Selman Akbulut and M. Firat Arikan- Stabilizations via Lefschetz Fibrations and Exact Open Books

8/3/2019 Selman Akbulut and M. Firat Arikan- Stabilizations via Lefschetz Fibrations and Exact Open Books

1/23

a r X i v : 1 1 1 2

. 0 5 1 9 v 1 [ m a t h . G T

] 2 D e c 2 0 1 1

STABILIZATIONS VIA LEFSCHETZ FIBRATIONS ANDEXACT OPEN BOOKS

SELMAN AKBULUT AND M. FIRAT ARIKAN

Abstract. We show that if a contact open book ( , h ) on a (2n +1)-manifold M (n 1)is induced by a Lefschetz bration : W D 2 , then there is a one-to-onecorrespondencebetween positive stabilizations of ( , h ) and positive stabilizations of . More precisely,any positive stabilization of ( , h ) is induced by the corresponding positive stabilizationof , and conversely any positive stabilization of induces the corresponding positivestabilization of ( , h ). We dene exact open books as boundary open books of exactLefschetz brations, and show that any exact open book carries a contact structure.Moreover, we prove that there is a one-to-one correspondence (similar to the one above)between convex stabilizations of an exact open book and convex stabilizations of thecorresponding exact Lefschetz bration. We also show that convex stabilization of exactLefschetz brations produces symplectomorphic manifolds.

1. Introduction

In the last decade the correspondence given by Giroux [ 11], between contact structuresand open book decompositions have led to many developments in understanding the rela-tions between the contact geometry and the topology of the underlying odd dimensionalclosed manifolds. This correspondence is much stronger in dimension three and has beenused as a bridge between four dimensional geometries and topology, leading much progressin understanding of different types of llability and Lefschetz type brations.

One of the main features used in the above correspondence is positive stabilization.Namely, if we positively stabilize an open book ( , h) carrying a contact structure on aclosed 3-manifold M , then the resulting open book still carries . Such stabilizations canbe interpreted as taking the contact connect sum of ( M, ) with ( S 3, st ) where st is theunique tight (Stein llable) contact structure on the 3-sphere S 3. In terms of open books,this corresponds to taking the Murasugi sum (or plumbing) of ( , h) with the open book(H + , C ) on S 3 where H + is the positive (left-handed) Hopf band and C denotes theright-handed Dehn twist along the core circle C in H + .

To get analogous statements for higher dimensions, one can replace ( H + , C ) with its

generalization OB , which is an open book carrying the standard contact structure 0 on(2n + 1)-sphere S 2n +1 and obtained from a certain Milnor bration. The pages of OB arediffeomorphic to the closed tangent unit disk bundle D(T S n ) over S n and its monodromyis the (generalized) right-handed Dehn twist along the zero section (see below). Then

Date : December 5, 2011.1991 Mathematics Subject Classication. 58D27, 58A05, 57R65.Key words and phrases. Contact & symplectic structures, open book, Lefschetz bration, stabilization.The rst author is partially supported by NSF FRG grant DMS- 0905917.The second author is partially supported by NSF FRG grant DMS-1065910.

1
http://arxiv.org/abs/1112.0519v1http://arxiv.org/abs/1112.0519v1http://arxiv.org/abs/1112.0519v1http://arxiv.org/abs/1112.0519v1http://arxiv.org/abs/1112.0519v1http://arxiv.org/abs/1112.0519v1http://arxiv.org/abs/1112.0519v1http://arxiv.org/abs/1112.0519v1http://arxiv.org/abs/1112.0519v1http://arxiv.org/abs/1112.0519v1http://arxiv.org/abs/1112.0519v1http://arxiv.org/abs/1112.0519v1http://arxiv.org/abs/1112.0519v1http://arxiv.org/abs/1112.0519v1http://arxiv.org/abs/1112.0519v1http://arxiv.org/abs/1112.0519v1http://arxiv.org/abs/1112.0519v1http://arxiv.org/abs/1112.0519v1http://arxiv.org/abs/1112.0519v1http://arxiv.org/abs/1112.0519v1http://arxiv.org/abs/1112.0519v1http://arxiv.org/abs/1112.0519v1http://arxiv.org/abs/1112.0519v1http://arxiv.org/abs/1112.0519v1http://arxiv.org/abs/1112.0519v1http://arxiv.org/abs/1112.0519v1http://arxiv.org/abs/1112.0519v1http://arxiv.org/abs/1112.0519v1http://arxiv.org/abs/1112.0519v1http://arxiv.org/abs/1112.0519v1http://arxiv.org/abs/1112.0519v1http://arxiv.org/abs/1112.0519v1http://arxiv.org/abs/1112.0519v1http://arxiv.org/abs/1112.0519v1http://arxiv.org/abs/1112.0519v1


2/23

2 SELMAN AKBULUT AND M. FIRAT ARIKAN

one can dene a positive stabilization of an open book ( , h) carrying a contact structure on an (2n + 1)-dimensional closed manifold M 2n +1 by taking the Murasugi sum of ( , h) with OB , along a properly embedded Lagrangian n-ball L in with Legendrianboundary and a ber in D(T S n ). Again this amounts to taking the contact connect sum

of (M 2n +1

, ) with ( S 2n +1

, 0) and stabilized open book still carries [11]. In terms of contact surgery and Weinstein handles, a positive stabilization corresponds to performing(resp. attaching) a pair of subcritical and critical surgeries (resp. Weinstein handles)which cancels each other (see [18] for a proof).

One of the missing part of this picture is the relation of such operations to Lefschetzbrations. The aim of the present work is to provide some results to ll this gap. Throughout the paper, the base space of any Lefschetz bration is assumed to be the 2-disk,and we focus only on the open books which are induced by Lefschetz brations. Westudy the open book OB (which is induced by a certain Lefschetz bration LF on thestandard (2 n + 2)-ball) in Section 2 where we also recall positive stabilizations of openbooks and the characterization of Lefschetz brations. In Section 3, we explicitly denea process, called positive stabilization on Lefschetz brations and show that there is aone-to-one correspondence between positive stabilizations of open books and Lefschetzbrations. Exact open books are introduced in Section 4 as boundaries of exact Lefschetzbrations. After recalling Weinstein handles and isotropic setups briey in Section 5,we will get a similar correspondence for exact open books and exact Lefschetz brationsin Section 6, where we also dene convex stabilization as an exact symplectic version of positive stabilization. We remark that any observation we will make here is also true fordimensions 3 and 4, and so the work done here can be thought as canonical generalizationsof the corresponding 3- and 4-dimensional results to higher dimensions.

2. Preliminaries

2.1. The open book OB and the associated Lefschetz bration LF . Consider thepolynomial P on the complex space C n +1 (for n 1) given by

P (z1,...,z n +1 ) = z21 + z22 + + z

2n +1 .

It is clear that the only critical point of P occurs at the origin, and so the intersection of the zero set Z (P ) of P with the sphere

S 2n +1 = {|z1 |2 + |z2 |2 + + |zn +1 |2 = 2},

where > 0 is small enough, is a smooth manifold K of dimension 2n 1. K is amember of a family known as Brieskorn manifolds introduced in [ 3]. It is known by [15]that the complement S 2n +1 \ K of K bers over the unit circle S 1 C via the map : S 2n +1 \ K S 1 given by

(z1,...,z n +1 ) =P (z1,...,zn +1 )

|P (z1,...,zn +1 )|.

Let OB be the open book on S 2n +1 determined by the pair ( K, ). For any ei S 1,the Milnor ber (or the page of OB ) F := 1(ei ) is parallelizable and has the homotopytype of S n [15], and indeed it can be identied with the total space of the tangent bundleT S n of the n-sphere S n (e.g. [9], p. 81). By considering the closure F as the closedtangent unit disk bundle D(T S n ) over S n , we can identify the binding K of OB as the


3/23

STABILIZATIONS VIA LEFSCHETZ FIBRATIONS AND EXACT OPEN BOOKS 3

tangent unit sphere bundle S (T S n ) over S n . For our purposes we identify T S n with thecotangent bundle T S n by using the natural duality, and assume that each page F of OB is diffeomorphic to the cotangent unit disk bundle D(T S n ), and so the binding K isdiffeomorphic to the cotangent unit sphere bundle S (T S n ).

Now we dene the function : B 2n +2 D 2 from (2n + 2)-ball

B 2n +2 = {|z1 |2 + |z2 |2 + + |zn +1 |2 2} C n +1

onto the unit disk D 2 C by restricting P and then normalizing by 2, that is,

=12

P |B 2 n +2 .

By denition (see [12], for instance) is a local model for a Lefschetz bration over theunit disk having only one singular ber over the origin. Also regular bers 1(z), z = 0,are diffeomorphic to D(T S n ) because is a topological locally trivial bration on D 2 \{ 0}

[13] (e.g. [9], Chapter 3). Therefore, denes a Lefschetz bration LF on B 2n +2 whichinduces the open book OB on the boundary sphere S 2n +1 . By denition, the monodromyof LF is the monodromy of OB . According to [8, 12] this monodromy is (up to isotopy)equal to the right-handed Dehn twist

: D(T S n ) D(T S n )

along the vanishing cycle which is the zero section (a copy of S n ) in D(T S n ). To describe precisely, identify the interior of a page with T S n and write the points in T S n as(q , p ) R n +1 R n +1 such that |q | = 1 and q p . Then

(q , p ) = cos g(|p |) |p | 1 sin g(|p |)

| p | sin g(|p |) cos g(|p |)

q

pwhere g is a smooth function that increases monotonically from to 2 on some interval,and outside this interval g is identically equal to or 2. Observe that is the antipodalmap on the zero section S n { 0} = {(q , p ) | | q | = 1 , p = 0 }, while it is the identity mapfor |p | large. Note that as abstract open book OB is determined by the pair ( D(T S n ), ).

Now let z j = x j + iy j for j = 1 ,...,n + 1. Then with respect to the complex coordinatesz = ( z1,...,zn +1 ), the standard Stein structure on C n +1 (and hence on B 2n +2 ) is denedby the pair

(J 0, 0) = ( i i, | z| 2).

This denes the standard symplectic (indeed K ahler ) form

0 = d(d0 J 0) =n +1

j =1

dx j dy j

whose Liouville vector eld 0 (i.e., satisfying L 0 0 = 0) is given by

0 =12

n +1

j =1

(x j x j + y j y j ).


4/23


Then on the boundary sphere S 2n +1 , the 1-form

0 = 0 0 =12

n +1

j =1

(x j dy j y j dx j ) =i4

n +1

j =1

(z j dz j z j dz j )

is a contact form (i.e., 0 (d 0)n | S 2 n +1 > 0). The codimension one plane distributionkernel 0 = Ker( 0) is called the standard contact structure on S 2n +1 .

The compatibility between contact structures and open books is dened as follows:

Denition 2.1 ([11]). We say that a contact structure = Ker( ) on M is carried by (or supported by ) an open book (B, f ) on M (where B is the binding), if the followingconditions hold:

(i) (B, |T B ) is a contact manifold.(ii) For every t S 1, the page X = f 1(t) is a symplectic manifold with symplectic

form d.(iii) If X denotes the closure of a page X in M , then the orientation of B induced by

its contact form |T B coincides with its orientation as the boundary of ( X,d ).

The open book OB has been studied before, but since it is one of the main buildingblocks of the present paper and for completeness, here we discuss its important aspect:

Lemma 2.2. The open book OB carries the standard contact structure 0 on S 2n +1 in-herited from the standard Stein structure on B 2n +2 .

Proof. The rst condition of compatibility (Denition 2.1) immediately follows from [14]where they show that the restriction of 0 is a contact form on Brieskorn manifolds, andso, in particular, on the binding K . To check the second one, consider the vector eld

R =4i2

n +1

j =1

z j z j = R0 + R1; R0 =2i2

n +1

j =1

(z j z j z j z j ), R1 =2i2

n +1

j =1

(z j z j + z j z j ).

Observe that R0 |S 2 n +1 is the Reeb vector eld of the contact form 0 |S 2 n +1 . (That is, wehave (R0) = 1 , R 0 d 0 = 0 on S 2n +1 .) The ow of R is computed as

ht (z) = ( e4it/2

z1,...,e 4it/2

zn +1 )

which is a 1-parameter group of diffeomorphisms ht : C n +1 \ Z (P ) C n +1 \ Z (P ). Nowconsider the bration : C n +1 \ Z (P ) S 1 given by

(z) =P (z)

|P (z)| .

Then h t maps each ber 1(y) diffeomorphically onto the ber 1(ei y), and also thereis a diffeomorphism 1(y) = 1(y) R as shown in Chapter 9 of [15]. Furthermore, h tmaps S 2n +1 \ K diffeomorphically onto itself for all t. Hence, we conclude that h t mapseach ber 1(y) diffeomorphically onto the ber 1(ei y), but this means, in particular,that the Reeb vector eld R0 |S 2 n +1 is transverse to every page of the open book OB (notethat R1 does not live in T S 2n +1 ). So for any page F , the rank of d 0|F is maximal whichis equivalent to saying that d 0 is a symplectic form on F .


5/23


For the third condition, on T B 2n +2 |S 2 n +1 we compute 0( 0, R0) = 1, so { 0, R0} forma non-degenerate pair with respect to 0 = d 0. Therefore, for a xed page

F = {z | (z) = ei } S 2n +1 \ K of the bration , the tangent bundle T B 2n +2

restricted to F

is decomposed as

T B 2n +2 |F = 0 T S2n +1 |F = 0 R0 T F

where we use the fact that the Liouville vector eld 0 is transverse to S 2n +1 (which is of contact type) and that R0 is transverse to F . This shows that ( F , 0 |F ) is a symplecticsubmanifold of (B 2n +2 , 0) and, in particular, that the orientation on F given by 0 |F isinherited from the orientation on B 2n +2 given by 0.

Write = 0 |K , F = F . To nish the proof, we need to check that the orientation onF = K given by the form (d )n 1 coincides with the one induced by the orientationon F given by the volume form ( d 0 |F )n : We showed above that the latter orientation onF is inherited from the one on B 2n +2 given by the standard Stein structure ( J 0, 0). Notethat the orientation on ( S 2n +2 , 0) given by the volume form 0 (d 0)

n is also comingfrom this Stein structure. Moreover, the orientation on ( K, := Ker( )) (S 2n +1 , 0)determined by (d )n 1 matches up with the one inherited (as a contact submanifold)from (S 2n +1 , 0). Hence, the mentioned two orientations on K must coincide.

2.2. Positive stabilization of open books. We rst recall the plumbing or 2-Murasugisum of two contact open books (i.e., open books carrying contact structures): Let ( M i , i)be two closed contact manifolds such that each i is carried by an open book ( i , h i)on M i . Suppose that Li is a properly embedded Lagrangian ball in i with Legendrianboundary L i i . By the Weinstein neighborhood theorem each Li has a standardneighborhood N i in i which is symplectomorphic to ( T D n , d can ) where can = pdqis the canonical 1-form on R n R n with coordinates ( q , p ). Then the plumbing or 2-

Murasugi sum (P ( 1 , 2; L1, L2), h) of (1, h1) and ( 2, h2) along L1 and L2 is the openbook on the connected sum M 1# M 2 with the pages obtained by gluing is together alongN i s by interchanging q -coordinates in N 1 with p -coordinates in N 2, and vice versa. Todene h, extend each h i to h i on the new page by requiring h i to be identity map outsidethe domain of hi . Then the monodrodmy h is dened to be h2 h1. Without abuse of notation we will drop the tilde sign, and write h = h2 h1.

The following terminology was given in [11]. We describe it in a slightly different wayso that it ts into the notation of the present paper.

Denition 2.3 ([11]). Suppose that ( , h) carries the contact structure = Ker( ) on a(2n + 1)-manifold M . Let L be a properly embedded Lagrangian n-ball in a page ( , d )such that L is a Legendrian ( n 1)-sphere in the binding ( , |

). Then

the positive (or standard ) stabilization S OB [(, h); L] of (, h) along L is the open book(P ( , D(T S n ); L, D ), h) where D = D n is any ber in D(T S n ).

2.3. Characterization of Lefschetz brations. Here we recall the handle decompo-sition of Lefschetz brations as described in [12]: Let : W D 2 C be a givenLefschetz bration with a regular ber X 2n and monodromy h. Consider the base disk asD 2 = {z C : |z| 2}. We may assume that 0 D 2 and the points on D 2 are regularvalues and that all the critical values {1, 2, . . , } of are roots of unity. Such a iscalled a normalized Lefschetz bration . Since every Lefschetz bration can be normalized,


6/23


throughout the paper all Lefschetz brations will be assumed to be normalized. Denea Morse function F : W [0, 4] R given by F (x) = |(x)|2. Then outside of the setF 1(0) F 1(4), F has only nondegenerate critical points of index n + 1 (see [1]). Since| i | = 1 for all i, the map has no critical values on the set D t = {z C : |z| t} for

t < 1 and henceF 1([0, t]) = 1(D t ) = X D 2 for t < 1.

On the other hand, for t > 1, 1(D t ) is diffeomorphic to the manifold obtained fromX D 2 by attaching handles of index n + 1, via the attaching maps

j : S n D n +1 (X D 2) = X S 1, j = 1 , 2,...,.Let j : n +1 be the framing of the j -th handle, where k denotes the trivial

bundle of rank k, and denotes the normal bundle of the attaching sphere j (S n { 0})in (X D 2).Fact 2.4 ([12]). The embeddings j may be chosen so that for each j = 1 , 2,..., thereexists z j such that j (S n { 0}) 1(z j ) = X .

So, set j : S n X to be the embedding dened by restricting j to S n { 0}. Let 1 denote the normal bundle of S n = j (S n ) in X corresponding to the embedding j ,and consider as the normal bundle of j (S n ) in F 1(1 ). Clearly, = 1 (as thenormal bundle of X in W is trivial). Let denote the tangent bundle of S n .Fact 2.5 ([12]). For each j = 1 , 2,...,, there exists a bundle isomorphism j : 1such that the framing j of the (n + 1)-handle corresponding to j coincides with

j .

That is, j is given by the composition

n +1

j id 1

.Denition 2.6 ([12]). S n = j (S n ) is called a vanishing cycle of . The bundle isomor-phism j : 1 is called a normalization of j . The pair ( j ,

j ) is called a normalized

vanishing cycle.Let D(T S n ) denote the closed tangent unit disk bundle of S n . By the tubular

neighborhood theorem and the canonical isomorphism D(T S n ) = D(T S n ), we can applythe right-handed Dehn twist to a tubular neighborhood of j (S n ) in X , and we canextend , by the identity, to a self-diffeomorphism of X which we denote by

(j , j ) : X

X.

Up to smooth isotopy (j , j ) Diff(X ) depends only on the smooth isotopy class of the embedding j and the bundle isomorphism j .

Denition 2.7 ([12]). (j , j ) is called the right-handed Dehn twist with center ( j , j ).We will make use of the following theorem.

Theorem 2.8 ([12], [8]). The Lefschetz bration : W D 2 is uniquely determined by a sequence of vanishing cycles (1, 2,...,) and a sequence of their normalizations(1,

2,...,

). The monodromy of the bration is equal to

2 1 Diff(X )where j = (j , j ) is the right-handed Dehn twist with center ( j ,

j ).


7/23


Remark 2.9. Recall the right-handed Dehn twist : D(T S n ) D(T S n ) given ex-plicitly before. With respect to the coordinates ( q , p ) on R 2(n +1) consider the canonical1-form can = p dq on D(T S n ) R 2(n +1) . Then one can compute

can = can + |p |d(g(|p |))

which implies that the difference can can is exact. Therefore, is a symplecto-morphism of the symplectic manifold ( D(T S n ), d can ). As a result, if a regular ber X of a Lefschetz bration : W D 2 equipped with a symplectic structure , then themonodromy h of is a symplectomorphism of ( X, ). That is,

h = 2 1 Symp(X, )

where j = (j , j ) is the right-handed Dehn twist with center ( j , j ) as in Theorem 2.8.

Notation 2.10. For our purposes it is convenient to dene a notation for Lefschetzbrations. Let the quadruple ( ,W,X,h ) denote the Lefschetz bration : W D 2 onW with a regular ber X and the monodromy h. For instance, according to this notationwe have LF = ( , B 2n +2 , D(T S n ), ).

For completeness we give the following basic well-known fact as a denition:

Denition 2.11. Let (,W,X,h ) be any (normalized) Lefschetz bration. The pairs( 1(0), | W ) and (X, h ) are both called the induced open book (or sometimes theboundary open book ) on W .

3. Positive stabilization of Lefschetz fibrations

Now we dene a process on Lefschetz brations as a counterpart of positive stabi-lization on open books. We will use Weinstein handles introduced in [19]. Using the

symplectization model near convex boundaries, these handles can be glued to symplecticmanifolds along isotropic spheres to obtain new ones, and they give elementary symplecticcobordisms between contact manifolds. We will briey explain them later.

Denition 3.1. Let (,W,X,h ) be a Lefschetz bration which induces a contact openbook on W . Suppose that L (X, ) is a properly embedded Lagrangian n-ballwith a Legendrian boundary L X on a page of the induced open book. Thenthe positive stabilization S LF [(,W,X,h ); L] of (,W,X,h ) along L is a Lefschetz bra-tion ( , W , X , h ) described as follows:

(I) X is obtained from X by attaching a Weinstein n-handle H = D n D n along the

Legendrian sphere L X .(II) h = (, ) h where (, ) is the right-handed Dehn twist with center ( , )

dened as follows: (S n ) is the Lagrangian n-sphere S = D n { 0} L L inthe symplectic manifold ( X = X H, ) where is obtained by gluing andstandard symplectic form on H . If 1 denote the normal bundle of S in X , thenthe normalization : 1 is given by the bundle isomorphisms

=

T S = T X /TS = 1.


8/23


Remark 3.2. W is, indeed, diffeomorphic to W (see the proof of Theorem 3.3 below).Also in h = (, ) h, we think of h as an element in Diff (X ) by trivially extending overH . Moreover, the isomorphism T S T X /TS exists because S is Lagrangian in (X , )(the core D n { 0} of H is Lagrangian). Finally, note that there is a strong analogy

between S OB [(X, h ); L] and S LF [(,W,X,h ); L]. On the one hand, we haveS OB [(X, h ); L] = (P (X, D(T S n ); L, D ), h)

which means that we are plumbing the open book ( X, h ) on a given manifold M withthe open book OB = ( D(T S n ), ) on S 2n +1 . Therefore, S OB [(X, h ); L] is an openbook on the connected sum M # S 2n +1 M . On the other hand, we may regardS LF [(,W,,,X,h ); L] as the result of (informally speaking) Lefschetz plumbing of (,W,X,h ) with LF . Indeed, one can see that S LF [(,W,X,h ); L] is a Lefschetz brationon the boundary connect sum W # b B 2n +2 W .

We are now ready to prove

Theorem 3.3. Any positive stabilization S LF [(,W,X,h ); L] of a Lefschetz bration (,W,X,h ) with a contact boundary open book induces the open book S OB [(X, h ); L]. Con-versely, if a contact open book (X, h ) is induced by a Lefschetz bration (,W,X,h ), then any positive stabilization S OB [(X, h ); L] of (X, h ) is induced by S LF [(,W,X,h ); L].

Proof. By denition of S LF , the ber X is obtained from X by attaching 2 n-dimensionalWeinstein n-handle H along L X . Since every ber over D 2 is gaining H , we areactually attaching a (2 n + 2)-dimensional handle

H = H D 2 = D n D n +2

to W along L W . Say the resulting manifold is W , that is W = W H . Byextending the monodromy h (but calling it still h) trivially over H , we get an extended

Lefschetz bration : W D2

on W , i.e., we get (, W, X

, h). Note that ( , W, X

, h)is determined by Theorem 2.8. So far what we explained is the content of Stage (I)in Denition 3.1. In Stage (II), composing the monodromy h with (, ) correspondsto attaching an (2 n + 2)-dimensional handle H (so called a Lefschetz handle) withindex n + 1 to W along the Lagrangian sphere S in the ber ( X , ) of (, W, X , h).By Theorem 2.8, we know that ( , W, X , h) extends over the handle H and we get theLefschetz bration S LF [(,W,X,h ); L] = ( , W , X , h ) on the resulting manifold

W = W H = W H H .We immediately see that {H , H } form a canceling pair in the smooth category as theattaching sphere of H intersects the belt sphere of H transversely once, and so W

is diffeomorphic to the original manifold W (indeed W

= W # bB 2n +2 ). Therefore, theopen book (X , h ) induced by S LF [(,W,X,h ); L] is an open on the original boundaryW . Next we need to see that ( X , h ) is indeed isomorphic (as an abstract open book) toS OB [(X, h ); L]. To this end, rst observe that in the plumbing ( P (X, D(T S n ); L, D ), h)we are embedding a tubular neighborhood N (D ) of D in D(T S n ) into the page X in sucha way that the intersection N (D )X is a tubular neighborhood of the Legendrian sphereL( S n 1). Considering L as the equator of the zero section S n { 0} D(T S n ), it isclear that the part D(T S n ) \ N (D ) of D(T S n ) which is not mapped into X (during theplumbing) is the trivial bundle D(T D n ) = D n D n . Note that the canonical symplectic


9/23


structure on D(T S n ) restricts to the standard symplectic structure on D(T D n ) whichimplies that D(T S n ) \ N (D ) is the Weinstein handle H glued to X along L. Hence, thepage of the open book S OB [(X, h ); L], that is, the resulting page of the plumbing, is X .Also if we keep track of the vanishing cycle S n { 0} D(T S n ) in the above discussion,

we immediately see that it corresponds to the Lagrangian n-sphere S = C L L whereC = D n { 0} is the (Lagrangian) core disk of the Weinstein handle H which means thatthe right-handed Dehn twist coincides with , described in Denition 3.1. Composingwith h, we get h = h Thus, S OB [(X, h ); L] and (X , h ) are isomorphic. This provesthe rst statement.

For the second statement we basically follow the same steps in a different order: If S OB [(X, h ); L] is a given stabilization, then by the above discussion we know that the newpage is equal to X = X H . By assumption ( X, h ) is induced from (,W,X,h ). Soby attaching H = H D 2 (thickening of H ) to W , each ber of gains the handle H ,and we get (, W, X , h) on W . Since = (, ) , h = (, ) h = h. Therefore, we

have S OB [(X, h ); L] = (X

, h

). Moreover, composing with h (in the open book level)corresponds to attaching a Lefschetz handle H to W whose normalized vanishing cycleis (, ). Therefore, we obtain ( , W , X , h ) on W = W H ( W ). It is now clearthat S OB [(X, h ); L] is induced by S LF [(,W,X,h ); L].

4. Exact Open Books

We will dene exact open books as boundary open books induced by exact Lefschetzbrations. To this end, recall that a contact manifold ( M, ) is called strongly symplec-tically lled by a symplectic manifold ( X, ) if there exist a Liouville vector eld of dened (at least) locally near X = M such that is transverse to M and = . Such

a boundary is called convex . An exact symplectic manifold is a compact manifold X withboundary, together with a symplectic form and a 1-form satisfying = d, such that | X is a contact form which makes X convex. In such a case there is a Liouville vectoreld of such that = . We will write exact symplectic manifolds as triples of theform (X,, ). Also the pair ( , ) (or sometimes the triple ( ,, )) will be called anexact symplectic structure on X .

Let : E 2n +2 D 2 be a differentiable ber bundle, denoted by ( , E ), whose bersare compact manifolds with boundary. The boundary of such an E consists of two parts:The vertical part vE ; = 1(D 2), and the horizontal part h E := zD 2 E z whereE z = 1(z) is the ber over z D 2. The following denitions can be found in [16] wherea more general setting is used.

Denition 4.1 ([16]). An exact symplectic bration ( ,E ,, ) over the disk D 2 is adifferentiable ber bundle ( , E ) equipped with a 2-form and a 1-form on E , satisfying = d, such that

(i) each ber E z with z = |E z and z = |E z is an exact symplectic manifold,(ii) the following triviality condition near h E is satised: Choose a point z D 2

and consider the trivial bration : E := D 2 E z D 2 with the forms , which are pullbacks of z , z , respectively. Then there should be a ber-preservingdiffeomorphism : N N between neighborhoods N of h E in E and N of h E


10/23


in E which maps h E to h E , equals the identity on N E z , and = and = .

Denition 4.2 ([16]). An exact Lefschetz bration over D 2 is a tuple ( ,E,,,J 0, j 0)which satises the following conditions:

(i) : E D 2 is allowed to have nitely many critical points all of which lie in theinterior of E .

(ii) is injective on the set C of its critical points.(iii) J 0 is an integrable complex structure dened in a neighborhood of C in E such

that is a Kahler form for J 0.(iv) j 0 is a positively oriented complex structure on a neighborhood of the set (C ) in

D 2 of the critical values.(v) is (J 0, j 0)-holomorphic near C .

(vi) The Hessian of at any critical point is nondegenerate as a complex quadraticform, in other words, has nondegenerate complex second derivative at each its

critical point.(vii) (, E \ C,,) is an exact symplectic bration over D 2 \ (C ).

Remark 4.3. As pointed out in [ 16], one can nd an almost complex structure on J onE agreeing with J 0 near C and a positively oriented complex structure j on D 2 agreeingwith j 0 near (C ) such that is (J, j )-holomorphic and (, J )|Ker( ) is symmetric andpositive denite everywhere. The existence of ( J, j ) is guaranteed by the fact that thespace of such pairs (J, j ) is always contractible, and in particular, always nonempty.Furthermore, once we xed ( J, j ), we can modify by adding a positive 2-form on D 2 sothat it becomes symplectic and tames J everywhere on E .

For completeness and our future use, let us summarize the discussion in the last remark

and make additional observations about exact Lefschetz brations in the following lemma.Lemma 4.4. For any exact Lefschetz bration ( ,E,,,J 0, j 0), there exists an exact 2-form = d on the total space E and a pair (J, j ) as in Remark 4.3 such that

(i) is (J, j )-holomorphic and is symplectic and tames J everywhere on E ,(ii) (E, , ) is an exact symplectic manifold with convex boundary (E, |E ),

(iii) each regular ber (E z , z , z) is an exact symplectic submanifold of (E, , ).

Proof. The rst statement follows by Remark 4.3. More precisely, consider the positive2-form dr d on D 2 where r is the radial and is the angular coordinates. Then it isstandard to check that the form = + (dr d) is symplectic and tames J on E ,and also that is (J, j )-holomorphic. For the second one, we have

= + (dr d) = d + (d(rd )) = d + d(rd ) = d( + (rd )) ,

and so is exact with a primitive = + (rd ). Let V z be the Liouville vector eldof z . By the local triviality condition near h E , these V z s glue together (smoothly) andgives a Liouville vector eld V for near h E . Now, consider the collar neighborhood N of vE in E which is projected (by ) onto an annular region of the form (1 , 1] S 1 D 2for 0 < < 1. Consider the vector eld r/r in T D2| (1 , 1] S 1 . Taking small enough,we can nd a lift H of r/r which is a vector eld in TE |N . Then (H ) = r/r bydenition. Also note that H can never be tangent to the bers of in N because if H| p


11/23


were tangent to the ber E z over the point z = re i D 2 (with r = 0) at some point pE z , then this would give the contradiction

0 = (H| p) = r/r | (r, ) = 0 .

(The rst equality follows from the fact that kills the ber directions.) So, in particular,H is transverse to vE and is pointing out from the boundary. Now set = V + H .

Clearly, is nonvanishing vector eld in a collar neighborhood of E = h E vE . Alsoalong the boundary E , it is transverse and outward pointing. Moreover, it follows fromstandard computations that L = and = . As a result, is a Liouville vectoreld of and |E is a contact form which makes E the convex boundary of the exactsymplectic manifold ( E, , ).

For the last part, we simply observe that |E z = |E z = z and |E z = |E z = zwhich shows that each regular ber ( E z , z , z ) is an exact symplectic submanifold of thetotal space ( E, , ).

Notation 4.5. Since they will not be considered in our discussions, from now on we dropJ 0 and j 0 from our notation. Also we will assume that = d is already modied as inthe proof of Lemma 4.4 and that its Liouville vector eld is already constructed andgiven to us. Furthermore, we also want to specify the regular ber and the monodromyin our notation as before. Therefore, we introduce the following:

Let ( ,E,,, ,X,h ) denote an exact Lefschetz bration over the disk D 2 with thefollowing properties:

(i) The underlying smooth Lefschetz bration is ( ,E,X,h ) with h Symp(X, X ).(ii) The Liouville vector eld of is transverse to E and outward pointing.

(iii) (E ,, ) is an exact symplectic manifold with convex boundary ( E, |E ).(iv) (X, |X , |X ) is an exact symplectic submanifold of ( E ,, ).

Note that any exact Lefschetz bration over the disk D 2 admits such representation.

Denition 4.6. If an open book is induced by an exact Lefschetz bration, then it issaid to be an exact open book .

Theorem 4.7. The exact open book (X, h ) induced by a given exact Lefschetz bration ( ,E,,, ,X,h ) caries the contact structure = Ker( | E ) on E .

Proof. We need to show that all three conditions in Denition 2.1 hold. Assuming(,E,,, ,X,h ) is normalized (and so z0 = (0 , 0) is a regular value), the binding of (X, h ) is the boundary of the regular ber E z0 = 1(z0). We know that ( E z0 , z0 |E z 0 )

is the convex boundary of ( E z0 , z0 , z0 ) which is an exact symplectic submanifold of (E ,, ). Since z0 |E z 0 = ( |E z 0 )|E z 0 = |E z 0 , we conclude that |E z 0 is a contactform on the binding E z0 , so the rst condition follows.

For the second one, each regular ber E z of ( ,E,,, ,X,h ) is an exact symplecticsubmanifold of (E ,, ) with the symplectic form z = |E z = d |E z . In particular, anypage X of the boundary open book ( X, h ) equips with the symplectic structure d |X asbeing a regular ber of .

To check the orientation condition, we need to specify the dimensions. Say E hasdimension 2n + 2, and so the page X and the binding B have dimensions 2n and 2n 1,


12/23


respectively. For simplicity, write = |B and = |X (= d |X ). Let R be the Reebvector eld of and let be the Lioville vector eld of pointing out from B. To nishthe proof, we need to check that at a given point p X = B the orientation on T p Bgiven by the form p (d

p)n 1 coincides with the one induced by the orientation on

T p X given by the volume form (

p)n

:Consider the contact structure := Ker( ) which is a symplectic subbundle (withrank 2n 2) of = Ker( ), and the decomposition (see [10], for instance)

T p B = R p

p.

From their denitions, we have ( , R ) > 0 which means that { , R } is a nonde-generate pairing with respect to . Also since they are both transverse to , we get thedecomposition

T p X = p R p

p.

Choose a symplectic basis {u1, v1,...,u n 1, vn 1} for the symplectic subspace ( p,

p)giving the orientation on p determined by (

p)n 1, that is, we have

( p)n 1(u1, v1,...,u n 1, vn 1) > 0.

Since (X, ) is a symplectic manifold and p( p, R

p) > 0, we get a symplectic basis

{ p , R p , u1, v1,...,u n 1, vn 1}

for the symplectic space ( T p X, p) giving the orientation on T p X determined by ( )n ,

equivalently, ( p)n ( p , R

p , u1, v1,...,u n 1, vn 1) > 0. Then the induced orientation on

the subspace T p B T p X is determined by the oriented basis{R p , u1, v1,...,u n 1, vn 1}

( p is outward pointing normal direction at pB = X ). Now, using the fact = d ,

it is not hard to see that p (d

p)n 1(R p , u1, v1,...,u n 1, vn 1) > 0.

5. Isotropic Setups and Weinstein Handles

In this section we briey recall the isotropic setups and Weinstein handles introduced in[19]. Using them we will continue to study exact Lefschetz brations in the next section.

5.1. Isotropic setups. Let (M, = Ker( )) be a (2n +1)-dimensional contact manifold.Any subbundle of the symplectic bundle ( ,d ) has a symplectic orthogonal

.Therefore, if Y is an isotropic submanifold of M , then d |Y = 0 (as |Y = 0), and so

T Y (T Y )

from which we obtain the quotient bundle,

CSN (M, Y ) = ( T Y )

/ TY which is called the conformal symplectic normal bundle of Y . Moreover, if N (M, Y )denotes the normal bundle of Y in M , then we have the decomposition

N (M, Y ) = T M |Y / T Y = T M |Y / Y Y / (T Y )

(TY )

/ T Y = RY T Y CSN (M, Y )


13/23


where RY is the Reeb vector-eld R of restricted to Y . If we further assume that Y is asphere, then RY T Y has a naturally trivialization. Hence, as pointed out in [19], anygiven trivialization of CSN (M, Y ) determines a framing on Y (that is, the trivializationof the normal bundle N (M, Y )), and the latter can be used to perform a surgery on M

along Y . Moreover, the resulting contact structure on the surgered manifold agrees withthat of M away from Y . Such an elementary surgery can be achieved also by attaching aWeinstein handle by making use of isotropic setups which we recall next.

A quintuple of the form ( P,,,M,Y ) is called an isotropic setup if (P, ) is a sym-plectic manifold, is a Liouville vector eld, M is a hypersurface transverse to (hencea contact manifold), and Y is an isotropic submanifold of M . The following propositionis the basic tool enabling us to attach Weinstein handles.

Proposition 5.1 ([19]). Let (P 1, 1, 1, M 1, Y 1), (P 2, 2, 2, M 2, Y 2) be two isotropic se-tups. Suppose that a given diffomorphism Y 1 Y 2 is covered by a symplectic bundleisomorphism

CSN (M 1, Y 1) CSN (M 2, Y 2).Then there exist neighborhoods U j of Y j in M j and an isomorphism of isotropic setups

: (U 1, 1|U 1 , 1 |U 1 , M 1 U 1, Y 1) (U 2, 2|U 2 , 2|U 2 , M 2 U 2, Y 2)

which restricts to the given map Y 1 Y 2, and induces the given bundle isomorphism.

5.2. Weinstein handles. Denote the coordinates on R 2n +2 = R 2(n +1) by

(x0, y0, x1, y1,...,x n , yn )

and consider the standard symplectic structure on R 2n +2 as

0 =n

j =0

dx j dy j .

We will focus on two special Weinstein handles that we need for the present paper.Namely, let H n and H n +1 be the (2n + 2)-dimensional Weinstein handles in R 2n +2 withindexes n and n + 1, respectively. These handles are dened as follow: Consider

n = x02

x 0

y02

y0

+n

j =1

2x j

x j+ y j

y j

, n +1 =n

j =0

2x j

x j+ y j

y j

which are the negative gradient vector elds of the Morse functions

f n =x204

+y204

+n

j =1

x2 j 12

y2 j , f n +1 =n

j =0

x2 j 12

y2 j

respectively. We have the contractions k = k 0, for k = n, n + 1, given as

n = x02

dy0 +y02

dx0 +n

j =1

( 2x j dy j y j dx j ) , n +1 =n

j =0

( 2x j dy j y j dx j )

from which we compute that L k 0 = d( k 0) = 0. Therefore, n , n +1 are bothLiouville vector elds of 0. Next, consider the unstable manifold

E k = {x0 = = xn = y0 = = yn k = 0 },


14/23


and the hypersurface X = f 1k ( 1) which is of contact type. The pull back of k on E kis zero, and so the descending sphere

S k 1 = E k X

is isotropic (Legendrian if k = n + 1) in the contact manifold ( X , k |X ). Similarly,we have the stable manifold E 2n +2 k+ = {yn k+1 = = yn = 0} and the hypersurfaceX + = f 1k (1) intersecting each other along the ascending sphere

S 2n +1 k = E 2n +2 k+ X +which is a submanifold of the contact manifold ( X + , k |X + ).

R 2 n +2 k

R k

X = f 1k ( 1)

X = f 1k ( 1)

S k 1

S k 1

ow of k

Figure 1. Weinstein handle H k (shaded) and the ow of k transverse to H k .

The Weinstein handle H k is the region bounded by a neighborhood (which can betaken arbitrarily small) of the descending sphere S k 1 in X together with a connectingmanifold S 2n +1 k D k depicted in Figure 1. It follows (see [19]) that we can choose in such a way that k is everywhere transverse to the boundary H k . Now we state themain theorem of [19] which tells us, in particular, when we can attach Weinstein handlesand how the symplectic structure extends over the handle.

Theorem 5.2 ([19]). Let Y be an isotropic sphere in the contact manifold M with a trivialization of CSN (M, Y ). Let M be the manifold obtained from M by elementary surgery along Y . Then the elementary cobordism P from M to M obtained by attaching


15/23


a Weinstein handle to M [0, 1] along a neighborhood of Y carries a symplectic structureand a Liouville vector eld which is transverse to M and M . The contact structureinduced on M is the given one, while that on M differs from that on M only on thespheres where the surgery takes place.

One important fact about gluing symplectic manifolds is not mentioned rigorouslybefore (at least in [ 19]). For our purposes it is convenient to state it as a lemma:

Lemma 5.3. Gluing two exact symplectic manifolds using an isomorphism of isotropicsetups results in an exact symplectic manifold.

Proof. Let (P 1, 1, 1) and ( P 2, 2, 2) be two exact symplectic manifold, and supposethat (as in Proposition 5.1) there exists an isomorphism of isotropic setups

: (U 1, 1|U 1 , 1 |U 1 , M 1 U 1, Y 1) (U 2, 2|U 2 , 2|U 2 , M 2 U 2, Y 2)

which restricts to a given map Y 1 Y 2, and induces a given bundle isomorphism

CSN (M 1, Y 1) CSN (M 2, Y 2).

Let P be the manifold obtained by gluing ( P 1, 1, 1) and (P 2, 2, 2) using the isomor-phism . This exactly means that along the gluing region we are gluing is, is (and so is) together using . Therefore, on the gluing region either of ( 1, 1, 1) or (2, 2, 2)denes an exact symplectic structure. Observe that on P \ P 2 (resp. on P \ P 1) the triple(1, 1, 1), (resp. ( 2, 2, 2)) denes an exact symplectic structure. Hence, P equipswith the exact symplectic structure which we write as ( 1 2, 1 2, 1 2).

6. Convex Stabilizations

Our observation via isotropic setups and Weinstein handles is the fact that we canperform certain positive stabilizations, which will be called convex stabilizations, onexact Lefschetz brations. Convex stabilizations will be dened explicitly at the end of the section where a summary of results and some corollaries are also presented in thisnew terminology. The main theorem of this section is

Theorem 6.1. Any positive stabilization of an exact Lefschetz bration along a properly embedded Legendrian disk is also an exact Lefschetz bration.

Proof. Let ( ,E,,, ,X,h ) be a given exact Lefschetz bration. We have alreadychecked in the proof of Theorem 3.3 that a positive stabilization S LF [( ,E,X,h ); L] of the underlying Lefschetz bration ( ,E,X,h ) is an another Lefschetz bration which wedenoted by ( , E , X , h ). So all we need to check is that the exact symplectic structure(,, ) extends over the handles H , H which we used to construct ( , E , X , h ) sothat we get an exact symplectic structure ( , , ) on E .

At this point one should ask why the Legendrian disk L given on a page X of theboundary exact open book (which carries = Ker( | E ) by Theorem 4.7) is also La-grangian on the page ( X,d ) (so that S LF [( ,E,X,h ); L] makes sense). We can checkthis as follows: From the basic equality

d(u, v) = Lu (v) L v(u) + ([u, v])


16/23


we immediately see that d(u, v) = 0 for all u, v T L (see Chapter III in [ 2] for a discus-sion on integrable submanifolds of contact structures). This shows that L is Lagrangianon the page ( X,d ).

Consider the 2 n-dimensional Weinstein handle H (of index n) used in Denition 3.1

and in the proof of Theorem 3.3. Taking the coordinates on R2n

H as (x1, y1,...,x n , yn ),we can symplectically embed H into H n by the map(x1, y1,...,x n , yn ) (0, 0, x1, y1,...,x n , yn ).

Indeed, we can trivially ber H n over D 2 with bers diffeomorphic to H by constructingit in a different way as follows: Our new model for H n will be H D 2. Consider thestandard symplectic form H = n j =1 dx j dy j on H whose Liouville vector eld H , thecorresponding Morse function f H and the contraction H = H H are

H =n

j =1

2x j

x j+ y j

y j

, f H =n

j =1

x2 j y2 j2

, H =n

j =1

( 2x j dy j y j dx j ) .

Let (r, ) be the radial and the angle coordinates on D 2-factor in H D 2. If pr1 (resp. pr2) denotes the projection onto H -factor (resp. D 2-factor), then, similar to the proof of Lemma 4.4, the modication

0 := pr1(H ) + pr

2(rdr d)is a symplectic form on the total space H n = H D 2 of the bration pr2 : H n D 2, andindeed is equivalent to the standard symplectic form 0. Considering H and r/ 2 /ras vector elds in T (H D 2) = T H T D2, it is straightforward to check that

0 := H r/ 2 /ris the Liouville vector eld of 0 (satisfying L 0 0 = 0) which gives the contraction

0

:= 0 0

= H r2

/ 2 d.Note that H is transverse to h H n = H D 2 and r/ 2 /r is transverse to vH n =H S 1, and so 0 is everywhere transverse to H n . It follows that each ber

H z := pr 12 (z) H (z D2)

is an 2n-dimensional Weinstein handle (of index n) and equips with the exact symplecticform 0z := 0|H z with the primitive 0z := 0 |H z and whose Liouville vector eld 0z := 0 |H z is transverse to H z and satises 0z

0z = 0z .

As a result, we obtain a trivial (no singular bers) Lefschetz bration ( pr2, H n , H, id)over D 2. One should note that this is not an exact Lefschetz bration because neitherH n nor H z is convex, but it can be glued to an exact Lefschetz bration along the

convex part, which we will denote by CX H n , of its boundary to construct a new exactLefschetz bration as we will see below. To describe CX H n , we rst observe that theboundary of H is decomposed into its convex and concave parts as

H = CX H CV H where CX H S n 1 D n is the tubular neighborhood of descending sphere S n 1 in thehypersurface f 1H ( 1) from which H points outward, and CV H = H S n

1 D n isthe connecting manifold from which H points inward. Then we get the decomposition

H n = ( H D 2) (H S 1) = ( CX H D 2) ( CV H D 2) (H S 1)


17/23


from which we deduce that

CX H n = CX H D 2 and CV H n = ( CV H D 2) (H S 1).

An easy way to understand this decomposition is given schematically in Figure 2.

x0

y0

H

CV H D 2

CX H D 2

H

R 2 n

CV H

H H

H H H

CX H

D2

r/ 2 /r

H S 1

ow of 0 = H r/ 2 /r

Figure 2. A schematic picture of the convex and concave parts of H nand the ow of 0 = H r/ 2 /r in R 2n R 2

Lemma 6.2. The handle H can be replaced by the handle H n . Moreover, the exact symplectic structure (,, ) on E extends over the handle H n .

Proof. We will replace the handle H = H D 2 used in the proof of Theorem 3.3 with theWeinstein handle H n = H D 2. Here we do the replacement in such a way that the newber E z X over z D 2 is obtained from the ber X = E z by attaching the Weinsteinhandle H z along the Legendrian sphere S z := S n 1 E z which we consider as a copy of the boundary L of the Legendrian (and so Lagrangian) ball L of the stabilization. Moreprecisely, we proceed as follows:

As S z is Legendrian in (E z , z), its conformal symplectic normal bundle CSN (E z , S z)is zero (i.e., has rank zero). Similarly, the descending sphere S n 1z is Legendrian in( CX H z , 0z) and so CSN ( CX H z , S n

1z ) is also zero. Therefore, by Proposition 5.1, we

can nd neighborhoods U z of S n 1z in H z and V z of S z in E z and an isomorphism of isotropic setups

z : (U z , 0z |U z , 0z |U z , CX H z U z , S n 1

z ) (V z , z |V z , z |V z , E z V z , S z)

which restricts to the map f z : S n 1z S z given in stage (I) of Denition 3.1. (Here f zis the embedding of the attaching sphere of H z .) Now using Theorem 5.2 we attach each


18/23


H z to corresponding E z using the isomorphism z and obtain the new ber X equippedwith exact symplectic structure

(z , z , z) := ( z z 0z , z z 0z , z z

0z).

(Note that z z 0z = d( z z

0z) and we use Lemma 5.3 to obtain this structure.)Next, we x a copy of S z0 E z0 (with z0 intD 2) of the Legendrian sphere L

in a xed regular ber E z0 . Since z0 is a regular value of , we may assume that E z0is the binding of the (exact) open book induced by . Since this open book carriesthe contact structure = Ker( ) on E (which we know by Theorem 4.7), the binding(E z0 , z0 := Ker( z0 )) is a contact submanifold manifold of ( E, ), and so z0 is asubbundle of (|E z 0 , d |E z 0 ) and we have (see for instance [10])

T E |E z 0 = TE z0 (z0 )

where (z0 )

= CSN (E,E z0 ) is the symplectically orthogonal complement of z0 in(|E z

0, d |E z

0

). ((z0 )

is also called the conformal symplectic normal bundle of E z0 inE .) The latter equality implies that CSN (E,E z0 ) can be identied with the classicalnormal bundle N (E,E z0 ) of E z0 in E . But we know, by denition of open books,that the binding has a trivial normal bundle, so CSN (E,E z0 ) = E z0 D 2. Then fromthe inclusions S z0 E z0 E we have

CSN (E,S z0 ) = CSN (E,E z0 )|S z 0 CSN (E z0 , S z0 ) = S z0 D2.

(Recall CSN (E z0 , S z0 ) is zero as S z0 is Legendrian in E z0 .)

For S n 1, we follow not the same but similar lines: We x a copy S n 1z0 CX H z0 in axed ber H z0 (with z0 intD 2). The restriction of 0z0 onto

CX H z0 is a contact form

making CX

H z0 convex, and S n 1z0 is Legendrian in (

CX

H z0 , 0z0 | CX H z 0 ). So we have

CSN ( CX H z0 , S n 1

z0 ) = 0 .

Also ( CX H z0 , 0z0 | CX H z 0 ) is a contact submanifold manifold of ( CX H n , 0| CX H n ). Then

from the inclusions S n 1z0 CX H z0 CX H n we see that

CSN ( CX H n , S n 1z0 ) = CSN ( CX H n , CX H z0 )|S n 1z 0 CSN (

CX H z0 , S n 1

z0 ) = S n 1z0 D

2.

Now we will show that all the above individual attachments are indeed pieces of theattachment of the Weinstein handle H n to E along S z0 by nding an isotropic setup whichagrees with each individual ber-wise gluing. To this end, note that we have the map

f z0 : S n 1z0 S z0 given in stage (I) of Denition 3.1. Dene the map

: CSN ( CX H n , S n 1z0 ) = S n 1z0 D

2 S z0 D 2 = CSN (E,S z0 )

by the rule( p,z) = ( f z0 ( p), z).

Clearly, is a bundle map and covers f z0 , and so by Proposition 5.1, we can nd neigh-borhoods U of S n 1z0 in H n and V of S z0 in E and an isomorphism of isotropic setups

n : (U, 0|U , 0|U , CX H n U,S n 1z0 ) (V, |V , |V , E V, S z0 )


19/23


which restricts to f z0 and the bundle map . We may assume that CX H n U = CX H n ,that is, n attaches H n to E along the whole convex part CX H n = CX H D 2 of itsboundary. Now consider the boundaries

CX H n = CX H z0 D 2 and h E =zD 2

E z = E z0 D 2.

For each z D 2, by attaching H z to E z using z , we glue CX H z0 { z} CX H n withE z0 { z} h E and also we map S n 1z onto S z by f z . Therefore, attaching all H zs toE zs along zs denes a smooth map S n 1z0 D

2 S z0 D 2 which is identity on theD 2-factor and maps S n 1z0 onto S z0 via f z0 , and so it coincides with . Hence, we concludethat overall effect of attaching all H z s to E z s using zs on E is equivalent to attachingWeinstein handle H n to E using n .

By Lemma 5.3 we know that the resulting manifold E := E n H n has an exactsymplectic structure (, , ) obtained by gluing those on E and H n . In other words,

(, , ) = ( n 0, n 0, n 0).

Also, clearly, extends over H n and we get a Lefschetz bration : E D 2 with regularber X and monodromy h (original h which is trivially extended over H ). To check that(, , ) restricts to (z , z , z ) on each new regular ber E z X , we proceed as follows:For each z D 2, by taking U z (resp. V z) small enough, we can guarantee that the union

zD 2U z resp.

zD 2V z

lies in the collar neighborhood of CX H n (resp. h E ) where we have the local trivialitycondition (as described in the denition of exact symplectic bration). By using these localtrivialities, we combine all the exact symplectic structures ( z z 0z , z z 0z , z z 0z)together, and surely the resulting structure must be ( n 0, n 0, n 0) because(z , z , z )s (resp. (0z , 0z , 0z)s) patch together and give ( ,, ) (resp. ( 0, 0, 0)).This completes the proof of Lemma 6.2.

So far we have constructed an exact Lefschetz bration ( , E, , , , X , h) on

E := E n H n ,in other words, we extended ( ,, ) over the handle H by showing that H can bereplaced by H n . Next, we want to extend ( , , ) over H by showing that H can bereplaced by the Weinstein handle H n +1 .

Remark 6.3. Although Weinstein handles are attached along the convex part of their

boundaries (according to the convention of present paper which coincides with the onein [19]), we actually need to reverse the direction of the Liouville vector eld of theWeinstein handle when it is being attached to a convex boundary of a symplectic manifold.Otherwise it is impossible to match the Liouville directions of the symplectizations usedin the gluing. Since we have attached H n to E along the whole CX H n by matching 0 = H + r/ 2 /r with , we have to now consider

CV H n = ( CV H D 2) (H S 1)

as a subset of the convex part of the boundary E .


20/23


Lemma 6.4. The Lefschetz handle H can be replaced by the Weinstein handle H n +1 .

Proof. Recall that H is attach to E along the Lagrangian n-sphere S on a page (X , d |X )of the boundary exact open book ( X , h) carrying the contact structure = Ker( | E )

on E . Say S

E 0 ( X

) for some 0 S 1

= D2

. From its construction (given inDenition 3.1) and the notation introduced above, S is the union

S = L f 0 D

of the Lagrangian n-disk L (E 0 , 0 = d 0 ) and the Lagrangian core disk D ( D n )of the 2n-dimensional Weinstein handle ( H 0 , d 00 ). Note that H 0 H is the ber (over0 S 1) of the trivial bration H S 1. By assumption L is Legendrian in ( E, ). Onthe other hand, the contact form | E restricts to a contact form

:= ( | E )|H S 1 = ( 0 0)|H S 1 = 0 |H S 1 =

d2

H =d2

+n

j =1

(2x j dy j + y j dx j )

on a convex part H S 1 E . Observe that the core disk D H 0 = H { 0} is givenby the set

{x1 = x2 = = xn = 0 , = 0(constant) },and so clearly = 0 on D which means that D is Legendrian in (H S 1, ) ( E, | E ). Therefore, the n-sphere S is also Legendrian in ( E, | E ) which impliesthat CSN ( E, S ) = 0. Moreover, we also have CSN ( CX H n +1 , S n ) = 0 as S n is Leg-endrian in ( CX H n +1 , n +1 | CX H n +1 ) by denition. Then by Proposition 5.1, we can ndneighborhoods U of S n in H n +1 and V of S in E and an isomorphism of isotropic setups

n +1 : (U , 0|U , n +1 |U , CX H n +1 U , S n ) (V , |V , |V , E V , S )

which restricts to the embedding : S n

S determined by Denition 3.1. Now byTheorem 5.2 attaching H n +1 to E using n +1 results in an exact symplectic manifold

E := E n +1 H n +1equipped with the exact symplectic data

( , , ) = ( n +1 0, n +1 n +1 , n +1 n +1 ).

Again we may assume that CX H n +1 U = CX H n +1 , that is, n +1 attaches H n +1 toE along the whole convex part CX H n +1 of its boundary. Note that the step we justexplained replaces H with the Weinstein handle H n +1 . From the bundle isomorphisms

1 = T S = T S (T E/ )|S we see that the framings on the normal bundle N ( E, S ) which are used to attach H andH n +1 coincide, and so attaching H and H n +1 are topologically the same. Therefore, weknow by Theorem 2.8 that when we add H n +1 to (, E, , , , X , h), we can extend theunderlying topological Lefschetz bration (, E, X , h) over H n +1 and get the Lefschetzbration

S LF [( ,E,X,h ); L] = ( , E , X , (, ) h)where (, ) is the right-handed Dehn twist described in Denition 3.1. The proof of Lemma 6.4 is now complete.


21/23


To be able to say that we have constructed an exact Lefschetz bration

( , E , , , , X , (, ) h)on E , it remains to check that the exact symplectic structure ( , , ) restricts to an

exact symplectic structure on every new regular ber E

z . Note that this time we arenot changing the diffeomorphism type of the regular ber, that is E z E z X . The

Weinstein handle H n +1 is attached to E along the neighborhood

E V N (E 0 , S ) [0, 1]

of the attaching sphere S E 0 in E where we identify the interval [0 , 1] with a closedarc in S 1 \ { pt} such that 0 < 0 < 1. Consider the mapping torus

X [0, 1]/ (x, 0) (h(x), 1)

of the open book (X , h) on E and the inclusion

N (E 0 , S ) [0, 1]X [0, 1].

Observe that attaching H n +1 to E along the attaching region N (E 0 , S ) [0, 1] results ina new mapping torus

X [0, 1]/ (x, 0) (((, ) h)(x), 1)for the open book (X , (, ) h) on the new boundary ( E , := Ker( |E )) obtainedfrom the corresponding elementary (contact) surgery on ( E, ) along the Legendriansphere S . To get this new mapping torus, we are just gluing two copies of X equippedwith the exact symplectic structure ( X , dX , X ) using the symplectomorphism

(, ) h Symp(X , X ).Therefore, attaching H n +1 does not change the exact symplectic structures of regular

bers. But of course, it does change the structure of the Lefschetz bration: Relative to : E D 2, the new Lefschetz bration : E = E H n +1 D 2 has one more criticalpoint (and so one more singular ber) located at the origin in the Weinstein handle H n +1 .

We conclude that ( , , ) restricts to(z ,

z ,

z) = ( z , z , z)

on each regular ber E z E z X of . Hence, we have an exact Lefschetz bration

( , E , , , , X , (, ) h) as claimed. This nishes the proof of Theorem 6.1.

We have the following consequence of Theorem 6.1:

Corollary 6.5. Any positive stabilization of an exact open book along a properly embedded Legendrian disk is also an exact open book.

Proof. By denition if (X, h ) is an exact open book, then there exist an exact Lefschetzbration ( ,E,,, ,X,h ) which induces (X, h ) on the boundary. Let L be any properlyembedded Legendrian (and so Lagrangian) disk in ( X, ). Then by Theorem 3.3 we knowthat the stabilization S OB [(X, h ); L] is induced by S LF [( ,E,X,h ); L]. Then Theorem 6.1implies that there exists an exact Lefschetz bration ( , E , , , , X , h ) whose under-lying topological Lefschetz bration is S LF [( ,E,X,h ); L]. In particular, S OB [(X, h ); L]is induced by an exact Lefschetz bration ( , E , , , , X , h ), and hence, it is exactby denition.


22/23


After all, the following denitions make sense and t into the frame very well.

Denition 6.6. (i) A convex stabilization S CLF [( ,E,,, ,X,h ); L] of an exactLefschetz bration ( ,E,,, ,X,h ) is dened to be the positive stabilizationS LF [( ,E,X,h ); L] where L is a properly embedded Legendrian disk on X .

(ii) A convex stabilization S COB [(X, h ); L] of an exact open book (X, h ) is dened tobe the positive stabilization S OB [(X, h ); L] where L is a properly embedded Leg-endrian disk on X .

The theorem that we state next can be considered as the exact symplectic version of Theorem 3.3. It summarizes some of the results that we have shown in the language of convex stabilizations. The proof is a straight forward combination of previous statementsand denitions, and so will be omitted.

Theorem 6.7. S CLF [( ,E,,, ,X,h ); L] induces the (exact) open book S COB [(X, h ); L].Conversely, if an (exact) open book (X, h ) is induced by (,E,,, ,X,h ), then any convex stabilization S COB [(X, h ); L] of (X, h ) is induced by S CLF [( ,E,,, ,X,h ); L].

Combining the results we get so far, we know that a convex stabilization of an exactLefschetz bration produces an another exact Lefschetz bration on a manifold which hasthe same diffeomorphism type with the original one. One can see that these manifoldsare, indeed, symplectomorphic:

Theorem 6.8. Let (E , , ) be the total space of S CLF [( ,E,,, ,X,h ); L], Then (E , , ) is symplectomorphic to (E ,, ) In other words, the pair

{H n , H n +1 }used in the construction of S CLF [( ,E,,, ,X,h ); L] is a symplectically canceling pair.

Proof. We have already observed in the proof of Theorem 3.3 that {H n , H n +1 } is a can-celing pair in smooth category (as the belt sphere of H n intersects the attaching sphereof H n +1 transversely once). Moreover, Lemma 3.6b in [ 6] (see also Lemma 3.9 in [18])implies that two Weinstein handles form a symplectically canceling pair if they form acanceling pair in smooth category and their Morse-index difference is one. As a result,we conclude that {H n , H n +1 } is a canceling pair in symplectic category as well.

As an immediate corollary, we have

Corollary 6.9. Let (resp. ) be the contact structure on E (resp. E ) induced by the exact symplectic structure of ( ,E,,, ,X,h ) (resp. ( , E , , , , X , h ) =S C

LF [( ,E,,, ,X,h ); L]). Then (E, ) is contactomorhic (E , ).

Finally, as an application, we verify a well-known result for the class of exact openbooks and their convex stabilizations. Namely,

Corollary 6.10. Let be a contact structure carried by an exact open book (X, h ). Then any convex stabilization S COB [(X, h ); L] of (X, h ) carries .

Proof. By assumption, there exist an exact Lefschetz bration ( ,E,,, ,X,h ) whichinduces (X, h ) on the boundary. Note that, by Theorem 4.7, (,, ) induces onE . Theorem 6.7 implies that S COB [(X, h ); L] is induced by S CLF [( ,E,,, ,X,h ); L].


23/23


Moreover, again by Theorem 4.7, we know that S COB [(X, h ); L] carries the contact structureinduced by the exact symplectic structure on S CLF [( ,E,,, ,X,h ); L]. Now the proof follows from Corollary 6.9.

References[1] A. Andreotti and T. Frankel, The second Lefschetz theorem on hyperplane section , Global Analysis,

Princeton University Press (1969), 1-20.[2] D. Blair, Contact Manifolds in Riemannian Geometry , Lecture Notes in Mathematics 509, Springer-

Verlag, 1976.[3] E. Brieskorn, Beispiele zur Differentialtopologie von Singularit aten , Invent. Math. 2 (1966), 1-14.[4] K. Cieliebak and Y. Eliashberg Symplectic geometry of Stein manifolds (Incomplete draft) (2008)[5] Deligne and Katz, Groupes de Monodromie en Geometrie Algebrique , Lecture Notes in Math. 340,

Springer-Verlag.[6] Y. Eliashberg, Symplectic geometry of plurisubharmonic functions , Gauge theory and symplectic

geometry (Montreal, PQ, 1995), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 488, KluwerAcad. Publ., Dordrecht, 1997, With notes by Miguel Abreu, pp. 4967. MR 1461569 (98g:58055)

[7] Y. Eliashberg and M. Gromov, Convex Symplectic Manifolds , Proceedings of Symposia in PureMathematics, vol. 52, Part 2, 135162 (1991).[8] Deligne and Katz, Groupes de Monodromie en Geometrie Algebrique , Lecture Notes in Math. 340,

Springer-Verlag.[9] A. Dimca Singularities and Topology of Hypersurfaces , Springer-Verlag, 1992.[10] H. Geiges, Contact geometry , Handbook of differential geometry. Vol. II, 315382, Elsevier/North-

Holland, Amsterdam, 2006.[11] E. Giroux, Geometrie de contact: de la dimension trois vers les dimensions superieures , Proceedings

of the International Congress of Mathematicians, Vol. II (Beijing), Higher Ed. Press, (2002), pp.405414. MR 2004c:53144

[12] A. Kas, On the handlebody decomposition associated to a Lefschetz bration , Pasic Jornal of Math.(1980) vol. 89, No. 1, 89-104

[13] T. Le D ung, Some remarks on the relative monodromy , Real and complex singularities (Oslo 1976),

Sijthoff and Noordhoff, Amsterdam, 1977, pp. 397-403.[14] R. Lutz, C. Meckert Structures de contact sur certaines spheres exotiques , C. R. Acad. Sci. Paris

Ser. A-B 282 (1976), no. 11, Aii, A591A593[15] J. Milnor, Singular Points of Complex Hypersurfaces , Princeton Univ. Press, 1968.[16] P. Seidel, A long exact sequence for symplectic Floer cohomology , Topology 42 (2003), p. 1003-1063.

(arXiv:math/0105186 v2)[17] I. Torisu, Convex contact structures and bered links in 3-manifolds , Internat. Math. Res. Notices

(2000), no. 9, 441454. MR MR1756943 (2001i:57039)[18] O. Van Koert, Lecture notes on stabilization of contact open books , (arXiv:1012.4359 v1)[19] A. Weinstein, Contact surgery and symplectic handlebodies , Hokkaido Math. J. 20 (1991), no. 2,

241251.

Department of Mathematics, Michigan State University, Lansing MI, USA

E-mail address : [email protected]

Max Planck Institute for Mathematics, Bonn, GERMANYE-mail address : [email protected]
http://arxiv.org/abs/math/0105186http://arxiv.org/abs/1012.4359http://arxiv.org/abs/1012.4359http://arxiv.org/abs/math/0105186

Selman Akbulut and M. Firat Arikan- Stabilizations via Lefschetz Fibrations and Exact Open Books

Documents

Transcript of Selman Akbulut and M. Firat Arikan- Stabilizations via Lefschetz Fibrations and Exact Open Books