecture) If 1205871(119872) is finite

then119872 is elliptic

(2) (Sfsc) If 1205871(119872) is infinite and contains a nontrivial

normal cyclic subgroup then119872 is a Seifert fiber space(Sfsc)

6 ISRN Geometry

(3) If 1205871(119872) is infinite and contains no nontrivial normal

cyclic subgroup then119872 is geometrizable

With the strong torus theorem [31] the third conjecturereduces to conjecture 2 and also to consider the two caseswhen 120587

1(119872) does not containZoplusZ and when119872 contains an

incompressible torus In the latter case the JSJ decompositiontheorem applies A manifold modeled on one of the sevennonhyperbolic geometries has with a few exceptions wellunderstood a 120587

1containing Z oplus Z Finally the conjecture

3 reduces to the conjecture 2 and to the following conjecture

(4) (Hyperbolization conjecture) If 1205871(119872) is infinite and

contains no Z oplus Z then119872 is hyperbolic

Thurston has proved the conjecture to be true for Hakenmanifolds by proving the following

Thurstonrsquos Hyperbolization Theorem Any Haken manifoldwith infinite 120587

1that contains no incompressible torus is

hyperbolicIndeed a Haken 3-manifold with finite 120587

1is a ball and

it follows from Thurstonrsquos hyperbolization theorem togetherwith the Seifert fiber space theorem for Hakenmanifolds thatthe geometrization conjecture is true for Haken 3-manifolds

To prove the geometrization conjecture in all cases itsuffices to prove the orthogonalization conjecture the Sfscand the hyperbolization conjecture for non-Haken orientable3-manifolds

So that the Sfsc appears as one of three parts ofthe geometrization conjecture that the conjecture is truefor orientable irreducible 3-manifolds whose 120587

1contains

an infinite cyclic normal subgroup It is the one usuallyconsidered as the easiest one and has been the first to beproved The two other parts have been proved by the workof Perelman et al following the Hamilton program makinguse of the Ricci flow (cf [32])

3 History of the Proof of the Sfs Theorems

The progress in proving the conjecture has been made asfollows

31 The Haken Orientable Case 1967ndash1979 In 1967 Wald-hausen and Feustel [10 29 30] show that a Haken 3-manifoldhas a 120587

1with nontrivial center if and only if it is a Seifert

fiber space with an orientable base It motivates the Sfscand solves the Haken case when the cyclic normal subgroupis centralThe result is announced byWaldhausen in [10] anda proof appears in two papers of Feustel [29 30]

In 1975 Gordon and Heil [11] show partially the Sfscin the Haken case a Haken manifold 119872 whose 120587

1contains

an infinite cyclic normal subgroup is either a Seifert space oris obtained by gluing two copies of a twisted 119868-bundle alonga nonorientable surface So that they reduce the remainingHaken cases to this case of 3-manifolds

In 1979 Jaco and Shalen [12] and independentlyMacLachlan (unpublished) achieve the proof for theremaining Haken 3-manifolds

32 The Non-Haken Orientable Case 1979ndash1994 Using theHaken case already proved it suffices to restrict oneself to theclosed 3-manifolds The proof has proceeded in several mainsteps

In 1983 Scott [13] by generalizing a result of Waldhausenin the Haken case shows a strong result of rigidity for Seifertfiber spaces

Let 119872 and 119873 be two closed orientable irreducible 3-manifolds with 119873 a Seifert fiber space with infinite 120587

1

If 1205871(119872) and 120587

1(119873) are isomorphic then 119872 and 119873 are

homeomorphicNote that with the sphere theorem and classical argu-

ments of algebraic topology 119872 and 119873 are 119870(120587 1) and theconditions ldquo120587

1(119872) and 120587

1(119873) which are isomorphicrdquo can be

replaced by ldquo119872 and119873 which have the same homotopy typerdquoThat reduces the proof of the Seifert fiber space theorem

to a similar statement involving the fundamental group Theconjecture becomes as follows

(Sfsc) If 119872 is a closed orientable 3-manifold and Γ =

1205871(119872) contains an infinite cyclic normal subgroup then Γ is

the 1205871of a closed orientable Seifert fiber space

Scott remarks also that Γ is the group of a closedorientable Seifert fiber space if and only if ΓZ is the funda-mental group of a 2-dimensional closed orbifold (eventuallynonorientable) an alternate proof is given in the Lemma 153of [20] So finally he reduces the proof of the Seifert fiberspace theorem to the proof of the following conjecture


1205871(119872) contains an infinite cyclic normal subgroup then ΓZ

is the fundamental group of a (closed) 2-orbifoldLate 1980s Mess [14] proves in an unpublished paperLet 119872 be closed orientable and irreducible with Γ =

1205871(119872) containing an infinite cyclic normal subgroup119862Then(i) the covering of 119872 associated to 119862 is homeomorphic

to the open manifold D2 times 1198781

(ii) the covering action of Γ119862 on D2 times 1198781 gives rise to an

almost everywhere defined action onD2times1 followingits terminology Γ119862 is coarse quasi-isometric to theEuclidean or hyperbolic plane

(iii) in the case where Γ119862 is coarse quasi-isometric to theEuclidean plane then it is the group of a 2-orbifoldhence (with the result of Scott) 119872 is a Seifert fiberspace

(iv) in the remaining case where Γ119862 is coarse quasi-isometric to the hyperbolic plane Γ119862 induces anaction on the circle at infinity which makes Γ119862 aconvergence group

A convergence group is a group 119866 acting by orientationpreserving homeomorphism on the circle in such a way thatif 119879 denotes the set of ordered triples (119909 119910 119911) isin 119878

1times 1198781times

1198781 119909 = 119910 = 119911 with 119909 119910 119911 appearing in that order on 119878

1 in thedirect sense the action induced by119866 on119879 is free and properlydiscontinuous

With the work of Mess the proof of the Sfsc reduces tothat of

(Convergence groups conjecture) Convergence groupsare fundamental groups of 2-orbifolds

ISRN Geometry 7

In 1988 Tukia shows (see [15]) that some of the conver-gence groups (when 119879Γ is noncompact and Γ has no torsionelementswith ordergt3) are Fuchsian groups and in particularare 120587orb1

of 2-orbifoldsIn 1992 Gabai shows (see [17]) independently from

other works and in full generality that convergence groupsare Fuchsian groups They act on 119878

1 up to conjugacy inHomeo(1198781) as the restrictions on 119878

1= 120597H2 of their natural

action on H2 That proves the Sfst

In 1994 At the same time Casson and Jungreis showthe cases left remained by Tukia (see [16]) That provesindependently from Gabai the Sfst

The Seifert fiber space theorem is proved and then followthe center conjecture the torus theorem and one of three ofthe geometrization conjectures

In 1999 later Bowditch obtains a different proof of theSfsc (see [20]) Its proof generalizes to other kinds of groupas PD(3) groups (groups of type FP with cohomologicaldimension ⩽ 3 whose cohomology groups satisfy a relationsomewhat analogous to Poincare duality for 3-manifoldsit is conjectured that they coincide with 120587

1of closed P2-

irreducible 3-manifolds with infinite 1205871 see [20])

In 2000 in his thesis Maillot extends the techniques ofMess to establish a proof of the Sfsc in the more generalcases of open 3-manifolds and of 3-orbifolds This resultalready known as a consequence of the Sfsc (as provedby Mess et al) and of the Thurston orbifolds theorem hasthe merit to be proved by using none of these results and tostate as a corollary a complete proof of the Sfsc and of thetechniques of Mess

In 2003 the argument of Mess is pursued by Maillotin [19] where he reduces to groups quasi-isometric to aRiemannian plane and he shows in [18] that they are virtuallysurface groups (and hence Fuchsian groups)

33 The Nonorientable Case 1992ndash94 The solution to theSfsc in the nonorientable case goes back to Whittenimproved by Whitten and Heil

In 1992 Whitten proves in [21] the Sfst for nonori-entable and irreducible 3-manifolds which are not a fakeP2times1198781 andwhose120587

1does not contain anyZ

2lowastZ2 He obtains

in particular the Sfst in the P2-irreducible case

In 1994 Heil and Whitten in [22] characterise thosenonorientable irreducible 3-manifolds which do not containa fakeP

2times119868 and whose 120587

1contains a normalZ subgroup and

possibly Z2lowastZ2 they call them Seifert bundlesmodP they

constitute a larger class than Seifert bundles Consider themanifoldP obtained from two copies ofP2times119868 by a connecteddisk summand that is by gluing aD2times119868 alongD2times0 andD2times1to one boundary components of each copies of P2 times 119868 (seeFigure 2) It is a nonorientable 3-manifold that is irreduciblewith a boundary made of two P2 and one Klein bottle Itsfundamental group is the infinite dihedral group Z

2lowast Z2

which contains a unique infinite cyclic normal subgroupTheKlein bottle in the boundary contains a ldquofiberedrdquo annulus thatis made of two rectangular bands with vertices119860 119861 119862 and119863(as in Figure 2) whose 120587

1embeds in 120587

1(P) onto the normal

A

AB

CD B

CD

P2times I

D2times I

P2times I

Figure 2 The nonorientable 3-manifold P and a fibered annulus inthe Klein bottle component of 120597P

Z so that it can be fibered with fibers that generate the cyclicinfinite normal subgroup

A Seifert bundle mod P is either P or is obtained froma Seifert bundle by gluing on its boundary a finite numberof copies of the manifold P along fibered annuli and suchthat the fibers map onto fibers Therefore the 120587

1contains

an infinite cyclic normal subgroup From the irreducibilityof P and the incompressibility of the fibered annuli followsthat Seifert bundles mod P with the same exceptions as inTheorem 2 are all irreducible

Theorem 9 Let119872 be a nonorientable irreducible 3-manifoldwhich does not contain a fake P

2times 119868 and such that 120587

1(119872)

contains a nontrivial cyclic normal subgroup Then119872 is eitherP2times 119868 or a Seifert bundlemodP

They deduce in the nonorientable case the torus theo-rem ldquoif 119872 is an irreducible nonorientable 3-manifold with1205871(119872) sup Z oplusZ then119872 contains an incompressible torus or

Klein bottlerdquoThe orientation covering space 119873 of a Seifert bundle

modP 119872 becomes a Seifert bundle 119873 once all spheres inits boundary have been filled up with balls The coveringinvolution of 119873 extends to 119873 with isolated fixed points andquotient an orbifold 119872 obtained from 119872 by filling up allprojective planes in its boundary with cones over P2 Theorbifold 119872 is modeled in the sense of orbifolds onto one ofthe 6 Seifertic geometries and finally Seifert bundles mod P

are precisely those nonorientable 3-manifolds that are said tobe modeled (with a noncomplete metric) onto one of the 6geometries S3 E3 S2 times R H2 times R 119873119894119897 and the universalcover of 119878119871

2R

34 With the Poincare Conjecture In 2003ndash06 the work ofPerelman (2003 et al) proves the Poincare conjecture as acorollary of the orthogonalization conjecture Hence eachsimply connected closed 3-manifold is a 3-sphere and thereexists no fake ball or fakeP

2times119868 As already said one can simply

avoid in the Sfs theorem the hypothesis of irreducibilityThemore general statement of the theorem becomes as follows

8 ISRN Geometry

Theorem 10 Let 119872 be a 3-manifold whose 1205871is infinite and

contains a nontrivial cyclic normal subgroup After filling upall spheres in 120597119872with balls one obtains either a connected sumof P2times 119868 with itself or with P

3 or a Seifert bundle modP

Moreover 119872 is a Seifert bundle exactly when 1205871(119872) does not

contain any Z2or equivalently when 120597119872 does not contain any

P2

35 3-Manifold Groups with Infinite Conjugacy Classes 2005It is shown in [4] that in both cases of 3-manifold groupsand of PD(3)-groups the hypothesis in the Sfs and Seifertbundles theorems ldquocontains a nontrivial cyclic subgrouprdquo canbe weakened into ldquocontains a nontrivial finite conjugacy classrdquoor equivalently into ldquohas a Von Neumann algebra which is nota factor of type II-1rdquo The results become

(1) Let 119866 be either an infinite 1205871of a P2-irreducible 3-

manifold or a PD(3)-group that contains a nontrivialfinite conjugacy class Then 119866 is the 120587

1of a Seifert

bundle(2) Let 119866 be an infinite 120587

1of a irreducible 3-manifold If

119866 contains a nontrivial finite conjugacy class then 119866

is the 1205871of a Seifert bundle mod P

The proof of these results makes use of the Sfs theoremsfor 3-manifolds as well as for PD(3) groups

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] H Seifert and WThrelfall Lehrbuch der Topologie 89 Teubner1934

[2] E H Spanier Algebraic Topology McGraw-Hill New York NYUSA 1966

[3] J Hempel 3-Manifolds Princeton University Press PrincetonNJ USA 1976

[4] H Seifert ldquoTopologie dreidimensionaler gefaserter RaumerdquoActa Mathematica vol 60 no 1 pp 147ndash288 1933 An englishtranslation byW Heil in H Seifert andWThrelfall A Textbookof Topology Pure and Applied Mathematics no 89 AcademicPress 1980

[5] H Seifert ldquoTopology of 3-dimensional fibered spacesrdquo ActaMathematica vol 60 no 1 pp 147ndash288 1933

[6] D B A Epstein ldquoPeriodic flows on three-manifoldsrdquo Annals ofMathematics vol 95 pp 66ndash82 1972

[7] P Scott ldquoThe geometries of 3-manifoldsrdquo The Bulletin of theLondon Mathematical Society vol 15 no 5 pp 401ndash487 1983

[8] J L Tollefson ldquoThe compact 3-manifolds covered by 1198782 times R1rdquoProceedings of the American Mathematical Society vol 45 pp461ndash462 1974

[9] P Orlik Seifert Manifolds vol 291 of Lecture Notes in Mathe-matics Springer Berlin Germany 1972

[10] F Waldhausen ldquoGruppen mit Zentrum und 3-dimensionaleMannigfaltigkeitenrdquo Topology vol 6 pp 505ndash517 1967

[11] C M Gordon and W Heil ldquoCyclic normal subgroups offundamental groups of 3-manifoldsrdquoTopology vol 14 no 4 pp305ndash309 1975

[12] W Jaco and P B Shalen Seifert Fibered Spaces in 3-Manifoldsvol 21 ofMemoirs of the AMS no 220 1979

[13] P Scott ldquoThere are no fake Seifert fibre spaces with infinite 1205871rdquo

Annals of Mathematics vol 117 no 1 pp 35ndash70 1983

[14] G Mess ldquoCenters of 3-manifold groups and groups which arecoarse quasi-isometric to planesrdquo 1988 Unpublished

[15] P Tukia ldquoHomeomorphic conjugates of Fuchsian groupsrdquoJournal fur die Reine und Angewandte Mathematik vol 391 pp1ndash54 1988

[16] A Casson and D Jungreis ldquoConvergence groups and Seifertfibered 3-manifoldsrdquo Inventiones Mathematicae vol 118 no 3pp 441ndash456 1994

[17] D Gabai ldquoConvergence groups are Fuchsian groupsrdquo Annals ofMathematics vol 136 no 3 pp 447ndash510 1992

[18] S Maillot ldquoQuasi-isometries of groups graphs and surfacesrdquoCommentarii Mathematici Helvetici vol 76 no 1 pp 29ndash602001

[19] S Maillot ldquoOpen 3-manifolds whose fundamental groups haveinfinite center and a torus theorem for 3-orbifoldsrdquo Transac-tions of the American Mathematical Society vol 355 no 11 pp4595ndash4638 2003

[20] B H Bowditch ldquoPlanar groups and the Seifert conjecturerdquoJournal fur die Reine und Angewandte Mathematik vol 576 pp11ndash62 2004

[21] W Whitten ldquoRecognizing nonorientable Seifert bundlesrdquo Jour-nal of Knot Theory and Its Ramifications vol 1 no 4 pp 471ndash475 1992

[22] W Heil and W Whitten ldquoThe Seifert fiber space conjectureand torus theorem for nonorientable 3-manifoldsrdquo CanadianMathematical Bulletin vol 37 no 4 pp 482ndash489 1994

[23] P de la Harpe and J P Preaux ldquoGroupes fondamentaux desvarietes de dimension 3 et algebres drsquooperateursrdquo Annales de laFaculte des Sciences de Toulouse vol 16 no 3 pp 561ndash589 2007

[24] R Kirby ldquoProblems in low dimensional topologyrdquo 1995httpmathberkeleyedu7Ekirby

[25] KMurasugi ldquoRemarks on torus knotsrdquo Proceedings of the JapanAcademy vol 37 p 222 1961

[26] L Neuwirth ldquoA note on torus knots and links determined bytheir groupsrdquo Duke Mathematical Journal vol 28 pp 545ndash5511961

[27] G Burde and H Zieschang ldquoEine Kennzeichnung der Torus-knotenrdquoMathematische Annalen vol 167 pp 169ndash176 1966

[28] F Waldhausen ldquoOn the determination of some bounded 3-manifolds by their fundamental groups alonerdquo in Proceedings ofthe International Symposium on Topology and Its Applicationspp 331ndash332 Herceg Novi Montenegro 1968

[29] C D Feustel ldquoOn the torus theorem and its applicationsrdquoTransactions of the American Mathematical Society vol 217 pp1ndash43 1976

[30] C D Feustel ldquoOn the torus theorem for closed 3-manifoldsrdquoTransactions of the American Mathematical Society vol 217 pp45ndash57 1976

ISRN Geometry 9

[31] P Scott ldquoA new proof of the annulus and torus theoremsrdquo TheAmerican Journal of Mathematics vol 102 no 2 pp 241ndash2771980

[32] L Bessieres G Besson M Boileau S Maillot and J PortildquoGeometrisation of 3-manifoldsrdquo EMS Tracts in Mathematicsvol 13 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Review ArticleA Survey on Seifert Fiber Space Theorem

Jean-Philippe Preacuteaux

Laboratoire drsquoAnalyse Topologie et Probabilites UMR CNRS 7353 Aix-Marseille Universite 39 rue FJoliot-Curie13453 Marseille Cedex 13 France

Correspondence should be addressed to Jean-Philippe Preaux preauxcmiuniv-mrsfr

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

Academic Editors G Martin and J Porti

Copyright copy 2014 Jean-Philippe PreauxThis is an open access article distributed under theCreativeCommonsAttribution Licensewhich 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 itsgeneralizations

1 Introduction

The reader is supposed to be familiar with the topology of 3-manifolds Basic courses can be found in reference books (cf[1ndash3])

In the topology of low-dimensional manifolds (dimen-sion at most 3) the fundamental group or 120587

1 plays a central

key role On the one hand several of the main topologicalproperties of 2 and 3-manifolds can be rephrased in term ofproperties of the fundamental group and on the other hand inthe generic cases the120587

1fully determines their homeomorphic

type That the 1205871generally determines their homotopy

type follows from the fact that they are generically 119870(120587 1)

(ie their universal covering space is contractible) that thehomotopy type determines generically the homeomorphismtype appears as a rigidity property or informally becausethe lack of dimension prevents the existence of too manymanifolds which contrasts with higher dimensions

This has been well known since a long time for surfacesadvances in the study of 3-manifolds have shown that itremains globally true in dimension 3 For example onecan think of the Poincare conjecture the Dehn and spheretheorems of Papakyriakopoulos the torus theorem therigidity theorem for Haken manifold or Mostowrsquos rigiditytheorem for hyperbolic 3-manifolds and so forth It providesa somehow common paradigm for their study much linkedto combinatorial and geometrical group theory which hasdeveloped into an independent discipline low-dimensionaltopology among the more general topology of manifolds

The first reference book on the subject originated inthe annotated notes that the young student Seifert tookduring the courses of algebraic topology given by ThrelfallIn 1933 Seifert introduces a particular class of 3-manifoldsknown as Seifert manifolds or Seifert fiber spaces Theyhave been since widely studied well understood andhave given a great impact on the modern understand-ing of 3-manifolds They suit many nice properties mostof them being already known since the deep work ofSeifert

Nevertheless one of their main properties the so-called Seifert fiber space theorem has been a long stand-ing conjecture before its proof was completed by a hugecollective work involving Waldhausen Gordon and HeilJaco and Shalen Scott Mess Tukia Casson and Jun-greis and Gabai for about twenty years It has becomeanother example of the characteristic meaning of the 120587

1for

3-manifoldsThe Seifert fiber space conjecture characterizes the Seifert

fiber spaces with infinite 1205871in the class of orientable

irreducible 3-manifolds in terms of a property of theirfundamental groups they contain an infinite cyclic normalsubgroup It has now become a theorem of major importancein the understanding of (compact) 3-manifolds as we furtherexplain

We review here the motivations and applications forthe understanding of 3-manifolds of the Seifert fiber spacetheorem its generalizations for nonorientable 3-manifoldsand PD(3) groups and the various steps in its proof

Hindawi Publishing CorporationISRN GeometryVolume 2014 Article ID 694106 9 pageshttpdxdoiorg1011552014694106

2 ISRN Geometry






119894=0119903119894(119911)times[0 1]

















ISRN Geometry 3





with angle 21205873

A

B

C

O

A

B

C

O

(a)




A

B

A

B

(b)




1









3










1(119872) is finite



1 997888rarr 119873 997888rarr 1205871(119872) 997888rarr 120587

orb1

(119861) 997888rarr 1 (1)








1to



follows


1is infinite and



4 ISRN Geometry




2-irreducible 3-









2-






subgrouprdquo








1is the infinite
















1turn






ISRN Geometry 5







1has



1of an orientable




















1is





2R of 119878119871

2R (cf [7])









6 ISRN Geometry














1is a ball and




1contains








1contains






1

If 1205871(119872) and 120587




1(119872) and 120587


















1times 1198781times





ISRN Geometry 7










1of closed P2-














2lowast Z2


1embeds in 120587


A

AB

CD B

CD

P2times I

D2times I

P2times I




1contains




1(119872)






2R




8 ISRN Geometry






P2




1of a Seifert








References
































ISRN Geometry 9










Volume 2014




Journal of











Journal of


Function Spaces






Algebra









2 ISRN Geometry






119894=0119903119894(119911)times[0 1]

















ISRN Geometry 3





with angle 21205873

A

B

C

O

A

B

C

O

(a)




A

B

A

B

(b)




1









3










1(119872) is finite



1 997888rarr 119873 997888rarr 1205871(119872) 997888rarr 120587

orb1

(119861) 997888rarr 1 (1)








1to



follows


1is infinite and



4 ISRN Geometry




2-irreducible 3-









2-






subgrouprdquo








1is the infinite
















1turn






ISRN Geometry 5







1has



1of an orientable




















1is





2R of 119878119871

2R (cf [7])









6 ISRN Geometry














1is a ball and




1contains








1contains






1

If 1205871(119872) and 120587




1(119872) and 120587


















1times 1198781times





ISRN Geometry 7










1of closed P2-














2lowast Z2


1embeds in 120587


A

AB

CD B

CD

P2times I

D2times I

P2times I




1contains




1(119872)






2R




8 ISRN Geometry






P2




1of a Seifert








References
































ISRN Geometry 9










Volume 2014




Journal of











Journal of


Function Spaces






Algebra









ISRN Geometry 3





with angle 21205873

A

B

C

O

A

B

C

O

(a)




A

B

A

B

(b)




1









3










1(119872) is finite



1 997888rarr 119873 997888rarr 1205871(119872) 997888rarr 120587

orb1

(119861) 997888rarr 1 (1)








1to



follows


1is infinite and



4 ISRN Geometry




2-irreducible 3-









2-






subgrouprdquo








1is the infinite
















1turn






ISRN Geometry 5







1has



1of an orientable




















1is





2R of 119878119871

2R (cf [7])









6 ISRN Geometry














1is a ball and




1contains








1contains






1

If 1205871(119872) and 120587




1(119872) and 120587


















1times 1198781times





ISRN Geometry 7










1of closed P2-














2lowast Z2


1embeds in 120587


A

AB

CD B

CD

P2times I

D2times I

P2times I




1contains




1(119872)






2R




8 ISRN Geometry






P2




1of a Seifert








References
































ISRN Geometry 9










Volume 2014




Journal of











Journal of


Function Spaces






Algebra









4 ISRN Geometry




2-irreducible 3-









2-






subgrouprdquo








1is the infinite
















1turn






ISRN Geometry 5







1has



1of an orientable




















1is





2R of 119878119871

2R (cf [7])









6 ISRN Geometry














1is a ball and




1contains








1contains






1

If 1205871(119872) and 120587




1(119872) and 120587


















1times 1198781times





ISRN Geometry 7










1of closed P2-














2lowast Z2


1embeds in 120587


A

AB

CD B

CD

P2times I

D2times I

P2times I




1contains




1(119872)






2R




8 ISRN Geometry






P2




1of a Seifert








References
































ISRN Geometry 9










Volume 2014




Journal of











Journal of


Function Spaces






Algebra









ISRN Geometry 5







1has



1of an orientable




















1is





2R of 119878119871

2R (cf [7])









6 ISRN Geometry














1is a ball and




1contains








1contains






1

If 1205871(119872) and 120587




1(119872) and 120587


















1times 1198781times





ISRN Geometry 7










1of closed P2-














2lowast Z2


1embeds in 120587


A

AB

CD B

CD

P2times I

D2times I

P2times I




1contains




1(119872)






2R




8 ISRN Geometry






P2




1of a Seifert








References
































ISRN Geometry 9










Volume 2014




Journal of











Journal of


Function Spaces






Algebra









6 ISRN Geometry














1is a ball and




1contains








1contains






1

If 1205871(119872) and 120587




1(119872) and 120587


















1times 1198781times





ISRN Geometry 7










1of closed P2-














2lowast Z2


1embeds in 120587


A

AB

CD B

CD

P2times I

D2times I

P2times I




1contains




1(119872)






2R




8 ISRN Geometry






P2




1of a Seifert








References
































ISRN Geometry 9










Volume 2014




Journal of











Journal of


Function Spaces






Algebra









ISRN Geometry 7










1of closed P2-














2lowast Z2


1embeds in 120587


A

AB

CD B

CD

P2times I

D2times I

P2times I




1contains




1(119872)






2R




8 ISRN Geometry






P2




1of a Seifert








References
































ISRN Geometry 9










Volume 2014




Journal of











Journal of


Function Spaces






Algebra









8 ISRN Geometry






P2




1of a Seifert








References
































ISRN Geometry 9










Volume 2014




Journal of











Journal of


Function Spaces






Algebra









ISRN Geometry 9










Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









a survey on seifert fiber space theorem

Documents