aa r X i v : . [ m a t h . G T ] J un CONTACT STRUCTURES, DEFORMATIONS AND TAUT FOLIATIONS
JONATHAN BOWDEN
Abstract.
Using deformations of foliations to contact structures as well as rigidity proper-ties of Anosov foliations we provide infinite families of examples which show that the spaceof taut foliations in a given homotopy class of plane fields need not be path connected.Similar methods also show that the space of representations of the fundamental group ofa hyperbolic surface to the group of smooth diffeomorphisms of the circle with fixed Eulerclass is in general not path connected. As an important step along the way we resolve thequestion of which universally tight contact structures on Seifert fibered spaces are deforma-tions of taut or Reebless foliations when the genus of the base is positive or the twistingnumber of the contact structure in the sense of Giroux is non-negative. Introduction
In their book on confoliations Eliashberg and Thurston [7] established a fundamental linkbetween the theory of foliations and contact topology, by showing that any foliation that isnot the product foliation on S × S can be C -approximated by a contact structure. Theproof of this result naturally leads to the study of confoliations, which are a generalisationof both contact structures and foliations. Recall that a smooth cooriented 2-plane field ξ = Ker( α ) on an oriented 3-manifold M is a confoliation if α ∧ dα ≥
0. For the mostpart interest has focussed on the contact case, where the study of deformations and isotopyare equivalent in view of Gray’s Stability Theorem. On the other hand many questions inthe deformation theory of foliations or more generally confoliations remain to a large extentunexplored.Rather than considering general confoliations, we will focus on questions concerning thetopology of the space of foliations. In contact topology one has a tight vs. overtwisteddichotomy, which is in some sense mirrored in the theory of foliations by Reebless foliationsand those with Reeb components. In analogy with 3-dimensional contact topology whereone seeks to understand deformation classes of tight contact structures, we will be primarilyconcerned with studying the topology of the space of Reebless and taut foliations and thecontact structures approximating them.It is well known that every contact structure is isotopic to a deformation of a foliation byEtnyre [9] (see also Mori [29]). More precisely, Etnyre showed that for any contact structure ξ there is a smooth 1-parameter family ξ t such that ξ is integrable and ξ t is a contactstructure isotopic to ξ for t >
0. The foliations that Etnyre considers are constructed bycompleting the foliation given by the pages of an open book supporting the contact structure ξ to a genuine foliation by inserting Reeb components in a neighbourhood of the bindingand spiralling accordingly. This led Etnyre to ask whether every universally tight contactstructure on a manifold with infinite fundamental group is a deformation of a Reeblessfoliation. By considering the known criteria for the existence of Reebless foliations on small Date : July 2, 2015.
Seifert fibered spaces, it is easy to see that this is false in general. This was first observedby Lekili and Ozbagci [22]. Nevertheless it is still an interesting problem to determine whichcontact structures can be realised as deformations of Reebless foliations, a problem whichwas already raised by Eliashberg and Thurston in [7]. Furthermore, the counter examplescoming from small Seifert fibered spaces are not completely satisfactory, since the obviousnecessary condition for a manifold to admit a Reebless foliation is that it admits universallytight contact structures for both orientations, and for small Seifert manifolds this is in factequivalent to the existence of a Reebless foliation (cf. Proposition 6.4).In contrast to the case of small Seifert manifolds Etnyre’s original question has a positiveanswer for Seifert fibered spaces whose bases have positive genus. There are two casesdepending on whether the twisting number t ( ξ ) of the contact structure ξ is positive or not(cf. Definition 5.3). Theorem A.
Let ξ be a universally tight contact structure on a Seifert fibered space withinfinite fundamental group and t ( ξ ) ≥ , then ξ is isotopic to a deformation of a Reeblessfoliation. If g > and t ( ξ ) < , then ξ is isotopic to a deformation of a taut foliation. The proof of Theorem A involves examining the Giroux-type normal forms for universallytight contact structures of Massot [25], [26] and considering foliations that are well adapted tothese normal forms. The cases of negative and non-negative twisting are treated separately,with the former being reduced to the t ( ξ ) = − Question 1.1.
Does every irreducible -manifold with infinite fundamental group that admitsboth positive and negative universally tight contact structures necessarily admit a (smooth)Reebless foliation? Until recently there was little known about the topology of the space of foliations on a3-manifold. For the class of horizontal foliations on S -bundles Larcanch´e [21] showed thatthe inclusion of the space of horizontal integrable plane fields into the space of all integrableplane fields is homotopic to a point and in particular its image is contained in a single pathcomponent in the space of all integrable plane fields. She also showed that any integrableplane field that is sufficiently close to the tangent distribution T F of a taut foliation F canbe deformed to T F through integrable plane fields. In her PhD thesis Eynard-Bontempsshowed that a much more general result holds. In particular, she proved the followingtheorem, which mirrors Eliashberg’s h -principal for overtwisted contact structures. Theorem 1.2 (Eynard-Bontemps [10]) . Let F and F be smooth oriented taut foliations ona -manifold M whose tangent distributions are homotopic as (oriented) plane fields. Then T F and T F are smoothly homotopic through integrable plane fields. The foliations that Eynard-Bontemps constructs use a parametric version of a constructionof Thurston, first exploited by Larcanch´e, that allows foliations to be extended over solidtori using foliations that contain Reeb components. In view of this it is natural to ask
ONTACT STRUCTURES, DEFORMATIONS AND TAUT FOLIATIONS 3 whether any two horizontal foliations are in fact homotopic through horizontal foliations ormore generally whether any two taut foliations whose tangent plane fields are homotopicare homotopic through taut or even Reebless foliations. Since any horizontal foliation on an S -bundle is essentially determined by its holonomy representation, the former question isthen related to the topology of the representation space Rep( π (Σ g ) , Diff + ( S )) consideredwith its natural C ∞ -topology. Concerning the topology of this space we prove the following: Theorem B.
Let
Comp ( e ) denote the number of path components of Rep e ( π (Σ g ) , Diff + ( S )) with fixed Euler class e = 0 such that e divides g − = 0 and write g − n e . Then thefollowing holds: Comp ( e ) ≥ n g + 1 . This should be compared with results of Goldman [15] who showed that the space ofrepresentations of π (Σ g ) to the n -fold cover of PSL (2 , R ) with Euler number satisfying2 g − n e has precisely n g components. In particular, the proof of Theorem B shows thatthe images of these components in Rep e ( π (Σ g ) , Diff + ( S )) remain distinct. The idea behindthe proof of this theorem is very simple: a smooth family of representations ρ t correspondsto a smooth family of foliations F t via the suspension construction and one then deformsthis family to a family of contact structures using a parametric version of Eliashberg andThurston’s perturbation theorem. Deformations of contact structures correspond to isotopiesvia Gray’s Stability Theorem and this then gives an isotopy of contact structures, which thendistinguish path components in the representation space.In general, however, there is no parametric version of Eliashberg and Thurston’s per-turbation theorem, since in general the contact structure approximating a foliation is notunique. On the other hand under certain additional assumptions, that are for instance truefor horizontal foliations on non-trivial S -bundles, Vogel has shown a remarkable uniquenessresult for the isotopy class of a contact structure approximating a foliation, which implies inparticular that this isotopy class is in fact a C -deformation invariant in certain situations. Theorem 1.3 (Vogel [33]) . Let F be an oriented C -foliation without torus leaves. Assumefurthermore that F is neither a foliation by planes nor by cylinders only. Then there is a C -neighbourhood U of T F in the space of oriented plane fields so that all positive contactstructures in U are isotopic. If one considers only deformations that are C ∞ , or continuous in the C -topology wouldeven suffice, then one can give a comparatively simple argument using linear perturbationsto deform families of foliations to contact structures in a smooth manner (cf. Section 4).Such deformations then provide the desired obstructions used to prove Theorem B and thissuffices for our purposes.We also present a second independent proof of Theorem B, which uses the rich structuretheory of Anosov foliations instead of contact topology. Matsumoto [28] has shown that anyrepresentation in Rep e ( π (Σ g ) , Diff + ( S )) with maximal Euler class e = ± (2 g −
2) is topo-logically conjugate to a Fuchsian representation given by a cocompact lattice in
PSL (2 , R )and Ghys [11] showed that this conjugacy can be assumed to be smooth. Furthermore, thespace of Fuchsian representations of π (Σ g ) can be identified with Teichm¨uller space and isthus contractible. The suspension foliations corresponding to Fuchsian representations areAnosov in the sense that they are diffeomorphic to the weak stable foliation of the Anosovflow given by the geodesic flow of some hyperbolic metric on the unit cotangent bundle JONATHAN BOWDEN ST ∗ Σ g . By considering fiberwise coverings it is easy to construct Anosov representationswith non-maximal Euler classes. In general not every horizontal foliation lies in the samecomponent as an Anosov foliation. We do however obtain the following analogue of Ghys’result, which answers a question posed to us by Y. Mitsumatsu. Theorem C.
Any representation ρ ∈ Rep ( π (Σ g ) , Diff + ( S )) that lies in the C -path com-ponent of an Anosov representation ρ An is itself Anosov. In particular, it is conjugate to adiscrete subgroup of a finite covering of P SL (2 , R ) and is injective. Since non-injective representations always exist in the case of non-maximal Euler class thisimmediately implies the existence of more than one path component in the representationspace for any non-maximal Euler class that admits Anosov representations. By using certainconjugacy invariants (cf. Theorem 9.8) it is then easy to recover the precise estimates ofTheorem B.Of course not every taut foliation on an S -bundle is horizontal so this theorem still leavesopen the question of whether taut foliations are always deformable through taut foliations.The first example of a pair of oriented taut foliations that are homotopic as foliations but notas (oriented) taut foliations is due to Vogel [33]. By considering foliations on certain smallSeifert fibered manifolds we obtain an infinite family of examples that have the additionalproperties that they still cannot be deformed to one another through taut foliations evenif one forgets orientations or if one considers only diffeomorphism classes of foliations, incontrast to the situation in Vogel’s example. Theorem D.
There exist an infinite family of manifolds M n each admitting a pair of tautfoliations F , F that are homotopic as oriented foliations but not as taut foliations. Further-more, the same result holds true for unoriented foliations or if one considers diffeomorphismclasses of foliations. Since the manifolds considered in Theorem D are non-Haken the notions of tautness andReeblessness coincide, so in particular it follows that any deformation of foliations betweenthe foliations F and F must contain Reeb components.Further examples of taut foliations that cannot be joined by a path in the space of Reeblessfoliations are given by using the special structure of foliations on the unit cotangent bundleover a closed surface of genus at least 2. In particular, we show that the weak unstablefoliation of the geodesic flow F hor on ST ∗ Σ g cannot be smoothly deformed to any tautfoliation with a torus leaf F T without introducing Reeb components (Corollary 8.17). Onecan view this fact as a generalisation of the result of Ghys and Matsumoto concerninghorizontal foliations of ST ∗ Σ g , in that it shows that the path component of an Anosovfoliation in the space of all Reebless foliations contains only Anosov foliations. This is perhapsslightly surprising since for the product foliation on Σ g × S one can spiral along any verticaltorus γ × S to obtain smooth deformations that introduce incompressible torus leaves. Onthe other hand, although there exists no smooth deformation through taut foliations, onecan construct a taut deformation between F hor and F T through foliations that are only ofclass C (Proposition 8.18). Thus these examples highlight once more the difference betweenfoliations of class C and those of higher regularity. Outline of paper:
In Section 2 we recall some basic definitions and constructions of folia-tions and contact structures and in Section 3 we review some basic facts about Seifert fiberedspaces and horizontal foliations. Section 4 contains the relevant versions of Eliashberg and
ONTACT STRUCTURES, DEFORMATIONS AND TAUT FOLIATIONS 5
Thurston’s results on deforming foliations to contact structures and Section 5 contains back-ground on horizontal contact structures and normal forms. In Sections 6 and 7 we proveTheorem A first for negative twisting numbers and then in the non-negative case. Section 8contains our main results concerning deformations of taut foliations and finally in Section 9we analyse components of the representation space of a surface group that contain Anosovrepresentations, yielding an alternative proof of Theorem B.
Acknowledgments:
We thank T. Vogel for his patience in explaining many wonderful ideaswhich provided the chief source of inspiration for the results of this article. We also thank Y.Mitsumatsu for his stimulating questions and H. Eynard-Bontemps, S. Matsumoto and thereferee for helpful comments. The hospitality of the Max Planck Institute f¨ur Mathematikin Bonn, where part of this research was carried out, is also gratefully acknowledged. Thisresearch was also partially supported by DFG Grant BO4423/1-1.
Conventions:
Unless otherwise specified all manifolds, contact structures and foliations aresmooth and (co)oriented and all manifolds are closed.2.
Foliations and contact structures
In this section we recall some basic definitions and constructions for foliations and contactstructures. For a more in depth discussion of foliations on 3-manifolds we refer to the bookof Calegari [3].2.1.
Foliations:
A codimension-1 foliation F on a 3-manifold M is a decomposition of M into connected injectively immersed surfaces called leaves that is locally diffeomorphic tolevel sets of the projection of R to the z -axis. We will always assume that all foliations aresmooth and cooriented unless otherwise specified. One can then define a global non-vanishing1-form α by requiring that Ker( α ) = T F = ξ ⊂ T M.
By Frobenius’ Theorem a cooriented distribution is tangent to a foliation if and only if α ∧ dα ≡ ξ is called integrable . An important example of a foliation is the Reebfoliation. Example . Consider D × S with coordinates (( r, θ ) , φ ). Choose a non-negative function γ ( r ) on [0 ,
1] that is infinitely tangent to a constant map at the end points,is decreasing on the interior and has γ (0) = 1 , γ (1) = 0. Then F Reeb is defined as the kernelof the following form α = γ ( r ) dφ + (1 − γ ( r )) dr. This foliation has a unique compact leaf given by ∂D × S and the foliation on int ( D ) × S is by parabolic planes. A solid torus with such a foliation will be called a Reeb component .General foliations are very flexible - they satisfy an h -principal due to Wood [34] - andin particular every 2-plane field is homotopic to the tangent distribution of a foliation. Amore geometrically significant class of foliations are those that are taut . Here a foliation istaut if every leaf admits a closed transversal. Note that any foliation that contains a Reebcomponent is not taut, since the boundary leaf of the Reeb component is separating andcompact. Thus taut foliations fall into the more general class of Reebless foliations, i.e. those
JONATHAN BOWDEN
Figure 1.
A piece of the Reeb foliationthat contain no Reeb component. The existence of a Reebless foliation puts restrictions onthe topology of M due to the following theorem that is usually attributed Novikov, althoughthe statement about incompressibility may be due to Thurston. Theorem 2.2 (Novikov) . Let F be a Reebless foliation on a -manifold. Then all leavesof F are incompressible, π ( M ) = 0 and all transverse loops are essential in π ( M ) . Inparticular, π ( M ) is infinite. It follows from Novikov’s theorem that a foliation is Reebless if and only if all its torusleaves are incompressible. We also have the following criterion for tautness, which followsfrom Novikov’s notion of dead end components combined with the Poincar´e-Hopf Theorem.
Theorem 2.3.
Let F be a foliation on a -manifold M . If no oriented combination of torusleaves of F is null-homologous in H ( M ) , then F is taut. It will be important to modify foliations in various situations below and we will repeatedlymake use of a spinning construction which introduces toral leaves into foliations that aretransverse to an embedded torus.
Construction . Let F be a foliation on a manifold obtained bycutting a closed manifold M open along an embedded torus M = M \ T × ( − ǫ, ǫ )and assume that F is transverse on the boundary components T − , T + of M . We furthermoreassume that F is linear on the boundary so that it is given as the kernel of closed 1-forms α − and α + respectively. Letting z be the normal coordinate on T × ( − ǫ, ǫ ) we then definea foliation as the kernel of the following 1-form α = ρ ( − z ) α − + ρ ( z ) α + + (1 − ρ ( | z | )) dz. Here ρ is a non-decreasing function that is positive for z >
0, satisfies ρ ( z ) = 1 near ǫ andis identically zero otherwise so that ρ vanishes to infinite order at the origin.Note that spiralling along an embedded torus T has the effect of introducing a closed torusleaf. Furthermore, if we consider the foliation given by cutting open a manifold along anembedded torus transverse to a foliation F such that the induced foliation on T is linear,then we take α + = α − so that foliation obtained by spiralling can be obtained through asmooth 1-parameter deformation of foliations. In this case we will say that the resulting ONTACT STRUCTURES, DEFORMATIONS AND TAUT FOLIATIONS 7
Figure 2.
A cross-section of the foliation obtained after spiralling. The lefthand figure shows the unstable case α + = α − and the right hand figure showsthe example α + = − α − , which is then stable. The torus leaf is represented bythe thick (red) line.foliation is obtained from F by spinning along the torus T . If the induced foliation F | T on T is not linear and is without 2-dimensional Reeb components, then one can still spinalong T to obtain a foliation that is only of class C as long as we choose the direction ofspinning to be transverse to F | T . Finally observe that if T is a compressible torus given asthe boundary of a tubular neighbourhood of a closed transversal, then spiralling along T hasthe effect of introducing a Reeb component having T as a closed leaf. In this case spinningalong T corresponds to turbulisation . This in particular shows that Reeblessness and hencetautness are not deformation invariants of foliations.2.2. Contact structures:
In addition to foliations we will also consider totally non-integrableplane fields or contact structures . Here a contact structure ξ is a distribution such that α ∧ dα is nowhere zero for any defining 1-form with ξ = Ker( α ). Unless specified our contact struc-tures will always be positive with respect to the orientation on M so that α ∧ dα >
0. If α only satisfies the weaker inequality α ∧ dα ≥
0, then ξ is called a (positive) confoliation .There is a fundamental classification of contact structures into those that are tight andthose that are not. Definition 2.5 (Overtwistedness) . A contact structure ξ on manifold M is called over-twisted if it admits an embedded disc D ֒ → M such that T D | ∂D = ξ | ∂D . If a contact structure ξ admits no such disc then it is called tight . A contact structure is universally tight if its pullback to the universal cover f M → M is tight.2.3. Topology on the space of plane fields:
We wish to approximate foliations by contactstructures. For this we consider plane fields as sections of the oriented Grassmann bundle of M , which can be identified with the unit cotangent bundle ST ∗ M after a choice of metric.We then say that two plane fields are C k -close, if they C k -close as sections of this bundle. Inthe context of approximating foliations by contact structures it is most natural to considerthe tangent distribution of a foliation rather than the foliation itself. In view of this wewill speak about convergence of sequence of foliations {F n } in the strong sense that T F n converges in the C k -topology. In particular, a C -foliation will be a foliation that is tangentto a continuous 2-plane field. JONATHAN BOWDEN Seifert manifolds and horizontal foliations
Seifert manifolds:
A Seifert manifold is a closed 3-manifold that admits a locally free S -action. These manifolds are well understood and can all be built using the followingrecipe: Let R be an oriented, compact, connected surface (with boundary) of genus g andlet R i = ∂ i R for 0 ≤ i ≤ r denote its oriented boundary components. We then obtain aSeifert manifold by gluing solid tori W i = D × S to the i -th boundary component of R × S in such a way that the oriented meridian m i = ∂D maps to − α i [ R i ] + β i [ S ] in homology,where S is oriented to intersect R positively and α i = 0.The obvious S -action on R × S extends to a locally free S -action on M in a naturalway and the numbers ( g, β α , ..., β r α r ) are called the Seifert invariants of M . This S -actionhas a finite number of orbits that have non-trivial stabilisers, which are called exceptionalfibers . These exceptional fibers correspond to the cores of those solid tori W i for which theattaching slope β i α i is not integral. The Seifert invariants are not unique, as one can add andsubtract integers so that the sum P β i α i remains unchanged to obtain equivalent manifolds.This then corresponds to a different choice of section on R × S with respect to which theSeifert invariants were defined. However, the Seifert invariants can be put in a normal formby requiring that b = β α ∈ Z and that0 < β α ≤ β α ≤ ... ≤ β r α r < . This normal form is then unique, except for a small list of manifolds (see [16]). Note thataccording to our conventions a Seifert fibered space M with normalised Seifert invariants( g, b, β α , ..., β r α r ) is an oriented manifold. The Seifert fibered space M considered with theopposite orientation has normalised Seifert invariants ( g, − b − r, − β α , ..., − β r α r ). Warning:
The conventions for Seifert manifolds differ greatly in the literature. Here wefollow the conventions of [25] and [6], which differ from those of [17] and [23].Given a Seifert manifold M there is a natural fiberwise branched n -fold covering givenby quotienting out the n -th roots of unity Z n ⊂ S . The Seifert invariants of the quotientmanifold can then be easily determined in terms of those of M and we note this in thefollowing proposition for future reference. Proposition 3.1 (Fiberwise branched covers) . Let M be a Seifert manifold with Seifertinvariants ( g, b, β α , ..., β r α r ) , where α i , β i are coprime. Then there is a fiber preserving branched n -fold covering map M p −→ M ′ , where the quotient space M ′ has (unnormalised) Seifertinvariants ( g, nb, nβ α , ..., nβ r α r ) . The branching locus of p is a (possibly empty) subset of theexceptional fibers and the branching order around the i -th singular fiber is gcd ( n, α i ) .Proof. Let M = ( R × S ) ∪ W ∪ ... ∪ W r be the decomposition associated to the descriptionof M via its Seifert invariants ( g, b, β α , ..., β r α r ). We set M ′ = M/ Z n where Z n ⊂ S denotesthe n -th roots of unity and M p −→ M ′ is the quotient map. The quotient manifold has anatural ( S / Z n )-action and thus M ′ is again Seifert fibered and the map is fiber preserving.Furthermore, the decomposition of M gives a decomposition M ′ = ( R × S ) ∪ W ′ ∪ ... ∪ W ′ r suchthat the restriction of the map p to R × S is the product of the standard n -fold cover S → S with the identity on R . Under the covering map the meridian class m i = − α i [ R i ] + β i [ S ] ONTACT STRUCTURES, DEFORMATIONS AND TAUT FOLIATIONS 9 given by the i -th solid torus W i maps to − α i [ R i ] + nβ i [ S ], which must then be a multipleof the meridian class m ′ i along which W ′ i is attached: m ′ i = − α i gcd( nβ i , α i ) [ R i ] + nβ i gcd( nβ i , α i ) [ S ] . The divisibility of p ∗ ( m i ), which is gcd( nβ i , α i ) = gcd( n, α i ) since α i , β i are coprime, thencorresponds to the branching index of p over the i -th exceptional fiber. (cid:3) Horizontal Foliations:
We next discuss horizontal foliations on Seifert manifolds re-ferring to [6] for further details. Here a foliation on a Seifert fibered space is called horizontal ,if it is everywhere transverse to the fibers of the Seifert fibration. A horizontal foliation F ona Seifert fibered space is equivalent to a representation e ρ : π ( M ) → g Diff + ( S ), such that thehomotopy class of the fiber is mapped to a generator of the centre of g Diff + ( S ), which acts on R as the group of 1-periodic diffeomorphisms. One then has M = ( e B × R ) / e ρ , where e B denotesthe universal cover of the quotient orbifold of M , and the horizontal foliation on the productdescends to F . The representation e ρ then descends to a representation of the orbifold fun-damental group of the base to the ordinary diffeomorphism group ρ : π orb ( B ) → Diff + ( S ).In all but a few cases a Seifert manifold admits a horizontal foliation if and only if it admitsone with holonomy in PSL (2 , R ), in the sense that the image of the holonomy map in ρ liesin PSL (2 , R ). Moreover, an examination of the proof of ([6], Theorem 3.2) and its analoguefor PSL (2 , R )-foliations shows that it is always possible to ensure that the holonomy aroundsome embedded curve in the base is hyperbolic provided that the base has positive genus.We note this in the following proposition. Proposition 3.2 (Existence of horizontal foliations [6]) . Let M be a Seifert fibered spacewhose base has genus g , then M admits a horizontal foliation if − g − r ≤ − b − r ≤ g − . In this case the horizontal foliation can be taken so as to have holonomy in PSL (2 , R ) andso that some embedded curve in the base whose holonomy is hyperbolic. If g > then theconverse also holds. Thus in most cases the existence of a horizontal foliation on M is the same as the existenceof a flat connection on M thought of as an orbifold PSL (2 , R )-bundle. In the case of genuszero, one has slightly more elaborate criteria for the existence of a PSL (2 , R )-foliation. Theorem 3.3 ([20], Theorem 1) . Let M be a Seifert manifold with normalised invariants (0 , b, β α , , ..., β r α r ) . Then M admits a horizontal foliation with holonomy in PSL (2 , R ) if andonly if one of the following holds: • − r ≤ − b − r ≤ − • b = − and P ri =1 β i α i ≤ or b = 1 − r and P ri =1 β i α i ≥ r − . Perturbing foliations
In their book on confoliations, Thurston and Eliashberg showed how to perturb foliationsto contact structures. In its most general form, their theorem shows that any 2-dimensionalfoliation F that is not the product foliation on S × S can be C -approximated by bothpositive and negative contact structures. Under additional assumptions on the holonomy ofthe foliation this perturbation can actually be realised as a deformation . That is, there is a smooth family ξ t of plane fields, such that ξ is the tangent plane field of F and ξ t iscontact for all t >
0. Moreover, if every closed leaf has linear holonomy or if the foliation isminimal with some holonomy, then F can be linearly deformed to a contact structure. Herea linear deformation is a 1-parameter family of 1-forms α t such that Ker ( α ) = T F and ddt α t ∧ dα t (cid:12)(cid:12)(cid:12)(cid:12) t =0 > . This latter condition is then equivalent to the existence of a 1-form β such that h α, β i = α ∧ dβ + β ∧ dα > . Note further that h f α, f β i = f h α, β i so that the condition of being linearly deformable depends only on the foliation and not onthe particular choice of defining 1-form. Theorem 4.1 (Eliashberg-Thurston [7]) . Let F be a C -foliation that is not without holo-nomy. (1) If all closed leaves admit some curve with attracting holonomy. Then T F can besmoothly deformed to a positive resp. negative contact structure. (2) If all closed leaves have linear holonomy, then this deformation can be chosen to belinear.Remark . Foliations without holonomy are very special and can be C -approximated bysurface fibrations over S . Thus the assumption that the foliation has some holonomy canbe replaced by the topological assumption that the underlying manifold does not fiber over S . Examples of manifolds which cannot fiber are non-trivial S -bundles over surfaces ofgenus at least 2, or more generally Seifert fibered spaces with non-trivial Euler class andhyperbolic quotient orbifolds, and rational homology spheres.In general it is not possible to deform families of foliations to contact structures in asmooth manner. However, if a smooth family of foliations F τ admits linear deformations forall τ in some compact parameter space K , then the fact that h α, β i > Proposition 4.3 (Deformation of families) . Let F τ be a smooth family of foliations that isparametrised by some compact space K and suppose that each foliation in the family admits alinear deformation. Then F τ can be smoothly deformed to a family of positive resp. negativecontact structures ξ ± τ . Another consequence of the convexity of the linear deformation condition is that the isotopyclasses of any two linear deformations determining the same orientation is unique. This is animmediate consequence of Gray’s Stability Theorem and we record this fact in the following:
Proposition 4.4.
Any two positive, resp. negative linear deformations of a foliation areisotopic.
ONTACT STRUCTURES, DEFORMATIONS AND TAUT FOLIATIONS 11 Horizontal contact structures
Horizontal contact structures on Seifert manifolds, like horizontal foliations, may be thoughtof as connections with a certain curvature condition. As opposed to the flat case where thehorizontal distribution is integrable, the distribution in question is contact if and only if theholonomy around the boundary of any embedded disc in the base is negative. To be precisethis means that for the induced connection on the R -bundle given by unwrapping the S -fibers over the disc the holonomy h around the boundary of the disc satisfies h ( x ) − x < Theorem 5.1 (Honda [17], Lisca-Mati´c [23]) . A Seifert manifold with normalised invariants ( g, b, β α , ..., β r α r ) carries a (positive) contact structure transverse to the Seifert fibration if andonly if one of the following holds: • − b − r ≤ g − • g = 0 , r ≤ and − b − P β i α i < • g = 0 and there are relatively prime integers < a < m such that β α > m − am , β α > am and β i α i > m − m , for i ≥ . Remark . The final condition is the realisability condition of [6], which is equivalent tothe existence of a horizontal foliation by Naimi [30]. For g >
Classification results.
A given Seifert manifold can admit several isotopy classesof horizontal contact structures. An important invariant of contact structures on Seifertmanifolds is the “enroulement” or twisting number as introduced by Giroux. For this recallthat a Legendrian knot K in a contact manifold ( M, ξ ) inherits a canonical framing givenby taking a vector field of unit normals along K that are also tangent to ξ . After choosinga reference framing this gives an integer which is called Thurston-Bennequin number tb ( K ) of the Legendrian knot K . Definition 5.3 (Giroux [14]) . Let ξ be a contact structure on a Seifert fibered space. The twisting number t ( ξ ) of ξ is the maximal Thurston-Bennequin number of a knot thatis smoothly isotopic to a regular fiber, where the Thurston-Bennequin number is measuredrelative to the canonical framing coming from the base. Generalising results of Giroux [14] from S -bundles to the case of general Seifert fiberedspaces Massot [25] has shown that a contact structure can be isotoped to a horizontal one ifand only if it is universally tight and has negative twisting number: Theorem 5.4 ([25] Theorem A) . Let ξ be a contact structure on a Seifert fibered space. Then ξ can be made horizontal via an isotopy if and only if it is universally tight and t ( ξ ) < . An important step in the proof of Theorem 5.4 is to show that any contact structure with t ( ξ ) < normal form . More precisely, given a Seifertfibered space described as M = ( R × S ) ∪ W ∪ ... ∪ W r we say that ξ is in normal form ifit is tangent to the S -fibers on c M = R × S . Lemma 5.5 ([25] Proposition 5.5) . Let ξ be a contact structure on a Seifert fibered spacewith t ( ξ ) < . Then ξ can be isotoped into normal form. In applications it will be important to use a slightly more precise version of this result,that follows from the way that Lemma 5.5 is proved.
Lemma 5.6 ([25] pp. 1757–8) . Let ξ be a universally tight contact structure on a Seifertmanifold M and let F be a regular fiber that is Legendrian and satisfies tb ( F ) = t ( ξ ) < .Then ξ can be brought into normal form by an isotopy that fixes neighbourhoods of theexceptional fibers.Sketch of proof. First assume that F is a regular fiber over a base point p in R that isLegendrian and realises t ( ξ ). Then by the Weinstein Neighbourhood Theorem for Legendrianknots we can assume after an isotopy with support near F that the contact structure isvertical near F and is given as the kernel of a 1-form α n = cos ( nθ ) dx − sin ( nθ ) dy , where θ denotes the fiber coordinate and ( x, y ) are coordinates on a neighbourhood of p . Here n = − t ( ξ ) and it is essential that this number is positive so that the form α n determines apositive contact structure. We then consider a bouquet of circles B = γ ∨ · · · ∨ γ k basedat p in R onto which R contracts. One then uses Giroux’s Flexibility Theorem to makethe contact structure vertical near γ i × S by isotopies with support disjoint from a smallneighbourhood of F . Again at this point it is essential that the twisting is negative in orderto apply Giroux’s Flexibility Theorem for surfaces (in this case annuli) with Legendrianboundary. These isotopies all have support in a small neighbourhood N of B × S . Bystretching N to fill out all of c M we obtain the desired isotopy, which in total has supportdisjoint from neighbourhoods of the exceptional fibers. (cid:3) Now any vertical contact structure ξ on c M = R × S is given as the pullback of the canonicalcontact structure ξ can under a fiberwise n -fold covering c M → ST ∗ R that we denote p ξ (cf.[14] Proposition 3.3). The map p ξ is defined by sending a point x to the image of ξ x underthe projection π : R × S → R considered as an element in ST ∗ R . Note that the number n corresponds to − t ( ξ ).Under the additional assumption that the contact structure is universally tight one canfurther restrict the possibilities for the contact structures on the solid torus neighbourhoodsof the exceptional fibers. For in this case the contact structures on each of the solid tori W i is also universally tight. Thus according to classification results of Giroux (cf. [25] Lemma3.4) there are at most two possibilities for each ξ | W i up to isotopy relative to ∂W i . The twopossibilities are distinguished by the fact that the are isotopic relative to the boundary tocontact structures that are positively resp. negatively transverse to the fibers on int ( W i ). Infact, these choices must be made coherently due to the following: Lemma 5.7 ([25] Proposition 6.1) . A contact structure ξ on a Seifert fibered space M =( R × S ) ∪ W ∪ ... ∪ W r in normal form is universally tight if and only if the restrictions ξ | W i are isotopic relative to ∂W i to ones that are all either positively or negatively transverse tothe S -fibers on int ( W i ) .In particular, there are at most two universally tight extensions of ξ | R × S which are isotopicas unoriented contact structures and they can both be made horizontal after a suitable isotopy. This lemma is extremely useful as it means that determining a universally tight contactstructure with t ( ξ ) < ONTACT STRUCTURES, DEFORMATIONS AND TAUT FOLIATIONS 13 determining it on c M . In order to state Massot’s classification of universally tight contactstructures with negative twisting in terms of these normal forms it is convenient to describethe contact structure on R × S in a slightly different fashion.For this one notes that the choice of section b s in c M used to compute the normalisedSeifert invariants gives a section in ST ∗ R via the covering map p ξ . This section then givesan identification of ST ∗ R and c M with R × S and with respect to these identifications themap p ξ is the product of the identity with the standard n -fold cover of S up to fiberwiseisotopy.Moreover, under this identification the canonical contact structure is isotopic to the kernelof some 1-form α λ = cos( θ ) λ + sin( θ ) λ ◦ J, where λ is a non-vanishing 1-form on R and J is an almost complex structure. The contactstructure on c M is then given by the kernel of the 1-form α λ,n = cos( nθ ) λ + sin( nθ ) λ ◦ J. Note that the isotopy class of Ker( α λ,n ) is independent of the choice of J . Furthermore, anyhomotopy of λ through non-vanishing 1-forms induces an isotopy of the contact structureKer( α λ,n ) and the homotopy class of λ as a non-vanishing 1-form is called the R -class ofthe normal form. The other important invariant of λ is given by its indices on the boundarycomponents of R and the collection of these indices ( x , · · · , x r ) ordered according to thetori W i is called the multi-index of the normal form. The various indices correspond todifferent fiberwise homotopy classes of maps p ξ : ∂W i → S ⊆ ∂R, which can in turn be identified with Z as a torsor.We saw above that there are restrictions on the ways to extend a vertical contact structureon R × S to one that is universally tight. On the other hand there are simple arithmeticcriteria to determine when this is possible. The following is a special case of ([25], TheoremB), which is stated only for Seifert fibered spaces whose invariants are in normal form, butits proof (cf. [25] p. 1758) holds without this assumption. Lemma 5.8.
Let M be a Seifert fibered space with (not necessarily normalised) Seifertinvariants ( g, b, β α , ..., β r α r ) and assume that b + r X i =1 l β i α i m = 2 − g. Then there is a universally tight contact structure given by a normal form with multi-index ( b, ⌈ β α ⌉ , ..., ⌈ β r α r ⌉ ) . In particular, ξ can be assumed to be horizontal on the complement of R × S . We now describe Massot’s classification which divides into two cases. The first is theflexible case, where the specific normal form is not important, and the second is the rigidcase where it contains essential information about the isotopy class of the contact structure.Note that in the latter case one requires the additional assumption that the base has genus g > Theorem 5.9 (Flexible Case, [25] Theorem D) . Let ξ, ξ ′ be universally tight contact struc-tures on a Seifert fibered space with normalised Seifert invariants ( g, b, β α , ..., β r α r ) . If − b − r < g − and t ( ξ ) = t ( ξ ′ ) = − , then ξ and ξ ′ are isotopic as unoriented contact structures,i.e. they are isotopic after possibly swapping the orientations of one of the plane fields. Theorem 5.10 (Rigid Case, [25] Theorem E) . Let ξ be a universally tight contact structureon a Seifert fibered space with normalised Seifert invariants ( g, b, β α , ..., β r α r ) , where g > .Assume furthermore that either t ( ξ ) = − n < − or − b − r = 2 g − and t ( ξ ) = − . Thenthe R -class of the normal form is unique.Moreover, the multi-index of this normal form is ( nb, ⌈ nβ α ⌉ , ..., ⌈ nβ r α r ⌉ ) . In the case where the base has genus g = 0, the second part of Theorem 5.10 still holds.The following is a special case of [25] Proposition 8.2. Note that Massot uses the notation e = − b − r at this point. Theorem 5.11 ([25] Proposition 8.2) . Let ξ be a universally tight contact structure on aSeifert fibered space with normalised Seifert invariants ( g, b, β α , ..., β r α r ) , where g = 0 . Assumefurthermore that either t ( ξ ) = − n < − or − b − r = 2 g − and t ( ξ ) = − . Then themulti-index of any normal form is ( nb, ⌈ nβ α ⌉ , ..., ⌈ nβ r α r ⌉ ) . In particular, there is at most oneisotopy class of contact structures with t ( ξ ) = − n up to orientation reversal of plane fields. Realising contact structures as branched coverings.
Contact structures on S -bundles over surfaces with twisting − n can be realised as coverings of contact structureswith twisting − Definition 5.12 (Contact branched cover) . Let ( M ′ , ξ ′ ) be a contact manifold with ξ ′ = Ker ( α ′ ) and let M p −→ M ′ be a branched covering of -manifolds such that the image of thebranching locus L ⊂ M under p is a transverse link. Choose a -form β on M with supportin a neighbourhood of L such that p ∗ α ′ ∧ dβ > and β = 0 along L. The pull-back contact structure p ∗ ξ ′ is then defined as the kernel of the following -form forany ǫ > sufficiently small: α = p ∗ α ′ + ǫ β. Remark . Since the conditions imposed on β are convex, it follows that p ∗ ξ ′ is welldefined up to isotopy in view of Gray’s Stability Theorem.Now if M admits a contact structure ξ with twisting number − n , then ξ is in fact isotopicto the pullback of a contact structure ξ ′ with twisting number − n -fold fiberwisebranched cover. Proposition 5.14.
Let M be a Seifert manifold admitting a contact structure with twistingnumber t ( ξ ) = − n < − , then there is a fiberwise branched covering M p −→ M ′ and acontact structure ξ ′ on M ′ with twisting − such that ξ is isotopic to p ∗ ξ ′ .Proof. Let ( g, b, β α , ..., β r α r ) be the normalised Seifert invariants of M and let M p −→ M ′ be the n -fold fiberwise branched cover given by Proposition 3.1. Since t ( ξ ) = − n by assumption,the contact structure ξ admits a normal form with associated 1-form α λ,n = cos( nθ ) λ + sin( nθ ) λ ◦ J. ONTACT STRUCTURES, DEFORMATIONS AND TAUT FOLIATIONS 15
By Theorem 5.10 the indices of λ are ( nb, ⌈ nβ α ⌉ , ..., ⌈ nβ r α r ⌉ ) and Poincar´e-Hopf implies nb + r X i =1 l nβ i α i m = 2 − g. It then follows by Lemma 5.8 that the Seifert manifold M ′ , which has Seifert invariants( g, nb, nβ α , ..., nβ r α r ), admits a contact structure ξ ′ with normal form given by α λ = cos( θ ) λ + sin( θ ) λ ◦ J, which is in particular transverse to the branching locus of the map p . The pullback of thecontact structure ξ ′ can then be perturbed in a C ∞ -small fashion near the branching locus,where the contact structure is transverse, to obtain p ∗ ξ ′ . Since there is a unique way toextend the pullback p ∗ ξ ′ | c M to a contact structure on all of M , which can be made positivelytransverse by Lemma 5.7 and p ∗ ξ ′ is isotopic to a positively transverse contact structure, weconclude that ξ is isotopic to p ∗ ξ ′ . (cid:3) We next note that any two contact structures with twisting number − − b − r < g −
2. In the other case we have:
Proposition 5.15.
Let ξ, ξ ′ be universally tight contact structures on a Seifert fibered spacewith normalised Seifert invariants ( g, b, β α , ..., β r α r ) and assume that − b − r = 2 g − , thetwisting numbers satisfy t ( ξ ) = t ( ξ ′ ) = − and g > . Then any two normal forms of ξ and ξ ′ are contactomorphic by a diffeomorphism with support disjoint from the exceptional fibers.Proof. Let α λ and α λ ′ be the 1-forms associated to the normal form of ξ and ξ ′ respectivelyon c M = R × S ⊂ M . After possibly replacing λ with − λ , we may assume that both ξ and ξ ′ are isotopic to positively transverse contact structures. By Theorem 5.10 the indices of both λ and λ ′ must then agree on ∂R , since by assumption − b − r = 2 g −
2. This is equivalentto the restrictions of the maps p ξ , p ξ ′ : c M → ST ∗ R being fiberwise isotopic on the boundary of c M . Furthermore, since t ( ξ ) = t ( ξ ′ ) = − p ξ and p ξ ′ agree near ∂ c M . It follows that p ξ ◦ p − ξ ′ is a diffeomorphism of c M that extends toall of M so that ξ and ξ ′ are contactomorphic. (cid:3) Remark . If M admits an orientation preserving diffeomorphism that reverses the ori-entation on the fibers, then any oriented horizontal contact structure is contactomorphic tothe contact structure given by reversing the orientation of the plane field. In this case theabove proposition in fact holds for contactomorphism classes of oriented contact structures.Examples of such manifolds are given by Brieskorn spheres Σ( p, q, r ) ⊂ C , in which casethe conjugation map on C yields the desired self-diffeomorphism.6. Deformations of taut foliations on Seifert manifolds
In this section we consider the problem of determining which contact structures on Seifertmanifolds are deformations of taut foliations. The obvious necessary condition for a contactstructure to be a perturbation of a taut foliation is that it is universally tight. We will show that in most cases a universally tight contact structure ξ with negative twisting on a Seifertfibered manifolds is indeed a deformation of a taut foliation.In fact by Proposition 5.14 it suffices to consider contact structures with twisting number −
1, in which case it is fairly easy to construct the necessary foliations at least when thegenus of the base is at least one. The genus zero case is more subtle as not every contactstructure with negative twisting can be a perturbation of a taut foliation. We first note somepreliminary lemmas.
Lemma 6.1.
Let ξ = p ∗ ξ ′ be a contact structure on a Seifert fibered space which is thepullback of a contact structure ξ ′ under a fibered branched cover M p −→ M ′ . Assume that ξ ′ is isotopic to a linear deformation of a taut foliation through an isotopy that is transverse tothe branching locus of p . Then ξ is also a linear deformation of a taut foliation.Proof. Let α ′ be a defining form for ξ ′ and let α t be a smooth family of non-vanishing 1-formsso that Ker( α ) is integrable and tangent to a taut foliation and α t is contact for t >
0. Afterapplying an initial isotopy, we may also assume that Ker( α ) = ξ ′ and that the entire familyis transverse to the branching locus L of p . Then p ∗ α t is a deformation of a taut foliationthat is contact away from L for t >
0, where it is closed. We let β be a 1-form as in Definition5.12 such that p ∗ α ∧ dβ > β = 0 along L . Then e α t = p ∗ α t + t ǫβ provides the desiredlinear deformation since ddt e α t ∧ d e α t (cid:12)(cid:12)(cid:12)(cid:12) t =0 = p ∗ (cid:18) ddt α t ∧ dα t (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) t =0 + ǫ ( p ∗ α ∧ dβ + β ∧ p ∗ dα ) > ǫ > (cid:3) We now come to the main result of this section.
Theorem 6.2.
Let ξ be a universally tight contact structure with negative twisting number t ( ξ ) = − n on a Seifert manifold and assume that the base orbifold has genus g > . Then ξ is a deformation of a taut foliation. Moreover, if n > or − g ≤ − b in the case that n = 1 , then this foliation can be taken to be horizontal and the deformation linear.Proof. By Proposition 5.14 there is a fiberwise branched covering M p −→ M ′ and a horizontalcontact structure ξ ′ so that ξ is isotopic to p ∗ ξ ′ and t ( ξ ′ ) = −
1. For convenience we assumethat both ξ and ξ ′ are in normal form and that p ∗ ξ ′ = ξ . We let ( g, nb, nβ α , , ..., nβ r α r ) denotethe unnormalised Seifert invariants of M ′ . By Theorem 5.1 we have that − b − r ≤ g − n > nb + r X i =1 l nβ i α i m = 2 − g so that the normalised invariants ( g, b ′ , β ′ α ′ , ..., β ′ r α ′ r ) of M ′ satisfy − b ′ − r = 2 g −
2. We considerseveral cases:
Case 1: n > : In this case − b ′ − r = 2 g − M ′ . Proposition 3.2 then gives a horizontal PSL (2 , R )-foliation F on M ′ with hyperbolic holonomy around some embedded curve γ . We may then applypart (2) of Theorem 4.1 to deform the foliation linearly to a horizontal contact structure ξ hor on M ′ . The characteristic foliation on the torus T γ corresponding to γ is Morse-Smale and ONTACT STRUCTURES, DEFORMATIONS AND TAUT FOLIATIONS 17 has two closed orbits each intersecting a fiber in a point. This is then stable under a suitablysmall linear deformation. In particular, T γ is convex in the sense of Giroux and has a dividingset with two components, each of which intersects the fiber once. Thus applying Giroux’sFlexibility Theorem there is an isotopy with support in a neighbourhood of T γ so that thetorus becomes ruled or in standard form (cf. [18] Corollary 3.6), that is the S -fibers becomeLegendrian. These Legendrian fibers then have Thurston-Bennequin number − t ( ξ hor ) ≥ −
1. The opposite inequality holds for all horizontal contactstructures by Theorem 5.4 and we conclude that t ( ξ hor ) = −
1. We may then isotope ξ hor into normal form through an isotopy that is fixed near the exceptional fibers of M ′ byLemma 5.6. Since all contact structures with twisting number − ξ hor agrees with that of ξ ′ afterapplying a suitable diffeomorphism. Note that the diffeomorphism given in the proof ofProposition 5.15 can be chosen with support disjoint from the singular locus. It follows that ξ ′ is isotopic to a deformation of a taut foliation. Since this isotopy was chosen to satisfythe hypotheses of Lemma 6.1 the result follows in this case. Case 2: n = 1 and − b − r = 2 g − : The argument from the previous case gives the result by taking M = M ′ . Case 3a: n = 1 and b ≤ g − < − b − r : Proposition 3.2 gives a horizontal
PSL (2 , R )-foliation F on M with hyperbolic holonomyaround some embedded curve γ , which can be linearly deformed to a horizontal contactstructure ξ hor . This contact structure must again have t ( ξ hor ) = − ξ by Theorem 5.9. Case 3b: n = 1 and b > g − : In this case we must have t ( ξ ) = −
1, since g > b > M up to changingthe orientation of the plane field by Theorem 5.9. Thus it will suffice to show that somehorizontal contact structure is a deformation of a taut foliation. To this end we let γ bea homologically essential simple closed curve in the base orbifold B , which exists by ourassumption that g >
0. We cut M open along the torus T γ which is the preimage of γ in M and take any horizontal foliation on the complement of T γ whose holonomy is conjugate to arotation on the two boundary components of B \ γ . We may assume that the rotation anglesare distinct, unless M = T , in which case all tight contact structures are deformations ofsome product foliation (cf. [7] Proposition 2.3.1).We then spiral this foliation along the torus T γ (cf. Section 2) to obtain a foliation with aunique torus leaf that is non-separating. For convenience we then insert a product foliationto obtain a foliation F γ with a single stack of torus leaves all of which are non-separating,meaning that F γ is in particular taut. If α − , α + denote closed forms defining the foliationon the boundary components of a tubular neighbourhood T × [ − ,
1] of T γ , then F γ is givenas the kernel of the following 1-form: α = ρ ( − z ) α − + ρ ( z ) α + + (1 − ρ ( | z | )) dz, where z denotes the second coordinate in T × [ − ,
1] and ρ is a non-decreasing function suchthat: ρ ( z ) = ( , if z ≤ , if z ≥ . We write α − = cos( θ − ) dx − sin( θ − ) dyα + = cos( θ + ) dx − sin( θ + ) dy , − π < θ − , θ + < . Here we identify the fiber direction of M with the y -coordinate. After possibly swappingthe orientation of the z coordinate, and hence of M , we may assume that θ − < θ + . We canthen deform F γ to a confoliation that is contact near T γ . On T × [ − ,
1] this is given by thefollowing explicit deformation α t = α + t (cos( f ( z )) dx − sin( f ( z )) dy )for a non-decreasing function f : R → [0 , π ] that is constant outside of [ − , − , ) and satisfies f ( z ) = ( θ − , if z ≤ − θ + , if z ≥ . A simple calculation shows that ξ t = Ker( α t ) is a positive confoliation that is contact on T × ( − , ) for t >
0. Furthermore, the form cos( f ( z )) dx − sin( f ( z )) dy is positive on S -fibers so that ξ t is horizontal for t ∈ (0 , − b < − g by assumption, M can admit ahorizontal contact structure for one and only one orientation by Theorem 5.1, so the change oforientation made above does not affect anything. The resulting confoliation is then transitiveand can thus be C ∞ -perturbed to a contact structure which is by construction horizontal.By ([7] Proposition 2.8.3), this perturbation can then be altered to a deformation. (cid:3) Under the assumption that the genus of the base g = 0 we have the following special case ofTheorem 6.2 that will be used below: Proposition 6.3.
Let ξ be a universally tight contact structure on a Seifert manifold M withnegative twisting number − n and suppose that ξ is isotopic to a vertical contact structureand the base orbifold is hyperbolic. Then ξ is a linear deformation of a horizontal (and hencetaut) foliation.Proof. The fact that ξ is isotopic to a vertical contact structure gives a natural n -fold covering M p ξ −→ ST ∗ B, where ST ∗ B is the unit cotangent bundle of the base orbifold of M , which is in turn acompact quotient of PSL (2 , R ). The cotangent bundle ST ∗ B carries a canonical contactstructure ξ can which descends from a left-invariant one on PSL (2 , R ) and p ∗ ξ ξ can = ξ . It iseasy to see that this contact structure is a linear deformation of a taut foliation by consideringthe linking form on the Lie algebra of PSL (2 , R ) and thus the same holds on the quotient ST ∗ B (cf. [2], Example 3.1). The proposition then follows by pulling back under p ξ . (cid:3) ONTACT STRUCTURES, DEFORMATIONS AND TAUT FOLIATIONS 19
Note that there can be no direct analogue of Theorem 6.2 in the genus zero case. Foras a consequence of Theorem 5.1, there are Seifert manifolds that admit horizontal contactstructures, but no taut foliations. A particularly interesting case is that of small Seifertfibered spaces which are those having 3 exceptional fibers and base orbifold of genus 0. Inthis case any universally tight contact structure must have negative twisting number, whichis equivalent to being isotopic to a horizontal contact structure. Furthermore, swapping theorientation of M has the effect of changing b to − b + 3. Thus inspection of the criteriaof Theorem 5.1 shows that M admits a horizontal contact structure in both orientationsif and only if its invariants are realisable and hence this is equivalent to the existence of ahorizontal foliation. For Seifert fibered spaces whose bases are of genus g = 0 the existence ofa taut foliation is equivalent to that of a horizontal foliation. We summarise in the followingproposition, which is proved by Lisca and Stipsicz [24] using Heegard-Floer homology, ratherthan using Theorem 5.1 which can be proven by completely elementary methods (cf. [17]). Proposition 6.4.
Let M be a Seifert fibered space over a base of genus g = 0 with infinitefundamental group. Then the following are equivalent: (1) M admits a universally tight contact structure with negative twisting number in bothorientations (2) M admits a horizontal contact structure in both orientations (3) M admits a horizontal foliation (4) M admits a taut foliation.If M is small, the assumption on the twisting number can be removed in (1) and taut can bereplaced by Reebless in (4). Note that it is not clear whether any given horizontal contact structure on a small Seifertfibered space is the deformation of a horizontal foliation in the case that both exist. However,in all likelihood this ought to be the case.7.
Deformations of Reebless foliations on Seifert manifolds
In this section we show that all universally tight contact structures ξ with t ( ξ ) ≥ S -bundles.In the following a very small Seifert fibered space is a Seifert fibered space that admitsa Seifert fibering with at most 2 exceptional fibers and whose base orbifold has genus g =0. Note that a very small Seifert fibered space is either a Lens space (including S ) or S × S . The Lens spaces do not admit Reebless foliations by Novikov’s Theorem and theonly Reebless foliation on S × S is the product foliation, which cannot be perturbed toany contact structure. Thus it is natural to rule out such spaces when showing that certaincontact structures are deformations of Reebless foliations. Furthermore, a folklore result ofEliashberg and Thurston [7] states that a perturbation of a Reebless foliation is universallytight. Unfortunately, as pointed out by V. Colin [5] the proof given loc. cit. contains a gap,and one only knows that there exists some perturbation that is universally tight. But in anycase it is a reasonable assumption to make when considering which contact structures aredeformations of Reebless foliations. We first recall the existence of normal forms for universally tight contact structures withnon-negative twisting.
Theorem 7.1 ([26] Theorem 3) . Let ξ be a universally tight contact structure on a Seifertmanifold M that is not very small and is not a T -bundle with finite order monodromy. If t ( ξ ) ≥ , then ξ is isotopic to a contact structure that is horizontal outside a (non-empty)collection of incompressible pre-Lagrangian vertical tori T = ⊔ Ni =1 T i , on which the contactstructure is itself vertical.Conversely, any two contact structures ξ , ξ that are vertical on a fixed collection of pre-Lagrangian vertical tori T and horizontal elsewhere are isotopic as unoriented contact struc-tures. With the aid of the normal form described above it is now a simple matter to show thefollowing.
Theorem 7.2.
Let ξ be a universally tight contact structure on a Seifert fibered space M with t ( ξ ) ≥ and assume that M is not very small. Then ξ is a deformation of a Reeblessfoliation.Proof. First assume that M is not a torus bundle with finite order monodromy. Then byTheorem 7.1 we may assume after a suitable isotopy that ξ is horizontal away from a finitecollection of tori T = ⊔ Ni =1 T i where ξ is vertical. Now let F be any foliation which has theincompressible tori T i as closed leaves and is horizontal otherwise. We also require that thesign of the intersection of any fiber with F agrees with that of ξ on M \ T and that theholonomies around curves on either side of the images of each T i in the base orbifold B of M are rational rotations. Such foliations can easily be constructed by taking any horizontalfoliation on the components of M \ T that has the correct co-orientation and then spirallinginto the torus leaves. Note that all torus leaves are incompressible so that F is Reebless.We first thicken the foliation near each torus leaf by inserting a stack of torus leaves T i × [ − , ξ ′ which is contact on N i = T i × [ − , ]. On N i thisdeformation is given by the explicit formula α t = dz + t (cos( f ( z )) dx − sin( f ( z )) dy ) . Here z denotes the normal coordinate and f is a monotone non-decreasing function suchthat f ( z ) = ( θ − , if z ≤ − θ + , if z ≥ , where θ − , θ + are the (negative) angles of the foliations on the negative resp. positive sideof T i . Since the foliation is positively transverse on one side of the torus and negativelytransverse on the other, we may furthermore assume that θ − < θ + and that the interval[ θ − , θ + ] contains either 0 or π but not both and thus ξ ′ becomes vertical precisely once oneach T i × [ − , ξ ′ can then be deformed to a contact structure which ishorizontal on M \ T . By Theorem 7.1 this contact structure is then isotopic to ξ . Finally thetwo step deformation of F can be achieved via a single deformation in view of ([7] Proposition2.8.3). ONTACT STRUCTURES, DEFORMATIONS AND TAUT FOLIATIONS 21 If M is a torus bundle with finite order holonomy, then the universally tight contactstructures are classified (see [13] [19]) and it is easy to see that they are all deformations ofsome T -fibration. (cid:3) Remark . Although the foliations in Theorem 7.2 are in general only Reebless, one cangive sufficient conditions so that they are taut. For by replacing dz with − dz in the modelused to define the foliation near the vertical tori T i , one can arrange that the torus leaveshave any given orientation. In particular, if one can orient the vertical tori T i of the normalform associated to ξ in such a way that no collection of these tori is null-homologous, thenthe foliation F constructed above is taut in view of Theorem 2.3.8. Topology of the space of taut and horizontal foliations
The topology of the space of representations Rep( π (Σ g ) , PSL (2 , R )) for a closed surfacegroup of genus g ≥ G the representation space of asurface group π (Σ g ) is { ( φ , ψ , ..., φ g , ψ g ) ∈ G g | g Y i =1 [ φ i , ψ i ] = 1 } . In the case of
PSL (2 , R ) the connected components of the representation space correspondto the preimages under the map given by the Euler classRep( π (Σ g ) , PSL (2 , R )) e −→ [2 − g, g − . Moreover, the quotient of the connected component with maximal Euler class under thenatural conjugation action is homeomorphic to Teichm¨uller space and is hence contractible.On the other hand the topology of the representation space Rep( π (Σ g ) , Diff + ( S )) endowedwith the natural C ∞ -topology, which can be interpreted as the space of foliated S -bundlesafter quotienting out by conjugation, is not as well understood. It would perhaps be naturalto conjecture that the map induced by the inclusion G = PSL (2 , R ) ֒ → Diff + ( S )induces a weak homotopy equivalence on representation spaces or at least a bijection on pathcomponents. It is known that both representation spaces are path connected in the case ofthe maximal component (cf. [11], [28]). Indeed, results of Matsumoto and Ghys show thatany maximal representation is smoothly conjugate to one that is Fuchsian.On the other hand, we will show that this is not the case for the space of representationswith non-maximal Euler class. The basic observation is that the cyclic n -fold cover G n of G = PSL (2 , R ) also acts smoothly on the circle via Z n -equivariant diffeomorphisms so thatthere is a natural map Rep( π (Σ g ) , G n ) −→ Rep( π (Σ g ) , Diff + ( S )) . In general the images of these maps lie in different path components for different values of n and fixed Euler class.We shall need some preliminaries concerning the relationship between horizontal foliationson S -bundles and their holonomy representations. For this we shall identify the universal cover g Diff + ( S ) of Diff + ( S ) with the group of diffeomorphisms of R that are periodic withrespect to integer translations. We then consider g Rep e ( π (Σ g ) , g Diff + ( S )) = { ( φ , ψ , ... , φ g , ψ g ) ∈ g Diff + ( S ) g | g Y i =1 [ φ i , ψ i ] = Tr − e } , where Tr − e denotes a translation by an integer − e , where e is the Euler number of theassociated oriented S -bundle M ( e ). Note that the space g Rep e ( π (Σ g ) , g Diff + ( S )) can beidentified with the space of representations Rep( π ( M ( e )) , g Diff + ( S )) that send the fiberclass [ S ] to the translation Tr . The natural map g Rep e ( π (Σ g ) , g Diff + ( S )) −→ Rep e ( π (Σ g ) , Diff + ( S ))is an abelian covering map, whose fiber can be identified with H (Σ g , Z ) as a torsor. Finallywe let Fol hor ( M ( e )) denote the space of horizontal foliations on the bundle M ( e ) with Eulerclass e , which then inherits a natural topology as a subspace of the space of sections ofthe oriented Grassmannian bundle, which can in turn be identified with the unit cotangentbundle ST ∗ M ( e ). After the choice of a base point as well as a standard generators a i , b i of π (Σ g ) and a trivialisation of M ( e ) over a neighbourhood of the bouquet of circles a ∨ b ∨ . . . ∨ a g ∨ b g one obtains a mapFol hor ( M ( e )) −→ g Rep e ( π (Σ g ) , g Diff + ( S )) . This is obtained by considering the holonomy around a loop in Σ g , which naturally gives apath in Diff + ( S ) with respect to the chosen trivialisation, and the homotopy class of thispath then gives an element in g Diff + ( S ). Conversely given any element in g Rep e ( π (Σ g ) , g Diff + ( S ))one can construct foliations with the given holonomy in a continuous manner. Lemma 8.1.
The map Fol hor ( M ( e )) −→ g Rep e ( π (Σ g ) , g Diff + ( S )) admits a section. More-over, any two foliations with the same associated holonomy representations are related by abundle automorphism that is isotopic to the identity.Proof. We consider the standard cell structureΣ g = ( a ∨ b ∨ . . . ∨ a g ∨ b g ) ∪ D , where a i , b i denote representatives of the standard generators of π (Σ g ). Now the bundle M ( e ) is trivial over a neighbourhood N of the 1-skeleton Σ (1) g and we choose a trivialisa-tion N × S . Note that N can be described as the union of strips b a i × [ − ǫ, ǫ ] , b b i × [ − ǫ, ǫ ]attached to a small disc D containing the base point p , where b a i , b b i are closed intervalscontained in a i , b i respectively that are disjoint from the base point p . Given an element e ρ ∈ g Rep( π (Σ g ) , g Diff + ( S )) we let { φ it } t ∈ [0 , be the linear path joining e ρ ( a i ) to the identity.We define { ψ it } t ∈ [0 , for e ρ ( b i ) in the same way. By reparametrising, we may assume thatthese paths are constant near the end points. We then define a foliation on N × S bypushing forward the product foliation using the paths φ it , ψ it on the parts of M ( e ) lying over b a i × [ − ǫ, ǫ ] , b b i × [ − ǫ, ǫ ] and extending by the trivial product foliation on D × S .We must then extend this over the 2-cell D . Since the foliation is already defined overa neighbourhood of the 1-skeleton, we only need to extend it over a slightly smaller disc D ′ contained in the interior of D . To this end we choose a trivialisation of M ( e ) over D ′ andnote that the foliation induces a loop γ t in Diff + ( S ) given as the holonomy around ∂D ′ ONTACT STRUCTURES, DEFORMATIONS AND TAUT FOLIATIONS 23 for some given basepoint q near p . The loop γ t is contractible since e ρ was an element in g Rep e ( π (Σ g ) , g Diff + ( S )) and thus lifts to a loop e γ t in g Diff + ( S ). This loop then extends to amap over D ′ using linear paths to the identity. Furthermore, the composition D ′ → g Diff + ( S ) → Diff + ( S )determines a fiber preserving isotopy Φ ρ of D ′ × S so that the pushforward of the productfoliation (Φ ρ ) ∗ F prod extends the foliation on N × S . By construction we may assume thatthe foliation is the pullback of the suspension foliation determined by γ t on ∂D ′ × S viaradial projection near the boundary. Thus we may assume that the resulting foliation F ρ on M ( e ) is smooth. The map ρ
7→ F ρ then defines the desired section, since the entireconstruction depends continuously on ρ .Now suppose F , F have the same holonomy representations. After an initial fiber pre-serving isotopy, we may assume that the foliations agree over a small disc D near the basepoint p . Then since the holonomy representations agree, there are fiberwise automorphismsover D ∪ Σ (1) g that are the identity near D so that the induced 1-dimensional foliations over( a ∨ b ∨ . . . ∨ a g ∨ b g ) \ D agree. After a further fiberwise isotopy we may then assumethat the foliations agree over a neighbourhood N of the 1-skeleton. Using the contractibilityof g Diff + ( S ) one can then extend this isotopy over the 2-cell D relative to the boundary,which then gives the desired fiberwise isotopy. (cid:3) Remark . Note that Lemma 8.1 holds with respect to the C k -topology for any 0 ≤ k ≤ ∞ ,where we consider the space of horizontal foliations as a subset of the space of sections ofthe unit cotangent bundle of the total space M ( e ). This is because the construction usesonly linear paths in g Diff + ( S ) and certain (fixed) cut-off functions, so that the tangent planefields of the associated foliations depends continuously on the holonomy. In addition, theproof of Lemma 8.1 easily extends to show that the fiber of the holonomy map is in fact(weakly) contractible. We remark that this is a special feature of codimension-1 foliationsand need not hold in higher codimension. Remark . We also note that the action of the full group of bundle automorphismsAut( M ( e )) on Fol hor ( M ( e )) descends to an action on g Rep e ( π (Σ g ) , g Diff + ( S )), which in turncorresponds to the action by the group of deck transformations Aut( M ( e )) / Aut ( M ( e )) ofthe covering map to Rep e ( π (Σ g ) , Diff + ( S )) which can be identified with H (Σ g , Z ).We have the following useful consequence of Lemma 8.1. Corollary 8.4.
Let F be a foliation on M ( e ) with holonomy e ρ and let e ρ t be a C k -continuouspath of representations in g Rep e ( π (Σ g ) , g Diff + ( S )) with e ρ = e ρ . Then there is a family offoliations F t whose tangent distributions depend continuously on t in the C k -norm for any ≤ k ≤ ∞ and such that F = F .Similarly, if e ρ n is a sequence converging in the C k -sense to e ρ then there is a sequenceof horizontal foliations F n so that the tangent distributions T F n converge to T F in the C k -sense.Proof. By Lemma 8.1 as well as Remark 8.2 there is a C k -continuous path of horizontalfoliations F t on M ( e ) whose holonomy representations are precisely e ρ t . Moreover, by thesecond part of Lemma 8.1 the original foliation F is fiberwise diffeomorphic to F by somediffeomorphism ϕ so that the desired path is given by applying ϕ to the path F t . The proofin the case of a sequence of representations follows mutatis mutandis . (cid:3) Now that we have clarified the relationship between holonomy representations and horizontalfoliations, we may now show that the space of representations Rep( π (Σ g ) , Diff + ( S )) is ingeneral not path connected. Theorem 8.5.
Let
Comp ( e ) denote the number of path components of the space of repre-sentations Rep e ( π (Σ g ) , Diff + ( S )) with fixed Euler class e = 0 such that e divides g − > and write g − n e . Then the following holds: Comp ( e ) ≥ n g + 1 . Proof.
By ([14] Th´eor`eme 3.1) there are horizontal contact structures ξ and ξ n with twistingnumber − − n respectively on the S -bundle M ( e ) with Euler class e . Furthermore,this contact structure ξ n can be made vertical, i.e. tangent to the S -fibers. Both contactstructures ξ and ξ n are linear deformations of a horizontal foliation by Theorem 6.2 andwe let ρ , ρ n be the associated holonomy representations in Rep( π (Σ g ) , Diff + ( S )). Assumethat ρ t is a smooth family of representations joining ρ to ρ n . We first lift this path to g Rep e ( π (Σ g ) , g Diff + ( S )) and then let F t denote the smooth family of foliations given byLemma 8.1. Note that the foliations we obtain in this way agree with the original foliationsup to fiber preserving automorphism of the total space of the associated bundle M ( e ). Inparticular, the twisting numbers of the contact structures obtained by linear perturbationof F and F agree with those of ξ resp. ξ n .Since each foliation in the family F t cannot have any closed leaves and M ( e ) does not fiberover S , we may perturb the family linearly to a 1-parameter family of contact structuresby Proposition 4.3. It then follows from Proposition 4.4 that ξ is isotopic to ξ n , which is acontradiction. Thus both ρ and ρ n lie in distinct components of Rep( π (Σ g ) , Diff + ( S )).The vertical contact structure ξ n determines a fiberwise n -fold cover of the unit cotangentbundle ST ∗ Σ g . By ([14] Lemme 3.9) the isotopy class of the associated n -fold covering is adeformation invariant of ξ n . Since all foliations are only well defined up to fiber preservingautomorphisms of M ( e ), it follows that only the fiberwise isomorphism class of the coveringis a deformation invariant of the associated foliation and hence of ρ n . Moreover, isomor-phism classes of fiberwise n -fold coverings are in one to one correspondence with elementsin H (Σ g , Z n ) and it follows that the numbers of path components of representations whoseperturbations have twisting number − n is at least n g . From this we conclude that Comp ( e ) ≥ n g + 1 . (cid:3) Remark . For the sake of concreteness let us consider the representations ρ d , ρ st given bya (2 d )-fold fiberwise cover of the suspension of a Fuchsian representation in PSL (2 , R ) andthe stabilisation of some Fuchsian representation respectively. More precisely, ρ st is the com-position of a Fuchsian representation of a surface of Euler characteristic d (2 − g ) = 2 − g ′ and a collapse map Σ g → Σ g ′ which collapses all but g ′ handles to a point. These represen-tations have the same Euler class but lie in different components of Rep( π (Σ g ) , Diff + ( S ))since the corresponding contact structures have twisting − n = − d in the first case and − Remark . Let G n denote the n -fold covering of PSL (2 , R ). The proof of Theorem 8.5 showsthat for fixed n the components of the representation spaces Rep( π (Σ g ) , G n ) as computedby Goldman [15] are distinguished by their contact perturbations. This applies both in the ONTACT STRUCTURES, DEFORMATIONS AND TAUT FOLIATIONS 25 case that n divides 2 g − T -bundles over S with Anosov monodromyof a certain kind, the stable and unstable foliations F s , F u cannot be deformed to one an-other through foliations without torus leaves. This uses Ghys and Sergiescu’s classificationresults [12] for foliations without closed leaves on such manifolds. However, F s and F u canbe deformed to one another through taut foliations: one first spins both foliations along afixed torus fiber to obtain foliations F ′ s , F ′ u with precisely one closed torus leaf T . On thecomplement of T one has a foliation by cylinders on T × (0 ,
1) intersecting each fiber ina linear foliation. It is then easy to construct a deformation between F ′ s and F ′ u throughfoliations with one torus leaf which is homologically non-trivial. Thus we conclude that onecan indeed deform F s to F u through taut foliations.In view of this, it remains to find taut foliations that cannot be deformed to one anotherthrough taut foliations, although their tangent distributions are homotopic. We give twotypes of examples of this phenomenon: the first uses deformations and contact topology andthe second uses the special structure of taut foliations on cotangent bundles. Theorem 8.8.
The space of taut foliations is in general not path connected on small Seifertfibered spaces.Proof.
We let M = − Σ(2 , , k −
1) be the link of the complex singularity z + z + z k − = 0taken with the opposite orientation, which has Seifert invariants (0 , − , , , k − k − ). As notedin ([25], p. 1746) the Seifert manifold M admits a vertical contact structure ξ vert that hastwisting number − (6 k −
7) if k >
1, which is then a linear deformation of a taut foliation F by Proposition 6.3. One further checks that the following holds(2) − n + l n m + l n m + l n (5 k − k − m = 2if and only if n = 6 l − ≤ l ≤ k −
1. By Theorem 5.11 this is a necessary condition forthe existence of a horizontal contact structure on M with twisting number − n . Moreover, thequotient space of the (6 l − M p −→ M ′ l given by Proposition 3.1 has normalisedinvariants (0 , − , , , k − l k − ) and thus admits a horizontal foliation F l by Theorem 3.3. Since F l cannot have any closed leaves and M ′ l does not fiber over S , the foliation F l can belinearly deformed to a horizontal contact structure ξ l . Now the corresponding necessarycondition for the existence of a horizontal contact structure on M ′ l with twisting number t ( ξ l ) is obtained by substituting n = − (6 l − t ( ξ l ) into equation (2) and it follows that − (6 l − t ( ξ l ) = 6 l ′ − , for some l ≤ l ′ ≤ k − . We observes that the (negative) twisting number of a contact structure is sub-multiplicativeunder covering maps. Thus, if 6 l − k −
7, then we deduce that − t ( p ∗ ξ l ) ≤ − (6 l − t ( ξ l ) < k − ξ ′ = p ∗ ξ l cannot be isotopic to ξ vert . Note that 6 l − k − l such that l > ( √ k − M is non-Haken all taut foliations are without closed leaves. Thus any path of tautfoliations joining F to F ′ = p ∗ F l can be deformed to an isotopy between ξ vert and ξ ′ byPropositions 4.3 and 4.4, which yields a contradiction if 6 l − k − (cid:3) Remark . Since the arguments above only used the arithmetic properties of the Seifertinvariants, they could also be applied to other small Seifert fibered manifolds. We also observethat the uniqueness results of Vogel [33] give alternative proofs of Theorems 8.5 and 8.8. Ashis results only assume C -closeness they yield that the conclusions about path componentsalso hold with respect to the weaker C -topology. All results also remain true for foliationsthat are only of class C as this suffices for Theorem 4.1 and its various consequences.Since the foliations in Theorem 8.8 are by construction horizontal, their tangent distribu-tions are homotopic as oriented plane fields. Thus by [21], they are homotopic as foliations(cf. also [10]). The construction of such a homotopy of integrable plane fields introducesmany Reeb components, so it is natural to ask whether this is necessary. The first exampleof this kind ([33], Example 9.5) was given by considering the oriented horizontal foliations F hor and F hor on the Brieskorn sphere Σ(2 , ,
11) taken with the positive orientation. Usingclassification results of contact structures on Σ(2 , , F hor cannotbe homotopic to F hor through oriented taut foliations, since the horizontal contact structuresobtained as perturbations ξ, ξ are not isotopic. However, since all Brieskorn spheres admitorientation preserving diffeomorphisms that reverse the orientation of the Seifert fibration(Remark 5.16), these contact structures are in fact contactomorphic so that one cannot de-duce disconnectedness for either diffeomorphism classes of taut foliations or for the space ofunoriented taut foliations in this example.On the other hand since the twisting number of a contact structure is a contactomorphisminvariant and does not depend on the orientation of the plane field, the proof of Theorem8.8 shows that the space of taut foliations is also disconnected even if one only considers diffeomorphism classes of unoriented foliations yielding the following Theorem 8.10.
There exist an infinite family of manifolds M n each admitting a pair of tautfoliations F , F that are homotopic as oriented foliations but not as taut foliations. Further-more, the same result holds true for unoriented foliations or if one considers diffeomorphismclasses of foliations. Since the manifolds Σ(2 , , k −
1) are non-Haken the notions of tautness and Reeblessnesscoincide and it follows that any homotopy between F and F in Theorem 8.10 must haveReeb components. Moreover, these examples show that the space of taut foliations in a givenhomotopy class of plane fields can have more than one equivalence class up to deformationand diffeomorphism, but nonetheless we can only distinguish finitely many such equivalenceclasses.It seems much more difficult to find examples where the number of equivalence classesis infinite. If instead one only considers deformation classes of taut foliations themselveswithout quotienting out by the action of the diffeomorphism group of the manifold as well,then it is possible to give examples where the number of components is infinite. This uses notonly the special structure of foliations on the unit cotangent bundle of a hyperbolic surfacebut also the structure of a foliation near torus leaves. Torus leaves and Kopell’s Lemma:
The behaviour of a foliation near a torus leaf iswell understood and is nicely described in Eynard-Bontemps’s thesis [10]. The fundamentalresult that puts restrictions on the behaviour of a foliation of class at least C near a torusleaf is the following lemma of Kopell. ONTACT STRUCTURES, DEFORMATIONS AND TAUT FOLIATIONS 27
Lemma 8.11 (Kopell) . Let f and g be commuting diffeomorphisms mapping [0 , intoitself (not necessarily surjectively) that fix the origin and are of class C and C respectively.Assume that f is contracting. Then either g has no fixed point in (0 , or g = Id . One also knows that torus leaves occur in a finite number of stacks in the following sense(cf. Eynard-Bontemps [10], Thurston [32]).
Lemma 8.12.
Let F be a -dimensional foliation on a closed -manifold M . Then either F is a foliation by tori on a T -bundle over S or there is a finite collection N i = T i × [0 , c i ] of foliated I -bundles over T in M with c i ≥ so that T i × { } and T i × { c i } are torus leavesand such that M \ ∪ N i contains no further torus leaves. Now for any stack N i as in Lemma 8.12, one has an induced holonomy homomorphism ρ F defined near each end of N i . Since there are no torus leaves outside of the stacks N i , itfollows with the help of Kopell’s Lemma that the holonomy f around some loop near saythe upper end of N i must be contracting on the outside of N i . Then by a result of Szekeres[31] one knows that f is the time 1 map of a flow generated by a C -vector field u ( z ) ∂ z that is smooth away from the fixed point c i (cf. [10] Th´eor`eme 1.1). Once again by Kopell’sLemma it follows that the entire image of ρ F is generated by elements contained in theflow generated by u ( z ) ∂ z . One can then conjugate the foliation to one whose characteristicfoliation is linear on tori near the ends of a stack of torus leaves (cf. [10], Lemme 5.21). Lemma 8.13.
Let F be a smooth foliation on T × [0 , ǫ ] having only L = T × { } as aclosed leaf and let ( x, y, z ) denote standard coordinates on T × [0 , ǫ ] . Then there is a C -diffeomorphism φ mapping T × [0 , ǫ ] into itself that fixes L and is smooth on T × (0 , ǫ ] suchthat the image of F under φ is defined by the kernel of the -form dz + u ( z )( a dx + b dy ) for some function u ( z ) ≥ that is positive away from L . For a stack of torus leaves we let λ ± = ( a ± , b ± ) be the asymptotic slope near the positiveresp. negative end of a stack normalised so that || λ ± || = 1. Note that near the negative endwe take coordinates of the form T × [ − ǫ, λ − and λ + do not coincidethen the stack of leaves is stable in the sense that any foliation in a C -neighbourhood of F has a closed torus leaf in a neighbourhood of N i . If a stack of tori has arbitrarily smallperturbations that are without closed leaves then the stack is called unstable .The final ingredient we shall need is Thurston’s straightening procedure for foliations on S -bundles (see also [3] p. 178 ff.). Theorem 8.14 (Thurston [32]) . Let F be a foliation on an S -bundle over a surface Σ g ofgenus g ≥ without closed leaves. Then F is isotopic to a horizontal foliation. Furthermore,if F is already horizontal on a vertical torus T , then this isotopy can be made relative to T .Remark . We note that if a vertical torus T is merely transverse to the foliation thenwe can assume after a suitable isotopy that the foliation is in fact horizontal on T . For sinceany foliation without closed leaves is isotopic to a horizontal one, it follows that [ e L ] · [ S ] = 1for any leaf e L of the foliation given by pulling back under covering induced by the universalcover of the base, when F is suitably oriented. This means that after a suitable isotopy allclosed leaves of the induced non-singular foliation F | T intersect each fiber positively (so thatthis intersection is non-empty). In particular, F | T has no 2-dimensional Reeb components and no closed orbit can be isotopic to a fiber. Consequently one can apply an isotopy of thefoliation with support near T so that F is horizontal on T .We are now ready to prove that the space of taut foliations on ST ∗ Σ g has infinitely manycomponents. Before giving the proof we clarify our orientation conventions concerning theEuler class of the (unit) cotangent bundle of a surface. Any real oriented vector bundleof rank-2 determines a unique complex line bundle. For the tangent bundle of an orientedsurface this is equivalent to the choice of an almost complex structure that is compatiblewith the orientation and the first Chern class of this complex line bundle T C Σ g agrees (moreor less by definition) with the Euler class of the tangent bundle as an oriented rank-2 bundle.Thus it is natural to identify T ∗ Σ g with the dual of the complex bundle so that the Eulernumber satisfies e = h e ( ST ∗ Σ g ) , [Σ g ] i = h c ( T ∗ C Σ g ) , [Σ g ] i = h− c ( T C Σ g ) , [Σ g ] i = 2 g − . Theorem 8.16.
The space of taut foliations on ST ∗ Σ g has at least Z g components if g ≥ ,which are all pairwise homotopic as foliations.Proof. By Giroux [14], Honda [19] all horizontal contact structures on ST ∗ Σ g are contac-tomorphic and can be made vertical. Furthermore, their isotopy classes are parametrisedby H (Σ g , Z ) ∼ = Z g , where H (Σ g , Z ) is identified with the set [Σ g , S ] of homotopy classof maps Σ g → S which in turn acts on isotopy classes of contact structures via gaugetransformations. Choose ξ vert , ξ ′ vert non-isotopic vertical contact structures. Such contactstructures are linear deformations of foliations F , F ′ by Theorem 6.2. In fact, identifying avertical contact structure with the canonical contact structure on ST ∗ Σ g it is easy to see thatthey are linear deformations of foliations that are descended from left invariant foliations on PSL (2 , R ) (cf. [2]).Now suppose F t is a smooth family of taut foliations joining F and F ′ . Note that thecondition of having no closed leaves is an open condition: for if F s has no closed leavesthen by Theorem 8.14 it is isotopic to a horizontal foliation, and since the Euler number e ( ST ∗ Σ g ) = 2 g − = 0 by assumption, any foliation sufficiently close to F s is also withoutclosed leaves. Thus the set of t for which F t is without closed leaves is open and non-empty since both foliations F and F ′ are without closed leaves, or equivalently the set ofvalues of t for which F t has a closed leaf is closed (and possibly empty). In view of this we let0 < t < F t has closed leaves, which must exist. Otherwisewe could linearly perturb the deformation by Proposition 4.3 to obtain a contradiction, sinceby assumption the contact deformations ξ vert and ξ ′ vert of F , F ′ respectively are not isotopic.Note that all the closed leaves of F t are incompressible tori. There is then a finite collectionof embeddings N i = T i × [0 , c i ] so that the foliation contains no closed leaves outside theunion of the N i and both T i × { } and T i × { c i } are closed leaves by Lemma 8.12. After anisotopy we may assume that the T i are vertical tori (i.e. they are tangent to the S -fibers)and we let γ i denote their image curves in Σ g .All stacks of torus leaves must be unstable by our assumption on t , otherwise all foliations F t with t sufficiently close to t would again have closed (torus) leaves. In particular, theasymptotic slopes of tori near both ends of N i must agree. We let N ′ i be a slight thickeningof N i . We then cut out N ′ i and reglue along the boundary to obtain a smooth foliation F ′′ .We let T i be the vertical torus corresponding to the stack N i and note that the foliation F ′′ restricts to a linear foliation on each T i . ONTACT STRUCTURES, DEFORMATIONS AND TAUT FOLIATIONS 29
We first claim that F ′′ cannot have any closed leaves. Since each closed leaf in the originalfoliation was contained in one of the sets N i , any closed leaf of F ′′ must intersect at leastone of the tori T i . We suppose that L is a closed leaf of F ′′ and will derive a contradiction.The intersection of L with each torus T i (when non-empty) is a collection of parallel circlesthat are isotopic to the fibers of M = ST ∗ Σ g . Let A be the closure of one of the annularcomponents of L \ ( ∪ i T i ) in the completion c M of M \ ( ∪ i T i ) that is obtained by addinga positive and a negative boundary torus T ± i for each T i . The foliation F ′′ on M \ ( ∪ i T i )extends to a foliation b F on c M in a natural way. Since the characteristic foliation on each T ± i is linear, the holonomy of the foliation b F around a boundary component of ∂A is trivial.Thus a neighbourhood of A in c M is also foliated by annuli. Using the compactness ofthe space of compact leaves it follows that the maximal 1-dimensional family { A t } t ∈ [0 , ofannuli containing A then contains the boundary tori which A intersects. Hence the family { A t } t ∈ [0 , fibers an annulus bundle over S inside c M and this is the case for each componentof L \ ( ∪ i T i ). After reidentifying the boundary tori T + i and T − i this would give a descriptionof M as a union of annulus bundles over S glued along boundary tori. But then M = ST ∗ Σ g would be fibered by tori, which is obviously a contradiction.Thus as F ′′ cannot have any closed leaves, we may apply Thurston’s straightening pro-cedure relative to each T i to obtain a horizontal foliation such that the intersection of F ′′ with T i is linear. This is equivalent to the fact that the holonomy around γ i is conjugate toa rotation contradicting ([28], Theorem 2.2) and we conclude that no foliation in the familycan have closed leaves. It follows that the family cannot exist and that F and F ′ cannot bedeformed to one another through taut foliations. Since there are Z g different isotopy classesof contact structures, there are at least this many deformation classes of taut foliations.Finally since the foliations we are considering are by construction horizontal, their tangentdistributions are homotopic as plane fields and thus by Larcanch´e [21] they are homotopicas integrable plane fields. (cid:3) In fact, the proof of Theorem 8.16 shows that if a family of taut foliations F t on ST ∗ Σ g contains a foliation which does not have closed leaves, then the same is true for the entirefamily. This observation also applies to families of Reebless foliations. Furthermore, sincea foliation on ST ∗ Σ g without closed leaves is isotopic to the suspension foliation given by aFuchsian representation in view of [11], we deduce the following corollary. Corollary 8.17.
Let F hor be a horizontal foliation on the unit cotangent bundle of a hy-perbolic surface ST ∗ Σ g . Then any foliation in the path component of F hor in the space ofReebless foliations is isotopic to a foliation given by the suspension of a Fuchsian represen-tation. It is easy to construct taut foliations F T with a single vertical torus leaf on any S -bundleas long as the base has positive genus and we may assume that the tangent distribution ofsuch a foliation is homotopic to a horizontal distribution. In view of Corollary 8.17 therecan be no Reebless deformation between F T and any horizontal foliation, even if one allowsdiffeomorphisms of either foliation. The same applies to any pair of diffeomorphic horizontalfoliations whose contact perturbations are not isotopic.The arguments above also apply to deformations of C -foliations that are only continuouswith respect to the C -topology. Note, however, that the foliations F T and F hor can in factbe deformed to one another through taut foliations that are only of class C . This follows by first spinning the horizontal foliation F hor along the vertical torus T , which can be donein a C -manner. The remainder of the foliation is determined by a representation of a freegroup to g Diff + ( S ). Joining any two such representations arbitrarily and spiralling into T then gives the desired deformation. We note this in the following proposition: Proposition 8.18.
There exist pairs of smooth taut foliations which can be deformed to oneanother through taut C -foliations, but not through taut C ∞ -foliations. Anosov foliations
In this section we give an alternative approach to the results obtained above that usesthe classification of Anosov foliations of Ghys [11] and results of Matsumoto [27], [28]. Wewill call a representation
Anosov if its associated suspension foliation is diffeomorphic to theweak stable foliation of an Anosov flow. Recall that a flow Φ tX generated by a vector field X on a closed 3-manifold M is Anosov if the tangent bundle splits as a sum of line bundlesthat are invariant under the flow T M = E u ⊕ E s ⊕ X such that for some choice of metric and C, λ > t > || (Φ tX ) ∗ ( v u ) || ≥ C − e λt || v u || and || (Φ tX ) ∗ ( v s ) || ≤ Ce − λt || v s || , where v u ∈ E u , v s ∈ E s . The line fields E u , E s are called the strong unstable resp. stablefoliations of the flow and the foliations F u , F s tangent to the integrable plane fields E u ⊕ X , E s ⊕ X are called the weak unstable resp. stable foliations of the flow. In general, the weak stableresp. unstable foliations need not be smooth and this puts strong restrictions on the possibleflows as we will see below. However, as we are interested in smooth foliations we will alwaysassume that both the weak stable and unstable foliations are smooth .An important property of Anosov flows and foliations is their structural stability. C -stability of the Anosov condition goes back to Anosov’s original paper [1]. Moreover, thedynamics of an Anosov representation in terms of its translation numbers also turn out tobe C -stable. Definition 9.1 (Translation number) . Let e φ ∈ ^ Homeo + ( S ) be considered as a -periodicdiffeomorphism of R . The translation number is defined astr ( e φ ) = lim n →∞ e φ n ( x ) n . If φ ∈ Homeo + ( S ) then the rotation number rot( φ ) is defined as the image of tr( e φ ) in S = R / Z for any lift of φ to ^ Homeo + ( S ).The prime examples of Anosov foliations come from the weak stable foliations of geodesicflows on hyperbolic surfaces and suspensions of linear Anosov maps on T . If one assumesthat the weak stable foliation of an Anosov flow is required to be sufficiently regular, thenGhys [11] has shown that these are the only possibilities up to taking finite covers. Thisresult is usually only stated in the case of the unit tangent bundle of a hyperbolic surface([11] Theorem 5.3). A more general result that classifies smooth Anosov foliations on all S -bundles can easily be gleaned from Ghys’ arguments, but as there is no clear statement ONTACT STRUCTURES, DEFORMATIONS AND TAUT FOLIATIONS 31 of this in [11] we include a brief account of the main steps to obtain this classification (cf.Theorem 9.4 below).If a smooth Anosov flow has sufficiently smooth weak stable foliation (in fact C , wouldsuffice), then it admits a transverse projective structure. Theorem 9.2 ([11] Theorem 4.1) . Let X generate a smooth Anosov flow on a closed -manifold whose weak stable foliation F s is smooth. Then F s admits a transverse projectivestructure. Using this projective structure one then gets a topological classification of Anosov flowswhose weak stable foliations are smooth by results of Barbot.
Theorem 9.3 ([11] Theorem 4.7) . Let X generate a smooth Anosov flow on a closed -manifold whose weak stable foliation F s is smooth. Then F s is topologically conjugate tothe weak stable foliation of the suspension of a linear Anosov diffeomorphism on T or toa homogeneous Anosov flow on ^ P SL (2 , R ) / e Γ for some co-compact lattice e Γ lying in theuniversal cover of PSL (2 , R ) . As a consequence of Theorem 9.3 we obtain that if the foliation in question is a suspensionfoliation on an S -bundle, then the holonomy representation is topologically conjugate tothe inclusion of a lattice π (Σ g ) ∼ = Γ ֒ → G n , where G n denotes some n -fold cover of PSL (2 , R ).In order to obtain a smooth classification of Anosov flows/foliations Ghys then observesthat the techniques used to prove Theorem 9.2 apply more generally to flows that are only topologically conjugate to an Anosov flow whose weak stable foliation is smooth (cf. [11]Section 5). In this way there is a version of Theorem 9.2 which holds under the weakerassumption that a suspension foliation is only topologically conjugate to an Anosov foliationand this yields the following smooth classification result. Theorem 9.4 (Smooth Anosov Foliations on S -bundles [11]) . Let F be a suspension folia-tion on an S -bundle whose holonomy map is topologically conjugate to one that is Anosov.Then the holonomy map of F is smoothly conjugate to a suspension of a co-compact latticein a finite cover of PSL (2 , R ) .In particular, any Anosov foliation on an S -bundle is smoothly conjugate to the suspensionof a co-compact lattice in some finite cover of PSL (2 , R ) .Sketch of proof. One first alters the smooth structure on the underlying S -bundle so thatthe foliation F becomes the weak stable foliation of a smooth flow that is topologicallyconjugate to one that is Anosov and has smooth weak stable foliation (cf. [11] pp. 177–8).The resulting weak stable foliation is then topologically conjugate to F and we denote it by F ′ .The arguments used to obtain a transverse projective structure for F ′ (Markov Partitions,etc.), then also give a transverse projective structure on F (cf. [11] pp. 179–181). It followsfrom the classification of projective structures on S as described in the proof of ([11] Lemma5.1) that the holonomy group of F is then smoothly conjugate to a subgroup of G n (which a priori may not be a lattice). By Theorem 9.3 the holonomy group is also topologicallyconjugate to a lattice in G n and this implies that the holonomy map is injective and itsimage Γ ⊆ G n contains no (non-trivial) elliptic elements. The same is then true of the projection Γ ⊆ PSL (2 , R ) of Γ and it follows from standard results about subgroups of PSL (2 , R ) that Γ is either discrete or elementary (i.e. it has a finite orbit when acting onthe closure of the Poincar´e disc). If the subgroup Γ were elementary, then after taking afinite index subgroup the action would fix a point on the boundary of the disc. Thus (up toconjugacy) Γ is contained in the subgroup of upper triangular matrices and is hence solvable,which is absurd. It then follows that Γ, and hence π (Σ g ) ∼ = Γ, is a discrete subgroup, whichnecessarily acts co-compactly and the classification follows. (cid:3) It is customary to call foliations that are given associated to a left-invariant Anosov flow ona left quotient of a Lie group by a co-compact lattice algebraic Anosov and in the sequel weadopt this terminology.
Remark . We note that Theorem 9.4 readily generalises to horizontal foliations on anarbitrary Seifert fibered space M yielding a similar classification result. In this situation, onecan consider the holonomy map associated to the foliation as a representation of the orbifoldfundamental group π orb ( B ) of the base (which is necessarily hyperbolic), by considering theholonomy around curves in the base that avoid the orbifold points. Assuming that F istopologically conjugate to a smooth Anosov foliation one can alter the smooth structureexactly as above so that the flow becomes smooth and then the arguments of ([11] Section5) sketched above provide a transverse projective structure. Thus, just as in the case of an S -bundle, the holonomy representation is smoothly conjugate to one with values in some G n . One then applies Theorem 9.3 and argues exactly as above, after first taking a torsionfree subgroup of finite index, to deduce that the resulting holonomy map must then be givenby the inclusion of a co-compact lattice in G n .As a consequence of the special structure of Anosov flows on S -bundles, the rotationnumbers of an Anosov representation are stable under deformations. We are grateful to S.Matsumoto for suggesting a simplified proof of the following lemma. Lemma 9.6.
Let ρ An ∈ Rep e ( π (Σ g ) , Diff + ( S )) be an Anosov representation. Then for anyrepresentation ρ that is sufficiently C -close to ρ An the rotation numbers of ρ ( γ ) are rationalfor all γ ∈ π (Σ g ) .Proof. After lifting ρ An to g Rep e ( π (Σ g ) , g Diff + ( S )) we consider the foliation F given byLemma 8.1. Let F = F s , F u be the weak stable and unstable foliations of the Anosov flowgenerated by the normalised vector field X generating F u ∩ F s . For ρ sufficiently C -closeto ρ An we choose lifts to g Rep e ( π (Σ g ) , g Diff + ( S )) that are also C -close. Then the tangentdistributions of the associated foliation T F ρ remains C -close to T F s (cf. Corollary 8.4) andhence the foliation F ρ remains transverse to F u , since transversality is an open condition.Moreover, the normalised vector field X ′ generating the intersection F ρ ∩ F u is C -close to X .By Theorem 9.4 the foliation F s is smoothly conjugate to an algebraic Anosov foliationgiven by the suspension of a lattice in some covering of PSL (2 , R ). Hence we can assumethat the flow Φ Xt is conjugate to a covering of the geodesic flow on the unit tangent bundle ST Σ g for some choice of hyperbolic metric on Σ g = H / Γ. We consider the geodesic flow onthe unit tangent bundle of H as well as its associated weak unstable and stable foliations e F u , e F s . Note that the projection to the base induces an isometry on the leaves of e F u and e F s ONTACT STRUCTURES, DEFORMATIONS AND TAUT FOLIATIONS 33 respectively. These coverings fit into the following commutative diagram: π ∗ ST H / / (cid:15) (cid:15) M ( e ) π (cid:15) (cid:15) ST H (cid:15) (cid:15) / / ST Σ g (cid:15) (cid:15) H / / Σ g = H / Γ . The induced flow on a leaf L of the weak unstable foliation e F u consists of geodesics thatare all asymptotic in forward time to the same point on ∂ ∞ H . In particular, the anglebetween the flow lines on L and the boundary of an ǫ -neighbourhood N ǫ ( σ ) with respect tothe hyperbolic metric of any (geodesic) flow line σ on L is constant and the flow points into N ǫ ( σ ) along the boundary. We consider an isometric lift L of L to π ∗ ST H . Assume that σ lies on a leaf L and projects to a closed lift γ in M ( e ) of a k -fold cover of a simple closedgeodesic γ in ST Σ g . Note that in this case the image of the leaf L in M ( e ) is a cylinder. Wefactor the covering map of M ( e ) above, by first taking the cyclic covering corresponding tothe element [ γ ] ∈ Σ g : L ⊆ π ∗ ST H / / (cid:15) (cid:15) M ( e ) = π ∗ ST H / Z / / (cid:15) (cid:15) M ( e ) (cid:15) (cid:15) H / / H / Z / / Σ g = H / Γ . Then the image of N ǫ ( σ ) is an annular neighbourhood N ǫ = N ǫ ( σ ) / Z of a lift γ ′ that projects diffeomorphically to γ , so that the vector field X has angle uniformly bounded away from0 along ∂N ǫ . Since X ′ is C -close to X on M ( e ) the same is true for the lifted vector fieldon the covering M ( e ). Thus the first return map of the flow on N ǫ is a contraction so that X ′ also has a closed orbit in N ǫ , and this closed orbit is isotopic to γ ′ . It follows that therotation number of the holonomy given by the k -th iterate γ k is 1 and thus rot( ρ ( γ )) mustbe rational.Hence the lemma holds for holonomies around simple closed geodesics in the base. How-ever, every free homotopy class of loops in Σ g has a simple geodesic representative afterpassing to some finite cover. We observe that the arguments are not affected by taking finitecovers. Thus we can assume that every element in π (Σ g ) is conjugate to one that is rep-resented by a simple closed geodesic and since the rotation number is conjugation invariantthe lemma follows. (cid:3) In order to exploit the topological stability of Anosov foliations, we will need a result ofMatsumoto that characterises the conjugacy type of a representation in terms of translationnumbers (cf. [27] Theorem 1.1).
Lemma 9.7.
Let ρ , ρ ∈ Rep (Γ , Homeo + ( S )) for an arbitrary finitely generated group Γ and assume for all g ∈ Γ there are lifts ] ρ ( g ) , ] ρ ( g ) to ^ Homeo + ( S ) such that the translationnumbers satisfy tr ( ^ ρ ( g ) ^ ρ ( g )) − tr ( ^ ρ ( g )) − tr ( ^ ρ ( g )) = tr ( ^ ρ ( g ) ^ ρ ( g )) − tr ( ^ ρ ( g )) − tr ( ^ ρ ( g )) for all g , g ∈ Γ . If in addition rot ( ρ ( s k )) = rot ( ρ ( s k )) for some generating set h s k i ⊂ G ,then ρ and ρ have the same bounded integral Euler class and the representations are semi-conjugate. If the actions are minimal, then they are in fact conjugate. With the aid of this lemma we obtain the following theorem, which answers a question posedto us by Y. Mitsumatsu. For the statement we let a i , b i ∈ π (Σ g ) be standard generators forthe fundamental group. Theorem 9.8.
Let ρ t be a C -continuous path in Rep ( π (Σ g ) , Diff + ( S )) such that ρ isAnosov. Then ρ t consists entirely of Anosov representations.Moreover, the space of Anosov representations Rep An ⊂ Rep ( π (Σ g ) , Diff + ( S )) has finitelymany C -path components and the map Rep An −→ ( Z k ) g , ρ ( rot ( ρ ( a i )) , rot ( ρ ( b i ))) gi =1 induces a bijection of path components, where k is such that the Euler number of the under-lying S -bundles satisfies k e = 2 g − .Proof. Let ρ t be a C -continuous path starting at an Anosov representation ρ . We set S An = { t | ρ τ is Anosov for all 0 ≤ τ ≤ t } . We first show that S An is open. Assume that ρ s is Anosov for all τ ≤ s ∈ [0 , ρ t ( g ) are rational for any g ∈ π (Σ g ) and t sufficiently close to s (independently of g ). Thus by the continuity of the rotation number, these rotation numbersmust be constant for t close to s . We choose a lift ] ρ t ( g ) of the path ρ t ( g ) to g Diff + ( S ) foreach g ∈ π (Σ g ). Then ] ρ t ( g ) has the same translation number as ] ρ s ( g ) for all elements g . Itfollows that tr ( ^ ρ t ( g ) ^ ρ t ( g )) ≡ tr ( ^ ρ t ( g g )) = tr ( ^ ρ s ( g g )) ≡ tr ( ^ ρ s ( g ) ^ ρ s ( g )) mod Z and we conclude by continuity of translation numbers that tr ( ^ ρ t ( g ) ^ ρ t ( g ))) is constant forall t sufficiently close to s . It follows that the hypotheses of Lemma 9.7 are satisfied for ρ s and ρ t where | s − t | < ǫ is sufficiently small and both representations have the same boundedintegral Euler class and are thus semi-conjugate.Furthermore, the action on S induced by ρ = ρ t either has a finite orbit, an exceptionalminimal set or it is minimal. Note that the first case is ruled out since the Euler classis non-zero. If the action had an exceptional minimal set K ⊂ S , then the semi-properleaves of the associated suspension foliation must have infinitely many ends by Duminy’sTheorem (cf. [4]). Moreover, since the actions are semi-conjugate, we conclude that theAnosov foliation given by the suspension of ρ s also has a leaf with infinitely many ends. Butthe weak stable foliation of an Anosov flow can have only leaves with 1 or 2 ends yieldinga contradiction. Thus in fact ρ itself must be minimal and hence again by Lemma 9.7 itis topologically conjugate to ρ s . Finally by Theorem 9.4 the representation ρ = ρ t is thensmoothly conjugate to an (algebraic) Anosov representation and the openness of the set S An follows.We next show that S An is also closed. Let ( t n ) be a sequence of elements in S An convergingto s . By construction we can assume that this convergence is monotone, i.e. t n ր s . Nowby the arguments above the rotation numbers of ρ t ( g ) are constant for each g ∈ π (Σ g ) and ONTACT STRUCTURES, DEFORMATIONS AND TAUT FOLIATIONS 35 t < s and it follows that ρ t ( g ) is also constant on the interval [0 , s ]. Taking lifts it againfollows that: tr ( ^ ρ t ( g ) ^ ρ t ( g )) ≡ tr ( ^ ρ s ( g ) ^ ρ s ( g )) mod Z , for all t ∈ [0 , s ] . Then arguing exactly as above we deduce that ρ s is topologically, and hence smoothly,conjugate to an Anosov representation. Consequently the set S An is both open and closedand it is obviously non-empty. We conclude that the path ρ t is wholly contained in Rep An ,showing that this set indeed consists of C -path components of Rep( π (Σ g ) , Diff + ( S )).Due to Theorem 9.4 and results of Goldman [15] we can now describe the C -path com-ponents of the space of Anosov representations explicitly. Any Anosov representation issmoothly conjugate to an embedding of a discrete subgroup in the k -fold cover G k of PSL (2 , R ) by Theorem 9.4, where k is determined by the Euler class of the representation.The number of components of Rep max ( π (Σ g ) , G k ) is finite by [15] and are distinguishedby elements in H ( π (Σ g ) , Z k ). In particular, the path components of Rep max ( π (Σ g ) , G k )can be distinguished by the rotation numbers on the images of generators a i , b i . To seethis note that the rotation number of an algebraic Anosov representation lies in the k -throots of unity Z k ⊂ S so that the rotation numbers ρ An ( a i ) , ρ An ( b i ) are constant on C -components. This concludes the proof, since the maps { a , b , ..., a g , b g } → Z k given byrot( ρ An ) correspond precisely to the elements H ( π (Σ g ) , Z k ) that distinguish componentsof Rep max ( π (Σ g ) , G k ). (cid:3) As a consequence of Theorem 9.8 we obtain the following extension of Ghys and Matsumoto’sglobal stability statement about conjugacy classes of representations in Rep( π (Σ g ) , Diff + ( S ))for the case of maximal Euler class [11], [28] to other topological components. Corollary 9.9.
Any representation ρ ∈ Rep ( π (Σ g ) , Diff + ( S )) that lies in the C -path com-ponent of an Anosov representation ρ An is smoothly conjugate to an embedding of a discretesubgroup in some finite cover of PSL (2 , R ) and is topologically conjugate to ρ An . In particu-lar, it is an injective discrete co-compact representation.Remark . Theorem 9.8 and its corollary also remain true for Anosov representations ofany hyperbolic orbifold group π orb ( B hyp ) in view of Remark 9.5. Since both Theorem 9.8and its corollary hold with respect to the C -topology, they yield a proof of the C -versionof Theorem 8.5 without using contact topology in the form of Vogel’s results [33]. Remark . The arguments used in the proof of Theorem 9.8 imply that Corollary 8.17holds for any Anosov foliation on a Seifert fibered space. One can use the closedness of thespace of Anosov representations instead of the results of Matsumoto [28] to rule out the firstinstance of unstable stacks of torus leaves and the remainder of the proof holds verbatim .If we restrict ourselves to the C -topology then we obtain a slight strengthening of Theorem9.8 which then distinguishes connected components rather than just path components. Theorem 9.12.
The space of Anosov representations
Rep An ⊂ Rep ( π (Σ g ) , Diff + ( S )) isboth open and closed with respect to the C -topology. It has finitely many connected com-ponents which are distinguished by the rotation numbers of the images of a set of standardgenerators a i , b i ∈ π (Σ g ) .Proof. The openness follows immediately from the C -stability of the Anosov condition.Closedness follows from the fact that any sequence of Anosov representations ( ρ n ) converging to some ρ can (after possibly passing to a subsequence) be assumed to all be topologicallyconjugate to a fixed Anosov representation by Theorem 9.4 and Matsumoto [28]. Thusrotation numbers and translation numbers of lifts are automatically constant and one canargue exactly as in the proof of Theorem 9.8 to deduce closedness. (cid:3)
It would be interesting to know whether the above results also hold for (non-smooth)topological actions on S , which would involve showing some sort of stability statements forlaminations that are Anosov in some suitable sense. In fact, by results of Ghys, topologicalactions on S are classified up to semi-conjugacy by the bounded integral Euler class e Z b . Itthen seems to be an open problem to determine the imageRep( π (Σ g ) , Homeo + ( S )) e Z b −→ H b ( π (Σ g ) , Z ) . Note that the map to real bounded cohomology given by the real bounded Euler class e R b iscontinuous with respect to the weak- ∗ topology on the vector space H b ( π (Σ g ) , R ). One alsohas a natural map Rep( π (Σ g ) , Homeo + ( S )) Φ g −→ ( S ) g = T g given by the rotation numbers on the standard generators of π (Σ g ). Since the the boundedintegral Euler class of an action on S is determined by its real Euler class together with therotation numbers on generators by Matsumoto [27], one can identify the image of e R b × Φ g with the image of e Z b , which then inherits a topology in a natural way. It would be interestingto understand whether this image is connected or not, with respect to the weak- ∗ topology,which would then provide insights into the topology of the representation spaces in whichwe are interested. In fact one knows that the image consists of classes that admit cocyclerepresentatives taking only the values 0 ,
1. The straight line between any two such cocycles tz + (1 − t ) z obviously gives a continuous path in H b ( π (Σ g ) , R ). However, it is not clear,and perhaps very unlikely, that this path lies in the image of the bounded Euler class. References [1] D. Anosov,
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