数理解析研究所講究録
第 1764 巻 2011 年 139-148

139( A certain surgery construction of contact 3-manifolds )( Jiro ADACHI
Department of Mathematics,
Hokkaido University

1Introduction

The construction and classification of contact manifolds is still a basic
problem in differential topology. It was shown by Weinstein [W2] andEliashberg [E] that the manifold obtained from a contact manifold bya certain handle surgery carries a contact structure. In order to definesurgery of contact manifolds, they use symplectic and Stein handles. In
this note, a new construction method, contact round surgery, is introduced. In other words, symplectic round handle is introduced. Round
handle itself has some geometric meanings. Therefore, it may have somemeaning in contact topology to consider round surgery of contact manifolds. An attempt to apply round handle theory to contact topology isin [Ad2].
In this note, round surgery of contact 3-manifolds is introduced. In
other words, 4-dimensional symplectic round handles are constructed.
The cases of general dimensions are discussed in [Ad3]. The situation israther complicated in that case.
We should remark that a 4-dimensional symplectic round handle appeared in [Ga]. That symplectic round handle is different from the roundhandle defined in this paper. Although Gay’s symplectic round handle
is not exact, the symplectic round handle in this paper is exact. Because

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there is a global Liouville vector field on the round handle in this paper.
Unfortunately, no application of symplectic round handle is given in [Ga].
These days, Weinstein‘s contact surgery is one of important tools incontact topology. In terms of contact Dehn surgery, Weinstein contact
surgery along a Legendrian knot is a (-l)-surgery. Attaching the symplectic handle on the concave end of the trivial cobordism corresponds to( l)-surgery. Ding and Geiges [DGe] proved that every closed orientablecontact manifold is constructed by these two operations. And now, theseare good tools to deal with Heegaard Floer homology (see [OS]). Thenthe contact round surgery may also be a good tool in contact topology.
The result is described as follows.
$s$

$+$

Theorem A. Let $M$ be a convex contact type subset of the boundarywith respect to a Liouvilleof a 4-dimensional symplectic manifold
vector field $X$ defied $nearM\subset W$ . Let $L\subset M$ be an Legendnan link withtwo components with respect to the contact structure induced on $M$ from$X$ and. Then the Liouville vector field $X$ and the symplectic structure
extend to the manifold obtained from $W$ by attaching a round handle
and the modified boundary is also convex.
of index along$(W, \omega)$$\omega$$\omega$$k$$\tilde{S}_{f}^{k}$

We should remark that the framing of the attachment is restricted to theso called contact framing of Legendrian knots. As a corollary, we obtainthe following.

Corollary B. Let $(M, \xi)$ be a contact 3-manifold, and $L\subset(M, \xi)a$
Legendrian link with two components. Then the manifold obtained from
$M$ by a round surgery alongwith respect to contact framing of has
a contact structure. If $(M, \xi)$ is strongly symplectically fillable, then theobtained contact manifold is also strongly symplectically fillable.
$L$

2$L$

Round handle and Round surgery

Round handle was introduced by Asimov [As] to study the Morse-Smaleflow (see also [M]). As well as for such a study, round handle itself is a141
useful tool for the study of manifolds and some structures on manifolds.
Round handle and round handle decomposition are defined as follows.
Let $M$ be a manifold of dimension with boundary.
$\partial M\neq\emptyset$$n$Definition. A round handle of dimension
is defied as a pair$n$and index$k$attached to $M$$R_{k}=(D^{k}\cross D^{n-k-1}\cross S^{1}, f)$of a product of an $(n-1)$ -dimensional disk $D^{k}\cross D^{n-k-1}$ with a circle
and an attaching embedding $f:\partial_{-}(D^{k}\cross D^{n-k-1}\cross S^{1})arrow\partial M$ , whereis the attaching region.

$\partial_{-}(D^{k}\cross D^{n-k-1}\cross S^{1})$$:=\partial D^{k}\cross D^{n-k-1}\cross S^{1}$

Round handles are used to study flow manifolds. Flow manifold isdefined as follows. Let
be a pair of a manifold $M$ with a
specific unionof connected components of the boundary. Thepairis called a flow manifold if there exists a non-singularvector field on $M$ which looks inward on
and outward on. The following property of flow manifolds is proved by Asimov.
$(M, \partial_{-}M)$$\partial_{-}M$$\partial M$$(M, \partial_{-}M)$$\partial_{-}M$$\partial_{+}M:=$$\partial M\backslash \partial_{-}M$

Theorem 2.1 (Asimov, [As]). Let
be a compact flow manifold whose dimension is greater than 3. Then, $M$ has a round handle
decomposition.
$(M, \partial_{-}M)$

By using round handles in stead of ordinary handles, round surgery
is defined. In other words, a round surgery corresponds to attachinga round handle to a cobordism. Let $M$ be a manifold of dimension. A round surgery of index is defined as the operation removing anembedded int
from $M$ and regluingby the identity mapping of.
In this paper we construct certain symplectic structures on these roundhandles.
$n$

$k$

$(\partial D^{k}\cross D^{n-k}\cross S^{1})$$\partial D^{k}\cross\partial D^{n-k}\cross S^{1}$$D^{k}\cross\partial D^{n-k}\cross S^{1}$1423Symplectic round handle

3.1

Liouville vector fields and symplectizations

A contact manifold appears as a hypersurface in a symplectic manifold.
This relation is given by the so called Liouville vector field. A vector field$X$ on a symplectic manifold
is called a Liouville vector field if the. The
Lie derivative along it preserves the symplectic structure:
following property is well known (see [Wl] for example).
$(M,\omega)$$L_{X}\omega=\omega$

Lemma 3.1. Let $X$ be a Liouville vector field on a symplectic manifold. If $M\subset W$ is a hypersurface tmnsverse to $X$ , then the pullback
where $i:M\mapsto W$ is the inclusion
is a contact form on
mapping.
$(W, \omega)$$i^{*}(X_{\lrcorner}\omega)$$M_{\rangle}$

Such hypersurface $M\subset(W,\omega)$ transverse to $X$ is said to be of contact
type. When a contact type hypersurface is a boundary of a symplectic
manifold, it is said to be convex (resp. concave) if the Liouville vector
field looks outward (resp. inward) there.
From another point of view, the induced contact structure depends
more on the Liouville vector field than on the hypersurface. The following
lemma implies this property (see [W2]).
Lemma 3.2. Let $X$ be a Liouville vector field on a syrnplectic manifold. And let $M_{j}\subset W_{f}j=0,1$ , be hypersurfaces with the inclusion mappings $i_{j}:M_{i}\mapsto W$ . Assume that there exists a diffeomorphism$f:M_{0}arrow M_{1}$ following the integml curvesof X. Then is contactomor,
phic with respect to the induced contact structures$j=0,1$ .
$(W_{)}\omega)$$f$$\xi_{j}=ker(i_{j}^{*}(X_{\lrcorner}\omega))$

One of the most typical examples is the symplectization of a contact
manifold. In other words, any contact manifold with a contact form isrealized as a contact type hypersurface in some symplectic manifold. Let$(M, \xi)$ be a contact manifold with a contact form
on $M$ . The 2-form$:=d(e^{t}\alpha)=e^{t}(dt\wedge\alpha+d\alpha)$ is a symplectic structure on, where$\alpha$$\omega$$M\cross \mathbb{R}$143
is a coordinate of . The symplectic manifoldis called
the symplectization of. The Liouville vector field on
is, which is transverse to, and the induced
contact on $M=M\cross\{0\}$ is . This implies thatis realized as acontact type hypersurface in the symplectization.
On the other hand, a tubular neighborhood of any contact type hypersurface is symplectomorphic to the tubular neighborhood of the hypersurface in its symplectization (see [Adl], [Ge] for example).
$t$$\mathbb{R}$$(M\cross \mathbb{R}, \omega)$$(M, \alpha)$$(M\cross \mathbb{R}, \omega)$$X=\partial/\partial t$$M=M\cross\{0\}\subset M\cross \mathbb{R}$$(M, \alpha)$$\alpha$$(M\cross \mathbb{R}, \omega=d(e^{t}\alpha))$Lemma 3.3. Let $M$ be a compact contact type hypersurface in a symplectic manifoldwith a Liouville vector field X. Let denote thecontact form
induced on $M$ , where $i:M\mapsto W$ is the inclusion.
Then there exists a local symplectomorphism between neighborhoods of$M=M\cross\{0\}\subset(M\cross \mathbb{R}, d(e^{t}\alpha))$ and $M\subset(W, \omega)$which maps
to$X$ , and $M$
to $M$ identically.
$(W, \omega)$$\alpha$$i^{*}(X_{\lrcorner}\omega)$$\partial/\partial t$

This lemma is important in the attaching procedure of symplectic roundhandles in Subsection 3.3.

3.2

The model symplectic round handle

The model round handle is taken as a subset inwith some
symplectic structure whose attaching region is of concave contact type
and the belt region is of convex contact type. Note that like Weinstein‘s$2n$ -dimensional symplectic
handle is defined for index $k=1,2,$(see [W2]), the 4-dimensional symplectic round handle is defined only forindex $k=1$ .
First of all, we need the following symplectic structure and Liouville
vector field on. The standard symplectic structureon
is given as$\mathbb{R}^{3}\cross S^{1}$$\ldots,$$\mathbb{R}^{3}\cross S^{1}$$n$$\mathbb{R}^{3}\cross S^{1}$$\omega_{0}$$\omega_{0}:=(dp\wedge dq)+dz\wedge d\phi$where $(p, q, z, \phi)$ are coordinates of,$\mathbb{R}^{3}\cross S^{1}=\mathbb{R}^{2}\cross \mathbb{R}\cross S^{1}$. Set a vector

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field

$X_{1}$

on$\mathbb{R}^{2n-1}\cross S^{1}$$X_{1}$

as(3.1)$:=(-q \frac{\partial}{\partial q}+2p\frac{\partial}{\partial p})+z\frac{\partial}{\partial z}$(see the dotted curves and arrows in Figure 1). It is the Liouville vector

field for the symplectic structure . Note that the vector field
gradient vector field of the function$\omega_{0}$$f_{1}(p, q, z, \phi)$$X_{k}$

is the(3.2)$:=(- \frac{1}{2}q^{2}+p^{2})+\frac{1}{2}z^{2}$

with respect to the standard Euclidean metric. It is a Morse-like function
whose critical loci are not isolated points but circles.
Now, the model symplectic round handle is defined as follows. In
above andorder to define it, we use two functions$f_{1}$$g_{1}(p, q, z, \phi)$

on

(3.3)

$:=-Aq^{2}+B(p^{2}+z^{2})$, where $A,$ $B$ are arbitrary positive constants. The 4-dimensionalmodel symplectic round handle is defined as a domain in the symplecticspace
above bounded by the following two hypersurfaces:$\mathbb{R}^{3}\cross S^{1}$$(\mathbb{R}^{3}\cross S^{1}, \omega_{0})$$W_{-}:=\{x=(p, q, z, \phi)\in \mathbb{R}^{3}\cross S^{1}|f_{1}(x)=-1\}$$V_{c}:=\{x=(p, q, z, \phi)\in \mathbb{R}^{3}\cross S^{1}|g_{1}(x)=c\},$,$c>0$ .

These hypersurfaces do intersect and bound a domain if $B/A$ is sufficiently large.
Definition. The 4-dimensional model symplectic round handle
dex 1 is defined as$R_{1}^{0}:=\{x=(p, q, z, \phi)\in \mathbb{R}^{3}\cross S^{1}|f_{1}(x)\geq-1, g_{1}(x)\leq c\}$with the symplectic structure$\omega_{0}$$R_{1}^{0}$of in(3.4)(see Figure 1).

.
is homeomorphic toThis
The model symplectic round handle has the following properties.
$R_{k}^{0}$$D^{k}\cross D^{n-k-1}\cross S^{1}$

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a$\cross$$t$

Figure 1: Model round handle

Lemma 3.4. Let$(R_{1}^{0}, \omega_{0})$(1) The attaching region,field . At
$X_{1}$$\partial_{-}R_{1}^{0}$(2) The belt region. At$X_{1}$

be a model symplectic round handle.
is transverse to the Liouville vector
looks inward.

$\partial_{-}R_{1}^{0}\subset R_{1}^{0}$$X_{1}$

is transverse to the Liouville vector field
looks outward.

$\partial_{+}R_{1}^{0}\subset R_{1}^{0}$$\partial_{+}R_{1)}^{0}X_{1}$(3) The attaching coreis a Legendrian link with twocomponents in the contact manifold.
$\tilde{S}_{0}^{1}\cong S^{0}\cross S^{1}$$\partial_{-}R_{1}^{0}\subset W_{-}$(4)can be taken so that its attaching region is contained in an arbitmry small neighborhood ofin.

$R_{1}^{0}$$\tilde{S}_{0}^{1}$

3.3$W_{-}$

Attaching the model symplectic round handles

We attach the model symplectic round handle defined in the previous
subsection to a symplectic manifold in this subsection. The attachment
implies Theorem A.
We attach the model symplectic round handle
to the following
setup. Let
be a symplectic manifold with boundary, andconvex components with respect to a Liouville vector field $X$ defined near$M\subset W$ . Let
denote a contact form
on $M$ , where $i:M\mapsto W$
is the inclusion mapping, and the contact structure. We are$(R_{1}^{0}, \omega_{0})$$(W, \omega)$$M\subset\partial W$$i^{*}(X_{\lrcorner}\omega)$$\alpha$$\xi$$ker\alpha$146
along an Legendrian link $L\subset(M, \xi)$ with twogoing to attachcomponents.
as follows. It isWe attach the model symplectic round handlewell known that Legendrian knots have a unique standard contact tubularneighborhood. Therefore, there exists a local strict contactomorphism$\alpha_{1})arrow(U(L, M), \alpha)$ between suitable neighborhoods of$(R_{1}^{0},\omega_{0})$$(R_{1}^{0},\omega_{0})$$\varphi:(U(\tilde{S}_{0}^{1}, W_{-}),$. We may
and $L\subset M$ which satisfies
can be taken arbitrarily
sincesuppose
from the construction of the model symplectic round handle
close to(see Lemma 3.4). Then, from Lemma 3.3, the contactomorphism extends
to a symplectomorphism of neighborhoods. By this symplectomorphism,are glued symplectically.
andtwo symplectic manifolds$\varphi(\tilde{S}_{0}^{k})=L$$\tilde{S}_{0}^{k}\subset\partial_{-}R_{k}^{0}\subset W_{-}$$\partial_{-}R_{k}^{0}\subset U(\tilde{S}_{0}^{k}, W_{-})\subset W_{-}$$\partial_{-}R_{k}^{0}$$\tilde{S}_{0}^{k}$$(R_{k}^{0}, \omega_{0})$3.4$(W, \omega)$

Contact round surgery

Now, we define the contact round surgery as follows. Then we obtain
Corollary almost directly. Let $(M, \xi=ker\alpha)$ be a contact 3-manifold,
and $L\subset(M_{i}\xi)$ a Legendrian link with two components. In general,round surgery of an $(2n-1)$ -dimensional manifold $M$ is defined by attaching a round handle of dimension $2n$ to $M\cross[0,1]$ (see Subsection 2 for
precise definition). Then we need a symplectic structure on $M\cross[0,1]$ .
We take $M\cross[0,1]$ as a subset of the symplectizationof the given contact manifold $(M, \xi)$ . The induced contact structure on$M\cross\{i\}$ isfor both $i=0,1$ . Regarding as a Legendrian link in$(M\cross\{1\}, \xi)$ , we can attach a symplectic round handle of index 1 alongby Theorem A. Since the modified end is also convex from Theorem ,a contact structure is induced there. Thus we obtained a new contact
manifold. We call this operation a contact round surgery (see Figure 2).
$B$$(M\cross \mathbb{R},$ $d(e^{t}\alpha)$

$\xi$

$L$

$L$

$A$

Like Weinstein’s contact surgery, we can discuss the strong symplectic fillability by the contact round surgery. A contact manifold $(M, \xi)$is said to be strongly symplectically fillable if it is a convex contact type

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Figure 2: Contact round surgery

boundary of a compact symplectic manifold
and the induced contact structure is also . The manifold obtained from $(M, \xi)$ by a contact
round surgery and $(M, \xi)$ itself have a symplectic cobordism
constructed by attaching a symplectic handle. Note that $(M, \xi)$ is a convex
boundary of
and a concave end of). Therefore, they are
glued symplectically along $(M, \xi)$ . Then we obtain a symplectic filling of
the surgered contact manifold.
$(W, \omega)$$\xi$$(\tilde{W},\tilde{\omega})$$(W, \omega)$$\tilde{W},\tilde{\omega}$

References[Adl]

J. Adachi, Liouville setup and contact cobordism, Hokkaido Math. J. 25 (1996),

637-650.
[Ad2]J. Adachi, Round surgery and contact structures, (preprint).

[Ad3]

J. Adachi, Contact round surgery and symplectic round handlebodies, (preprint).

[As]

D. Asimov, Round handles and non-singular Mors e-Smale fiows, Ann. of Math.
(2) 102 (1975), 41-54.

[DGe]

F. Ding, H. Geiges, A Legendrian surgery presentation of contact 3-manifolds,
Math. Proc. Cambridge Philos. Soc. 136 (2004), 583-598.

[E]

Ya. Eliashberg, Topological characterization of Stein manifolds of dimension$>2$ , Internat. J. Math. 1 (1990), 29-46.

[Ga]

D. Gay, Four-dimensional symplectic cobordisms containing three-handles,
Geom. Topol. 10 (2006), 1749-1759.

[Ge]

H. Geiges, An introduction to contact topology, Cambridge Studies in Advanced
Mathematics 109, Cambridge University Press, Cambridge, 2008.

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[M]

J. Morgan, Nonsingular Morse-Smale flows on 3-dimensional manifolds, Topology 18 (1979), 41-53.

[OS]

B. Ozbagci and A. Stipsicz, Surgery on contact 3-manifolds and Stein surfaces,
Bolyai Society Mathematical Studies, 13, Springer-Verlag, Berlin, 2004.

[Wl]

A. Weinstein, On the hypotheses of Rabinowitz’ pe riodic orbit theorems, J. Differential Equations 33 (1979), 353-358.

[W2]

A. Weinstein, Contact surgery and symplectic handlebodies, Hokkaido Math. J.
20 (1991), 241-251.

Department of Mathematics,Hokkaido University,
Sapporo, 060-0810, Japan.
e-mail: j-adachi@math.sci.hokudai.ac.jp