A Reeb flow on the three-sphere without a disk-like global surface of section
AA REEB FLOW ON THE THREE-SPHERE WITHOUT A DISK-LIKEGLOBAL SURFACE OF SECTION
OTTO VAN KOERT
Abstract.
We show that there are Reeb flows on the standard, tight three-sphere that donot admit global surfaces of section with one boundary component. In particular, the Reebflows that we construct do not admit disk-like global surfaces of section. These Reeb flowsare constructed using integrable systems, and a connected sum construction that extends theintegrable system. Introduction
The purpose of this note is to show the existence of a Reeb flow on the standard, tight three-sphere that does not admit a global disk-like surface of section. The motivation for proving sucha statement comes from a famous theorem of Hofer, Wysocki and Zehnder, [5], which assertsthat a Reeb flow on a dynamically convex, tight three-sphere admits a global disk-like surface ofsection. The Reeb condition is clearly necessary since there are flows on the three-sphere withoutany periodic orbits; for example, Kuperberg, [8], has constructed such flows. The work of Hofer,Wysocki and Zehnder has been generalized in many ways by different people; we mention hereHryniewicz and Salom˜ao, [6, 7].Given a global surface of section for a Reeb flow, the resulting return map is conjugated toan area-preserving diffeomorphism. This makes disk-like global surfaces of section particularlyattractive, since a lot is known about area-preserving diffeomorphisms of an open disk. We mentionthe results of Franks proved in [2], which imply that an area-preserving diffeomorphism of the diskhas either one periodic point or infinitely many. The work of Hryniewicz and Salom˜ao, [7], showsthat one can weaken the condition of dynamical convexity by imposing linking conditions, so onemay wonder whether a disk-like global surface of section always exists for the tight three-sphere.We show that this is not the case.
Theorem 1.1.
There is a Reeb flow on S equipped with its standard, tight contact structure, thatdoes not admit a global surface of section with only one boundary component. In particular, thisReeb flow does not admit a disk-like global surface of section. To prove the theorem, we construct a dynamical system using the so-called book-connectedsum, a special case of the Murasugi sum. In this book-connected sum, we take a connected sumalong balls that intersect a periodic orbit, inducing the connected sum of these orbits. We retaincontrol on the behavior of the orbits by virtue of having many invariant sets and a planar globalsurface of section with four binding orbits. The proof is completed using symmetry of the linkingnumber and an elementary analysis of the pairs of orbits that can occur. The question on whetherthere is a Reeb flow on the tight S without any global surface of section is left open.The book-connected sum also gives a method to construct completely integrable Reeb flows, inthe sense of Arnold-Liouville, on a large class of contact three-manifolds. The construction is verysimple, and does not deserve the name “theorem”, so we will denote it by Observation 1.2.
Suppose that W is an oriented surface with boundary, and assume that ψ : W → W is the time-1 flow of an autonomous Hamiltonian such that ψ | ∂W = Id. Let Y denote thecontact open book Y = OB( W, ψ ). Then R × Y is a completely integrable system in the sense of Key words and phrases.
Global surfaces of section, Reeb flows, Integrable systems. a r X i v : . [ m a t h . S G ] F e b OTTO VAN KOERT
Arnold-Liouville . Furthermore, if ( W (cid:48) , ψ (cid:48) ) is another such pair, then the contact connected sum Y Y (cid:48) is still completely integrable.The projection to the R -coordinate of the symplectization and the Hamiltonian generating thesymplectomorphism ψ give a pair of Hamiltonians that are in involution, and the connected sumcan be performed in such a way that the Hamiltonians generating ψ and ψ (cid:48) patch together to anautonomous Hamiltonian as we will see in Lemma 2.6. Explicit formulas are given in Section 3.1.The assumption that the flow is generated by an autonomous Hamiltonian is restrictive, but itdoes allow ψ to be a product of Dehn twists along disjoint, separating curves.In section 2 we collect standard facts about open books, surfaces of section and linking numbers,together with a lemma involving a connected sum operation. The proof of the theorem is in section3, and the appendix contains details concerning connected sums of open books. Acknowledgements.
I thank Pedro Salom˜ao for helpful comments. I was supported by NRFgrant NRF-2016R1C1B2007662, which was funded by the Korean Government.2.
Setup
In the following, we will assume that Y is a 3-manifold with a non-vanishing vector field X . Definition 2.1. A global surface of section for a flow ϕ t , generated by X, is an embedded,connected surface S ⊂ Y with the following properties. (i): The boundary of the surface consists of periodic orbits of X (or an invariant set in higherdimensions). (ii): The vector field X is transverse to the interior ˚ S of the surface; we can and will assumethat S is oriented such that X is positively transverse to S . (iii): For every x ∈ Y \ ∂S there exists t + > t − < ϕ t + ( x ) ∈ ˚ S and ϕ t − ( x ) ∈ ˚ S .The main point of a global surface of section is that it can be used to discretize the dynamicsby defining the global return map τ : ˚ S → ˚ S by sending x (cid:55)→ ϕ t + ( x ) ( x ), where t + ( x ) > S as an obstruction to a global surface of section, solet us briefly collect some properties of the linking number in Y = S . Given knots γ and δ , choosea Seifert surface S γ for γ . Then define the linking number lk( γ, δ ) as the algebraic intersectionnumber S γ · δ . This is independent of the chosen Seifert surface. An immediate corollary of thisdefinition is the following lemma. Lemma 2.2.
If a periodic orbit δ of the flow ϕ t bounds a global surface of section, then δ linkspositively with every other periodic orbit γ of ϕ t , i.e. lk( δ, γ ) > . Lemma 2.3.
The linking number of two knots is symmetric, i.e. lk( γ, δ ) = lk( δ, γ ) . See for example [4, Corollary 4.5.3].2.1.
Giroux model for contact open books and global surfaces of section.
Global surfacesof section are closely related to open books, and Giroux’ original construction of contact open booksis particularly well-suited to our needs, because it gives a flow that is easy to understand. Weassume that we are given a Liouville domain (
W, dλ ) and a symplectomorphism ψ satisfying(1) ψ ∗ λ = λ − dU , where U is negative.(2) in a neighborhood of the boundary ψ is the identity, so ψ | ν W ( ∂W ) = Id. Remark . Given any compactly supported symplectomorphism, meaning condition (2) is satis-fied, one can always deform it to a symplectomorphism satisfying condition (1). See for example[11, Lemma 2.13]. We allow the integrals to be dependent on a “small” subset to obtain more interesting topology. For example,there is always a “3-atom A” in Y in the sense of Bolsinov-Fomenko’s book [1], section 3.5. REEB FLOW ON THE THREE-SPHERE WITHOUT A DISK-LIKE GLOBAL SURFACE OF SECTION 3
In our application, condition (1) is automatically satisfied. Indeed, we assume that ψ is thetime-1 flow of an autonomous Hamiltonian, so ψ = F l X H . We note that ddt F l X H t ∗ λ = F l X H t ∗ L X H λ = F l X H t ∗ ( ι X H dλ + dι X H λ ) = F l X H t ∗ ( − dH + dι X H λ ) . This is exact and W is compact, so by integration in t we can choose a primitive U that is negativeon W .Given the above setup, define the mapping torus M ( W, ψ ) := W × R / ( x, ϕ ) ∼ ( ψ ( x ) , ϕ + U ( x ) )The form α M := dϕ + λ descends to a well-defined contact form. We will consider the set B ( W ) := ∂W × D , which we will refer to as a neighborhood of the binding. The binding itself isthen the set ∂W × { } . We will denote the projection onto the first factor by p ∂ : B ( W ) → ∂W .Equip the neighborhood of the binding with the contact form α B := h ( r ) λ | ∂W + h ( r ) dϕ. (2.1)Here h and h are profile functions from [0 ,
1) to R whose behavior is indicated in Figure 1; near r = 0, h ∼ − r , and h ∼ r . For larger r , the function h is assumed to have exponentialdrop-off, and h should be constant. We glue the contact manifolds together to obtain a contact h r rh Figure 1.
Functions for the contact form near the bindingopen book OB(
W, ψ ) = B ( W ) (cid:97) M ( W, ψ ) / ∼ , where we identify collar neighborhoods of B ( W ) and M ( W, ψ ) using the map B ( W ) \ ∂W × D (1 / −→ M ( W, ψ ) , ( x ; r, ϕ ) −→ [ F l X − r ( x ) , ϕ ]Here X is the Liouville vector field on W . The above choice of h and h is used to show that thecontact forms glue together to give a globally defined contact form.On the complement of the binding we have a fiber bundle over the circle, given by ϑ : Y \ ∂W × { } −→ S = R / Z y (cid:55)−→ (cid:40) ϕ π if y = ( w ; r, ϕ ) ∈ ∂W × D ϕ − U y = [ w, ϕ ] ∈ M ( W, ψ )The Reeb vector fields on the two models are given by R M = ∂∂ϕ on M ( W, ψ ), and R B = 1 h ( r ) h (cid:48) ( r ) − h ( r ) h (cid:48) ( r ) (cid:18) h (cid:48) ( r ) R λ | ∂W − h (cid:48) ( r ) ∂∂ϕ (cid:19) on B ( W ) . (2.2)We conclude Lemma 2.5.
The binding is an invariant set under the Reeb flow, and each fiber of ϑ is a globalhypersurface of section. Furthermore, the set B ( W ) is a disjoint union of B × { } and sets of theform B × S , each of which are invariant under the Reeb flow. OTTO VAN KOERT
Book-connected sum of contact open books.
We describe the abstract model for thebook-connected sum in the contact setting. For a description in the smooth setting, see for example[10], section 7 and 8. Suppose that OB( W i , ψ i ) are abstract contact open books for i = 1 , j -th boundary component of the i -th page W i by b ji : this labeling is a choice that willplay a role later on. Take points p i in the first boundary component of each page, so p i ∈ b i . Definethe boundary connected sum W (cid:92) b ,b W by attaching a Weinstein 1-handle to W (cid:96) W alongDarboux balls in ∂W (cid:96) ∂W containing p and p , respectively. This is illustrated in Figure 2.Below we will write W (cid:92)W := W (cid:92) b ,b W to remove some clutter from the notation, but it isimportant to realize the dependence on the choice of the additional data, and we reintroduce thisnotational dependence when we need this dependence. To be more precise about the attaching W W P P T(cid:92) ∼ = Figure 2.
Boundary connected sum of pageslocus of the 1-handle, denote the attaching locus in W i by A i , and choose A i so small such that ψ i | A i = Id. We obtain embeddings ι j : W j −→ W (cid:92)W , which we use to extend the symplectomorphisms ψ and ψ to a symplectomorphism ψ (cid:92)ψ of W (cid:92)W . Put ψ (cid:92)ψ = ι ◦ ψ ( ι − ( x )) if x ∈ Im i ι ◦ ψ ( ι − ( x )) if x ∈ Im i Id otherwise.The 1-handle attachment induces a connected sum on the first binding component of W and W . We will denote the resulting binding component of W (cid:92)W by b = b b . Choose a collarneighborhood ν W (cid:92)W ( b ) such that ψ (cid:92)ψ restricts to the identity on this neighborhood. Points in W (cid:92)W \ ν W (cid:92)W ( b ) that do not lie in the image of ι or ι will be called tube points.This leads to the following distinguished sets:(1) A neighborhood of the connecting orbit, N ( b ) = N ( b b ) = { x ∈ B ( W (cid:92)W ) | p ∂ ( x ) ∈ b b . } . (2) Tube orbits, T = { [ x, ϕ ] ∈ M ( W (cid:92)W , ψ (cid:92)ψ ) | x is a tube point } . (3) Page orbits, consisting of points that lie in the mapping torus part, minus the tube, or ina neighborhood of a binding orbit that is not b = b b . In a formula, P j = { [ x, ϕ ] ∈ M ( W (cid:92)W , ψ (cid:92)ψ ) } | x ∈ Im ι j } ∪ (cid:91) b (cid:48) (cid:54) = b,b (cid:48) ⊂ Y j N ( b (cid:48) ) . These sets are invariant under the Reeb flow as we shall explain in Lemma 2.6. We will refer toOB( W (cid:92)W , ψ (cid:92)ψ ) as the abstract book-connected sum . This model for the connected sumalso has concrete model, meaning one for which open books are defined via a binding and a fiberbundle over the complement of the binding. In the concrete model, the connected sum is moreobvious; simply take small balls centered at two points in a binding component and perform theusual connected sum. A local model shows that the open book extends. We won’t make use ofthis description, though, and instead refer to the previously mentioned [10]. REEB FLOW ON THE THREE-SPHERE WITHOUT A DISK-LIKE GLOBAL SURFACE OF SECTION 5
Lemma 2.6.
The contact open book
OB( W (cid:92)W , ψ (cid:92)ψ ) , equipped with the contact form comingfrom the abstract book-connected sum, is contactomorphic to the contact connected sum of the openbooks, OB( W , ψ ) W , ψ ) . Furthermore, for the Reeb flow of the abstract book-connectedsum we have the following invariant sets(1) a neighborhood of the connecting orbit b = b b , N ( b ) , as defined in Section 2.2.(2) the set T consisting of tube orbits.(3) the sets P and P .If γ is an orbit in P i , then there is a Seifert surface S γ with S γ ⊂ P i ∪ N ( b ) . This Seifert surfacecan be chosen to intersect N ( b ) in finitely many disks whose image under p ∂ consist of finitelymany points. For intuition, the last statement says that S γ intersects N ( b ) in “flat” disks. Proof.
The first assertion is a standard fact, which we reprove in the appendix. One ingredientof the construction in the appendix is that the connected sum (or equivalently, the region of 1-handle attachment) is performed along Darboux balls that are contained in N ( b ) and in N ( b ),respectively. We will use this ingredient for the statement about the Seifert surfaces.To verify the statement on invariant sets, we use the explicit model from equation (2.2).(1) near the binding, the Reeb vector field is given by R B ; orbits have constant r -coordinate,and form therefore an invariant set. Furthermore, these sets, which are all diffeomorphicto ∂ ( W (cid:92)W ) × S , foliate N ( b ).(2) on the set T , the monodromy ψ (cid:92)ψ is defined to be the identity. The explicit form of M ( W (cid:92)W , ψ (cid:92)ψ ) shows that the return map on T is also the identity, so T is an invariantset.(3) it follows that the complement of N ( b ) and T is also an invariant set, and this is the union P ∪ P . This is a disjoint union as P ⊂ Y and P ⊂ Y .Let us now consider the Seifert surfaces. Take a periodic Reeb orbit γ in P . The latter setis disjoint from the Darboux balls used for the connected sum, so we may view P ⊂ Y . Hencewe regard γ as an orbit in Y . Choose a Seifert surface S (cid:48) γ ⊂ Y capping γ . We isotope thissurface such that it intersects the set N ( b ) in the desired way, so that p ∂ ( S (cid:48) γ ∩ N ( b )) consists ofonly finitely many points, which can be done with a standard transversality argument involving asurface and a link. Furthermore, we can arrange that these points do not lie in the Darboux ball ι ( D ), yielding a Seifert surface S γ ⊂ Y satisfying the claim. (cid:3) Remark . We point out that the abstract contact open book changes the contact form in aneighborhood of all orbits that intersect the balls used for the connected sum because of Equa-tion (2.1). Also, unlike the Weinstein model for the connected sum, the Reeb flow on the abstractbook-connected sum does not have a non-degenerate Lyapunov orbit, but a degenerate familyinstead. One can deform the contact form on the abstract contact open book or the return mapas explained in Remark 3.3 to obtain a non-degenerate Lyapunov orbit in the separating sphereof the connected sum. 3.
Proof of the theorem
Lemma 2.6 implies the following corollary which will be used to prove the theorem.
Corollary 3.1.
Consider the setup of Section 2.2 with the same notation. Suppose γ and γ areorbits in P and P , respectively. Then lk( γ , γ ) = 0 . Furthermore, if γ is an orbit in P or in P and τ is an orbit in T , then lk( γ, τ ) = 0 . Put differently, neither the P -orbits nor the T -orbits can be the boundary of a global surfacewhich has only one boundary component. OTTO VAN KOERT
Proof.
To see that the corollary holds, assume that γ ⊂ P . Lemma 2.6 gives a Seifert surface S γ for γ with S γ ⊂ P ∪ N ( b ). It follows that lk( γ, γ ) = S γ · γ = 0 for any periodic orbit γ that iscontained in P or in T , since S γ is disjoint from these sets. (cid:3) We take now three open books, each representing the standard tight S , namely Y i = OB( W i , ψ i ),where each page is a cylinder, so W i ∼ = ( I × S , rdϕ ), and each return map ψ i is a positive Dehntwist, i.e. a map of the form I × S → I × S , ( r, ϕ ) (cid:55)→ ( r, ϕ + tw ( r )), where tw is a smoothfunction that decreases from 2 π at the lower boundary to 0 at the upper boundary. We willdenote the boundary components of the i -th cylinder W i by its upper component u i = { } × S and lower component (cid:96) i = {− } × S , respectively.Form the book-connected sum Y = OB(( W (cid:92) u ,u W ) (cid:92) (cid:96) ,u W , ψ (cid:92)ψ (cid:92)ψ ) ∼ = OB( W (cid:92) u ,u ( W (cid:92) (cid:96) ,u W ) , ψ (cid:92)ψ (cid:92)ψ ) . The page of the new open book is illustrated in Figure 3. By the first statement of Lemma 2.6, we u u (cid:96) (cid:96) u (cid:96) Figure 3.
A page of an open book with four binding componentssee that Y is contactomorphic to ( S , ξ ) as ( S , ξ ) S , ξ ) ∼ = ( S , ξ ). We endow Y with theReeb flow from the construction from Section 2.2 and will argue by contradiction. Assume that S is a global surface of section for the Reeb flow with only one boundary component, which we call δ . By Lemma 2.6 the contact manifold Y is the disjoint union of invariant sets in the form of P -orbits, tube orbits T and the neighborhoods of the two connecting orbits, N ( u u ) and N ( (cid:96) u ). Obviously, δ must be contained in one of these invariant sets. Consider the set P , the page orbits in W corresponding to the first connected sum. This set contains (cid:96) , so itis not empty. By Corollary 3.1 applied to the first connected sum in Y Y Y ) we see thatlk( (cid:96) , δ ) = 0 unless(1) δ ⊂ P , or(2) δ ⊂ N ( u u ).See Figure 4. First assume δ ⊂ P . Applying again Corollary 3.1 we see that lk( δ, γ T ) = 0 for anytube orbit γ T contradicting Lemma 2.2. Hence we assume δ ⊂ N ( u u ). Now apply Lemma 2.6to the second connected sum in ( Y Y ) Y . We conclude that there is a Seifert surface S δ capping δ which is disjoint from orbits in the second tube. This implieslk( δ, γ T ) = S δ · γ T = 0for every periodic orbit in the second tube (which exist since the monodromy is the identity onthe tube). As the linking number is independent of the choice of Seifert surface, we get again acontradiction to Lemma 2.2. A positive Dehn twist is often also called a right-handed Dehn twist. However, with the conventions in thispaper, rdϕ as Liouville form, the positive twist actually turns to the left. Note that the argument here only involvesthe invariant sets and not the direction of the twists. However, negative Dehn twists will yield an overtwisted three-sphere.
REEB FLOW ON THE THREE-SPHERE WITHOUT A DISK-LIKE GLOBAL SURFACE OF SECTION 7 N ( u u ) P T T Figure 4.
The page and tube orbits
Remark . For the topology, the choice of bindings along which we performed the book-connectedsum is irrelevant. It does matter for the dynamics. For example, the above argument fails for Y = OB(( W (cid:92) u ,u W ) (cid:92) u u ,u W , ψ (cid:92)ψ (cid:92)ψ ) , because the long “transit” orbit u u u could bound a disk that links with all orbits.3.1. Some comments on integrable systems.
Let us first give explicit formulas for the inte-grals mentioned in the introduction. Suppose that Y = OB( W, ψ ), where ψ is the time-1 flow ofan autonomous Hamiltonian H : W → R . Then R × Y has two integrals that are in involution,namely H : R × Y −→ R , ( t, y ) (cid:55)−→ t, and H : R × Y −→ R , ( t, y ) (cid:55)−→ (cid:40) H ( w ) y = [ w, ϕ ] ∈ M ( W, ψ ) h ( r ) + H ( w ) − h (1) y = ( w ; r, ϕ ) ∈ B ( W ) . We note here that the condition ψ | ∂W = Id implies that H | ∂W is locally constant. However, H | ∂W does not need to be constant; its values can differ for the different boundary components. Theseformulas prove the first assertion of Observation 1.2. For the second assertion, we use the abstractbook-connected sum to perform the connected sum.We note that this construction only gives C ∞ -integrals due to the cutoff function involved inthe definitions of h and H . Remark . By the above, we can realize the flow of the proof of the main theorem as a completelyintegrable system. The way we described to do so yields a degenerate system. In this case theslope of the Liouville tori is constant across a large family of them. In Figure 5 we have sketchedlevel sets of a Hamiltonian generating a return map that is isotopic to the one of the above systemand that is non-degenerate in the sense of Kolmogorov. The critical points on the singular levelsets correspond to the Lyapunov orbits of the connected sum.
Figure 5.
Level sets of a Hamiltonian generating ψ (cid:92)ψ (cid:92)ψ Appendix: connected sum and book-connected sum
We will derive the first assertion of Lemma 2.6 from a sequence of more general statements.Consider a Liouville cobordism (
W, ω, X ), so ω is a symplectic form, and X is a globally defined Li-ouville vector field that is transverse to all boundary components. Denote the convex boundary of W by ∂ + W := { x ∈ ∂W | X ( x ) points outward } . The convex boundary admits the contact form α + := i X ω | ∂ + W . Put B := ∂ + W , and consider anembedded isotropic sphere S : S k − → B , so S ∗ α + = 0. We will impose the additional conditionthat the conformal symplectic normal bundle CSN B ( S ) is trivial. Here is a brief, non-intrinsicdescription of this bundle. First of all, choose a complex structure J that is compatible with( ξ = ker α + , dα + ). Denote the Reeb vector field of α + by R + . Then we have T B | S ∼ = (cid:104) R + (cid:105) ⊕ T S ⊕ JT S ⊕ CSN B ( S ) . Choose a trivialization (cid:15) : S × ( R n − k , ω ) → CSN B ( S ). As was first found by Weinstein, thisdata suffices for the attachment of symplectic handles . We refer to the book of Geiges, [3],chapter 6, for a detailed description.Denote a 2 n -dimensional Weinstein k -handle by H nk . We write˜ W | S,(cid:15) := W ∪ S,(cid:15) H nk . We define contact surgery of B along ( S, (cid:15) ) as the convex boundary ∂ + ( (cid:102) W | S,(cid:15) ). We will indicatehandle attachment by (cid:101) and surgery by , so ∂ + ( (cid:102) W | S,(cid:15) ) = ∂ + W | S,(cid:15) . (4.1)4.1. Relation with open books.
We will now describe the effect of some special subcriticalcontact surgeries on open books. We start with an open book whose monodromy is trivial; forthis, consider a stabilized Liouville domain ( W × D , ω W + ω D , X W + X D ). Here ω W and X W standfor the symplectic form and Liouville vector field on W , respectively. The subscript D is used forthe corresponding objects on D . We will take ω D = rdr ∧ dϕ and X D = r∂ r where ( r, ϕ ) arethe usual polar coordinates. We denote the Liouville forms by λ W = i X W ω W and λ D = i X D ω D ,respectively.We obtain a natural open book on the convex boundary of W × D after first “rounding” itscorners by the following procedure. Note that a neighborhood of the corner is given by ( − ε W , × ∂W × ( − ε D , × S . The neighborhoods ( − ε W , × ∂W in W and ( − ε D , × S are taken so smallsuch that X W and X D are outward pointing on each level set { t } × ∂W and { r } × S , respectively.We will parametrize the rounded corner explicitly: choose real-valued functions f, g defined on[0 ,
1] with the following properties • f satisfies f | [0 , / ≡
0, and is strictly decreasing on (1 / ,
1] with f (1) = − ε W . • g is a strictly increasing function on [0 , /
4) with g (0) = − ε D and g | [3 / , ≡ ψ : ∂W × (0 , × ∂D −→ W × D , ( w, t, z ) (cid:55)−→ ( F l X W f ( t ) ( w ) , F l X D g ( t ) ( z ))The Liouville form λ W + λ D pulls back under ψ to e f ( t ) i ∗ W λ W + e g ( t ) i ∗ D λ D . (4.2)Away from the rounded corner, the manifold ∂ + ( W × D ) consists of the disjoint union of( W \ ( − ε W , × ∂W ) × S and ∂W × D < − ε D . On the former set, the Liouville form restricts to the contact form λ W + dϕ : we identify this settogether with its contact form with the trivial mapping torus M ( W, Id) which is endowed withthe same contact form. On the set ∂W × D , the Liouville form restricts to λ W | ∂W + r dϕ .Together with the rounded corner, parametrized by ψ , we identify this contact manifold with aneighborhood of the binding which is equipped with a contact form that has the same form asdefined in Equation (2.1) (use Equation (4.2) to see this). REEB FLOW ON THE THREE-SPHERE WITHOUT A DISK-LIKE GLOBAL SURFACE OF SECTION 9
We hence get an abstract contact open book on the convex boundary. The choice of roundingdata only affects the profile functions in Equation (2.1) and for any two profile functions that areadmissible, a convex combination of them also is. From now on, we will implicitly round cornersand just write ∂ + ( W × D ) ∼ = OB( W, Id) . (4.3)In the following, we will decompose a Liouville domain as W = W c ∪ W m , where W m ∼ = ( − ε, × ∂W . We note that Equation (4.3) holds for W m as well, in which case we get an open book on anon-compact contact manifold. We now make a sequence of simple observations:(1) suppose we are given S : S k − → ∂W , an isotropic embedding into ∂W , together witha trivialization (cid:15) of its conformal symplectic normal bundle. The embedding S extendsto an embedding into ∂W × D , which we continue to denote by S . We also obtain atrivialization of the new larger conformal symplectic normal bundle by using ∂ x and ∂ y ,the tangent vectors to ∂W × { } in the disk direction D , to trivialize the new directionsin CSN ∂ + ( W × D ) ( S ). We denote this new trivialization by (cid:15) ⊕ (cid:15) D .By choosing an appropriate (large) attaching region for the handle, we obtain a sym-plectomorphism, namely the identity, (cid:94) W × D | S × Id ,(cid:15) ⊕ (cid:15) D ∼ = (cid:102) W | S,(cid:15) × D . (4.4)(2) by applying in succession Equation (4.3), observation (1), relating handle attachment tosurgery as in Equation (4.1), and Equation (4.3) again, we findOB( (cid:102) W ,
Id) ∼ = ∂ + ( (cid:102) W × D ) ∼ = ∂ + ( (cid:94) W × D ) ∼ = ∂ + ( W × D ) ∼ = OB( W, Id) . We now find (see also [11], Proposition 4.2)
Corollary 4.1.
Suppose that
OB(
W, ψ ) is a contact open book. Suppose that S : S k − → B = ∂W is an isotropic embedding into the binding of the open book, and assume that we are given atrivialization of its conformal symplectic normal bundle of the form (cid:15) B ⊕ (cid:15) D . Then OB(
W, ψ ) | S,(cid:15) B ⊕ (cid:15) D ∼ = OB( (cid:102) W | S,(cid:15) B , ˜ ψ ) , (4.5) where ˜ ψ equals ψ on the complement of the attached k -handle and is the identity on the k -handle.Proof. Decompose the Liouville filling W = W c ∪ W m , where W m is the “margin” of the page,i.e. a neighborhood of the (convex) boundary, where the monodromy ψ equals the identity. TheLiouville domain W c is the content of the page, i.e. the complement of the margin. We haveOB( W, ψ ) = M ( W c , ψ ) ∪ M ( W m , Id) ∪ B ( W ) = M ( W c , ψ ) ∪ OB( W m , Id)We will now apply observation (2) to OB( W m , Id). As the surgery takes place in a neighborhoodof the binding we haveOB(
W, ψ ) = M ( W c , ψ ) ∪ OB( W m , Id) ∼ = M ( W c , ψ ) ∪ OB( (cid:103) W m , Id) = OB( (cid:102)
W , ˜ ψ ) . This proves the claim. (cid:3)
As a special case we obtain the following.
Corollary 4.2.
There is a contactomorphism between the following contact manifolds, each pre-sented as abstract open books,
OB( W , ψ ) W , ψ ) ∼ = OB( W (cid:92)W , ψ (cid:92)ψ ) . (4.6)Indeed, for this, we take k = 1 in Corollary 4.1, put W = W (cid:96) W , and choose S : S → ∂ ( W (cid:96) W ) mapping one point of S to one boundary component of W and the other pointof S to a boundary component of W ; there is only one homotopy class of trivializations of theconformal symplectic normal bundle in this case. The left-hand side in (4.5) then reduces to 0-surgery along S , which is the contact connected sum of the open books. The right-hand side is acontact open book whose page is obtained by attaching a symplectic 1-handle to W = W (cid:96) W :this is the new page ˜ W . The monodromy extends as the identity on the symplectic 1-handle. Inother words, the right-hand side is the abstract book-connected sum. References [1] A. Bolsinov, A. Fomenko,
Integrable Hamiltonian systems: Geometry, Topology Classification , CRC press,ISBN 0-415-29805-9.[2] J. Franks,
Geodesics on S and periodic points of annulus homeomorphisms , Invent. Math. 108 (1992), no.2, 403–418.[3] H. Geiges, An introduction to contact topology , Cambridge Studies in Advanced Mathematics, 109. Cam-bridge University Press, Cambridge, 2008. xvi+440 pp. ISBN: 978-0-521-86585-2[4] R. Gompf, A. Stipsicz, , Graduate Studies in Mathematics, 20. AmericanMathematical Society, Providence, RI, 1999. xvi+558 pp.[5] H. Hofer, K. Wysocki, E. Zehnder,
The dynamics on three-dimensional strictly convex energy surfaces , Ann.of Math. (2) 148 (1998), no. 1, 197–289.[6] U. Hryniewicz,
Systems of global surfaces of section for dynamically convex Reeb flows on the 3-sphere , J.Symplectic Geom. 12 (2014), no. 4, 791–862.[7] U. Hryniewicz, P. Salom˜ao,
On the existence of disk-like global sections for Reeb flows on the tight 3-sphere ,Duke Math. J. 160 (2011), no. 3, 415–465.[8] K. Kuperberg,
A smooth counterexample to the Seifert conjecture , Ann. of Math. (2) 140 (1994), no. 3,723–732.[9] S. Lisi, A. Marinkovi´c, K. Niederkr¨uger,
Some properties of the Bourgeois contact structures , preprintarXiv:1801.00869[10] O. Manzoli Neto, S. Massago, O. Saeki,
Open book structures on ( n − -connected (2 n + 1) -manifolds J.Math. Sci. Univ. Tokyo 13 (2006), no. 4, 439–523.[11] O. van Koert,
Lecture notes on stabilization of contact open books , M¨unster J. Math. 10 (2017), no. 2,425–455.
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