Genus zero PALF structures on the Akbulut-Yasui plugs

TL;DR

This paper constructs genus zero PALF structures on Akbulut-Yasui plugs, detailing their monodromies and analyzing the effects of plug twists on Stein surfaces, advancing understanding of 4-manifold topology.

Contribution

It introduces explicit genus zero PALF structures on Akbulut-Yasui plugs and describes their monodromies as positive factorizations, providing new tools for 4-manifold studies.

Findings

Constructed genus zero PALF structures on Akbulut-Yasui plugs

Described monodromies as positive factorizations in the mapping class group

Analyzed monodromies of PALFs on Stein surfaces related by plug twists

Abstract

We construct a genus zero PALF structure on each of plugs introduced by Akbulut and Yasui and describe the monodromy as a positive factorization in the mapping class group of a fiber. We also examine the monodromies of PALFs on a certain pair of compact Stein surfaces such that one is obtained by applying a plug twist to the other.

Equations

$$\left(\begin{array}[]{cc}-8&1\\ 1&-2\\ \end{array}\right)and\left(\begin{array}[]{cc}-8&-3\\ -3&-3\\ \end{array}\right)$$

Full text

1. Introduction

The problem of classifying all differential structures defined on a given 4-manifold is an important problem in understanding the overall picture of a 4-manifold.

Akbulut and Yasui [4] introduced corks and plugs. Corks and plugs are compact Stein surfaces. Matveyev, Curtis-Freedman-Hsiang-Stong, and Akbulut-Matveyev’s theorem show that the study of exotic manifold pairs constructed using cork is important for the classification problem of the differential structure of 4-manifolds.

Theorem 1.1**.**

(Matveyev [12], Curtis-Freedman-Hsiang-Stong [7], Akbulut-Matveyev [2]) For every homeomorphic but non-diffeomorphic pair of simply connected closed 4-manifolds, one is obtained from the other by removing a contractible 4-manifold and gluing it via an involution on the boundary. Such a contractible 4-manifold has since been called a Cork. Furthermore, corks and their complements can always be made compact Stein 4-manifolds.

It is shown by Akbulut and Yasui [6] using cork that an infinite number of exotic Stein surface pairs embedded in X exist for any four-dimensional two-handle body X with $b_{2}(X)\geq 1$.

The plug generalizes the Gluck twist. The plug is also used to make exotic manifolds, as well as cork.

On the other hand, Loi and Piergallini [11] proved that every compact Stein surface admits a positive allowable Lefschetz fibration over $D^{2}$ (a PALF for short). Therefore we can investigate compact Stein surfaces in terms of positive factorizations in mapping class groups (see also Akbulut and Ozbagci [3], Akbulut and Arikan [1]).

Since corks and plugs are Stein surfaces, the study of the relationship between Stein surfaces and mapping class groups using PALFs plays an important role in classifying differential structures on 4-manifolds.

If a PALF is created from a Stein surface by the existing method ( [11], [3], [1] ), its genus will be large, and it will be complicated and difficult to handle as a mapping class group element.

Gompf [8] indicates that the Stein surface is compatible with Kirby calculus. In this paper, we use Kirby calculus to make the best PALF with genus zero from the Akbulut-Yasui plug.

One planar (i.e. genus zero) PALF on the Akbulut cork was made in the previous paper of the author [14], but in this paper, we made an infinite number of planar PALFs on the Akbulut-Yasui plugs. Being planar is also playing an important role in [10].

In this paper, we first construct a genus zero PALF structure on each of plugs introduced by Akbulut and Yasui [4] and describe the monodromy as a positive factorization in the mapping class group of a fiber.

Theorem 1.2**.**

For any $m\geq 1,n\geq 2$, the Akbulut-Yasui plug $(W_{m,n},f_{m,n})$ admits a genus zero PALF structure. The monodromy of the PALF is described by the factorization $t_{\alpha_{2n+m}}\cdots t_{\alpha_{1}}$, where $t_{\alpha}$ is a right-handed Dehn twist along a simple closed curve $\alpha$ on a fiber and $\alpha_{2n+m},\ldots,\alpha_{1}$ are simple closed curves shown in Figure 2.

Note that the genus of a PALF on the manifold $W_{m,n}$ in a known way (cf. [3] and [1]) is much more than zero. We obtained similar results for the Akbulut cork $W_{1}$ [14]. In the present paper, we construct a genus zero PALF on an infinite number of the Akbulut-Yasui plugs.

Furthermore, we show that example of two 4-manifolds $A$ and $B$ which is obtained from $A$ by plug twist of $A$. The manifolds $A$ and $B$ admit genus zero PALF structure, and have the following properties:

Theorem 1.3**.**

The manifolds $A$ and $B$ which are showed by the Kirby diagrams in the Figure 3 have the following properties.

1. (1)

$B$* is obtained by plug twist of $A$ along the Akbulut-Yasui plug $(W_{1,2},f_{1,2})$.* 2. (2)

$A,B$* admit genus zero PALF structure.* 3. (3)

The second Betti numbers of $A$ and $B$ are $2$, and the second homology groups of $A$ and $B$ are isomorphic. 4. (4)

The boundaries of $A$ and $B$ are diffeomorphic. 5. (5)

$A,B$* do not have isomorphic intersection numbers. Especially $A$ and $B$ are not homeomorphic.* 6. (6)

The monodromy representation of genus zero PALF structures which $W_{1,2}$ admits is $t_{\alpha_{4}}t_{\alpha_{3}}t_{\alpha_{2}}t_{\alpha_{1}}$, where $\alpha_{i}$ is a simple closed curve in diagram 4, and $t_{\alpha_{i}}$ is right-handed Dehn twist along $\alpha_{i}$. Monodromy representations of genus zero PALF structure of $A,B$ are $t_{\alpha_{4}}t_{\alpha_{3}}t_{\alpha_{2}}t_{\alpha_{1}}t_{\beta}$ and $t_{\alpha_{4}}t_{\alpha_{3}}t_{\alpha_{2}}t_{\alpha_{1}}t_{\gamma}$, where $\beta$ and $\gamma$ are simple closed curves in the diagrams 5 and 6.

2. Preliminaries

2.1. Mapping class groups

In this subsection, we review a precise definition of the mapping class groups of surfaces with boundary and that of Dehn twists along simple closed curves on surfaces.

Definition 2.1**.**

Let $F$ be a compact oriented connected surface with boundary. Let Diff${}^{+}(F,\partial F)$ be the group of all orientation-preserving self-diffeomorphisms of $F$ fixing the boundary $\partial F$ point-wise. Let Diff${}^{+}_{0}(F,\partial F)$ be the subgroup of Diff${}^{+}(F,\partial F)$ consisting of self-diffeomorphisms isotopic to the identity. The quotient group Diff${}^{+}(F,\partial F)/$ Diff${}^{+}_{0}(F,\partial F)$ is called the mapping class group of $F$ and it is denoted by Map$(F,\partial F)$.

Definition 2.2**.**

A positive $($or right-handed$)$ Dehn twist along a simple closed curve $\alpha$, $t_{\alpha}:F\rightarrow F$ is a diffeomorphism obtained by cutting $F$ along $\alpha$, twisting $360^{\circ}$ to the right and regluing.

2.2. PALF

Definition 2.3**.**

Let $M^{4}$ and $B^{2}$ be compact oriented smooth manifolds of dimensions $4$ and $2$. Let $f:M\rightarrow B$ be a smooth map. $f$ is called a positive Lefschetz fibration over $B$ if it satisfies the following conditions (1) and (2):

1. (1)

There are finitely many critical values $b_{1},\ldots,b_{m}$ of $f$ in the interior of $B$ and there is a unique critical point $p_{i}$ on each fiber $f^{-1}(b_{i})$, and 2. (2)

The map $f$ is locally written as $f(z_{1},z_{2})=z_{1}^{2}+z_{2}^{2}$ with respect to some local complex coordinates around $p_{i}$ and $b_{i}$ compatible with the orientations of $M$ and $B$.

Definition 2.4**.**

A positive Lefschetz fibration is called allowable if its all vanishing cycles are homologically non-trivial on the fiber. A positive allowable Lefschetz fibration over $D^{2}$ with bounded fibers is called a PALF for short.

The following Lemma is useful to prove Theorem 1.2.

Lemma 2.5** (cf. Akbulut-Ozbagci [3, Remark 1]).**

Suppose that a $4$-manifold $X$ admits a PALF. If a $4$-manifold $Y$ is obtained from $X$ by attaching a Lefschetz $2$-handle, then $Y$ also admits a PALF.

The Lefschetz $2$-handle is defined as follows.

Definition 2.6**.**

Suppose that $X$ admits a PALF. A Lefschetz $2$-handle is a $2$-handle attached along a homologically non-trivial simple closed curve in the boundary of $X$ with framing $-1$ relative to the product framing induced by the fiber structure.

2.3. Stein surfaces

In this subsection, we recall a definition of the Stein surfaces. The question of which smooth $4$-manifolds admit Stein structures can be completely reduced to a problem in handlebody theory.

Definition 2.7**.**

A complex manifold is called a Stein manifold if it admits a proper biholomorphic embedding to $\mathbb{C}^{n}$.

Definition 2.8**.**

Let $W$ be a compact manifold with boundary. The manifold $W$ is called a Stein domain if it satisfies following condition: There is a Stein manifold $X$ and a plurisubharmonic function $\varphi:X\rightarrow[0,\infty)$ such that $W=\varphi^{-1}([0,a])$ for a regular value $a$ of $\varphi$.

Definition 2.9**.**

A Stein manifold or a Stein domain is called a Stein surface if its complex dimension is $2$.

2.4. Plugs

In this subsection, we give the definition of the plug.

Definition 2.10**.**

(Akbulut-Yasui [4, Definition 2.2.]) Let $P$ be a compact Stein 4-manifold with boundary and $\tau:\partial P\rightarrow\partial P$ an involution on the boundary, which cannot extend to any self-homeomorphism of $P$. We call $(P,\tau)$ a Plug of $X$, if $P\subset X$ and $X$ keeps its homeomorphism type and changes its diffeomorphism type when removing $P$ and gluing it via $\tau$. We call $(P,\tau)$ a Plug if there exists a smooth 4-manifold $X$ such that $(P,\tau)$ is a plug of $X$.

Definition 2.11**.**

(Akbulut-Yasui [4, Definition 2.3.]) Let $W_{m,n}$ be a smooth 4-manifold given by Figure 1. Let $f_{m,n}:\partial W_{m,n}\rightarrow\partial W_{m,n}$ be the obvious involution obtained from first surgering $S^{1}\times D^{3}$ to $D^{2}\times S^{2}$ in the interiors of $W_{m,n}$, then surgering the other imbedded $D^{2}\times S^{2}$ back to $S^{1}\times D^{2}$ (i.e. replacing the dot in Figure 1).

Theorem 2.12**.**

(Akbulut-Yasui [4, Theorem 2.5(2)]) For $m\geq 1$ and $n\geq 2$, the pair $(W_{m,n},f_{m,n})$ is a plug.

3. Proofs of Theorems 1.2 and 1.3.

In this section, we give the proof of Theorem 1.2 and Theorem 1.3.

Proof of Theorem 1.2. Let $F_{m,n}$ be the compact oriented surface of genus zero with $2n+m$ boundary components and $\alpha_{1},\ldots,\alpha_{2n+m}$ the curves on $F_{m,n}$ shown in Figure 8 $(a)$. Note that Figure 2 and Figure 8 $(a)$ show the same PALF. We denote the right-handed Dehn twists along $\alpha_{1},\ldots,\alpha_{2n+m}$ by $t_{\alpha_{1}},\ldots,t_{\alpha_{2n+m}}$, respectively. Let $f:X_{m,n}\rightarrow D^{2}$ be a Lefschetz fibration over $D^{2}$ with monodromy representation $(t_{\alpha_{2n+m}},\ldots,t_{\alpha_{1}})$. Since each curve $\alpha_{i}$ is homologically non-trivial on $F_{m,n}$, we see that $f$ is a PALF with fiber $F_{m,n}$.

We now show that $X_{m,n}$ is diffeomorphic to $W_{m,n}$.

The Kirby diagram for $X_{m,n}$ corresponding to the monodromy representation $(t_{\alpha_{2n+m}},\ldots,t_{\alpha_{1}})$ is given by Figure 8 $(b)$. We slide the $-1$-framed $2$-handles over $-1$-framed $2$-handles and erase canceling $1$-handle/$2$-handle pairs to get Figure 8 $(c)$. We get Figure 8 $(d)$ by sliding the $-m$-framed $2$-handle over $-1$-framed $2$-handles and sliding the $-n$-framed $2$-handle over $-1$-framed $2$-handles and erasing canceling $1$-handle/$2$-handle pairs.

The Kirby diagram for $W_{m,n}$ is given by Figure 7 $(a)$. We slide the [math]-framed $2$-handle under the $1$-handle to get Figure 7 $(b)$.

Since Figure 7 $(b)$ and Figure 8 $(d)$ are the same, we conclude that $X_{m,n}$ is diffeomorphic to $W_{m,n}$, which implies the theorem. $\square$

*Proof of Theorem 1.3. * (1) The plug twist of $A$ along $(W_{1,2},f_{1,2})$ is represented by replacing the dot with 0 mutually by the definition of the Akbulut-Yasui plug. Therefore $B$ is obtained from $A$ by plug twisting along $(W_{1,2},f_{1,2})$.

(2) First, we transform the Kirby diagram of $A$ as in Figure 12. Figure 12 $(a)$ is the Kirby diagram of $A$. We slide the [math]-framed $2$-handle under the $1$-handle to get Figure 12 $(b)$. We get Figure 12 $(c)$ by creating canceling $1$-handle/$2$-handle pairs. We create canceling pairs to get Figure 12 $(d)$. Then we consider the $4$-manifold with a genus zero PALF structure as in Figure 5. The obvious Kirby diagram for this manifold is given by Figure 10. Therefore, the manifold $A$ admits a genus zero PALF structure.

Similarly, we transform the Kirby diagram of $B$ as in Figure 13. Figure 13 $(a)$ is the Kirby diagram of $B$. We slide the [math]-framed $2$-handle under the $1$-handle to get Figure 13 $(b)$. We get Figure 13 $(c)$ by creating canceling $1$-handle/$2$-handle pairs. We create canceling pairs to get Figure 13 $(d)$. Then we consider a $4$-manifold which admits a genus zero PALF structure as in Figure 6. The obvious Kirby diagram for this manifold is given by Figure 11. Therefore, the manifold $A$ admits a genus zero PALF structure.

(3)We give a handle decomposition of $A$ and $B$ with one [math]-handle, two $2$-handles as in Figure 3. Therefore, $H_{0}(A;\mathbb{Z})\cong H_{0}(B;\mathbb{Z})\cong\mathbb{Z}$, $H_{1}(A;\mathbb{Z})\cong H_{1}(B;\mathbb{Z})\cong\{0\}$, $H_{2}(A;\mathbb{Z})\cong H_{2}(B;\mathbb{Z})\cong\mathbb{Z}\oplus\mathbb{Z}$, and $H_{i}(A;\mathbb{Z})\cong H_{i}(B;\mathbb{Z})\cong\{0\}\ (i\geq 3)$. The second Betti numbers of $A$ and $B$ are equal to 2.

(4) By the Kirby diagrams of $A$ and $B$ (Figure 3), both of the boundaries of $A$ and $B$ are represented by integral surgery diagrams. Therefore the boundaries of $A$ and $B$ are diffeomorphic to each other.

(5) We transform the Kirby diagrams of the manifolds $A$ and $B$ by Kirby calculus in Figure 14 and Figure 15. We obtain the intersection matrices

[TABLE]

of $A$ and $B$ from the diagrams, respectively. The former is even and the latter is odd. Therefore $A$ and $B$ do not have isomorphic intersection form, especially $A$ and $B$ are not homeomorphic.

(6) The genus zero PALF structure on $W_{1,2}$ is obtained from a trivial surface bundle over $D^{2}$ by attaching Lefschetz $2$-handles along simple closed curves in Figure 4. The genus zero PALF structure $A$ (respectively $B$) is obtained from surface bundle over $D^{2}$ by attaching Lefschetz $2$-handles along simple closed curves in Figure 5 (respectively Figure 6). Therefore the monodromy representation of the PALF on $A$ (respectively $B$) is $t_{\alpha_{5}}t_{\alpha_{4}}t_{\alpha_{3}}t_{\alpha_{2}}t_{\alpha_{1}}t_{\beta}$ (respectively $t_{\alpha_{5}}t_{\alpha_{4}}t_{\alpha_{3}}t_{\alpha_{2}}t_{\alpha_{1}}t_{\gamma}$) (where $t_{\alpha_{i}}$ is right-handed Dehn twist along the simple closed curve $\alpha_{i}$). $\Box$

Remark 3.1**.**

Theorem 1.3 $(2)$ can be generalized in the same way for any $m\geq 1,n\geq 2,l\leq-2$ in Figure 16.

Acknowledgements**.**

The author would like to thank his adviser Hisaaki Endo for his helpful comments and his encouragement. The author wishes to thank Kouichi Yasui and Yuichi Yamada for their useful comments.