Exotic families of symplectic manifolds with Milnor fibers of $ADE$-type

The constructions of exotic Weinstein manifolds, i.e., diffeomorphic manifolds having different Weinstein structures, have been studied extensively. See, for example, [McL09] or [AS10]. In this paper, we investigate Weinstein manifolds which are exotic to the Milnor fibers of simple singularities.

We construct different Weinstein manifolds by attaching the same Weinstein handles to the same Weinstein manifold in different ways. However, as smooth handles, they are attached in the same way. Thus, the construction gives diffeomorphic, but different Weinstein manifolds.

We note that the idea has been used historically. See [May09], [MS10], and [AS10]. The main difference between the previous works and the current paper is that, for some particular Weinstein manifolds, we will identify plumbings and end connected sums procedures as the same smooth handle attachments. However, they are different as Weinstein handle attachments. Thus, one can show that, for example, the Milnor fiber of $A_{k+1}$ singularity and the end connected sum of the Milnor fiber of $A_{k}$ singularity and a cotangent bundle of the sphere are diffeomorphic but different as Weinstein manifolds.

What allows us to distinguish a plumbing and an end connected sum as Weinstein manifolds, is the number of connected components of the wrapped Fukaya categories. Since the end connected sum procedure increases the number of connected components, but the plumbing procedure does not, two procedures lead to the different Weinstein manifolds. Thus, we can distinguish different Weinstein manifolds without using vanishing wrapped Fukaya categories or without using symplectic homologies. The first is used in [May09], [MS10], and the second is used in [McL09], [AS10].

The present paper contains two generalizations of the above construction of exotic pairs. More specific results will be stated in Section 1.2.

In Sections 3 – 5, we observe that a plumbing procedure and an end connected sum procedure can be realized as the same smooth handle attachments, but different Weinstein handle attachments.

We note that the Weinstein manifolds given in Theorem 1.1 will be denoted by $X^{2n+2}_{k}$ and $Y^{2n+2}_{k}$ in Sections 3 – 5.

After proving Theorem 1.1, we generalize it in two different ways. The first generalization is to consider the end connected sums of multiple Milnor fibers of $A$ type.

The second generalization is obtained by comparing Milnor fibers of different simple singularities. The detailed results are given in Theorems 1.3 and 1.4.

In Section 7, $Q_{m}^{2n}$ will be defined as an end connected sum of the Milnor fiber of $A_{m}$ singularity and $T^{*}S^{n}$.

The decomposability of wrapped Fukaya categories cannot prove Theorems 1.3 and 1.4 completely. Instead, we compare the symplectic cohomologies of the given Weinstein manifolds.

This paper consists of seven sections. The first two sections are the introduction and the preliminaries. In Sections 3 – 5, we prove Theorem 1.1. Section 6 (resp. 7) contains the first (resp. second) generalization.

The authors appreciate Youngjin Bae and Hanwool Bae for the helpful discussions. We also appreciate Yanki Lekili for his comments on the draft of the present paper.

The first (resp. last) named author has been supported by a KIAS individual grant (MG079401) (resp. the Institute for Basic Science (IBS-R003-D1)).

Our main tool in the current paper is the notion of Lefschetz fibration, which is defined as follows:

We note that it would be more precise to use the term ‘Liouville Lefschetz fibration’ in Definition 2.1 because there are Lefschetz fibrations of other types. However, in this paper, this is the only type which we considered. Thus, we omit the adjective for convenience.

Let $\pi:E\to\mathbb{C}$ be a Lefschetz fibration defined on a Weinstein manifold $E$. Then, it is well-known that $\pi$ induces a decomposition of $E$ into two parts, one is a subcritical part, and the other is a collection of Weinstein critical handles. See [Sei08, Lemma 16.9] or [BEE12, Section 8]. We note that the subcritical part is given as a product of the regular fiber of $\pi$ and $\mathbb{C}$. In order to attaching critical handles to the subcritical part, one needs a collection of Legendrian attaching spheres which one can obtain from

• •

  the cyclic order of the critical values of $\pi$, and

• •

  the vanishing cycles corresponding to the critical values of $\pi$.

Conversely, if one has such decomposition data of $E$, then, one can construct a Lefschetz fibration $\pi$ defined on $E$. We give a brief explanation after Definition 2.2.

Based on the converse direction, one has an alternative definition for Definition 2.1.

Definitions 2.1 and 2.2 are interchangeable. In the rest of Section 2.1, we explain the reason for that. For more details, we refer the reader to [Sei08] and [BEE12, Section 8].

Let $E=(F;L_{1},\cdots,L_{m})$ be a given abstract Weinstein Lefschetz fibration. Then, one can construct a Weinstein domain as follows: First, we consider the product of $F$ and $\mathbb{D}^{2}$. We remark that the product is not a Weinstein domain, because the product is a manifold with corner. To be more precise, we should consider a Weinstein manifold which is the product of symplectic completions of $F$ and $\mathbb{D}^{2}$. Then, we should consider a corresponding Weinstein domain, i.e., a Weinstein domain whose symplectic completion is the product Weinstein manifold. However, for convenience, we consider $F\times\mathbb{D}^{2}$.

The vertical boundary $F\times\partial\mathbb{D}^{2}$ has a natural contact structure. Moreover, the vanishing cycle $L_{i}$ can be lifted to a Legendrian $\Lambda_{i}$ near $2\pi i/m\in S^{1}$. We do not give the lifting procedure explicitly, but it is easily achieved by using the product structure on $F\times\mathbb{D}^{2}$. We note that by assuming that the disk $\mathbb{D}^{2}$ has a sufficiently large radius, one could assume that the projection images of $\Lambda_{i}$ onto the $\partial\mathbb{D}^{2}=S^{1}$ factor are disjoint to each other.

Finally, one could attach critical Weinstein handles along $\Lambda_{i}$ for all $i=1,\cdots,m$. Then, the completion of the resulting Weinstein domain admits a Lefschetz fibration satisfying that the regular fiber is $F$, and that there are exactly $m$ singular values.

By [GP17], it is known that every Weinstein manifold admits a Lefschetz fibration. Also, it is known that, for a Weinstein manifold, there are infinitely many different Lefschetz fibrations. Then, it would be natural to ask that if one has two different Lefschetz fibrations of the same total space, is there any relation between these two Lefschetz fibrations? The above questions is partially answered as follows.

Let

be a given abstract Lefschetz fibration. It is known that there are four moves producing another abstract Lefschetz fibration from $(F;L_{1},\cdots,L_{m})$ so that the total spaces of two abstract Lefschetz fibrations are Weinstein homotopic. We list the four moves here.

• •

  Deformation means a simultaneous Weinstein deformation of $F$ and exact Lagrangian isotopy of $(L_{1},\cdots,L_{m})$.

• •

  Cyclic permutation is to replace the ordered collection $(L_{1},\cdots,L_{m})$ with $(L_{2},\cdots,L_{m},L_{1})$. In other words,

  $$(F;L_{1},\cdots,L_{m})\simeq(F;L_{2},\cdots,L_{m},L_{1}).$$

  The equivalence means that their total spaces are equivalent.

• •

  Hurwitz moves. Let $\tau_{i}$ denote the symplectic Dehn twist around $L_{i}$. Hurwitz move is to replace $(L_{1},\cdots,L_{m})$ with either $(L_{2},\tau_{2}(L_{1}),L_{3},\cdots,L_{n})$ or $(\tau_{1}^{-1}(L_{2}),L_{1},L_{3},\cdots,L_{m})$, i.e.,

  $$(F;L_{1},\cdots,L_{m})\simeq(F;L_{2},\tau_{2}(L_{1}),\cdots,L_{m})\simeq(F;\tau_{1}^{-1}(L_{2}),L_{1},\cdots,L_{m}).$$

• •

  Stabilization. Let $\operatorname{dim}F=2n-2$, or equivalently, the total space is of dimension $2n$. For a parameterized Lagrangian disk $D^{n-1}\hookrightarrow F$ with Legendrian boundary $S^{n-2}=\partial D^{n-1}\hookrightarrow\partial F$ such that $0=[\lambda]\in H^{1}(D^{n-1},\partial D^{n-1})$ where $\lambda$ is the Liouville $1$-form, replace $F$ with $\tilde{F}$, obtained by attaching a $(2n-2)$ dimensional Weinstein $(n-1)$-handle to $F$ along $\partial D^{n-1}$, and replace $(L_{1},\cdots,L_{m})$ with $(\tilde{L},L_{1},\cdots,L_{m})$, where $\tilde{L}\subset\tilde{F}$ is obtained by gluing together $D^{n-1}$ and the core of the handle. In other words,

  $$(F;L_{1},\cdots,L_{m})\simeq(\tilde{F};\tilde{L},L_{1},\cdots,L_{m}).$$

The goal of Section 2.3 is to define the notion of end connected sum. Since the notion of end connected sum is defined as an attachment of index $1$ Weinstein handle, we start the current subsection by reviewing the notion of Weinstein handle attachment.

In [Wei91], Weinstein explained how to attach a Weinstein handle to a Weinstein manifold. A rough explanation on that is the following: In order to a Weinstein handle $H$ to a Weinstein manifold $W$, one needs

• •

  an isotropic embedding of the attaching sphere of $H$ onto the asymptotic boundary $\partial_{\infty}W$, and

• •

  a conformal symplectic normal bundle of $\Lambda$ where $\Lambda$ is the isotropic image of the above embedding.

We note that a conformal symplectic normal bundle of $\Lambda$ means a conformal symplectic structure on the bundle $T\Lambda^{\perp^{\prime}}/T\Lambda$ where $\perp^{\prime}$ means the “symplectic orthogonal operation” in the tangent bundle of $W$. We refer the reader to [Wei91] for more details.

Let $W_{1}$ and $W_{2}$ be connected Weinstein manifolds of the same dimension $2n\geq 4$. Then, it is easy to check that there is a unique way, up to Weinstein homotopic, to construct a connected Weinstein manifold by attaching a Weinstein handle of index $1$ to $W:=W_{1}\cup W_{2}$, where $\cup$ means the disjoint union.

As mentioned in Section 2, from a Weinstein manifold $F$ and a cyclically ordered finite collection of exact Lagrangian spheres in $F$, one could construct another Weinstein manifold equipped with a Lefschetz fibration $\pi$ such that

• •

  the regular fiber of $\pi$ is $F$, and

• •

  the number of critical values of $\pi$ is the same as the number of exact Lagrangians in the cyclically ordered collection.

We will construct $X^{2n+2}_{k}$ and $Y^{2n+2}_{k}$ for $n\geq 2,k\geq 1$ by using the above method.

It is well-known that $A_{k}^{2n}$ admits a Lefschetz fibration $\rho$ such that

• •

  the regular fiber of $\rho$ is $T^{*}S^{n-1}$,

• •

  $\rho$ has $k+1$ singular values, and

• •

  the corresponding vanishing cycles for singular values are the zero section of the regular fiber $T^{*}S^{n-1}$.

The existence, and the properties of $\rho$ are well-known in the literature. We refer the reader to [KS02] and [Wu14] for more details.

Let $F=A_{2}^{2n}$. Since $F$ is a plumbing of two $T^{*}S^{n}$, one can find two exact Lagrangian spheres which are the zero sections of two $T^{*}S^{n}$. Let those Lagrangian spheres be denoted by $\alpha$ and $\beta$. Then, $\rho(\alpha)$ and $\rho(\beta)$ are given by curves connecting two singular values of $\rho$. Figure 1 is the picture of the base of $\rho$.

Since $\alpha$ and $\beta$ are Lagrangian spheres in $F$, there are generalized Dehn twists $\tau_{\alpha}$ and $\tau_{\beta}$ along $\alpha$ and $\beta$.

In Section 3.1, we constructed $X^{2n+2}_{k}$ and $Y^{2n+2}_{k}$ as total spaces of abstract Lefschetz fibrations. Before going further, we investigate the wrapped Fukaya categories of $X^{2n+2}_{k}$ and $Y^{2n+2}_{k}$ in Sections 3.2 and 3.3. By studying those wrapped Fukaya categories, we can have a hint for proving that $X^{2n+2}_{k}$ and $Y^{2n+2}_{k}$ are not equivalent as Weinstein manifolds.

Because of the length of the proof, we give a sketch of the proof in the present subsection.

In order to prove Lemma 3.7, we construct a Weinstein sectorial covering $\{\overline{Y}_{1},\overline{Y}_{2}\}$ of a Weinstein domain which is Weinstein homotopic to $Y^{2n+2}_{k}$. In the construction of the Weinstein sectorial covering, we need to use the notions of Weinstein homotopy, symplectic completion, Lagrangian skeleta, etc. The construction would take the most part of Section 3.3.

The constructed Weinstein sectorial covering $\{\overline{Y}_{1},\overline{Y}_{2}\}$ satisfies that $\overline{Y}_{1}\simeq A^{2n+2}_{2k}$ and $\overline{Y}_{2}\simeq T^{*}S^{n+1}$. The notion of equivalences here is Weinstein homotopic. Also, after a proper modification of the Liouville structures, the intersection $\overline{Y}_{1}\cap\overline{Y}_{2}$ is equivalent to $T^{*}[0,1]\times\mathbb{D}^{2n}$ whose wrapped Fukaya category vanishes.

By the result of [GPS18b], we conclude that

Then, the above equation proves Lemma 3.7.

We note that in the current subsection, we modify Weinstein structures via Weinstein homotopies. For the notion of Weinstein homotopies, see [CE12]. Also, we refer the reader to [Sta18].

Before studying $\textup{WFuk}(Y^{2n+2}_{k})$, we recall the construction of $Y^{2n+2}_{k}$. By Equation (3.2), $Y^{2n+2}_{k}$ is obtained by attaching $(2k+3)$ Weinstein critical handles to the subcritical part. Let $H_{1},\cdots,H_{2k+3}$ denote the critical handles which are attached to the subcritical part of $Y^{2n+2}_{k}$ such that

• •

  $H_{1},\cdots,H_{2k+1}$ are attached along the Legendrians induced from $\alpha$, and

• •

  $H_{2k+2}$ and $H_{2k+3}$ are attached along the Legendrians induced from $\beta$.

We discuss a decomposition of the subcritical part of $Y^{2n+2}_{k}$. Since the subcritical part is given as a product of the fiber $A_{2}^{2n}$ and a disk $\mathbb{D}^{2}$, we will decompose $A_{2}^{2n}$ and $\mathbb{D}^{2}$.

In order to decompose $A_{2}^{2n}$, we recall that $A_{2}^{2n}$ is a union of $T^{*}\alpha$ and $T^{*}\beta$, as mentioned in Section 3.1. This induces that $A_{2}^{2n}$ admits a Weinstein handle decomposition which consists of one $2n$-dimensional index $0$-handle and two $2n$-dimensional index $n$-handles. Moreover, the union of the unique $0$-handle and one $n$-handle is $T^{*}\alpha$, and the union of the $0$-handle and the other $n$-handle is $T^{*}\beta$. We use $h_{0},h_{\alpha},h_{\beta}$ to denote the index $0$-handle, the index $n$-handle making $T^{*}\alpha$, the index $n$-handle making $T^{*}\beta$ of the fiber $A_{2}^{2n}$, respectively.

For a decomposition of $\mathbb{D}^{2}$, we consider a Weinstein sectorial covering of $\mathbb{D}^{2}$. In order to do this, we start from a Weinstein handle decomposition of $\mathbb{D}^{2}$, consisting of three index $0$-handles and two index $1$-handles. Figure 2, $a)$ describes the handle decomposition.

From the Weinstein handle decomposition, the centers of two $1$-handles have curves as their unstable manifolds with respect to the Liouville flow. Then, those two curves divide $\mathbb{D}^{2}$ into three parts. This decomposition of $\mathbb{D}^{2}$ is given in Figure 2, $b)$. Let $T$ be the piece of the decomposition such that the boundary of $T$ contains the both unstable manifolds. Also, let $S_{1},S_{2}$ denote the other two pieces.

Based on the above arguments, the subcritical part $A_{2}^{2n}\times\mathbb{D}^{2}$ admits a decomposition into a union of nine pieces

One can observe that $h_{\alpha}\times(S_{1}\cup S_{2}\cup T)$ and $h_{\beta}\times(S_{1}\cup S_{2}\cup T)$ are equivalent to $(2n+2)$-dimensional index $n$-Weinstein handles, up to Weinstein homotopy. In other words, we can construct the subcritical part $A_{2}^{2n}\times\mathbb{D}^{2}$ by attaching two Weinstein handles $H_{\alpha}$ and $H_{\beta}$ to $h_{0}\times(S_{1}\cup S_{2}\cup T)$, instead of attaching $h_{\alpha}\times(S_{1}\cup S_{2}\cup T)$ and $h_{\beta}\times(S_{1}\cup S_{2}\cup T)$. The notation $H_{\alpha}$ (resp. $H_{\beta}$) denotes the Weinstein handle replacing $h_{\alpha}\times(S_{1}\cup S_{2}\cup T)$ (resp. $h_{\beta}\times(S_{1}\cup S_{2}\cup T)$).

We note that by attaching $H_{\alpha}$ and $H_{\beta}$ to $h_{0}\times(S_{1}\cup s_{2}\cup T)$, we obtain a Weinstein domain which does not admit a product Liouville structure. In other words, by replacing $h_{\alpha}\times(S_{1}\cup S_{2}\cup T)$ (resp. $h_{\beta}\times(S_{1}\cup S_{2}\cup T)$) with $H_{\alpha}$ (resp. $H_{\beta}$), we modify the Liouville structure on them, and the modified structures do not respect the product structure.

To summarize, we decompose the subcritical part as a union of the following five pieces,

We note that the decomposition is not a Weinstein handle decomposition. This is because the first three pieces are not Weinstein handles.

For the later use, we modify the Weinstein domain which we obtained by attaching five pieces in (3.3). We observe that the attaching regions of $H_{\alpha}$ and $H_{\beta}$ are contained in $\partial\big{(}h_{0}\times(S_{1}\cup S_{2}\cup T)\big{)}$. However, we can modify it so that the attaching region of $H_{\alpha}$ (resp. $H_{\beta}$) is contained in $\partial h_{0}\times S_{1}$ (resp. $h_{0}\times S_{2}$). We note that the modification does not change the symplectic completion of the resulting Weinstein domain, up to Weinstein homotopy.

By the above modification, one can observe that the resulting Weinstein domain, even after attaching critical handles, does not change up to Weinstein homotopy. However, the Lagrangian skeleton changes. Figure 4 describes the change on Lagrangian skeleta.

Thanks to the modification, we can set

Then, $\overline{Y}$ is equivalent to the original $Y^{2n+2}_{k}$ up to Weinstein homotopy. We note that Equations (3.5) and (3.6) make sense since the attaching region of $H_{\alpha}$ (resp. $H_{\beta}$) is contained in $h_{0}\times S_{1}$ (resp. $h_{0}\times S_{2}$).

This induces that

Moreover, since $\{\overline{Y}_{1},\overline{Y}_{2}\}$ is a Weinstein sectorial covering of $\overline{Y}$, the result of [GPS18b] and Equation (3.7) conclude that

In order to complete the proof, we consider the wrapped Fukaya categories of Weinstein sectors $\overline{Y}_{1},\overline{Y}_{2}$, and $\overline{Y}_{1}\cap\overline{Y}_{2}$.

For $\textup{WFuk}(\overline{Y}_{1}\cap\overline{Y}_{2})$, we observe that

Moreover, from the definition of $T$, one can easily check that as a Weinstein sector, $T$ is equivalent to $T^{*}[0,1]$. Thus, we have

Since $\textup{WFuk}(\mathbb{D}^{2n})$ is quasi-equivalent to the zero category, so is the wrapped Fukaya category of the intersection.

For $\textup{WFuk}(\overline{Y}_{1})$ (resp. $\textup{WFuk}(\overline{Y}_{2})$), we consider the convex completion of the Weinstein sector $\overline{Y}_{1}$ (resp. $\overline{Y}_{2}$). For the notion of convex completion, see [GPS20, Lemma 2.32].

By taking the convex completion of $\overline{Y}_{1}$ (resp. $\overline{Y}_{2}$), one obtains

together with a stop. Since the boundary of the Liouville sector $\overline{Y}_{1}$ (resp. $\overline{Y}_{2}$) is given by a product of $h_{0}\simeq\mathbb{D}^{2n}$ and an $1$ dimensional curve, the stop is a Legendrian disk. We note that the $1$ dimensional curve above is given in Figure 2, as a part of the boundary of $S_{1}\cup T$ (resp. $S_{2}\cup T$), contained in the interior disk.

Since the stop is a disk, we have the followings.

Thus, we consider $\textup{WFuk}(Y_{1})$ (resp. $\textup{WFuk}(Y_{2})$) instead of $\textup{WFuk}(\overline{Y}_{1})$ (resp. $\textup{WFuk}(\overline{Y}_{2})$).

In order to study $Y_{1}$ (resp. $Y_{2}$), we consider the construction of $Y_{1}$ (resp. $Y_{2}$). Equation (3.8) (resp. Equation (3.9)) means that $Y_{1}$ (resp. $Y_{2}$) is obtained by attaching $H_{\alpha}$ and critical handles $H_{1},\cdots,H_{2k+1}$ to a Weinstein domain

Since $S_{1}\cup S_{2}\cup T=\mathbb{D}^{2}$, and since $h_{0}$ is a $0$-handle, it is equivalent to attach $H_{\alpha}$ and $H_{1}\cdots,H_{2k+1}$ to the $2n+2$ dimensional index $0$-handle.

From the above argument, it is easy to check that $Y_{1}$ is a Weinstein domain which is equivalent to the total space of an abstract Lefschetz fibration

where the number of $\alpha$ above is $2k+1$. It is known that the total space of the abstract Lefschetz fibration is $A_{2k}^{2n+2}$. Thus, it concludes that

Similarly, $Y_{2}$ is equivalent to the total space of

which is $T^{*}S^{n+1}$.

It concludes that

i.e.,

∎