Compact Stein surfaces as branched covers with same branch sets

Let $\Sigma_{g,r}^{k}$ be an oriented, connected genus $g$ surface with $k$ marked points and $r$ boundary components. We denote the mapping class group of $\Sigma_{g,r}^{k}$ by $\mathcal{M}_{g,r}^{k}$. More precisely, $\mathcal{M}_{g,r}^{k}$ is the group of isotopy classes of orientation preserving self-diffeomorphisms of $\Sigma_{g,r}^{k}$ which fix the marked points setwise and the boundary pointwise. We also use the notations $\mathcal{M}_{g,r}$ if $k=0$, and $\mathcal{M}_{\Sigma_{g,r}^{k}}$ for $\mathcal{M}_{g,r}^{k}$. For a simple closed curve $\alpha$ in $\Sigma_{g,r}^{k}$, $t_{\alpha}\in\mathcal{M}_{g,r}^{k}$ denotes the right-handed Dehn twist along $\alpha$. Furthermore, for a simple arc $a$ connecting two distinct marked points in $\Sigma_{g,r}^{k}$, write $\tau_{a}\in\mathcal{M}_{g,r}^{k}$ for the right-handed half-twist along $a$. We will use the opposite notation to the usual functional one for the products in $\mathcal{M}_{g,r}^{k}$, i.e. $h_{1}h_{2}$ means that we apply $h_{1}$ first and then $h_{2}$. Moreover, for a subset $A\subset\Sigma_{g,r}^{k}$ and $h\in\mathcal{M}_{g,r}^{k}$, the notation $(A)h$ means the image of $A$ under $h$.

It is well known that the braid group $B_{m}$ on $m$ strands can be identified with the mapping class group $\mathcal{M}_{0,1}^{m}$ as follows (cf. [9, Section 3.2]): Consider an $m$-marked disk $\Sigma_{0,1}^{m}$ as the unit closed disk $\mathbb{D}_{m}\subset\mathbb{C}$ with $m$ marked points which lie on the real axis. Set $P_{1},P_{2},\cdots,P_{m}$ as the $m$ marked points, where $P_{1}<P_{2}<\cdots<P_{m}$. Define an arc $A_{i}$ on the real axis to be one with end points in $P_{i}$ and $P_{i+1}$. Then, the $i$-th standard generator $\sigma_{i}$ of $B_{m}$ can be identified with the right-handed half-twist $\tau_{A_{i}}$. In this article, under this identification, a simple arc with end points in the set of marked points represents the corresponding element of $B_{m}$ to the half-twist along the arc.

Let $D^{2}_{1}$ and $D^{2}_{2}$ be oriented $2$-disks.

We will review briefly braid monodromies of braided surfaces (see, for more details, [2, Section 3], [14, Chapter 16, 17], [26, §1, 2]). Before that, we recall a special basis for the fundamental group of a punctured disk. Let $Q$ be a set of $n$ points $x_{1},x_{2},\dots,x_{n}$ in the interior of an oriented $2$-disk $D^{2}$ with the standard orientation and let $x_{0}$ be a point in $\partial D^{2}$. Since the fundamental group $\pi_{1}(D^{2}-Q,x_{0})$ is a free group of rank $n$, we give a basis for this group as follows: Take a collection of oriented paths $s_{1},s_{2},\dots,s_{n}$ starting from $x_{0}$ to each $x_{i}$, respectively. Assume that $s_{i}$ and $s_{j}$, if $i\neq j$, are disjoint except $x_{0}$, and the arcs $s_{1},s_{2},\dots,s_{n}$ are indexed so that they appear in order as we move counterclockwise about $x_{0}$. By using the path $s_{i}$, connect $x_{0}$ to a small oriented disk around each $x_{i}$ with the same orientation of $D^{2}$. Then, we obtain an oriented loop $\gamma_{i}$ based at $x_{0}$, and $\gamma_{1},\gamma_{2},\dots,\gamma_{n}$ freely generate $\pi_{1}(D^{2}-Q,x_{0})$. The ordered $n$-tuple $(\gamma_{1},\gamma_{2},\dots,\gamma_{n})$ is called a Hurwitz system for $(Q,x_{0})$ (see Figure 1).

We now turn to the case of braided surfaces. Let $Q(p_{S}):=\{a_{1},a_{2},\dots,a_{n}\}\subset\rm{Int}\,D_{1}^{2}$ be the set of branch points of the branched covering $p_{S}:S\rightarrow D^{2}_{1}$. Fix a point $a_{0}$ in $\partial D^{2}_{1}$ and Hurwitz system $(\gamma_{1},\gamma_{2},\dots,\gamma_{n})$ for $(Q(p_{S}),a_{0})$. For each $\gamma_{i}$, the restriction of $pr_{1}$ to $pr_{1}^{-1}(\gamma_{i})$ induces a trivial disk bundle over $\gamma_{i}$. Since for any point $a\in\gamma_{i}\subset D_{1}^{2}-Q(p_{S})$, $p_{S}^{-1}(a)$ consists of $m$ points, each fiber $pr_{1}^{-1}(a)=\{a\}\times D_{2}^{2}=:D_{2}^{2}(a)$ of the disk bundle contains $m$ points, which are the intersection points of $D_{2}^{2}(a_{0})$ and $S$. Hence, we associate an element $\beta_{i}\in B_{m}$ to $\gamma_{i}$ as a motion of the set $D_{2}^{2}(a_{0})\cap S$ over $\gamma_{i}$. By this correspondence, we can define a homomorphism $\omega_{S}:\pi_{1}(D_{1}^{2}-Q(p_{S}))\rightarrow B_{m}$ by $\omega_{S}(\gamma_{i})=\beta_{i}$ for each $i$. This homomorphism $\omega_{S}$ is called a braid monodromy of $S$. The ordered $n$-tuple $(\omega_{S}(\gamma_{1}),\omega_{S}(\gamma_{2}),\dots,\omega_{S}(\gamma_{n}))$ is also called a braid monodromy of $S$. Since $p_{S}$ is a simple branched covering, each $\omega_{S}(\gamma_{i})$ is a conjugate $w_{j}^{-1}\sigma_{j_{i}}^{\varepsilon_{i}}w_{j}$ of $\sigma_{j_{i}}^{\varepsilon_{i}}$ for some $w_{j}\in B_{m}$ and $\varepsilon_{i}\in\{\pm 1\}$. It is known that, for a finite set $Q$ and representation $\omega:\pi_{1}(D^{2}_{1}-Q,a_{0})\rightarrow B_{m}$ as above, we can construct a braided surface of degree $m$ whose branch set is $Q$ and braid monodromy is $\omega$. Obviously, since $p_{S}$ is a branched covering, we consider a covering monodromy of $p_{S}$, i.e. a representation $\rho_{S}:\pi_{1}(D^{2}_{1}-Q(p_{S}),a_{0})\rightarrow\mathfrak{S}_{m}$, where $\mathfrak{S}_{m}$ is the symmetric permutation group of degree $m$. Note that each $\rho_{S}(\gamma_{i})\in\mathfrak{S}_{m}$ is a transposition because $p_{S}$ is simple. Furthermore, we also remark that $\omega_{S}$ is a lift of $\rho_{S}$ to $B_{m}$.

At the end of this subsection, we define a crucial notion to examine compact Stein surfaces by braided surfaces.

We will briefly review positive Lefschetz fibrations and their monodromies (see [12, Chapter 8]).

Let $X$ be an oriented, connected, compact $4$-manifold.

A positive Lefschetz fibration $f:X\rightarrow D^{2}$ can be described by the mapping class group $\mathcal{M}_{\Sigma}$ of the fiber $\Sigma$ of $f$. Let $a_{0}\in\partial D^{2}$ be a fixed base point. Take a Hurwitz system $(\gamma_{1},\gamma_{2},\dots,\gamma_{n})$ for $(Q(f),a_{0})$. We can consider a homomorphism $\eta_{f}:\pi_{1}(D^{2}-Q(f),a_{0})\rightarrow\mathcal{M}_{\Sigma}$ as follows: The positive Lefschetz fibration $f$ restricts to a fiber bundle $f|f^{-1}(\gamma_{i}):f^{-1}(\gamma_{i})\rightarrow\gamma_{i}$ for each $\gamma_{i}$. The monodromy of this fiber bundle is the right-handed Dehn twist $t_{\alpha_{i}}$ along a simple closed curve $\alpha_{i}$ in $f^{-1}(a_{0})\approx\Sigma$. The simple closed curve $\alpha_{i}$ is called a vanishing cycle of the singular fiber $f^{-1}(a_{i})$. Define $\eta_{f}:\pi_{1}(D^{2}-Q(f),a_{0})\rightarrow\mathcal{M}_{\Sigma}$ by $\eta_{f}(\gamma_{i})=t_{\alpha_{i}}$ for each $\gamma_{i}$ and call $\eta_{f}$ a monodromy of $f$. We also call the ordered $n$-tuple $(t_{\alpha_{1}},t_{\alpha_{2}},\dots,t_{\alpha_{n}})$ a monodromy of $f$. We say a positive Lefschetz fibration to be allowable if all of the vanishing cycles $\alpha_{1},\alpha_{2},\dots,\alpha_{n}$ are homologically non-trivial in the fiber. After this, we call a positive allowable Lefschetz fibration a PALF shortly.

The following theorem tells us that PALFs and positive braided surfaces are so important to study compact Stein surfaces.

Note that according to [19, Proposition 1, 2] and the proof of Theorem 2.4, for a given PALF $f:X\rightarrow D^{2}$, we can construct a simple branched covering $p:X\rightarrow D^{4}$ whose branch set is a positive braided surface $S$ so that $f=pr_{1}\circ p$ and $Q(p_{S})=Q(f)$. Conversely, for a given simple branched covering $p:X\rightarrow D^{4}$ whose branch set is a positive braided surface $S$, $f:=pr_{1}\circ p:X\rightarrow D^{2}_{1}$ is a PALF, and $Q(f)=Q(p_{S})$ (cf. Figure 2). Suppose $a\in D_{1}^{2}$ is a regular point of the above PALF $f=pr_{1}\circ p$. The point $a$ is also a regular point of $p_{S}$. Since $p$ is a simple branched covering branched along $S$, $p$ restricts to a simple branched covering $p|p^{-1}(D_{2}^{2}(a)):p^{-1}(D_{2}^{2}(a))\rightarrow D_{2}^{2}(a)$ whose branch set is $S\cap D_{2}^{2}(a)$. It is easy to check that $p^{-1}(D_{2}^{2}(a))$ is the regular fiber $f^{-1}(a)$ of $f$.

In order to show Theorem 1.1, we will discuss contact structures on the lens space $L(2N,1)$ via open books. Hence, we review contact structures on $L(2N,1)$ and their supporting open book decompositions (see [7], [24, Section 2] for more details).

To begin with, we recall a stabilization of a Legendrian knot. Let $L$ be a Legendrian knot in $(S^{3},\xi_{std})$. A positive (resp. negative) stabilization on $L$ is a Legendrian knot $L_{+}$ (resp. $L_{-}$) obtained from adding a zig-zag to $L$ as depicted in the left (resp. right) of Figure 3. If $L$ lies on a page of a supporting open book decomposition of $(S^{3},\xi_{std})$, we stabilize the open book and modify $L$ as shown in the bottom of Figure 3.

Let $(L(2N,1),\xi_{N,j})$ be a tight contact manifold obtained from the Legendrian surgery on the Legendrian knot $O_{N,j}$ shown in Figure 4. Since $O_{N,j}$ is a Legendrian knot with $tb=-2N+1$ and ${\rm{rot}}=2(N-j)$, $O_{N,1},O_{N,2},\dots,O_{N,N}$ are mutually not Legendrian isotopic by [6, THEOREM 1.1]. Thus, according to Honda’s classification of tight contact structures on $L(2N,1)$ (see [13, Theorem 2.1]), $\xi_{N,1},\xi_{N,2},\dots,\xi_{N,N}$ are mutually not isotopic tight contact structures. To obtain a supporting open book decomposition of $(L(2N,1),\xi_{N,j})$, we explain how $O_{N,j}$ is obtained from the Legendrian unknot $O$ with $tb=-1$. Repeat $j-1$ times positively and negatively stabilizing $O$ alternately and, after that, perform $2(N-j)$ times negatively stabilizing the resulting Legendrian knot. Hence, the corresponding open book decomposition of $(L(2N,1),\xi_{N,j})$ is given as in Figure 5. We write $(\Sigma_{0,2N},\varphi_{N,j})$ for this open book, where the monodromy $\varphi_{N,j}$ is given by