Weinstein trisections of trivial surface bundles.

A trisection, introduced in [3] has been studied by many authors. One of the kind of the field in trisection theory is a Weinstein trisection. Lambert-Cole reproved the Thom conjecture by using trisection of $\mathbb{C}P^{2}$ and inequality about an invariant induced by Khovanov homology. In the proof, he used a contact structure induced in the spine of trisection.

Recently, Lambert-Cole, Meier, and Starkston introduce a trisection adapted to a symplectic 4-manifold called a Weinstein trisection[7]. They also showed that every closed symplectic 4-manifold admits a Weinstein trisection. Weinstein trisection sometimes gives us a tool to study symplectic closed 4-manifolds and symplectic surfaces in them [5, 6].

There are some examples of Weinstein trisection. For example, Weinstein trisection of $\mathbb{C}P^{2}$ and $\mathbb{C}P^{2}\#\overline{\mathbb{C}P^{2}}$ is described in [7]. On the other hand, given a finite group $G$, there is a symplectic closed 4-manifold with fundamental group isomorphic $G$. This means that there are so many symplectic closed 4-manifolds. Therefore, it is very important to make an example of Weinstein trisection as the first step to studying symplectic closed 4-manifolds with Weinstein trisection. The main result of this paper is the following:

This implies that the trisection genus of $\Sigma_{g}\times\Sigma_{h}$ equals the Weinstein trisection genus of its.

This corollary immediately follows from the result in [10]. It states that the trisection genus of $\Sigma_{g}\times\Sigma_{h}$ is $(2g+1)(2h+1)+1$.

This paper is organized as follows: In Section 2, we review definitions of trisection and Weinstein trisection. After that, we introduce a trisection of $\Sigma_{g}\times\Sigma_{h}$ which is constructed in [10] in Section 3. Then, we show this trisection can be seen as a Weinstein trisection in Section 4.

In this section, we set up the notion and objects we use in this paper. Trisection of 4-manifolds is a decomposition of a four-manifold.

Symplectic 4-manifolds is a 4-manifold with non-degenerate closed 2-form $\omega$. We denote it by $(X,\omega)$. In the field of symplectic topology, symplectic manifolds are considered the same if they a symplectomorphic.

Since $\omega$ is a non-degenerate, we can obtain a volume form $\Omega=\omega\wedge\cdots\wedge\omega$ of $X$ after taking a wedge product n times. Hence in dimension two, the volume form of it is also a symplectic form so, volume-preserving diffeomorphism is the same as a symplectomorphism.

Lambert-Cole, Meier, and Starkston define a trisection of a symplectic closed 4-manifold adopted to the symplectic structure. The Weinstein domain, first introduced in [9], is a symplectic manifold with a contact boundary. More precisely, let $(X,\omega)$ be a compact symplectic manifold with boundaries. Then, $(X,\omega)$ is called a Weinstein domain if there exists a Morse function $f$ on $X$ and a gradient-like vector field $X_{f}$ of $f$ such that $X_{f}$ is Liouville vector field (i.e. $d(\iota_{X_{f}}\omega)=\omega$).

Lambert-Cole, Meier, and Starkston showed that any symplectic closed 4-manifold admits a Weinstein trisection by using a branched covering [7]. Weinstein domains induce a contact structure in their boundaries by a 1-form $\iota_{X_{f}}\omega$. Also, $H_{ij}$ has contact structures induced by a Weinstein structure of $X_{i}$ and $X_{j}$ [6].

And they give the following question.

In this paper, we answer this question for the $\Sigma_{g}\times\Sigma_{h}$. The known trisections of symplectic four-dimensional manifolds are known as Weinstein trisections. To positively affirm that there are no trisections that are not Weinstein trisections, we still have too few examples.

In this section we review the construction of a trisection of $\Sigma_{g}\times\Sigma_{h}$ in [10]. Let $X=\Sigma_{g}\times\Sigma_{h}$. First, we consider the decomposition of $\Sigma_{g}$. We can decompose any closed surface into three disks.

If $g=1$, this decomposition is illustrated in Figure 1. To obtain the case of a higher genus, remove a neighborhood of a vertex and glue a copy of it to itself along their boundary.

We take mutually disjoint disks $N_{i}$ in $\Sigma_{h}$ for $i=1,2,3$. Then we define the 1-handlebodies of a trisection as follows:

$$X_{i}=(B_{i}\times(\Sigma_{h}-Int(N_{i};\Sigma_{h}))\cup(B_{i+1}\times N_{i+1}).$$

To check $X_{i}$ is a 1-handlebody, we show that $(B_{i}\times(\Sigma_{h}-Int(N_{i};\Sigma_{h}))$ is a 1-handlebody. Since $(\Sigma_{h}-Int(N_{i};\Sigma_{h})$ is a punctured genus $h$ surface, this can be represented by a disk and $2h$ 1-handles attached to it. Since $B_{i}$ is a disk, $(B_{i}\times(\Sigma_{h}-Int(N_{i};\Sigma_{h}))$ is a 1-handlebody diffeomorphic to $\natural^{2h}S^{1}\times B_{3}$. The intersection $(B_{i}\times(\Sigma_{h}-Int(N_{i};\Sigma_{h}))$ and $(B_{i+1}\times N_{i+1})$ is a $(B_{i}\cap B_{i+1})\times N_{i}$. This is a disjoint $2g+1$ 3-balls in their boundaries. Since $(B_{i+1}\times N_{i+1})$ is a 4-ball, $X_{i}$ is a 1-handlebody of genus $2g+2h$.

In [10], Williams showed that $X=X_{1}\cup X_{2}\cup X_{2}$ is a trisection.

The direction of this paper involves examining the above trisection in more detail to demonstrate how it will become a Weinstein trisection. To show this, we consider the Weinstein structure of $\Sigma_{g}-int(B_{i})$ and $\Sigma_{h}-int(N_{i})$. This is constructed in Section 5. It is easy to show that each of $N_{i}$ is mutually ambient isotopic since they are disjoint disks. Also, we can show that each of $B_{i}$ is mutually ambient isotopic. Actually, in Figure 1, $B_{i}$ is sent to $B_{i+1}$ by ambient isotopy along the diagonal from bottom-left to top-right. We can extend this ambient isotopy to an arbitrary genus since we can construct a decomposition of $\Sigma_{g}=B_{1}\cup B_{2}\cup B_{2}$ by connecting summing a torus by taking the regular neighborhood of a vertex depicted in Figure 1. We consider this feature with a symplectic structure in the next subsection.

Let us consider a 1-handlebody

$$X_{i}=(\Sigma_{h}-int(N_{i}))\times B_{i}\cup N_{i+1}\times B_{i+1}$$

that constructs a trisection prescribed in Section 3. We show that each of $X_{i}$ admits a Weinstein structure with respect to a symplectic structure defined by a product structure of $\Sigma_{g}\times\Sigma_{h}$.

Let $g_{1}:\Sigma_{g}-B_{3}\rightarrow\mathbb{R}$ be a $J$-convex function such that $B_{1}$ contains only one critical point and its index is $0$. Then we define the symplectic structure $\omega_{g}$ on $\Sigma_{g}$ so that

$$\omega_{g}|_{\Sigma_{g}-B_{3}}=-dd^{\mathbb{C}}g_{1}.$$

By Lemma 3.5, there is a symplectomorphism $\varphi$ such that $\varphi(B_{i})=B_{i+1}$. Then we define the $J$-convex function $g_{2}$ and $g_{3}$ as follows:

$$g_{2}=g_{1}\circ\varphi^{-1}|_{\Sigma_{g}-B_{1}}:\Sigma_{g}-B_{1}\rightarrow\mathbb{R},$$

$$g_{3}=g_{2}\circ\varphi^{-1}i|_{\Sigma_{g}-B_{2}}:\Sigma_{g}-B_{2}\rightarrow\mathbb{R}.$$

Then we can define the Weinstein structures on $\Sigma_{g}-B_{i+2}$ for $i=1,2,3$ by Theorem 4.5.

Let $N_{1}$ be a sufficiently small disk in $\Sigma_{h}$ and $f_{1}:\Sigma_{h}-N_{1}\rightarrow\mathbb{R}$ a $J$-convex function. Also, we suppose that $N_{2}$ is a sufficiently small regular neighborhood of index $0$ critical point of $f_{1}$ and $N_{3}$ a disk in the interior of $\Sigma_{h}-(N_{1}\cup N_{2})$. Then we define the symplectic structure $\omega_{h}$ of $\Sigma_{h}$ so that

$$\omega_{h}|_{\Sigma_{h}-N_{1}}=-dd^{\mathbb{C}}f_{1}.$$

By Lemma 3.6, we can obtain the symplectomorphism $\varphi_{2}:(\Sigma,\omega_{h})\rightarrow(\Sigma_{h},\omega_{h})$ such that

$$\varphi(N_{1})=N_{2},\ \varphi(N_{2})=N_{3}.$$

Then, we obtain the $J$ convex function $f_{2}:\Sigma-N_{2}\rightarrow\mathbb{R}$ as follows:

$$f_{2}=f_{1}\circ\varphi_{2}^{-1}|_{\Sigma_{h}-N_{2}}:\Sigma_{h}-N_{2}\rightarrow\mathbb{R}.$$

Also, we can obtain a symplectomorphism $\varphi_{3}:(\Sigma,\omega_{h})\rightarrow(\Sigma_{h},\omega_{h})$ such that

$$\varphi(N_{2})=N_{3},\ \varphi(N_{3})=N_{1}.$$

by Lemma 3.6. Then, we can define the $J$-convex function $f_{3}:\Sigma-N_{3}\rightarrow\mathbb{R}$ as follows:

$$f_{3}=f_{2}\circ\varphi_{3}^{-1}|_{\Sigma_{h}-N_{3}}:\Sigma_{h}-N_{3}\rightarrow\mathbb{R}.$$

Finally, we can construct a Weinstein structure on $\Sigma_{h}-N_{i}$ for $i=1,2,3$ by Theorem 4.5.

In this section, we give an example of the Weinstein trisection of $S^{2}\times S^{2}$. It is a trivial $S^{2}$ bundle over $S^{2}$. We denote it $S_{1}\times S_{2}$. Also, $S_{2}$ has a decomposition described above. More precisely, we assume that $S_{2}$ is decomposed into three disks $B_{1}$, $B_{2}$, and $B_{3}$ as in Figure 1. We note that the Weinstein trisection of $S^{2}\times S^{2}$ is constructed in former articles [7]. But in this section, we will describe it slightly more explicitly.

We construct a Weinstein trisection of $S^{2}\times S^{2}$ as the following steps:

1. Step 1:

   For a given symplectic structure on $S_{1}-N_{i}$, we define a Morse function $f_{i}$ so that $N_{i+1}$ is a neighborhood of index $0$ critical point of $f_{i}$, it does not contain the other critical point and whose gradient flow is a Liouville vector field of the symplectic structure. Furthermore, each of $N_{i}$ is mutually symplectically isotopic to each other.

2. Step 2:

   For a given symplectic structure on $S_{2}-B_{i}$, we define a Morse function $g_{i}$ so that $B_{i}$ is a neighborhood of index $0$ critical point of $g$ and does not contain the other critical points of $g$, $B_{i+1}$ is a neighborhood of index $2$ critical point of $g$ and does not contain the other critical points of $g$ and whose gradient vector field is a Liouville vector field for the symplectic structure. Furthermore, each of $B_{i}$ is mutually symplectically isotopic to each other.

3. Step 3:

   $f+g$ is a Morse function on $X_{i}$ and a gradient vector field of it is a Liouville vector field for the products of the symplectic structure of $S_{1}$ and $S_{2}$ and $(X_{i},\omega,g+f,grad(g+f))$ will be a Weinstein domain.

To begin the steps above, we review the Kähler structure on $S^{2}$. First of all, we review the Fubini-study form of $\mathbb{C}P^{1}$.

Let $(z_{1},z_{2})$ be a homogenious coordinate of $\mathbb{C}P^{1}$. Then we can take a chart of $\mathbb{C}P^{1}$ by taking $\phi_{i}:\mathbb{C}P^{1}\to\mathbb{C}$ for $z_{j}\neq 0$

$$\phi_{i}(z_{1},z_{2})=\frac{z_{i}}{z_{j}}\ (i\neq j).$$

We denote this chart by $\{U_{i},\phi_{i}\}_{i=1,2}$. Then we define the Fubini-Study form by

$$\omega_{FS}=2\frac{\partial^{2}\log(1+|z|^{2})}{\partial z\partial\bar{z}}dz\wedge d\bar{z}$$

where $z=z_{1}/z_{2}$. It is known that this form gives us the Kähler form on $U_{1}$, and in this case, $f=\log(1+|z|^{2})$ gives a Kähler potential. The critical point of $f$ is only $z=0$. This means that $z_{1}=0$ since $z_{2}\neq 0$. Also, we note that this is a $J$-holomorphic function on $U_{1}$. Hence, $U_{1}$ is a Stein manifold with $J$-holomorphic function $f$.

Next, we review the identification between $S^{2}$ and $\mathbb{C}P^{1}$. We denote the polar coordinate of the unit sphere by the following (See Figure 2):

	$\displaystyle x=$	$\displaystyle\sin\theta_{1}\cos\theta_{2},$
	$\displaystyle y=$	$\displaystyle sin\theta_{1}\sin\theta_{2},$
	$\displaystyle z=$	$\displaystyle\cos\theta_{1}.$

Then we can represent a point on $S^{2}$ by $(x,y,z)\in S^{2}\subset\mathbb{R}^{3}$. We denote the stereographic projection $\phi_{N}:S^{2}\backslash p_{N}\to\mathbb{R}^{2}$ where $p_{N}=(0,0,1)$. Then we can define the diffeomorphism $\Phi:\mathbb{C}P^{1}\to S^{2}$ by

$$\Phi(z_{1},z_{2})=\begin{cases}{}\phi_{N}(z)&(z_{2}\neq 0)\\ p_{N}&(z_{2}=0)\end{cases}$$

where $z=z_{1}/z_{2}$. We sometimes use this diffeomorphism to define the function on $S^{2}$ below.

We identify $f\circ\Phi^{-1}$ and ${\Phi^{-1}}^{\ast}\omega_{FS}$ by $f$ and $\omega_{FS}$ respectively.

The author thanks Takahiro Oba for a very meaningful discussion and suggestion about research in symplectic topology, also giving comments on a draft of this paper.