A Boothby-Wang theorem for Besse contact manifolds
aa r X i v : . [ m a t h . S G ] J u l A BOOTHBY–WANG THEOREM FOR BESSE CONTACT MANIFOLDS
MARC KEGEL AND CHRISTIAN LANGE
Abstract.
A Reeb flow on a contact manifold is called Besse if all its orbits are periodic, possibly withdifferent periods. We characterize contact manifolds whose Reeb flows are Besse as principal S -orbibundlesover integral symplectic orbifolds satisfying some cohomological condition. Apart from the cohomologicalcondition this statement appears in the work of Boyer and Galicki in the language of Sasakian geometry [12].We illustrate some non-commonly dealt with perspective on orbifolds in a proof of the above result. Moreprecisely, we work with orbifolds as quotients of manifolds by smooth Lie group actions with finite stabilizergroups. By introducing all relevant orbifold notions in this equivariant way we avoid patching constructionswith orbifold charts.As an application, and building on work by Cristofaro-Gardiner–Mazzucchelli, we deduce a completeclassification of closed Besse contact 3-manifolds up to strict contactomorphism. Introduction
Recall that a contact manifold is a pair (
M, α ) of a smooth, 2 n + 1-dimensional manifold M and a 1-form α on M , the so-called contact form, such that α ∧ ( dα ) n is nowhere vanishing. In the following we alwaysassume M to be connected. The Reeb vector field R on M is the unique vector field satisfying α ( R ) ≡ dα ( R, · ) ≡
0; it generates the so-called Reeb flow φ αt : M → M [18].Natural examples of Reeb flows are geodesic flows on unit sphere bundles of Riemannian, or more generallyFinsler manifolds [17], and on Riemannian orbifolds with isolated singularities [33]. A contact manifold ( M, α )is called
Besse if all its Reeb orbits are periodic. By results of Sullivan and Wadsley the Reeb flow of a Bessecontact manifold is actually periodic itself [45, 43], see Section 2.3. If in addition all orbits have the sameminimal period, then the contact manifold (
M, α ) is called
Zoll . Zoll Reeb flows generalize the much studiedclass of geodesic flows of Zoll metrics [9], and are closely related to systolic geometry, see [7, 2, 3, 1] and thereferences therein.According to a classical result by Boothby and Wang Zoll contact manifolds (
M, α ) are completely un-derstood in terms of prequantization bundles over integral symplectic manifolds [10], cf. [6]. More precisely,a symplectic manifold (
W, ω ) with integral form [ ω/ π ] gives rise to principal S -bundles M → W , one foreach integral lift of − [ ω/ π ] to H ( M ; Z ), with connection 1-forms α on M which are contact and whoseReeb vector fields generate the S -action on M . Conversely, every Zoll contact manifold arises in this way.Historically, this result was one of the first general constructions for contact manifolds and is used in severalother constructions as well, see e.g. [13, 25].In the more general orbifold setting, which the Besse condition naturaly gives rise to, several special casesof a corresponding result have been investigated [46, 44, 11, 14]. A fully analogous result, as we later learned,seems to have been folklore for some time and is explicitly stated in a book by Boyer and Galicki in thelanguage of Sasakian geometry, see Theorem 6.3.8, Theorem 7.1.3 and Theorem 7.1.6 in [12]. A proof of ittogether with several applications will also appear in the upcoming thesis of G. Placini [38]. In the case whenthe total space is a manifold this result applies to what we call a Besse contact manifold and adds to severalfurther interesting characterizations [44, 15, 22]. For instance, a contact manifold is Besse if and only if itis almost regular, which is a local condition on the flow [44], see Section 2.3. Moreover, 3-dimensional, andconjecturally also higher-dimensional Besse contact manifolds admit an intriguing characterization in termsof their Reeb period spectrum [15]. Other spectral characterizations are discussed in [22].In the this context we will use the orbifold Boothby–Wang theorem to prove a complete classification ofclosed Besse contact 3-manifolds up to strict contactomorphism, see Theorem 1.4 and Corollary 1.6. Moreover, Date : July 14, 2020.2010
Mathematics Subject Classification.
Key words and phrases.
Besse contact manifolds, Boothby–Wang theorem, symplectic orbifold, orbibundles, periodic Reebflow. we add a cohomological condition to the orbifold Boothby–Wang correspondence which characterizes thosebundles whose total space is a manifold, and which may therefore be useful to detect new examples ofBesse contact manifolds. This characterization relies on a general cohomological characterization of orbifoldsamong manifolds and an application of the Gysin sequence. On the technical side we explain how to thinkof a smooth orbifold as being represented by an almost free action (i.e. with finite isotropy groups) of a Liegroup G on a manifold M and give equivariant interpretations of several orbifold notions like orbibundles. Weillustrate this viewpoint in a proof of the orbifold Boothby–Wang theorem that avoids patching constructionswith orbifold charts and does not refer to additional Sasakian structures. In this way the proof becomes moreglobally flavoured and focused on the essential objects.The first part of this correspondence reads as follows. Here the orbifold cohomology H ∗ orb ( O ) of anorbifold O = M//G is defined as the G -equivariant cohomology of M and does not depend on the specificrepresentation of O in terms of M and G , see Section 3.1. We comment on the other relevant notions below. Theorem 1.1 (cf. Thm. 4.3.15, Thm. 6.3.8 and Thm. 7.1.3 in [12]) . Let ( M n +1 , α ) be a Besse contact manifold.Then, after rescaling by a suitable constant, the Reeb flow of α has period π and α is the connection -formof a corresponding principal S -orbibundle π : M → O over a symplectic orbifold ( O , ω ) , with ω the curvatureform of α and − ω/ π representing the real Euler class e R ∈ H ( O ; R ) of π : M → O . Moreover, the integralEuler class e of π : M → O induces isomorphisms e ∪ · : H i orb ( O ; Z ) ∼ −→ H i +2orb ( O ; Z ) (1.1) for all i ≥ n + 1 . In the theorem O is represented by an almost free S -action on M , which is induced by the Reeb vectorfield R of α . Let us add some more clarifying remarks. • A principal S -orbibundle whose total space M is a manifold is the same as an almost free S -actionon M (for the general definition see Section 3). In the theorem the S -action is the one induced bythe Reeb flow. In this case a connection -form α is a 1-form satisfying α ( R ) ≡ dα ( R, · ) ≡ curvature form is the S -basic dα . Here S -basic means that the 2-form is S -invariant andvanishes in the S -direction R . The basic cohomology class of ω (see Section 3.2) can be canonicallyidentified with an element in H ( O ; R ) via the equivariant de Rham theorem, see Section 3.2. Theform ω being symplectic means, in our language, that it is a closed basic 2-form on M satisfying ω n = 0. • The orbifold O in the theorem can also be seen as a symplectic reduction (see [35, 27]) of thesymplectisation ( R > × M, d ( rα )) performed on a level set of the Hamiltonian H ( r, x ) = r . Wewill make this more precise in Remark 4.2. • Condition (1.1) actually characterizes principal S -orbibundles over any orbifold whose total space isa manifold in terms of their integral Euler class (defined in Section 3.1), see Theorem 1.3. Examplesfor which this condition is satisfied are discussed at the end of the introduction.In Theorem 1.1 the fact that the quotient is an orbifold to which dα descends as a symplectic form wasobserved by Weinstein in [46], generalizing a statement by Thomas that only holds over the regular part [44,Theorem 2]. The first part of Theorem 1.1 appears in the work of Boyer and Galicki on Sasakian geometry,see [11] and Theorem 6.3.8 and 7.1.3 in [12]. Their results are stated in terms of quasi-regular (i.e. almostregular, see Section 2.3) K-contact manifolds. A K-contact manifold is a metric contact manifold whose Reebvector field is Killing, see [11]. For any Besse contact form α one can average a compatible metric via the S -action induced by the Reeb to obtain a K-contact structure. Conversely, any quasi-regular K-contact formis Besse, see Section 2.3.The converse construction to Theorem 1.1 works as follows. Theorem 1.2 (cf. Thm. 7.1.6 in [12]) . Let ( O n , ω ) be a symplectic orbifold with integral symplectic form ω/ π ,i.e. [ ω/ π ] ∈ Im (cid:0) H ( O ; Z ) → H ( O ; R ) (cid:1) . Then for every integral lift e = e Z of − [ ω/ π ] in H ( O ; Z ) that satisfies the cohomological condition (1.1)for all sufficiently large i , there exists a principal S -orbibundle X → O with Euler class e and a connection -form α on X with the following properties: • X is a manifold BOOTHBY–WANG THEOREM FOR BESSE CONTACT MANIFOLDS 3 • α is a Besse- and an almost regular contact form (see Section 2.3), • the curvature form of α is ω , • the vector field R defining the principal S -action on X coincides with the Reeb vector field of α . We point out that the seemingly weaker cohomological condition in Theorem 1.2 already implies the onein Theorem 1.1. A special version of this statement for certain classes of symplectic orbifolds, which arefor instance K¨ahler and only have a single nontrivial isotropy group, first appeared in [44, Theorem 3].Apart from the cohomology condition the theorem is stated by Boyer and Galicki in [12, Thm 7.1.6] in thelanguage of almost K¨ahler orbifolds and K -contact structures. We stress however that the class [ ω/ π ] doesnot uniquely determine the bundle X as the formulation in [12] suggests. Only its integral lift e does, cf.Example 3.3. Moreover, a related construction of almost regular contact manifolds via fibered Dehn twists isgiven in [14, Theorem 6.5]. For an integral Euler class that does not satisfy condition (1.1) the constructionwould yield a contact orbifold with periodic Reeb flow (cf. [12, Thm. 7.1.6]), but we will not make this notionprecise here. Since the characterization in terms of (1.1) might be useful elsewhere, we explicitly state ithere, the main contribution being due to Quillen, cf. [39, Theorem 7.7, p. 568]. Theorem 1.3 (cf. Thm. 7.7 in [39]) . An n -dimensional orbifold O is a manifold if and only if H i orb ( O ; Z ) = 0 for all sufficiently large i . The total space of a principal S -orbibundle over O with integral Euler class e is amanifold if and only if condition (1.1) is satisfied for all sufficiently large i . In particular, an orbifold O canbe represented by an S -action on a manifold if and only if there exists some cohomology class e ∈ H ( O ; Z ) which satisfies this condition. For example, quaternionic weighted projective spaces (see e.g. [8]) have a cohomology ring generated byan element of degree 4, and can thus not be represented by an S -action on a manifold.A Besse contact manifold ( M, α ) induces an almost free S -action and an orientation on M . Conversely,one might ask which such actions, or in other words which Seifert fibrations on an odd-dimensional, orientablesmooth manifold M are induced by a Besse contact form α on M . For instance, the trivial fibration of S × S cannot be realized through a Besse contact form since such a form would induce an exact symplectic formon S which is impossible by Stokes theorem. The same argument shows that the existence of a closedhypersurface meeting the fibration everywhere transversely yields an obstruction.Using the orbifold Boothby–Wang result we deduce the following characterization, showing that in dimen-sion three the presence of a closed surface transversely to the fibration is virtually the only obstruction. Thefirst part of this result can also be found in [19, Proposition 5.5] in the context of geodesible vector fields.The second part about 3-dimensional Seifert fibrations provides in combination with [15, Theorem 1.5] acomplete classification of Besse contact 3-manifolds up to strict contactomorphism, see Corollary 1.6. Theorem 1.4.
A Seifert fibration M → B of a closed, orientable n +1 -dimensional manifold M can be realizedby a Reeb flow if and only if the corresponding Euler class in H ( B ; Z ) maps to a class in H ( B ; R ) thatcan be represented by a symplectic form. In the -dimensional case this is equivalent to each of the followingconditions(1) the real Euler class of the fibration M → B is nontrivial,(2) the fibration M → B is not finitely covered by a trivial fibration S × Σ → Σ over an orientablesurface Σ . In particular, in the 3-dimensional case condition (2) is satisfied if M has a finite fundamental group. Forinstance, all Seifert fibrations of lens spaces except two exceptional examples with nonorientable base [21]can be realized through a Reeb flow. Moreover, we remark that if M → B is a 3-dimensional Seifert fibrationwith real Euler class e R and Seifert invariants ( g ; ( a , b ) , · · · , ( a n , b n )) (see e.g. [29, 21, 19]), then by [19,Prop. 6.1], cf. Equation (4.1), the real Euler class vanishes if and only if n X i =1 b i a i = 0 . The following statement appears in [15, Theorem 1.5]. Here the prime period spectrum is the collection of allminimal periods of closed Reeb orbits.
Theorem 1.5 (Cristofaro-Gardiner–Mazzucchelli) . Let α and α be two Besse contact forms on a closed -manifold M . Then the prime period spectra of α and α coincide if and only if there exists a diffeomorphism ψ : M → M such that ψ ∗ α = α . M. KEGEL AND C. LANGE
The prime period spectrum of a Besse contact form is up to rescaling determined by the induced Seifertfibration. Indeed, by Sullivan’s and Wadsley’s contributions to Theorem 1.1, the prime period spectrumcan be read off from the period of the flow and the isotropy groups of the induced S -action. In termsof Seifert invariants ( g ; ( a , b ) , · · · , ( a n , b n )) the prime period spectrum is up to rescaling given as { π } ∪ S i = ni =1 { π/a i } ⊂ R . In particular, this result together with Theorem 1.4 provides the complete classificationof Besse contact 3-manifolds up to strict contactomorphism as claimed. Corollary 1.6.
The classification of closed Besse contact -manifolds up to strict contactomorphism coincideswith the classification of Seifert fibrations M → B of orientable, closed -manifolds satisfying condition (1) or (2) in Theorem 1.4. We also remark that Theorem 1.4 complements the results of Geiges and Gonzalo [20] on Seifert fibred3-manifolds which admit a contact form for which all fibres of the fibration are Legendrian, i.e. lie in thecontact distribution, cf. [20, Corollary 11].The cohomological condition (1.1) can for instance be satisfied in case of complex weighted projectivespaces CP na . These are defined as quotients of the unit sphere S n +1 ⊂ C n +1 by the almost free action of theunit circle S ⊂ C of the form z ( z , . . . , z n ) = ( z a z , . . . , z a n z n )for some weights a = ( a , . . . , a n ) ∈ ( N \{ } ) n . Indeed, their cohomology ring is computed in [27] to be H ∗ orb ( CP na ; Z ) ∼ = Z [ u ] / (cid:10) a · · · a n u n +1 (cid:11) , where u has degree two, and so any cohomology class e = ku ∈ H ( CP na ; Z ) with k and a · · · a n coprimesatisfies condition (1.1). In this case the total space X in Theorem 1.2 is a lens space. Other examples canbe obtained as subbundles of these. For instance, Besse Brieskorn contact manifolds occur in this way [32].In fact, we are not aware of examples that do not occur in this way. In the Zoll case other examples cannotexist: For a = · · · = a n = 1 and e = u we obtain the Hopf-bundle X = S n +1 → CP n and according to astatement by ´Alvarez Paiva and Balacheff in [7, Theorem 3.2.], which is based on the Boothby–Wang theoremand a result of Gromov–Tischler, every Zoll contact manifold occurs as the restriction of such a bundle to asymplectic submanifold of CP n for some n . Question 1.7.
Can every Besse contact manifold be realized as the restriction of some bundle X = S n +1 → CP na over a complex weighted projective space? Acknowledgements
We would like to thank O. Goertsches and L. Zoller for discussions about equivariantcohomology and related hints to the literature. It is our pleasure to thank H. Geiges for valuable remarks.We are grateful to D. Kotschick, G. Placini and the anonymous referee for useful comments, in particular,for drawing our attention to [12] and pointing out a mistake in Theorem 1.1.C.L. was partially supported by the DFG-grant SFB/TRR 191 Symplectic structures in Geometry, Algebraand Dynamics. 2.
Periodic Reeb flows
In this section we provide some geometric examples for periodic Reeb flows. Moreover, we recall whyalmost regular and Besse Reeb flows are in fact periodic flows. Let us begin with some geometric examples.2.1.
Geometric examples.
There is an infinite dimensional space of Zoll metrics on S , all of whose geodesicsare closed and have the same length [9], but their geodesic (Reeb) flows are all conjugated by a strictcontactomorphism to the one of the standard round metric [2]. This example can be generalized in twodirections. On the one hand it is the starting point for Katok’s construction for Zoll Finsler metrics on S [30, 47] whose Reeb flows [17] are not conjugated to the one of the round metric anymore. On the otherhand there exist Riemannian orbifolds with all geodesics closed and whose unit sphere bundle is a manifoldon which the geodesic flow defines a Besse Reeb flow [33]. For instance, the complex weighted projectivespaces CP na = S n +1 /S as defined in the Section 1 and endowed with the quotient metric have this propertywhen all the weights are coprime [8]. However, we note that by the main result of [8] in the simply connectedcase such examples can only exist in even dimensions. Another source of examples for Besse Reeb flows are BOOTHBY–WANG THEOREM FOR BESSE CONTACT MANIFOLDS 5 rational ellipsoids. For example, the standard 1-form λ = P i =1 , ( x i dy i − y i dx i ) on R restricts as a contactform to the boundary of any symplectic ellipsoid E ( a, b ) := (cid:26) π | z | a + π | z | b ≤ (cid:27) ⊂ C = R . When b/a is rational this contact form is Besse and this construction generalizes to higher dimensions.2.2.
Besse Reeb flows.
According to a result of Sullivan [43] every Reeb flow on a contact manifold (
M, α )with Reeb vector field R is geodesible, i.e. there exists a Riemannian metric g on M with respect to which allReeb orbits are unit-speed geodesics. In fact, any Riemannian metric g satisfying g ( R, R ) = 1 and R ⊥ g ker( α )has this property. For a simplified proof of this statement we refer the reader to [19, Prop. 3.3]. In case of aBesse contact manifold a result of Wadsley then shows that the Reeb flow is periodic [45]. Actually, this isnot quite what is stated in the main theorem of [45], but it follows from the proof, see [8, Prop. B.2].2.3. Almost regular Reeb flows.
A contact manifold (
M, α ) is called almost regular , if there exists somepositive integer k , and each point x ∈ M has a cubical coordinate neighborhood U = ( z, x , . . . , x n ) suchthat(1) each integral curve of the Reeb vector field R passes through U at most k times, and(2) each component of the intersection of an integral curve with U has the form x = a , . . . , x n = a n ,with a i constant.Since a closed almost regular contact manifold can be covered by finitely many of such neighborhoods, itimmediately follows that its Reeb flow is Besse [44, Theorem 1]. In particular, the Reeb flow of an almostregular closed contact manifold is periodic by the preceding section. This was first proven by Thomas in [44]modulo a small gap that was already present in the work of Boothby–Wang and fixed by Geiges in this case,see [18, footnote on p. 342 and Lemma 7.2.7]. Conversely, it follows from the slice theorem that a contactmanifold whose Reeb flow is induced by an almost free S -action is almost regular.Note that it is easy to write down examples of open almost regular contact manifolds without periodicReeb orbits. Take for example R × [0 ,
1] with standard contact form α = xdy + dz and identify points( x, y,
1) with ( x, y + 1 , Orbifolds
Roughly speaking a smooth (for us always effective) orbifold can be defined as a topological Hausdorffspace locally modeled by quotients of smooth actions of finite groups on smooth manifolds such that theactions satisfy certain compatibility conditions [40, 41, 16]. Although such a description is globally notalways possible, every smooth orbifold O can be represented by an almost free action of a compact Lie group G on a manifold M , see [4, Cor. 1.24] or Section 3.1 below. We denote such a representation as M//G .As a topological space O is just the quotient space M/G and all the additional data of O is encoded inthe action of G on M . We will take this viewpoint as a definition of a smooth orbifold. Two such actionsrepresent diffeomorphic orbifolds if and only if there exist invariant Riemannian metrics with respect towhich the quotient spaces endowed with the quotient metric, which measures the distance between orbits,are isometric. Here we can take this as a definition, but it coincides with the usual notion [34]. The dimensionof an orbifold represented by an effective action of G on M is the difference of the dimensions of M and G .If G can be chosen to be discrete, the orbifold is called developable. To prove the independence of severalnotions of the specific representation of an orbifold, we need the following pullback construction. Lemma 3.1.
Suppose a smooth orbifold O is represented as M //G and as M //G . Then there exists amanifold X with an almost free action of G × G such that the following equivariant diagram commutes G × G y X (cid:15) (cid:15) / / G y M (cid:15) (cid:15) G y M / / O where the upper and the left arrow are the quotient maps for the action of G and G , respectively. Inparticular, the actions of G and G on X are free. M. KEGEL AND C. LANGE
We emphasize that this construction differes from the usual pullback construction. Indeed, the actions of G and G on M and M , respectively, are in general not free, but the lifted actions of G and of G on X are so. The proof of Lemma 3.1 is given in the appendix, Section 5.3.1. Orbibundles and orbifold cohomology.
The notion of a principal S -orbibundle is defined in [41] (see [4]for the definition in terms of groupoids). Every principal S -orbibundle over an orbifold M//G lifts to a G principal S -bundle over M [4, Example 2.29]. By this we mean a principal S -bundle P over M to whichthe action of G extends in such a way that it commutes with the S -action. In this case the total space ofthe S -orbibundle over M//G is P//G . Conversely, a G principal S -bundle over M gives rise to a principal S -orbibundle over M//G in the sense of [41]. When M //G and M //G represent the same orbifold, thentwo such bundles are equivalent if and only if in the diagram of Lemma 3.1, the pulled back bundles to X are ( G × G )-equivariantly equivalent. Here we take the latter viewpoint of this equivalence as a definition.Moreover, it follows from Lemma 3.1 that a principal S -orbibundle whose total space M is a manifold is thesame as an almost free S -action on M .Recall that principal S -bundles over a manifold B are classified by elements in H ( B ; Z ) [28, Theo-rem 13.1]. An analogous correspondence holds for S -orbibundles which we now want to describe. UsingLemma 3.1 one can show that for an orbifold O = M//G the homotopy type of the Borel construction M × G EG := ( M × EG ) /G is independent of the specific representation of O in terms of M and G , andconstruct canonical isomorphisms between the corresponding cohomology rings, see the appendix, Section 5.Here EG is a contractible CW-complex on which G acts freely and the action of G on M × EG is the diagonalaction. The space B O := M × G EG is called a classifying space for O and the orbifold cohomology of O with respect to some coefficient ring R is defined as H ∗ orb ( O ; R ) := H ∗ G ( M ; R ) := H ∗ ( B O ; R ). For the sake ofconcreteness we can always assume G to be an orthogonal group O( m ) (by choosing M to be the orthonormalframe bundle of O with respect to some Riemannian metric, see [4, Cor. 1.24]). In this case BG can be takento be a direct limit of Grassmannians Gr m ( R k ) ⊂ Gr m ( R k +1 ) ⊂ . . . with the final topology [36].The equivariant Euler class e G ( P ) ∈ H G ( M ; Z ) of a G principal S -bundle P over M is defined as theEuler class of the principal S -bundle over B O obtained by pushing down the pulled back G principal S -bundle p ∗ P over M × EG to B O . Two equivalent bundle over equivalent representations of an orbifold giverise to Euler classes that are identified via canonical isomorphisms mentioned above, see Section 5, i.e. wecan also view e G ( P ) as an orbi-Euler class e orb ∈ H ( O , Z ) associated with a principal S -orbibundle over O . It is shown in [26, 37] that G principal S -bundles P over M are classified via their equivariant Eulerclass by elements in H ( O ; Z ). In fact, their result, which is formulated for complex line bundles, is moregeneral covering all smooth Lie group actions. In terms of orbifolds it can be phrased as follows. Theorem 3.2 (Hattori, Yoshida) . Principal S -orbibundles over an orbifold O are classified by elements in H ( O ; Z ) via their Euler class. Example 3.3.
Let O be the quotient of C by the action of a cyclic group Z k < U(1). Since C is contractible,the integral cohomology ring of O can be computed to be H ∗ orb ( C / Z k ; Z ) = H ∗ ( Z k ; Z ) ∼ = Z [ u ] / h ku i , where u has degree 2 [5, 27]. Hence, there are k isomorphism classes of principal S -orbibundles over C / Z k . The bundle with Euler class e = lu can be constructed as a quotient of C × S by the Z k -action ξ ( z, λ ) = ( ξz, ξ l λ ). We see that the total space of this bundle is a manifold if and only if l and k are coprimein accordance with condition (1.1). In particular, for k prime the only bundle whose total space is not amanifold is the trivial bundle S × O .3.2. Basic cohomology.
Let an orbifold O be represented by an almost free action of G on M . A differentialform τ on M is called ( G -)basic if it is G -invariant and vanishes when contracted with vertical vector fields,i.e. vector fields in the vertical distribution of the projection M → M//G . By the Cartan formula thedifferential of a basic form is again basic. The cohomology of the subcomplex of basic differential forms iscalled the basic G -cohomology of M and denoted as H ∗ bas G ( M ), see e.g. [23]. If the action of G on M isfree, then O = M//G is a manifold and H ∗ bas G ( M ) is canonically isomorphic to the de Rham cohomology H ∗ dR ( M ) [23, Prop. 2.5]. If the action is only almost free we still have the equivariant de Rham theoremsaying that H ∗ orb ( O ; R ) is canonically isomorphic to H ∗ bas G ( M ) [24, Thm. 2.5.1]. We mention that the latteris in turn isomorphic to the de Rham cohomology H ∗ dR ( O ) of O as defined in e.g. [40]. More precisely, the BOOTHBY–WANG THEOREM FOR BESSE CONTACT MANIFOLDS 7 equivariant de Rham isomorphism looks as follows. First, the usual de Rham theorem (applied to finite-dimensional approximations) yields an isomorphism H ∗ orb ( O ; R ) ∼ = H ∗ dR ( B O ). The de Rham cohomologyof B O is in turn canonically isomorphic to the basic G -cohomology H ∗ bas G ( M × EG ) by [23, Prop. 2.5](again applied to finite-dimensional approximations), where the isomorphism is induced by the projection M × EG → M × G EG . Finally, the isomorphism between H ∗ bas G ( M ) and H ∗ bas G ( M × EG ) is induced bythe G -equivariant projection p : M × EG → M . For the convenience of the reader we sketch an argumentwhy the latter map is indeed an isomorphism. Lemma 3.4.
The map p ∗ : H ∗ bas G ( M ) → H ∗ bas G ( M × EG ) is an isomorphism.Proof. Since the actions of G on M and M × EG are almost free, we have natural isomorphisms H ∗ bas G ( M ) ∼ = H ∗ ( C G ( M )) and H ∗ bas G ( M × EG ) ∼ = H ∗ ( C G ( M × EG )) induced by the map ω ⊗ ω between cochaincomplexes. Here C G ( M ) = ( S ( g ∗ ) ⊗ Ω( G )) G denotes the cochain complex of the Cartan model for theequivariant cohomology (see [23], Section 4 and Theorem 5.2 for the details). Hence, it suffices to prove theclaim for the map p ∗ : H ∗ ( C G ( M )) → H ∗ ( C G ( M × EG )) induced by the map 1 ⊗ p ∗ on the cochain level.The latter map respects a filtration which gives rise to a spectral sequence that computes the equivariantcohomology, see [23, Section A.3]). By the comparison theorem [23, Thm. A.22] it suffices to show that1 ⊗ p ∗ induces an isomorphism on the first pages of the spectral sequences which are S ( g ∗ ) ⊗ H ∗ ( M ) and S ( g ∗ ) ⊗ H ∗ ( M × EG ) [23, Thm. A.8], respectively. Going through the proof of [23, Thm. A.8] shows thatthis induced map is just the map 1 ⊗ p ∗ . Hence, the claim follows. (cid:4) We call elements in H ∗ orb ( O ; R ) and H ∗ bas G ( M ) which lie in the image of H ∗ orb ( O ; Z ) integral . In particular,the Euler class of a principal S -orbibundle over O gives rise to an integral class in H ∗ bas G ( M ), which we callthe real Euler class of the bundle. In the following subsection we describe this class in terms of differentialforms on M .3.3. Real Euler class.
Recall that a connection 1-form of a principal S -bundle P → M is a 1-form α on P such that L R α = 0 and α ( R ) = 1 holds, where R is the vector field on P generated by the S -action and L is the Lie derivative. A connection -form of a G principal S -bundle P → M is a connection 1-form of theunderlying principal S -bundle which is in addition G -basic. It is not difficult to show that every G principal S -bundle P → M admits a connection 1-form. We will, however, only be concerned with the case in whichthe total space P//G is a manifold and in this case one can more easily construct a connection 1-form for thealmost free S -action on P//G , and then pull it back to a connection 1-form of the G principal S -bundle P → M .As for principal S -bundles the differential dα of a connection 1-form of a G principal S -bundle P → M descends to a so-called curvature form ω on M (cf. [18, p. 340]) which, in the G -equivariant case, is in addition G -basic. In particular, we have an induced cohomology class − [ ω/ π ] ∈ H ∗ bas G ( M ). In the equivariant casewe can also think of ω as an ( S × G )-basic 2-form on P , or as an S -basic 2-form on P//G if it is a manifold.
Lemma 3.5.
The curvature form ω of a G principal S -bundle P → M determines an integral form − [ ω/ π ] ∈ H ∗ bas G ( M ) which coincides with the real Euler class e G R ( P ) of the bundle.Proof. By Lemma 3.4 and the definition of the Euler class it is sufficient to show the claim for the pulledback G principal S -bundle over M × EG . A connection 1-form and the corresponding curvature formdescend to the respective forms of the quotient principal S -bundle over M × G EG . Since the projection M × EG → M × G EG induces an isomorphism from the de Rham cohomology of M × G EG to the basiccohomology of M × EG , the lemma follows from its non-equivariant version applied to the S -bundle over M × G EG [28, Theorem 13.1]. (cid:4) Proof of the main results
Proposition 4.1. An n -dimensional orbifold O is a manifold if and only if H i orb ( O ; Z ) = 0 for all i sufficientlylarge.Proof. It follows from a result of Quillen [39, Theorem 7.7, p. 568] that O is a manifold if and only if H ∗ orb ( O ; Z p ) is finite dimensional for all primes p , see [8, Remark 3.4]. In the compact, orientable case a moreelementary argument for this statement is provided in [8, Section 3]. Now we observe that for any coefficientsall cohomology groups are finitely generated since B O can be approximated by finite-dimensional manifolds,see Section 3.1. Therefore the claim now follows from the cohomological universal coefficient theorem. (cid:4) M. KEGEL AND C. LANGE
Proof of Theorem 1.3.
The first part of the theorem is just the statement of Proposition 4.1. For the secondpart we represent O as M//G and the principal S -orbibundle as a G principal S -bundle over M . Thesecond claim then follows from the Gysin sequence [31, Section 2.2] applied to the corresponding principal S -bundle over ( M × ES ) /S , i.e. · · · → H i ( M ; Z ) → H i − ( O ; Z ) e ∪· −−→ H i +1orb ( O ; Z ) → H i +1 ( M ; Z ) → · · · , since H i ( M ; Z ) = 0 for all i > n + 1. (cid:4) Recall that for an orbifold O represented by an almost free action of G on M a symplectic form on a O isa closed basic 2-form ω on M which satisfies ω n = 0. In this case the pair ( M//G, ω ) is called a symplecticorbifold.
Proof of Theorem 1.1.
According to Sections 2.2 and 2.3 we can assume that the Reeb flow is periodic, and,after rescaling, has period 2 π . In other words, the R -action defined by the flow factors through an S -action.Since the Reeb vector field has no zeros and S has only finite proper subgroups, this action is almost free.Since dα is S -basic and dα n = 0, α is a connection 1-form and the quotient ( M//S , dα ) is a symplecticorbifold.Let X be the S × S -space obtained by applying Lemma 3.1 to two copies M and M of M withthe same given S -action, and write π : X → M for the projection map. In order to prove that [ dα ] ∈ H bas S ( M ) ∼ = H ( O ) represents the real Euler class of the principal S -orbibundle M → M//S , we onlyneed to show, given Lemma 3.5, that the images of [ dα ] and of [ dπ ∗ α ] in H (( M × S ) /S ; R ) and H (( X × ES × ES ) /S ; R ), respectively, are identified by the canonical ismorphism between H (( M × S ) /S ; R ) and H (( X × ES × ES ) /S ; R ) that is induced by the projection π , see the appendix, Section 5. Unrolling thedefinitions shows that this is implied by Stokes theorem. The last claim is an application of Theorem 1.3. (cid:4) Remark . In the introduction we claimed that the orbifold O in The-orem 1.3 can also be seen as a symplectic reduction of the symplectisation ( R > × M, d ( rα )) performedon a level set of the Hamiltonian H ( r, x ) = r . Indeed, extending the S -action on M trivially to the R > factor yields an action with action vector field (0 , R ) that leaves the Hamiltonian H invariant. Because of ι (0 ,R ) d ( rα ) = dr = dH , this action is Hamiltonian and H specifies its moment map µ : R > × M → R ∗ via H = h µ, i . Note that any r ∈ R > is a regular value of H . Hence, for any r ∈ R > the orbifold O arises asa symplectic reduction µ − ( r ) //S as claimed, cf. [27, Thm. 1.11],[35]. Proof of Theorem 1.2.
Let ( O = M//G, ω ) be a symplectic orbifold of dimension 2 n with integral symplecticform ω/ π . For every integral cohomology class e ∈ H ( O ; Z ) mapping to − [ ω/ π ] ∈ H G ( M ) andsatisfying condition (1.1) for sufficiently large i we get by Theorem 3.2 a unique principal S -orbibundle π : X → O with integral Euler class e and real Euler class − [ ω/ π ]. By condition (1.1) and Theorem 1.3 thespace X is a manifold, and so we can think of the bundle as an almost free S -action on X with X//S = O .In particular, we can assume that M = X and G = S . Let α ′ be a connection 1-form for this bundle. As inthe proof of Theorem 1.1 it follows with Lemma 3.5 that the corresponding curvature form ω ′ represents thereal Euler class of the bundle. Hence, there is a basic 1-form β on X such that ω − ω ′ = dβ . Set α = α ′ + π ∗ β .This is also a basic connection 1-form, and it satisfies dα = π ∗ β . Since ω is a symplectic form on O it followsthat α is a contact form on X . (cid:4) Let us make some remarks before we prove Theorem 1.4. For an almost free S -action on an orientable3-manifold M with base B = M//S and connection 1-form α integration induces a homomorphism: h· , [ B ] i : H S ( M ) → R , [ ω ]
7→ h [ w ] , [ B ] i := 12 π Z α ∧ ω. We mention that under the canonical identification of H S ( M ) with the orbifold de Rham cohomology of B = M//S this map amounts to integration over the base orbifold M//S as defined in [40]. In particular,this homomorphism is in fact an isomorphism [40, Thm. 3]. In the present special case this follows from thefact that H S ( M ) is isomorphic to H S ( M ; R ) ∼ = H ( M/S ; R ) ∼ = R and that one can easily construct anowhere vanishing basic 2-form ω on M , because of the orientability assumptions, for which α ∧ ω is thena volume form of M . Such an ω is a symplectic form on M//S . In particular, we see that a class [ ω ] in H S ( M ) can be represented by such a symplectic form if and only if h [ ω ] , [ B ] i 6 = 0, and hence if and only BOOTHBY–WANG THEOREM FOR BESSE CONTACT MANIFOLDS 9 if [ ω ] = 0 in H S ( M ). Moreover, we record that this property is invariant under finite coverings, i.e. ifˆ M → M is a finite covering, then the S -action on M is covered by an S -action on ˆ M , and a class in H S ( M ) has a symplectic representative if and only if its pull back in H S ( ˆ M ) has so.Finally, we also mention that if the fibration M → B has real Euler class e R and Seifert invariants( g ; ( a , b ) , · · · , ( a n , b n )) (see e.g. [29, 21, 19]), then by [19, Prop. 6.1] we have that h e R , [ B ] i = − n X i =1 b i a i . (4.1) Proof of Theorem 1.4.
The first part of the theorem is a direct consequence of Theorem 1.1 and Theorem 1.2.For the second part we first note that if M is 3-dimensional, then B is a closed 2-orbifold equipped with anatural orientation. There exists a finite orbifold covering ˆ B of B with torsion free H ( ˆ B ; Z ). Indeed, bythe universal coefficient theorem we have H ( ˆ B ; Z ) tor = H orb1 ( ˆ B ; Z ) tor = ( π orb1 ( ˆ B ) ab ) tor , and every closed 2-orbifold is either finitely covered by a simply connected 2-orbifold or by a surface [42,Thm. 2.5]. As observed above, a class in H ( B ; Z ) can be represented by a symplectic form if and only ifit pulls back to a non-trivial class in H ( ˆ B ; R ) ∼ = H ( ˆ B ; Z ). Since this image is the integral Euler class ofthe pulled back almost free S -action on ˆ M with quotient ˆ B , this happens if and only of this action is nottrivial. Hence, if a given Seifert fibration on M with orientable base is not covered by a trivial fibration, thenit can be realized by a Reeb flow. Conversely, assume that such a fibration can be realized by a Reeb flowand is finitely covered by a trivial fibration ˆ M ∼ = S × Σ → Σ. The base of this fibration ˆ B = Σ is a surfacecovering B . Moreover, the real Euler class of the corresponding S -bundle vanishes in contradiction to thefact that its preimage in H ( B ; R ) is nontrivial. (cid:4) Appendix
In this section we prove Lemma 3.1 and confirm the independence of the orbifold cohomology and theorbifold Euler class from specific representations.
Proof of Lemma 3.1.
Suppose we have two representations of O n in terms of two actions G y M and G y M . Then there are invariant Riemannian metrics with respect to which the corresponding quotientspaces M /G and M /G are isometric. Let Fr h ( M i ) be the principal O( n )-bundle over M i consistingof orthonormal n -frames in the horizontal distribution of the projection M i → O , i = 1 ,
2. The naturalO( n )-action on these spaces commutes with the free actions of G and G , respectively. The quotient spacesFr h ( M ) /G and Fr h ( M ) /G are naturally identified with the orthonormal frame bundle Fr( O ) of O . Weconsider the space ¯ X = { ( x, y ) ∈ Fr h ( M ) × Fr h ( M ) | G x = G y ∈ Fr( O ) } with the induced actions of G , G and the diagonal action of O( n ). Then the space X = ¯ X/ O( n ) with theinduced action of G × G satisfies all conditions in the lemma. (cid:4) Now let us look at the independence of the Borel construction of the specific representation M i //G i , i = 1 , O . In view of Lemma 3.1 it suffices to compare the Borel constructions of M //G and of X//G ,where G = G × G . In this case the natural projection from the Borel construction( X × EG × EG ) / ( G × G )to ( X × EG ) / ( G × G ) = ( M × EG ) /G defines a fibre bundle with contractible fibre EG . In particular,it induces a homotopy equivalence between these spaces and an isomorphisms in cohomology. Here we havetaken the independence of a specific classifying space EG for granted; if EG and ˜ EG are two different modelsof this classifying space, then so is EG × ˜ EG and the same argument as above shows the independence of EG .It remains to observe that the canonical isomorphism π ∗ : H (( M × EG ) /G ; Z ) → H (( X × EG × EG ) / ( G × G ); Z )induced by the projection maps the integral Euler class of a G principal S -bundle over M to the integralEuler class of the pulled back G principal S -bundle over X . Indeed, these are the Euler classes of a principal S -bundle over ( M × EG ) /G and its pulled back bundle over ( X × EG × EG ) / ( G × G ). This showsthe independence of the Euler class of the specific representation of O as claimed in Section 3.1. References [1] A. Abbondandolo and G. Benedetti. On the local systolic optimality of Zoll contact forms. arXiv:1912.04187, preprint ,2019.[2] A. Abbondandolo, B. Bramham, U. L. Hryniewicz, and P. A. S. Salom˜ao. A systolic inequality for geodesic flows on thetwo-sphere.
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