# Floer cohomology of Dehn twists
along real Lagrangian spheres

Patricia Dietzsch  Department of Mathematics
ETH Zürich
Rämistrasse 101, 8092 Zürich, Switzerland<EMAIL_ADDRESS>
###### Abstract.

We study the Floer cohomology of the Dehn twist along a real Lagrangian sphere
in a symplectic manifold endowed with an anti-symplectic involution. We prove
that there exists a distinguished element in the Floer group that is a fixed
point of the automorphism induced by the involution. Our methods of proof are
based on Mak-Wu’s cobordism and Floer-theoretic considerations.

##  1\. Introduction and Main Results

Let $(M,\omega)$ be a closed symplectic manifold and $S\subset M$ a Lagrangian
sphere with a parametrization $\iota\colon S^{n}\xrightarrow{\approx}S$.
Associated to $(S,\iota)$ there exists a distinguished symplectic isotopy
class represented by the Dehn twist. The Dehn twist $\tau_{S}$ is a
symplectomorphism compactly supported in a neighbourhood of $S$. Seidel proved
that the square of the Dehn twist, in some cases, is not symplectically, but
only smoothly isotopic to the identity [Sei97a], [Sei38]. To prove this result
Seidel established a Floer homology exact sequence

$\displaystyle\dots\to(\mathrm{HF}^{*}(S,N)\otimes\mathrm{HF}^{*}(Q,S))^{k}\to\mathrm{HF}^{k}(Q,N)\to\mathrm{HF}^{k}(Q,\tau_{s}(N))\to\dots$
(1)

for admissible Lagrangian submanifolds $Q$ and $N$ in $M$ [Sei03],[Sei08].
There is a distinguished element $A\in\mathrm{HF}^{*}(\tau^{-1}_{S})$ that
characterizes the map $\mathrm{HF}^{k}(Q,N)\to HF^{k}(Q,\tau_{S}(N))$ that
occurs in the sequence.

Due to the relevance of the above exact sequence it is thus natural to
investigate properties of the element $A$. The goal of this paper is to study
the element $A$ in the situation, where there exists an anti-symplectic
involution that preserves $S$.

We work in the following setting. $(M,\omega)$ is a closed symplectically
aspherical symplectic manifold. Unless otherwise explicitely stated, all
involved Lagrangian submanifolds are assumed to be closed, oriented and
relatively symplectically aspherical. Floer cohomology groups are
$\mathbb{Z}_{2}$-graded with coefficients in the universal Novikov field over
$\mathbb{Z}_{2}$. More details about these assumptions are given in section
3.1.

Let $c\colon M\to M$ be an anti-symplectic involution satisfying $c(S)=S$.
Consider the smooth involution $\iota^{*}c:=\iota^{-1}c\iota\colon S^{n}\to
S^{n}$. We assume that $\iota^{*}c$ is either smoothly isotopic to the
identity or to the reflection
$r(x_{1},x_{2},\dots,x_{n+1})=(-x_{1},x_{2},\dots,x_{n+1}).$ This assumption
is satisfied in the important geometric setting where $(M,c)$ is a real fiber
of a real Lefschetz fibration with one critical point and $S$ is the
corresponding vanishing sphere.

Under this assumption, our main result is

######  Theorem A.

$c$ induces an automorphism
$c_{*}\colon\mathrm{HF}^{*}(\tau_{S}^{-1})\to\mathrm{HF}^{*}(\tau_{S}^{-1})$
and $c_{*}(A)=A$.

######  Remark 1.1.

$c_{*}\colon\mathrm{HF}^{*}(\tau_{S}^{-1})\to\mathrm{HF}^{*}(\tau_{S}^{-1})$
is an involution of a vector space over a field with characteristic $2$. Any
such map has a fixed point because $(c_{*}-\mathrm{id})^{2}=0$, hence
$\mathrm{ker}(c_{*}-\mathrm{id})\neq 0$. The relevance of the second part of
Theorem A is therefore not merely the existence of a fixed point. It should
rather be understood as a special property of the element $A$.

###  1.1. Examples.

The assumption on the isotopy class of $\iota^{*}c$ is automatically satisfied
for $n=1,2,3$. As already mentioned, the assumption is equivalent to $M$ being
a real fiber of a real Lefschetz fibration. This is the content of the
following

######  Proposition A.

Let $M$ be a symplectic manifold, $S\subset M$ a Lagrangian sphere with
parametrization $\iota$ and $c\colon M\to M$ an anti-symplectic involution.
Then the following statements are equivalent:

  1. (i)

$c(S)=S$ and $\iota^{*}c\simeq\mathrm{id}$ or $\iota^{*}c\simeq r$.

  2. (ii)

There exists a real Lefschetz fibration (see Sections 2.2 and 2.3) $\pi\colon
E\to\mathbb{D}^{2}$ with real structure $c_{E}\colon E\to E$, real fiber
$M=\pi^{-1}(1)$ and vanishing sphere $(S,\iota)$ such that $c_{E}$ restricts
to a real structure on $M$ that is Hamiltonian isotopic to $c$.

Seidel computed Floer cohomology of products of disjoint Dehn twists on
surfaces of genus $\geq 2$ in [Sei96]. As a special case, his result yields a
$\mathbb{Z}$-graded isomorphism

$\displaystyle\mathrm{HF}^{*}(\tau_{S}^{-1})\cong\mathrm{H}^{*}(M\backslash
S;\Lambda).$ (2)

Later, Gautschi [Gau03] generalised Seidel’s result to diffeomorphisms of
finite type, still on surfaces. Recently Pedrotti [Ped22] proved a
$\mathbb{Z}_{2}$-graded version of (2) for rational, $W^{+}$-monotone
symplectic manifolds of dimension at least $4$. The $W^{+}$-condition is
explained in Seidel [Sei97b]. It is immediate that symplectically aspherical
manifolds are $W^{+}$-monotone.

It turns out that the automorphism $c_{*}$ on $HF(\tau_{S}^{-1})$ corresponds
to the (topologically induced) map $c^{*}$ on singular cohomology
$\mathrm{H}^{*}(M\backslash S;\Lambda)$. Namely, under the assumption that $M$
is $W^{+}$-monotone and that $c(S)=S$ the following diagram commutes:

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(3)

Together with Theorem A this allows us to deduce topological restrictions on
the element $A\in\mathrm{HF}^{*}(\tau^{-1}_{S})$ and sometimes enables us to
compute $A$. More concrete examples are explained in section 2.5.

###  1.2. Outline of Proof of Theorem A

We outline the proof of Theorem A.

We view the Dehn twist as a monodromy in the Lefschetz fibration $\pi\colon
E\to\mathbb{D}^{2}$ from Proposition A. Carrying a result by Salepci [Sal10]
over to the symplectic setting, one gets

######  Proposition B.

$\tau$ is Hamiltonian isotopic to $c\circ\tilde{c}$ for some anti-symplectic
involution $\tilde{c}\colon M\to M$. In particular, $c\tau_{S}c$ is
Hamiltonian isotopic to $\tau_{S}^{-1}$.

Floer-theoretic considerations yield a homomorphism

$c_{*}\colon\mathrm{HF}^{*}(\tau_{S}^{-1})\to\mathrm{HF}^{*}(\tilde{c}\tau_{S}\tilde{c}).$

Proposition B implies that $\tilde{c}\tau_{S}\tilde{c}\simeq\tau_{S}^{-1}$ and
therefore
$\mathrm{HF}^{*}(\tilde{c}\tau_{S}\tilde{c})\cong\mathrm{HF}^{*}(\tau_{S}^{-1})$.
It follows that $c$ induces an automorphism of
$\mathrm{HF}^{*}(\tau_{S}^{-1})$, which proves the first part of Theorem A.

To show that $c_{*}(A)=A$, we adopt the framework of Biran-Cornea [BC13],
[BC14], [BC17] and Mak-Wu [MW18] about Lagrangian cobordisms.

Let $M^{-}$ be the symplectic manifold $(M,-\omega)$. We denote by
$\Gamma_{\phi}\subset M\times M^{-}$ the graph of $\phi$ for a
symplectomorphism $\phi$ on $M$. This is a Lagrangian submanifold of $M\times
M^{-}$. For $\phi=\mathrm{id}$ it is the diagonal and we write
$\Delta:=\Gamma_{\mathrm{id}}$. In [MW18] the authors construct a Lagrangian
cobordism $V_{MW}\subset M\times M^{-}\times\mathbb{C}$ that has three ends:
$S\times S,\Delta$ and $\Gamma_{\tau_{S}^{-1}}$. We recall the construction of
$V_{MW}$ in section 5. By general results on Lagrangian cobordisms due to
Biran-Cornea this cobordism induces an exact triangle in
$D\mathcal{F}uk(M\times M^{-})$:

$S\times S$$\Delta$$\Gamma_{\tau_{S}^{-1}}$

The associated long exact sequence is

$\displaystyle\dots\to\mathrm{HF}^{k}(K,S\times
S)\to\mathrm{HF}^{k}(K,\Delta)\to\mathrm{HF}^{k}(K,\Gamma_{\tau_{S}^{-1}})\to\mathrm{HF}^{k+1}(K,S\times
S)\to\dots,$ (4)

where $K$ is an admissible Lagrangian submanifold in $M\times M^{-}$. For the
special case $K=Q\times N$, this sequence reduces to Seidel’s long exact
sequence (1). The middle map in sequence (4) can be understood as
$\mu^{2}(A,-)$ for the element

$A\in\mathrm{HF}^{0}(\Delta,\Gamma_{\tau_{S}^{-1}})\cong\mathrm{HF}^{0}(\tau_{S}^{-1}).$

Consider the symplectomorphism

$\displaystyle\Phi\colon M\times M^{-}\times\mathbb{C}$
$\displaystyle\longrightarrow M\times M^{-}\times\mathbb{C}$
$\displaystyle(x,y,z)$ $\displaystyle\longmapsto(c(y),c(x),z).$

$\Phi$ preserves the ends of the cobordisms $V_{MW}$. In particular, $\Phi$
induces an automorphism

$\Phi_{*}\colon\mathrm{HF}^{*}(\Delta,\Gamma_{\tau^{-1}_{S}})\to\mathrm{HF}^{*}(\Delta,\Gamma_{\tau^{-1}_{S}}).$

This automorphism corresponds to the action of $c$ on
$\mathrm{HF}(\tau^{-1}_{S})$, namely the following diagram commutes

(9)

We explain these isomorphisms and the commutativity of the diagram in section
3. A major step in the proof is the following

######  Theorem B.

$\Phi(V_{MW})$ is Hamiltonian isotopic to $V_{MW}$.

We show how this implies Theorem A. Denote by
$\bar{A}\in\mathrm{HF}^{*}(\Delta,\Gamma_{\tau^{-1}_{S}})$ the element
corresponding to $A\in\mathrm{HF}^{*}(\tau^{-1}_{S})$ under the natural
isomorphism
$\mathrm{HF}(\tau^{-1}_{S})\cong\mathrm{HF}(\Delta,\Gamma_{\tau^{-1}_{S}})$.
As a consequence of Theorem B, the cobordisms $V_{MW}$ and $\Phi(V_{MW})$
induce isomorphic triangles. In particular, the following diagram commutes:

for all $K$. It follows that $\Phi_{*}(\bar{A})=\bar{A}$ and hence
$c_{*}(A)=A$ by commutativity of diagram (9).

######  Remark 1.2.

  1. (1)

It can be seen from the proof that all is needed are well-defined Floer
cohomology groups, and applicability of Biran-Cornea’s [BC13],[BC14] and Mak-
Wu’s [MW18] framework. One could therefore easily weaken the asphericity
assumption to monotonicity conditions.

  2. (2)

The assumption that $M$ is closed is important for our arguments: The version
of Floer cohomology we use only works for compactly supported
symplectomorphisms. In general however, the monodromy in a Lefschetz fibration
with non-compact fibers, if it exists, is not compactly supported. We expect
that the results generalize to a non-compact framework, when working with an
appropriate version of Floer theory.

  3. (3)

In general we have a symplectic isotopy

$c\circ\tau_{(S,\iota)}\circ c\simeq\tau_{(S,c\circ\iota)}^{-1}.$

However, it is unknown how the Dehn twist depends on the parametrization of
the sphere. It is only known that if $\iota^{*}c$ is isotopic to an isometry,
then the Dehn twist associated to $c\circ\iota$ is symplectically isotopic to
the Dehn twist associated to $\iota$ [Sei97a, Remark 3.1]. This explains why
we make the assumption on the mapping class of $\iota^{*}c$.

  4. (4)

The second map in the long exact sequence (1) is

$\mu^{2}(a_{N},-)\colon\mathrm{HF}^{k}(Q,N)\to\mathrm{HF}^{k}(Q,\tau_{S}(N))$

for some element $a_{N}\in\mathrm{HF}^{0}(N,\tau_{S}(N))$. $a_{N}$ and
$A\in\mathrm{HF}^{*}(\tau_{S}^{-1})$ are related as follows. There is an
operation

$*\colon\mathrm{HF}^{*}(\tau_{S}^{-1})\otimes\mathrm{HF}^{*}(N,N)\to\mathrm{HF}^{*}(N,\tau_{S}(N)).$

If $e_{N}\in HF^{*}(N,N)$ denotes the unit, we have $A*e_{N}=a_{N}$. The fixed
point property $c_{*}(A)=A$ then implies

$\displaystyle\gamma(a_{N})=a_{c(N)},$ (10)

where $\gamma$ is the isomorphism

$\displaystyle\mathrm{HF}^{*}(N,\tau_{S}(N))\cong\mathrm{HF}^{*}(\tilde{c}(N),c(N))\cong\mathrm{HF}^{*}(c(N),\tau_{S}(c(N)).$

The construction of $a_{N}$ is explained in [Sei08, Sections 17a-17c]. $a_{N}$
comes from counting the number of holomorphic sections of a Lefschetz
fibration with moving boundary condition coming from moving $N$ via parallel
transport. The invariance property (10) can be proven directly in Seidel’s
framework, by observing that the holomorphic sections for boundary conditions
coming from $N$ and $c(N)$ are in bijection.

###  1.3. Organisation of the Paper.

The rest of this paper is organised as follows. In section 2 we explain the
construction of real Lefschetz fibrations and the decomposition of the
monodromy into two anti-symplectic involuions as stated in Propositions A and
B. In section 3 we fix the setting and collect the properties of Floer
cohomology we need. In section 4 we briefly recall Biran-Cornea’s Lagrangian
cobordism framework and how cobordisms induce cone decompositions. Section 5
recalls the construction of the Mak-Wu cobordism. In section 6 we prove
Theorem B about the symmetry of the cobordism. Section 7 contains some more
background material on Floer cohomology for the convenience of the reader. The
appendix contains some algebraic background on Fukaya categories.

##  2\. Dehn twist and real Lefschetz fibrations.

In this section we show Propositions A and B. This is based on work by Salepci
[Sal10] on real Lefschetz fibrations in the smooth setting. Since we keep the
discussion here relatively brief, we refer the interested reader to the
following references for a more detailed treatment of (real) Lefschetz
fibrations: [Sei08, BC17, Sal12, Kea14].

###  2.1. Dehn twist.

Let $S\subset M$ be a Lagrangian sphere together with an embedding
$\varphi\colon S^{n}\to M$ of the $n$-dimensional sphere $S^{n}$ with image
$S$. We refer to $(S,\varphi)$ as a parametrized Lagrangian sphere. 111Seidel
uses the word “framed sphere” for this situation in [Sei08]. The Dehn twist
$\tau_{S}$ along $S$ is a symplectomorphism compactly supported in a
neighbourhood of $S$. It is defined up to symplectic isotopy. The precise map
will depend on a Dehn twist profile function and on a Weinstein neighbourhood
of $S$. As explained in [Sei97a, Proposition 2.3] the symplectic isotopy class
of $\tau_{S}$ is independent of $\varphi$ in dimension $4$. In general
however, it might depend on the parametrization [Sei08, Remark 3.1]. We
briefly recall the definition, following closely the exposition in [MW18].

######  Definition 2.1.

Let $\epsilon>0$. A Dehn twist profile function is a smooth function

$\nu_{\epsilon}^{Dehn}\colon\mathbb{R}_{\geq 0}\longrightarrow\mathbb{R}$

satisfying

$\displaystyle\begin{cases}\nu_{\epsilon}^{Dehn}(r)=\pi-r\qquad\qquad\qquad\qquad\qquad&\text{for
}0\leq r<<\epsilon,\\\ 0<\nu_{\epsilon}^{Dehn}(r)<\pi\text{ and strictly
decreasing}\qquad&\text{for }0<r<\epsilon,\\\
\nu_{\epsilon}^{Dehn}(r)=0\qquad\qquad\qquad\qquad\qquad\qquad&\text{for
}r\geq\epsilon.\end{cases}$

Consider the canonical Riemannian metric on $S^{n}$. We have a canonical
isomorphism $T_{*}S^{n}\cong T^{*}S^{n}$ and we denote by $\norm{\xi}$ the
norm of the tangent vector identified with $\xi\in T^{*}S$. We denote by

$T_{r}^{*}S^{n}=\left\\{\xi\in T^{*}S^{n}\;\mid\;\norm{\xi}<r\right\\}$

the open subset of $T^{*}S^{n}$ consisting of cotangent vectors of norm
strictly less than $r$.

Let $V\subset M$ be a Weinstein neighbourhood of $S$ together with a
symplectic embedding

$\varphi\colon V\longrightarrow T^{*}S^{n}$

that identifies $S\subset V$ with the zero-section $0_{S^{n}}\cong S^{n}$ via
$\iota$ and $\varphi(V)=T_{\epsilon}^{*}S^{n}$ for some $\epsilon>0$.

Consider the continuous function $\sigma\colon
T^{*}S^{n}\longrightarrow\mathbb{R},\,\sigma(\xi)=\norm{\xi}$. This function
is not smooth on the zero-section $0_{S^{n}}$, but has a well-defined
Hamiltonian flow on the complement:

$\psi^{\sigma}_{t}\colon\left(T^{*}S^{n}\right)\backslash
0_{S^{n}}\longrightarrow\left(T^{*}S^{n}\right)\backslash 0_{S^{n}}.$

######  Definition 2.2.

The model Dehn twist on $T^{*}S^{n}$ is the diffeomorphism defined by

$\displaystyle\tau_{S^{n}}\colon T^{*}S^{n}$ $\displaystyle\longrightarrow
T^{*}S^{n},$ $\displaystyle\xi$
$\displaystyle\longmapsto\begin{cases}\psi_{\nu_{\epsilon}^{Dehn}(\sigma(\xi))}^{\sigma}(\xi)&\text{
for }\xi\notin 0_{S^{n}},\\\ -x\qquad\qquad&\text{ for }\xi=x\in
S^{n}.\end{cases}$

The Dehn twist in $M$ along $S$ is then given by copying the model Dehn twist
into $V$ via $\varphi$:

$\displaystyle\tau_{S}=\begin{cases}\varphi^{-1}\circ\tau_{S^{n}}\circ\varphi\qquad&\text{on
$V$}\\\ \mathrm{id}\qquad&\text{on $M\backslash V$}.\end{cases}$

###  2.2. The Dehn twist as a monodromy.

We adopt here the definition used in [BC17]. We denote by $\mathbb{D}^{2}$ the
closed unit disc viewed as a subset of $\mathbb{C}$. A Lefschetz fibration
with base $\mathbb{D}^{2}$ consists of

  1. (1)

a closed symplectic manifold $(E,\Omega_{E})$ endowed with an almost complex
structure $J_{E}$,

  2. (2)

a proper $(J_{E},i)$-holomorphic map $\pi\colon E\to\mathbb{D}^{2}$

such that

  1. (1)

$\pi$ has only finitely many critical points with distinct critical values,

  2. (2)

all the critical points of $\pi$ are ordinary double points, that is for every
critical point $p\in E$, there exists $J_{E}$-holomorphic coordinates around
$p$ such that in these coordinates
$\pi(z_{1},\dots,z_{n})=z_{1}^{2}+\dots+z_{n}^{2}$ holds.

For $p\in\mathbb{D}^{2}$ we denote by $E_{p}:=\pi^{-1}(\\{p\\})$ the fiber
above $p$. All regular fibers of $\pi$ are symplectic manifolds with
symplectic form induced from $\Omega_{E}$.

Given a symplectic manifold $(M,\omega)$ and a parametrized Lagrangian sphere
$S$, one can construct a Lefschetz fibration with smooth fiber $M$ and
vanishing sphere $S$ such that the Dehn twist is symplectically isotopic to
the monodromy around a critical point. We refer the reader to [Sei03, Section
1] and [Sei08, Section (16e)] for a detailed explanation. We only include a
very brief outline of the construction here. Consider the following local
model for $\epsilon>0$: Let
$Q\colon\mathbb{C}^{n+1}\to\mathbb{C},Q(z_{1},\dots,z_{n+1})=z_{1}^{2}+\dots+z_{n+1}^{2}$
and define the total space of the fibration to be

$E^{0}_{\epsilon}:=\left\\{z\in\mathbb{C}^{n+1}\,\big{|}\,\lvert
Q(z)\rvert\leq 1,\frac{|z|^{4}-|Q(z)|^{2}}{4}<\epsilon\right\\}.$

The fibration then is $\pi_{\epsilon}^{0}\colon
E^{0}_{\epsilon}\to\mathbb{D}^{2},\pi(z)=Q(z).$ The symplectic form on
$E^{0}_{\epsilon}$ is of the form $\Omega_{0}+\mathrm{d}\gamma$, where
$\Omega_{0}$ is the standard symplectic form on $\mathbb{C}^{n+1}$ and
$\gamma$ a certain $1$-form whose precise form is not relevant to us. Its
effect is, that the fibration $\pi^{0}_{\epsilon}$ is trivial near the
boundary. The smooth fibers are symplectomorphic to $T^{*}_{\epsilon}S^{n}$.
Consider the family of Lagrangian spheres

$\Sigma_{r}=\sqrt{r}S^{n}=\\{(\sqrt{r}z_{1},\dots,\sqrt{r}z_{n+1})\,\big{|}\,z\in
S^{n}\subset\mathbb{R}^{n+1}\\}\subset\left(E_{\epsilon}^{0}\right)_{r}$

for $r>0$. They are called vanishing cycles. The union
$\Sigma=\left(\cup_{r>0}\Sigma_{r}\right)\cup\\{0\\}$ is a Lagrangian disc in
$E^{0}_{\epsilon}$, called a Lefschetz thimble. There is an isomorphism

$\Phi\colon
E^{0}_{\epsilon}\backslash\Sigma\to\mathbb{D}^{2}\times(T_{\epsilon}^{*}S^{n}\backslash
S^{n}).$

The monodromy
$\tau\colon(\pi_{\epsilon}^{0})^{-1}({1})\to(\pi_{\epsilon}^{0})^{-1}({1})$
along $\partial\mathbb{D}^{2}$ is the Dehn twist along the vanishing cycle
$\Sigma_{1}$ [Sei03, Lemma 1.10]. To get the claimed Lefschetz fibration
$\pi^{0}\colon E^{0}\to\mathbb{D}^{2}$, one glues $E^{0}_{\epsilon}$ together
with the trivial fibration $\mathbb{D}^{2}\times(M\backslash V)$ via
$\varphi$.

Locally, each Lefschetz fibration looks like a model Lefschetz fibration
$E^{0}$. In particular, there is a notion of vanishing spheres in any
Lefschetz fibration. The monodromy $\tau\colon E_{p}\to E_{p}$ along a path
around the singularity is the Dehn twist along a vanishing cycle in $E_{p}$.
Usually, the monodromy in not supported near $S$. However, $\tau$ is
symplectically isotopic to the Dehn twist as defined in section 2.1.

###  2.3. Real Lefschetz fibrations.

A Lefschetz fibration $\pi\colon E\to\mathbb{D}^{2}$ is called real, if the
total space $E$ is endowed with an anti-symplectic involution $c_{E}\colon
E\to E$ that covers complex conjugation
$c_{\mathbb{C}}\colon\mathbb{D}^{2}\to\mathbb{D}^{2}$, meaning the diagram

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(11)

commutes. Consider the fiber $M=E_{1}$ over $1$. $c_{E}$ induces an anti-
symplectic involution on $c$ on $M$. The following Lemma shows that the
assumption of Theorem A is satisfied.

######  Lemma 2.3.

$c(S)=S$ and $\iota^{-1}c\iota\simeq\mathrm{id}$ or $\iota^{-1}c\iota\simeq
r$, where $\iota$ is the canonical framing of the vanishing sphere $S$.

###### Proof.

$c(S)=S$ follows from $c_{E}(0)=0$ and the fact that $c_{E}$ commutes with
parallel transport. For the second part, note that it is enough to consider
the model $Q\colon\mathbb{C}^{n+1}\rightarrow\mathbb{C}$. In that case,
$S=S^{n}\subset\mathbb{C}^{n+1}$ is a standard sphere. Note that $c_{E}$
restricted to the thimble $\Sigma=B^{n+1}(0)$ is a smooth extension of the
sphere $c|_{S^{n}}$ to the ball. Moreover, since parallel transport commutes
with $c_{E}$, it is a linear extension, in the sense that

$c_{E}(x)=c\left(\frac{x}{\norm{x}}\right)\norm{x}.$

It follows that $c_{E}|_{\mathbb{R}^{n+1}}$ is an orthogonal linear
transformation and hence $c|_{S^{n}}$ is an isometry. In particular,
$c|_{S^{n}}$ is smoothly isotopic to $\mathrm{id}$ or $r$. ∎

Proposition A states that the condition on $\iota^{*}c$ is equivalent to $M$
being the fiber of a real Lefschetz fibration. One direction is proven in
Lemma 2.3 above. We now prove the other direction.

###### Proof of Proposition A..

Suppose the tuple $(M,S,\iota,c)$ satisfy the conclusion of Lemma 2.3. We want
to construct a real Lefschetz fibration whose fiber is $M$, whose vanishing
sphere is $(S,\iota)$ and whose real structure restricts to a real structure
Hamiltonian isotopic to $c$.

First we endow the Lefschetz fibration $\pi^{0}_{\epsilon}\colon
E^{0}_{\epsilon}\to\mathbb{D}^{2}$ from the previous section with a real
structure. We consider two options:

$c_{1}(z_{1},\dots,z_{n+1})=(\overline{z_{1}},\dots,\overline{z_{n+1}})$

and

$c_{2}(z_{1},z_{2},\dots,z_{n+1})=(-\overline{z_{1}},\overline{z_{2}},\dots,\overline{z_{n+1}}).$

These are real structures on $E^{0}_{\epsilon}$.

By Proposition 6.2 there exists a Hamiltonian isotopy $\psi_{t}$ on $M$
supported in $V$ such that in the model $T^{*}_{\delta}S^{n}$
($\delta<\epsilon$ small enough) one has

$\psi_{1}c(q,p)=(q,-p)$

if $\varphi^{*}c\simeq\mathrm{id}$ and

$\psi_{1}c(q,p)=(r(q),-r(p))$

if $\varphi^{*}c\simeq r$. These two maps exactly correspond to $c_{1}$ and
$c_{2}$ on the fiber $(\pi^{0}_{\epsilon})^{-1}(1)$.

We now glue the fibration $\pi\colon E^{0}\to\mathbb{D}^{2}$ from two parts:
the trivial fibration

$\mathbb{D}^{2}\times(M\backslash\varphi^{-1}(T_{\delta}^{*}S^{n}))$

and the local model fibration $E_{\delta}^{0}$. On the first part, we define
$c(z,x):=(\overline{z},\psi_{1}c(x))$. On $E_{\delta}^{0}$ we define
$c(z):=c_{1}(z)$ or $c(z)=c_{2}(z)$. These definitions are compatible on the
glued region and hence descend to a real structure $c_{E^{0}}$ on $E^{0}$
satisfying $c_{E}|_{E_{1}}=\psi_{1}c$. ∎

###  2.4. Splitting of the monodromy into anti-symplectic involutions.

Let

$\pi\colon E\to\mathbb{D}^{2}$

be a real Lefschetz fibration with real structure $c_{E}\colon E\to E$ as
above. We assume that $p\in E$ is the unique critical point of $\pi$ and
$\pi(p)=0$. Let $M:=E_{1}:=\pi^{-1}(\\{1\\})$ and denote by $\tau\colon M\to
M$ the monodromy along the boundary loop $\gamma(t)=e^{2\pi it},t\in[0,1].$
The following result is due to Salepci [Sal10] in the smooth category. Here we
adapt it to the symplectic framework.

######  Lemma 2.4.

$\tau$ splits into a product of two anti-symplectic involutions on $M$. More
concretely, $\tau=c_{+}\circ c_{-}$ for two anti-symplectic involutions
$c_{\pm}\colon M\longrightarrow M$, where $c_{+}=(c_{E})|_{E_{1}}$.
Equivalently, we have $c\tau c=\tau^{-1}$.

###### Proof.

$\Omega_{E}$ defines a symplectic connection on the smooth part of $E$. Let us
denote by

$P_{\gamma(s);t}\colon E_{\gamma(s)}\to E_{\gamma(s+t)}$

the parallel transport for time $t$ along $\gamma$. Let $v\in E_{-1}$.
Consider the parallel lift $w(t)\in E_{e^{\pi i-\pi it}}$ of $x:=c_{E}(v)$
along the upper half $\eta^{+}$ of $\gamma$. Note that

$c_{E}\circ(P_{1;\frac{1}{2}})^{-1}\circ c_{E}(v)=c_{E}(w(1)).$

It is straight-forward to check that $v(t):=c_{E}(w(t))$ is actually a
parallel lift of $v$ along the lower half $\eta^{-}$ of $\gamma$. This uses
$(dc_{E})(H_{w(t)})=H_{c_{E}(w(t))}$. Hence,

$c_{E}\circ(P_{1;\frac{1}{2}})^{-1}\circ c_{E}=P_{-1,\frac{1}{2}}$

and the lemma follows:

$\tau=P_{-1;\frac{1}{2}}\circ
P_{1;\frac{1}{2}}=c_{E}\circ(P_{1;\frac{1}{2}})^{-1}\circ c_{E}\circ
P_{1;\frac{1}{2}}=c_{+}\circ c_{-},$

where $c_{+}=(c_{E})|_{M}$ and $c_{-}=(P_{1;\frac{1}{2}})^{-1}\circ c_{E}\circ
P_{1;\frac{1}{2}}$. ∎

This proves Proposition B. Alternatively, Proposition B can also be shown
directly from the definition of a model Dehn twist without going through real
Lefschetz fibrations.

###  2.5. Examples in $2$ dimensions.

######  Example 2.5 (Genus $2$ surface).

Let us consider the genus $2$ surface $\Sigma_{2}$. Take $S$ to be a
separating curve, going once around between the two holes, as in Figure 1.
Consider the Dehn twist $\tau_{S}$ around $S$.

$\alpha_{1}$$S$$\alpha_{2}$$\beta_{1}$$\beta_{2}$ Figure 1. Genus $2$ surface
with Lagrangian sphere $S$.

As in [Sei96] we can work over $\mathbb{Z}_{2}$ instead of the Novikov field,
and the Floer cohomology groups are $\mathbb{Z}$-graded.

$\tau_{S}$ splits into the product of two anti-symplectic involutions: Take
$c$ to be the anti-symplectic involution which is a reflection along $S$. It
is straight forward to check that $\tilde{c}:=c\circ\tau_{S}$ is an anti-
symplectic involution. In particular, we can write $\tau_{S}=c\circ\tilde{c}$.

Let us compute $c_{*}\colon HF^{*}(\tau^{-1}_{S})\to HF^{*}(\tau^{-1}_{S})$.
By the isomorphism (2) Floer cohomology of $\tau^{-1}_{S}$ is

$\displaystyle HF^{*}(\tau^{-1}_{S})$ $\displaystyle\cong
H^{*}(\Sigma\backslash S;\mathbb{Z}_{2})$ $\displaystyle\cong
H^{*}(\Sigma\backslash S;\mathbb{Z}_{2})$ $\displaystyle\cong H^{*}(S^{1}\vee
S^{1};\mathbb{Z}_{2})\oplus H^{*}(S^{1}\vee S^{1};\mathbb{Z}_{2})$
$\displaystyle\cong\mathbb{Z}_{2}[pt_{1}]\oplus\mathbb{Z}_{2}\alpha_{1}\oplus\mathbb{Z}_{2}\beta_{1}\oplus\mathbb{Z}_{2}[pt_{2}]\oplus\mathbb{Z}_{2}\alpha_{2}\oplus\mathbb{Z}_{2}\beta_{2}.$

In degree $0$, the matrix representing $c^{*}$ on $H^{0}(\Sigma\backslash
S;\mathbb{Z}_{2})$ with respect to the basis $[pt_{1}],[pt_{2}]$ is

$\displaystyle\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}.$

It follows from Theorem A that $A=[pt_{1}]+[pt_{2}]$.

######  Example 2.6 (higher genus surfaces).

Similarly, we can consider any surface $\Sigma$ of genus $g\geq 2$, $S$ a
separating circle in it that is the fixed point set of a reflection. Then
$HF^{0}(\tau_{S}^{-1})\cong\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}$, where each of
the two summands corresponds to one of the connected components of
$\Sigma\backslash S$. Theorem A implies $A=(1,1)$.

######  Example 2.7 (Torus).

Let $S$ be any non-contractible embedded circle in the torus $T^{2}$. Using
the long exact sequence (4) applied to $K=\Delta$ one computes

$HF^{0}(\tau_{S}^{-1})\cong H^{even}(T^{2};\Lambda)/H^{0}(S;\Lambda)\cong
H^{2}(T^{2};\Lambda)\cong\Lambda.$

For any anti-symplectic involution $c\colon T^{2}\to T^{2}$ satisfying
$c(S)=S$, it follows that $c_{*}=\mathrm{id}$.

##  3\. Floer cohomology

In this section we collect the main properties of Floer cohomology we need in
the sequel.

###  3.1. Setting.

We assume that $M$ is symplectically aspherical, that is for every smooth map
$u\colon S^{2}\to M$, its symplectic area vanishes:

$\int_{S^{2}}u^{*}\omega=0.$

Moreover, we assume that all involved Lagrangian submanifolds are relatively
symplectically aspherical, that is for every smooth map $u\colon D^{2}\to M$
satisfying $u(\partial D^{2})\subset L$, we have

$\int_{D^{2}}u^{*}\omega=0.$

In particular, $S\subset M$ is relatively symplectically aspherical. This is
automatic if $M$ is symplectically aspherical, unless $S$ has dimension $1$.
In the latter case, the condition is equivalent to $S$ being a non-
contractible circle. In this situation, Floer cohomology $\mathrm{HF}^{*}(f)$
for a symplectomorphism $f\in\mathrm{Symp}(M)$, and Lagrangian Floer
cohomology $\mathrm{HF}^{*}(L,K)$ for Lagrangians $L,K$ as above can be
defined over the universal Novikov field

$\Lambda=\left\\{\sum
a_{k}q^{\omega_{k}}\bigg{|}|a_{k}\in\mathbb{Z}_{2},\omega_{k}\in\mathbb{R},\lim\limits_{k\to\infty}\omega_{k}=\infty\right\\}.$

$\mathrm{HF}^{*}(f)$ and $\mathrm{HF}^{*}(L,K)$ are $\mathbb{Z}_{2}$-graded,
whenever $L$ and $K$ are oriented. We include a section about the definition
of these groups for convenience of the reader in section 7. For a more
detailed exposition, we refer the reader to [DS94, Sei97a, Lee05] for
$\mathrm{HF}^{*}(f)$ and to [Flo88, Oh93, Oh95] for $\mathrm{HF}^{*}(L,K)$ .

###  3.2. Conjugation invariance.

Let $f$ be a symplectomorphism on $X$ and $\varphi$ be an antisymplectic
diffeomorphism on $X$. We will make substancial use of the following fact,
which is an anti-symplectic version of the well-known conjugation invariance
of Floer cohomology (see e.g. [Sei38, section 3]). We include a proof in
section 7.1.

######  Proposition 3.1.

There is a canonical graded isomorphism

$(\varphi f^{-1})_{*}\colon\mathrm{HF}^{*}(f^{-1})\to\mathrm{HF}^{*}(\varphi
f\varphi^{-1}).$

If $\tau_{S}=c\circ\tilde{c}$ we can apply this result to $\varphi=\tilde{c}$
and $f=\tau_{S}$. We get an automorphism

$\displaystyle
c_{*}\colon\mathrm{HF}^{*}(\tau_{S}^{-1})\longrightarrow\mathrm{HF}^{*}(\tilde{c}\tau_{S}\tilde{c})\cong\mathrm{HF}^{*}(\tau_{S}^{-1}).$

This is induced by the chain-level map sending a generator $x$ to
$\tilde{c}\tau_{S}^{-1}(x)=c(x)$, concatenated with a continuation map.

###  3.3. Lagrangian Floer cohomology.

Note that for any symplectomorphism $f$ on a symplectically aspherical
symplectic manifold $M$, the graph $\Gamma_{f}$ is a relatively aspherical
Lagrangian manifold in $M\times M^{-}$. Also, products of relatively
aspherical Lagrangians in $M$ are relatively aspherical Lagrangians in
$M\times M^{-}$.

We endow the graph $\Gamma_{f}$ with the following orientation: Given a
positive basis $v_{1},\dots,v_{2n}$ of $T_{x}M$, then the basis
$(v_{1},Df_{x}(v_{1})),\dots,(v_{2n},Df_{x}(v_{2n}))$ of $T_{x}\Delta\subset
T_{x}M\oplus T_{x}M$ is defined to be positive if $(-1)^{\frac{n(n-1)}{2}}=1$
and negative otherwise, see [WW10]. Moreover, given an oriented Lagrangian
$N$, note that $f(N)$ has a canonical orientation.

Let $Q$ and $N$ be oriented Lagrangians in $M$. There are the following
canonical graded isomorphisms between Floer cohomology groups for Lagrangians
in $M\times M^{-}$ and Lagrangians in $M$:

  1. (1)

$\mathrm{HF}^{*}(Q\times N,\Gamma_{f^{-1}})\cong\mathrm{HF}^{*}(Q,f(N))$

  2. (2)

$\mathrm{HF}^{*}(Q\times N,Q^{\prime}\times
N^{\prime})\cong\mathrm{HF}^{*}(Q,Q^{\prime})\otimes\mathrm{HF}^{*}(N^{\prime},N)$

###  3.4. Floer cohomology as a special case of Lagrangian Floer cohomology.

Floer cohomology of a symplectomorphism $f$ can be viewed as Lagrangian Floer
cohomology of the pair $(\Delta,\Gamma_{f})$. This isomorphism is well-known,
see for instance [WW10], [MW18] and [LZ18, section 2.7]. Namely we have

######  Proposition 3.2.

There is a canonical graded isomorphism
$\Psi_{f}\colon\mathrm{HF}(f)\to\mathrm{HF}(\Delta,\Gamma_{f})$.

For the convenience of the reader we include a sketch of the proof in section
7.

Let $\varphi\colon M\to M$ be an anti-symplectic involution. Consider the
symplectomorphism

$\displaystyle\Phi^{\varphi}\colon M\times M^{-}\longrightarrow M\times M^{-}$
$\displaystyle(x,y)\longmapsto(\varphi(y),\varphi(x)).$

The map $(\varphi f^{-1})_{*}$ on $\mathrm{HF}^{*}(\tau_{S}^{-1})$ corresponds
to $\Phi^{\varphi}_{*}$ under the isomorphism of Proposition 3.2, i.e. the
following diagram commutes:

As a special case, we recover the commutative diagram (9) by setting
$f=\tau_{S}$ and $\varphi=\tilde{c}$.

##  4\. Lagrangian cobordisms

###  4.1. Definition of a Lagrangian cobordism.

In this section we recall the definition of Lagrangian cobordisms as studied
by Biran and Cornea in the series of papers [BC13, BC14, BC17]. Let
$(M,\omega)$ be a symplectic manifold. Consider the product symplectic
manifold $(M\times\mathbb{R}^{2},\omega\oplus\omega_{\mathrm{std}})$. Here,
$\omega_{\mathrm{std}})=\mathrm{d}x\wedge\mathrm{d}y$ denotes the standard
symplectic form on $\mathbb{R}^{2}$. We denote by $\pi\colon
M\times\mathbb{R}^{2}\to\mathbb{R}^{2}$ the projection to the plane. For
subsets $V\subset M\times\mathbb{R}^{2}$ and $Z\subset\mathbb{R}^{2}$, we
write $V|_{Z}:=V\cap\pi^{-1}(Z)$ for the restriction of $V$ over $Z$. A
Lagrangian submanifold $V\subset M\times\mathbb{R}^{2}$ is called a Lagrangian
cobordism if there exists $R>0$ such that

  1. (i)

$V|_{(-\infty,-R]\times\mathbb{R}}=\bigcup_{j=1}^{k_{-}}L_{j}\times(-\infty,-R]\times\\{j\\}$

for some closed Lagrangian submanifolds $L_{1},\dots,L_{k_{-}}\subset M$,

  2. (ii)

$V|_{[R,\infty)\times\mathbb{R}}=\bigcup_{j=1}^{k_{+}}L_{j}^{\prime}\times[R,\infty)\times\\{j\\}$

for some closed Lagrangian submanifolds
$L_{1}^{\prime},\dots,L_{k_{+}}^{\prime}\subset M$,

  3. (iii)

$V|_{[-R,R]\times\mathbb{R}}\subset\mathbb{R}^{2}\times M$ is compact.

$V$ is called a Lagrangian cobordism from the Lagrangian family
$(L_{j}^{\prime})_{j=1,\dots,k_{+}}$ to the Lagrangian family
$\left(L_{i}\right)_{i=1,\dots,k_{-}}$, denoted by

$(L_{j}^{\prime})_{j=1,\dots,k_{+}}\rightsquigarrow\left(L_{i}\right)_{i=1,\dots,k_{-}}.$

###  4.2. Lagrangian cobordisms induce cone decompositions.

We recall here how a cobordism gives rise to cone decompositions of its ends
in $\mathcal{DF}uk(M)$. Since we work with cohomology, rather than homology,
we write here a cohomological reformulation of Theorem A from [BC14].

######  Theorem 4.1 (Theorem A in [BC14]).

Let $V$ be an oriented cobordism from $L$ to the family
$(L_{1}[l-1],L_{2}[l-2]\dots,L_{l})$. Assume that all Lagrangians involved
(including $V$) are uniformly monotone. Then there exists a graded quasi-
isomorphism

$L\cong\mathrm{Cone}\left(\dots\mathrm{Cone}(\mathrm{Cone}(L_{1}\to L_{2})\to
L_{3})\to\dots\to L_{l}\right)$

in the derived Fukaya category $\mathcal{DF}uk(M).$

Here, we denote by $L[k],k\in\mathbb{Z}$ the Lagrangian $L$ with the same
orientation for even $k$, and with oppostite orientation for odd $k$. (The
theorem also holds in the context of $\mathbb{Z}$-gradings, see also [MW18].)

A special case occurs when there are only three Lagrangians involved, namely
$V$ has one right end, $L$, and two left ends, $L_{1}[1]$ and $L_{2}$. Then we
get

$L\cong Cone(L_{1}\xrightarrow{\varphi}L_{2}).$

As we explain further in the appendix, the morphism $\varphi$ is determined by
a unique element $\alpha_{V}\in HF^{0}(L_{1},L_{2})$. In particular, for any
Lagrangian $K$ we get a quasi-isomorphism of chain complexes

$\mathrm{CF}^{*}(K,L)\cong
Cone\left(\mathrm{CF}^{*}(K,L_{1})\xrightarrow{\mu^{2}(\alpha_{V},-)}\mathrm{CF}^{*}(K,L_{2})\right).$

Note that $\alpha_{V}$ is independent of $K$.

The associate long exact sequence in cohomology is

$\dots\to\mathrm{HF}^{k-1}(K,L)\to\mathrm{HF}^{k}(K,L_{1})\xrightarrow{\mu^{2}(\alpha_{V},-)}\mathrm{HF}^{k}(K,L_{2})\to\mathrm{HF}^{k}(K,L)\to\dots$

##  5\. Mak-Wu cobordism

We consider a symplectic manifold $(M,\omega)$ and a parametrized Lagrangian
sphere $S\subset M$. Mak-Wu [MW18] constructed a Lagrangian cobordism $V_{MW}$
with three ends: $S\times S,\Delta$ and $\Gamma_{\tau_{S}^{-1}}$. In this
section, we will recall the construction of this cobordism, which closely
follows [MW18].

###  5.1. The graph of the Dehn twist.

Following the principle that surgeries provide cobordisms with three ends
[BC13, Section 6], the Mak-Wu cobordism also arises as the trace of a surgery.
The first step therefore is to understand $\Gamma_{\tau_{S}^{-1}}$ as the
result of a surgery between $S\times S\subset M\times M^{-}$ and the diagonal
$\Delta\subset M\times M^{-}$ along the clean intersection
$\Delta_{S}:=(S\times S)\cap\Delta$. The surgery construction takes place
locally in a Weinstein neighbourhood of $S\times S$. We choose a very specific
neighbourhood, so that we can later compare it to $\Gamma_{\tau_{S}^{-1}}$.
Namely, consider the symplectic embedding

$\displaystyle\widetilde{\varphi}\colon V\times V$
$\displaystyle\longrightarrow T_{\epsilon}^{*}S^{n}\oplus
T_{\epsilon}^{*}S^{n}\subset T^{*}(S^{n}\times S^{n})$ $\displaystyle(x,y)$
$\displaystyle\longmapsto(\varphi(x),-\varphi(y))$

that identifies $S\times S$ with the zero-section in $T^{*}(S^{n}\times
S^{n})$. Note that

$\widetilde{\varphi}^{-1}(N_{\Delta_{S}}^{*})=\Delta\cap(V\times V),$

where

$N_{\Delta_{S}}^{*}:=\left\\{\alpha\in T^{*}(S^{n}\times S^{n})\,|\,\forall
v\in\Delta_{S}\colon\alpha(v)=0\right\\}.$

We will define a surgery model in $T^{*}(S^{n}\times S^{n})$ for surgery of
the zero-section and $N_{\Delta_{S}}^{*}$ along their intersection
$\Delta_{S}$. Then we will glue the surgery model into $V\times V$ via
$\tilde{\varphi}$. To define the surgery model, we need some auxiliary
functions:

######  Definition 5.1.

A $\lambda$-admissible function $\nu_{\lambda}\colon\mathbb{R}_{\geq
0}\longrightarrow[0,\lambda]$ is a smooth function satisfying

$\displaystyle\begin{cases}\nu_{\lambda}(0)=\lambda,\\\
\nu_{\lambda}^{-1}\text{has vanishing derivatives of all orders at
}\lambda,\\\ 0<\nu_{\lambda}(r)<\lambda\text{ and strictly
decreasing}&\text{for }0<r<\epsilon,\\\ \nu_{\lambda}(r)=0&\text{for
}r\geq\epsilon.\end{cases}$

Let $\pi_{2}\colon T^{*}(S^{n}\times S^{n})\cong T^{*}S^{n}\oplus
T^{*}S^{n}\to T^{*}S^{n}$ be the projection to the second summand. Consider
$\sigma_{\pi}\colon T^{*}(S^{n}\times S^{n})\to\mathbb{R}$ defined by
$\sigma_{\pi}(\xi)=\norm{\pi_{2}(\xi)}$. This has a well-defined Hamiltonian
flow on $T^{*}(S^{n}\times S^{n})\backslash\Delta_{S}$. Let $\lambda<\pi$.
Consider a $\lambda$-admissible function $\nu=\nu_{\lambda}$, and define the
following flow handle:

$\displaystyle
H_{\nu}=\left\\{\psi_{\nu(\sigma_{\pi}(\xi))}^{\sigma_{\pi}}(\xi)\in
T^{*}(S^{n}\times S^{n})\,\big{|}\,\xi\in
N_{\Delta_{S}}^{*}\backslash\Delta_{S},\sigma_{\pi}(\xi)\leq\epsilon\right\\}.$

$H_{\nu}$ can be glued to a part of $S\times S$ and $N^{*}_{\Delta_{S}}$,
resulting in a smooth Lagrangian in $T^{*}(S^{n}\times S^{n})$ that coincides
with $N_{\Delta_{S}}^{*}$ outside of $T_{\epsilon}^{*}S\oplus
T_{\epsilon}^{*}S^{n}$. We denote the resulting Lagrangian by

$(S^{n}\times S^{n})\\#_{\Delta_{S}}^{\nu}N_{\Delta S}^{*}.$

We finally glue this model surgery into $V\times V$:

$(S\times
S)\\#_{\Delta_{S}}^{\nu}\Delta:=(\tilde{\varphi})^{-1}\left((S^{n}\times
S^{n})\\#_{\Delta_{S}}^{\nu}N_{\Delta_{S}}^{*}\right)\cup\left(\Delta\backslash(V\times
V^{-})\right).$

Mak-Wu [MW18, Lemma 3.4] show that all such surgeries are Hamiltonian isotopic
for different choices of $\nu$. Moreover, the same construction works for
$\nu=\nu_{\epsilon}^{\text{Dehn}}$ (even though this is not admissible) and
the result is again Hamiltonian isotopic to any of the other surgeries. It’s
straight-forward to see that

$(S\times
S)\\#_{\Delta_{S}}^{\nu_{\epsilon}^{\text{Dehn}}}\Delta=\Gamma_{\tau_{S}^{-1}}$

and so any of the above surgeries is Hamiltonian isotopic to
$\Gamma_{\tau_{S}^{-1}}$. In particular, since $\Gamma_{\tau_{S}^{-1}}$ is
relatively symplectically aspherical, so is the surgery $(S\times
S)\\#_{\Delta_{S}}^{\nu}\Delta$.

######  Remark 5.2.

This version of surgery is a special case of $E_{2}$-flow surgery, introduced
in [MW18, section 2.3] in more general situations.

###  5.2. The cobordism.

$(S\times S)\\#_{\Delta_{S}}^{\nu}\Delta$ is related to $S\times S$ and
$\Delta$ via a cobordism. This follows from a construction called ”trace of a
surgery”, which is a surgery construction in one dimension higher. This was
first introduced in [BC13] for the case of a transverse surgery in a point. As
shown in [MW18], exactly the same construction works for the $E_{2}$-surgery
along clean intersections. We recall the construction in our special case.

Consider the symplectomorphism

$\displaystyle\tilde{\varphi}\times\mathrm{id}\colon V\times V\times
T^{*}\mathbb{R}$ $\displaystyle\longrightarrow T_{\epsilon}^{*}S^{n}\oplus
T_{\epsilon}^{*}S^{n}\oplus T^{*}\mathbb{R}\subset T^{*}(S^{n}\times
S^{n}\times\mathbb{R})$

and define the handle in the model $T^{*}(S^{n}\times S^{n}\times\mathbb{R})$
as follows:

$\hat{H}_{\nu}=\left\\{\psi_{\nu(\sigma_{\hat{\pi}}(\xi))}^{\sigma_{\hat{\pi}}}(\xi)\in
T^{*}(S^{n}\times S^{n}\times\mathbb{R})\,\big{|}\,\xi\in
N_{\Delta_{S}\times\\{0\\}}^{*}\backslash(\Delta_{S}\times\\{0\\}),\sigma_{\hat{\pi}}(\xi)\leq\epsilon\right\\},$

where $\sigma_{\hat{\pi}}\colon T^{*}(S^{n}\times
S^{n}\times\mathbb{R})\to\mathbb{R}$ is given by
$\sigma_{\hat{\pi}}(\xi_{1},\xi_{2},p)=\norm{((\xi_{2},p)}$. Here,
$\nu=\nu_{\lambda}$ is a $\lambda$-admissible function, as defined in
Definition 5.1. One computes

$\displaystyle\psi_{t}^{\sigma{\hat{\pi}}}(\xi_{1},\xi_{2},p)=\left(\xi_{1},\psi^{\sigma}_{\frac{t\norm{\xi}}{\sqrt{\norm{\xi}^{2}+p^{2}}}}(\xi_{2}),\psi^{\sigma^{\mathbb{R}}}_{\frac{t\lvert
p\rvert}{\sqrt{\norm{\xi}^{2}+p^{2}}}}(p)\right)$

So more concretely, $\hat{H}_{\nu}$ can be described as follows:

$\displaystyle\hat{H}_{\nu}=\left\\{\left(\xi,\psi^{\sigma}_{\nu\left(\sqrt{\norm{\xi}^{2}+p^{2}}\right)\frac{\norm{\xi}}{\sqrt{\norm{\xi}^{2}+p^{2}}}}(\xi),\psi^{\sigma^{\mathbb{R}}}_{\nu\left(\sqrt{\norm{\xi}^{2}+p^{2}}\right)\frac{\norm{p}}{\sqrt{\norm{\xi}^{2}+p^{2}}}}(p)\right)\,\Big{|}\,\begin{array}[]{l}\xi\in
T_{\epsilon}^{*}S,p\in\mathbb{R},\\\
\sqrt{\norm{\xi}^{2}+p^{2}}<\epsilon\end{array}\right\\}.$

Here, $\sigma\colon T^{*}S\longrightarrow\mathbb{R}$ is the Hamiltonian
function $\sigma(\xi)=\norm{\xi}$ we used earlier to define $\tau_{S}$ and
$\sigma^{\mathbb{R}}\colon T^{*}\mathbb{R}\longrightarrow\mathbb{R}$ is given
by $\sigma^{\mathbb{R}}(p)=|p|$.

The model handle $\hat{H}_{\nu}$ glues to a part of $(S^{n}\times
S^{n}\times\mathbb{R})\backslash\partial H$, which yields the model surgery
trace

$(S^{n}\times
S^{n}\times\mathbb{R})\\#_{\Delta_{S}\times\\{0\\}}N_{\Delta_{S}\times\\{0\\}}^{*}.$

Gluing this into $M\times M^{-}\times T^{*}\mathbb{R}$ via
$\tilde{\varphi}\times\mathrm{id}$ we get

$V:=(S\times S\times\mathbb{R})\\#_{\Delta_{S}\times\\{0\\}}\left(\Delta\times
i\mathbb{R}\right):=(\tilde{\varphi}\times\mathrm{id})^{-1}\left((S^{n}\times
S^{n}\times\mathbb{R})\\#_{\Delta_{S}\times\\{0\\}}N_{\Delta
S\times\\{0\\}}^{*}\right).$

$V\subset M\times M^{-}\times T^{*}\mathbb{R}$ is a Lagrangian submanifold.
Under the identification $T^{*}\mathbb{R}\cong\mathbb{C}$ via
$(q,p)\leftrightarrow q-ip$, $V$ satisfies

$\displaystyle V\cap\pi_{\mathbb{C}}^{-1}(\epsilon)=S\times
S\times\\{\epsilon\\},$ $\displaystyle
V\cap\pi_{\mathbb{C}}^{-1}(i\epsilon)=\Delta\times\\{i\epsilon\\},$
$\displaystyle V\cap\pi_{\mathbb{C}}^{-1}(0)=(S\times
S)\\#_{\Delta_{S}}^{\nu}\Delta.$

By taking half of $V$, extending it by a ray of $(S\times
S)\\#_{\Delta_{S}}^{\nu}\Delta$ at $0$ and smoothing it, and bending the ends,
as explained in [BC13, Section 6.1], we get a cobordism

$\tilde{V}\colon(S\times
S)\\#_{\Delta_{S}}^{\nu}\Delta\rightsquigarrow(S\times S,\Delta).$

As discussed in section 5.1, $(S\times S)\\#_{\Delta_{S}}^{\nu}\Delta$ is
Hamiltonian isotopic to $\Gamma_{\tau_{S}^{-1}}$. Gluing a corresponing
suspension to $\tilde{V}$ finally gives us the claimed cobordism

$V_{MW}\colon\Gamma_{\tau_{S}^{-1}}\rightsquigarrow(S\times S,\Delta).$

###  5.3. Floer theory.

Mak-Wu [MW18] explain how to put gradings on $S\times S$, $\Delta$,
$\Gamma_{\tau_{S}^{-1}}$ and on $V_{MW}$ such that $V_{MW}$ becomes a graded
cobordism from $\Gamma_{\tau_{S}^{-1}}$ to $(S\times S[1],\Delta)$. Here, we
only use $\mathbb{Z}/2$-gradings, but it follows from their proof, that
$V_{MW}$ is an oriented cobordism.

In the situation of symplectically aspherical manifolds, $V_{MW}$ is a
relatively symplectically aspherical Lagrangian in $M\times
M^{-}\times\mathbb{C}$ with relatively symplectically aspherical ends. More
precisely, assuming $\omega|_{\pi_{2}(M)}\equiv 0$ and
$\omega|_{\pi_{2}(M,S)}\equiv 0$ implies that
$(\omega\oplus-\omega)|_{\pi_{2}(M\times M^{-},S\times S)}\equiv 0$,
$(\omega\oplus-\omega)|_{\pi_{2}(M\times M^{-},\Delta)}\equiv 0$,
$(\omega\oplus-\omega)|_{\pi_{2}(M\times M^{-},\Gamma_{\tau_{S}^{-1}})}\equiv
0$ and $(\omega\oplus-\omega\oplus\omega_{\mathbb{C}})|_{\pi_{2}(M\times
M^{-}\times\mathbb{C},V_{MV})}\equiv 0$. The latter follows from an argument
very similar to the proof of the corresponding result on exactness and
monotonicity in [MW18, Lemma 6.2, 6.3].

Floer theory for $V_{MW}$ and the ends is therefore well-defined. The cone-
decomposition result 4.2 from Biran-Cornea [BC13], [BC14] therefore yields a
long exact sequence of graded Lagrangian Floer cohomology groups [MW18,
Theorem 6.4]:

$\displaystyle\dots\to\mathrm{HF}^{k}(K,$ $\displaystyle S\times
S)\xrightarrow{\mu^{2}(B,-)}\mathrm{HF}^{k}(K,\Delta)$
$\displaystyle\xrightarrow{\mu^{2}(A,-)}\mathrm{HF}^{k}(K,\Gamma_{\tau^{-1}})\xrightarrow{\mu^{2}(C,-)}\mathrm{HF}^{k+1}(K,S\times
S)\to\dots$

for any admissible Lagrangian submanifold $K\subset\left(M\times
M,\omega\oplus-\omega\right)$. This is precisely the sequence (4). As
indicated, the maps are given by $\mu^{2}$ operations with elements
$A\in\mathrm{HF}^{0}(\Delta,\Gamma_{\tau^{-1}})$, $B\in\mathrm{HF}^{0}(S\times
S,\Delta)$ and $C\in\mathrm{HF}^{1}(\Gamma_{\tau_{S}^{-1}},S\times S)$. $A,B$
and $C$ are independent of $K$.

######  Proposition 5.3.

If $2c_{1}(M)=0$ in $\mathrm{H}^{2}(M;\mathbb{Z})$ and $2c_{1}(M,S)=0$ in
$\mathrm{H}^{2}(M,S)$ then $A\neq 0$. 222The condition $2c_{1}(M,S)=0$ is
automatic for $n\geq 2$. For $n=1$ it’s equivalent to $S$ being a non-
contractible circle.

###### Proof.

Under the condition on the Chern class, everything becomes
$\mathbb{Z}$-graded, see [Sei00]. For $K=\Delta$, the sequence becomes

$\displaystyle\dots\to\mathrm{HF}^{k}(S,S)\to\mathrm{HF}^{k}(\mathrm{id})\xrightarrow{\Psi}\mathrm{HF}^{k}(\Delta,\Gamma_{\tau^{-1}})\to\mathrm{HF}^{k+1}(S,S)\to\dots.$

Assume by contradiction that $A=0$. Then $\Psi=0$ and hence we get
$\mathbb{Z}$-graded isomorphisms

$\displaystyle\mathrm{H}^{*}(S;\Lambda)\cong
QH^{*}(S)\cong\mathrm{HF}^{*}(S,S)\cong\mathrm{HF}^{*}(\mathrm{id})\cong\mathrm{QH}^{*}(M)\cong\mathrm{H}^{*}(M;\Lambda).$

This is a impossible. We conclude that $A\neq 0$. ∎

##  6\. Symmetry of the Mak-Wu cobordism

We assume that $\iota^{*}c\simeq\mathrm{id}$ or $\iota^{*}c\simeq r$. In this
section, we prove Theorem B, i.e. that $\Phi(V_{MW})$ is Hamiltonian isotopic
to $V_{MW}$, where $\Phi(x,y,z)=(c(y),c(x),z).$

###  6.1. Linear approximation of anti-symplectic involution.

Any map

$c_{0}\colon S^{n}\to S^{n}$

induces an anti-symplectic involution

$c_{0}^{*}\colon T^{*}S^{n}\to T^{*}S^{n}$

via $c_{0}^{*}(q,p)=(c_{0}(q),-p\circ(Dc_{0})_{c_{0}(q)})$. We assume that
$c_{0}=\mathrm{id}$ or $c_{0}=r$ (or more generally that $c_{0}$ is any
isometry). Note that we secretly identify $T^{*}S^{n}$ and $T_{*}S^{n}$ via
the canonical isomorphism $\alpha\colon T_{*}S^{n}\to T^{*}S^{n}$ coming from
the standard Riemannian metric. The following diagram commutes:

$\textstyle{T^{*}S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c_{0}^{*}}$$\textstyle{T^{*}S}$$\textstyle{TS\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{-Dc_{0}}$$\scriptstyle{\alpha}$$\scriptstyle{\cong}$$\textstyle{TS\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\scriptstyle{\cong}$

The following Lemma collects some properties of $c_{0}^{*}$.

######  Lemma 6.1.

Assume that $c_{0}=\mathrm{id}$ or $c_{0}=r$ (or more generally that $c_{0}$
is any isometry). Then $c_{0}^{*}\colon T^{*}S\to T^{*}S$ satisfies

  * •

$c_{0}^{*}(\xi)=-\phi_{s}^{\sigma}(-c_{0}^{*}(-\phi_{s}^{\sigma}(-\xi)))$

  * •

$\norm{c_{0}^{*}(-\phi_{s}^{\sigma}(-\xi))}=\norm{\xi}.$

###### Proof.

Let $\xi\in T_{x}^{*}S\cong T_{x}S$. Let $\gamma$ be the unique geodesic in
$S$ with $\gamma(0)=x$ and $\gamma^{\prime}(0)=-\xi$. Then
$\phi_{s}^{\sigma}(-\xi)=\gamma^{\prime}(s)$. Note that
$c_{0}^{*}(\gamma^{\prime}(s))=-dc(\gamma^{\prime}(s))=-(c\circ\gamma)^{\prime}(s)$.
Moreover, $(c\circ\gamma)^{\prime}(0)=dc(-\xi)=-c_{0}^{*}(-\xi)$, hence

$\displaystyle\phi_{s}^{\sigma}(c_{0}^{*}\phi_{s}^{\sigma}(-\xi))$
$\displaystyle=\phi_{s}^{\sigma}(-\phi_{s}^{\sigma}(-c_{0}^{*}(-\xi)))$
$\displaystyle=c_{0}^{*}(-\xi).$

$c_{0}^{*}$ commutes with the minus sign because it is linear. So the first
claim follows. For the second, note that both $\phi_{s}^{\sigma}$ and
$c_{0}^{*}$ both preserve the length induced by $g$. The latter follows from
$c_{0}$ being an isometry. ∎

The next proposition shows that any anti-symplectic involution $c$ on
$T^{*}S^{n}$ as before locally looks like $c_{0}^{*}$ for $c_{0}=\mathrm{id}$
or $c_{0}=r$.

######  Proposition 6.2.

Let $c\colon T^{*}S^{n}\rightarrow T^{*}S^{n}$ be an anti-symplectic
involution restricting to $\sigma\colon S^{n}\longrightarrow S^{n}$ where
either $\sigma\simeq\mathrm{id}$ or $\sigma\simeq r$. Then for every
$\eta_{2}>0$ there exists $\eta_{2}>\eta_{1}>0$ and a Hamiltonian isotopy

$\psi_{t}^{H}\colon T^{*}S^{n}\longrightarrow T^{*}S^{n}$

with $\operatorname{supp}H\subset T^{*}_{\eta_{2}}S^{n}$ such that

$c\psi^{H}_{1}=c_{0}^{*}\;\text{ on }\;T^{*}_{\eta_{1}}S^{n},$

where $c_{0}=\mathrm{id}$ or $c_{0}=r$.

###### Proof.

Consider the symplectomorphism $\psi:=c\circ\sigma^{*}$. Write in local
coordinates $\psi(q,p)=(u(q,p),v(q,p))$ with $u(q,p)\in S^{n}$ and $v(q,p)\in
T_{u(q,p)}S^{n}$. Since $c=T^{*}\sigma$ on $S^{n}$ we have $u(q,0)=q$.
Consider the following isotopy of symplectomorphisms $\psi_{t}\colon
T^{*}S^{n}\to T^{*}S^{n}$ between $\psi_{0}=\mathrm{id}$ and
$\psi_{1}(q,p)=\psi$:

$\displaystyle\psi_{t}(q,p)=\begin{cases}(u(q,tp),\frac{v(q,tp)}{t})\qquad\qquad\;t\neq
0\\\ (u(q,0),(\partial_{p}v(q,0))p)\qquad t=0\end{cases}$

$\psi_{0}(q,p)=(q,p)$ because as it can be seen from writing $D\psi_{(q,0)}$
in local coordinates:

$\displaystyle D\psi_{(q,0)}=\begin{pmatrix}&\mathrm{id}&\partial_{q}v(q,0)\\\
&\partial_{p}u(q,0)&\partial_{p}v(q,0)\end{pmatrix}$

and using that $D\psi_{(q,0)}$ is a symplectic matrix, we get

$\partial_{p}v(q,0)=\mathrm{id}.$

$\psi_{t}$ is a Hamiltonian isotopy: For $n\geq 2$ this is automatic. For
$n=1$ it follows from $\psi_{t}(S^{n})=S^{n}$. Concatenate $\psi_{t}$ with a
Hamiltonian isotopy $\sigma^{*}\simeq c_{0}^{*}$. Finally cut off the
Hamiltonian so that the resulting Hamiltonian $H$ has support in
$T^{*}_{\eta_{2}}S^{n}$. Clearly, $\psi_{1}^{H}=c\circ c_{0}^{*}$ on
$T^{*}_{\eta_{1}}S^{n}$ for $\eta_{1}$ small enough. In particular,
$c\psi_{1}^{H}=c_{0}^{*}$ on $T^{*}_{\eta_{1}}S^{n}$. ∎

###  6.2. The symmetry of the surgery part.

Let $V\subset M$ be a Weinstein neighbourhood of $S$ and $\varphi\colon V\to
T_{\delta}^{*}S^{n}$ a symplectomorphism for some $\delta>0$. Let
$0<\epsilon<\delta$ small enough, such that $c(U)\subset U$ for
$U:=\varphi^{-1}(T_{\epsilon}^{*}S^{n})$. Consider $\Phi$ as a map

$U\times U\times\mathbb{C}\to V\times V\times\mathbb{C}.$

This induces via $\widetilde{\varphi}\times\mathrm{id}$ the map

$\displaystyle\Phi\colon T_{\epsilon}^{*}S^{n}\times
T_{\epsilon}^{*}S^{n}\times\mathbb{C}$ $\displaystyle\to
T_{\delta}^{*}S^{n}\times T_{\delta}^{*}S^{n}\times\mathbb{C}$
$\displaystyle(\xi_{1},\xi_{2},z)$
$\displaystyle\mapsto(c(-\xi_{2}),c(\xi_{1}),z).$

Let $\psi_{t}\colon T^{*}S^{n}\to T^{*}S^{n}$ be the Hamiltonian isotopy from
Proposition 6.2. Consider the Hamiltonian isotopy

$\displaystyle\Psi_{t}\colon T_{\delta}^{*}S^{n}\times
T_{\delta}^{*}S^{n}\times T^{*}\mathbb{R}$ $\displaystyle\to
T_{\delta}^{*}S^{n}\times T_{\delta}^{*}S^{n}\times T^{*}\mathbb{R}$
$\displaystyle(\xi_{1},\xi_{2},p)$
$\displaystyle\mapsto(\psi_{t}(\xi_{1}),-\psi_{t}(-\xi_{2}),p).$

We consider surgery in $T_{\epsilon}^{*}S^{n}$, so that the handle
$\hat{H}_{\nu}$ is contained in $T_{\epsilon}^{*}S^{n}$. We claim that

  * •

$\Psi_{1}(\hat{H}_{\nu})=\Phi(\hat{H}_{\nu})$,

  * •

$\Psi_{t}|_{S\times S\times\mathbb{R}}=\mathrm{id}$,

  * •

$\Psi_{t}(N_{\Delta_{S}}^{*}\times\\{p\\})=N_{\Delta_{S}}^{*}\times\\{p\\}$
for any $p\in i\mathbb{R}$,

  * •

$\pi_{\mathbb{C}}\circ\Psi_{t}=\pi_{\mathbb{C}}$.

We check these properties:

  * •

Let $\xi\in T^{*}S$ and $(q,p)\in T^{*}\mathbb{R}$ such that

$\sqrt{\norm{\xi}^{2}+p^{2}}<\epsilon.$

We introduce the following abbreviations:

$s(\norm{\xi},\lvert
p\rvert)=\nu\left(\sqrt{\norm{\xi}^{2}+p^{2}}\right)\frac{\norm{\xi}}{\sqrt{\norm{\xi}^{2}+p^{2}}}$

and

$r(\norm{\xi},\lvert
p\rvert)=\nu\left(\sqrt{\norm{\xi}^{2}+p^{2}}\right)\frac{\lvert
p\rvert}{\sqrt{\norm{\xi}^{2}+p^{2}}}.$

Elements of $\hat{H}_{\nu}$ are of the form

$\alpha:=(\xi,\psi_{s(\norm{\xi},\lvert
p\rvert)}^{\sigma}(-\xi),(r(\norm{\xi},\lvert p\rvert)+q,p)).$

Therefore, elements of $\Psi_{1}(\hat{H}_{\nu})$ are of the form

$\Psi_{1}(\alpha)=(cc_{0}^{*}(\xi),-cc_{0}^{*}(-\psi_{s(\norm{\xi})}^{\sigma}(-\xi)),(r(\norm{\xi},\lvert
p\rvert)+q,p)).$

Put

$\zeta:=c_{0}^{*}(-\phi_{s}^{\sigma}(-\xi)).$

Then $\norm{\zeta}=\norm{\xi}$ by part $2$ of Lemma 6.1. By part $1$ of Lemma
6.1 we have the equality

$c_{0}^{*}(\xi)=-\psi_{s}^{\sigma}(-c_{0}^{*}(-\psi_{s}^{\sigma}(-\xi)))=-\psi_{s}^{\sigma}(-\zeta).$

Thus

$\Psi_{1}(\alpha)=(c(-\psi_{s(\norm{\zeta})}^{\sigma}(-\zeta)),-c(\zeta),(r(\norm{\zeta},\lvert
p\rvert)+q,p))$

which are precisely the elements of $\Phi(\hat{H}_{\nu})$.

  * •

${\Psi_{t}}(S\times S\times\mathbb{R})=S\times S\times\mathbb{R}$: For
$(x,y)\in S\times S$

$\displaystyle\Psi_{t}(x,y,p)$ $\displaystyle=(\psi_{t}(x),-\psi_{t}(-y),p)\in
S\times S\times\mathbb{R}$

because $\psi_{t}(S)=S$.

  * •

$\Psi_{t}(\xi,-\xi,p)=(\psi_{t}(\xi),-\psi_{t}(\xi),p)$ and thus
$\Psi_{t}(N_{\Delta_{S}}^{*}\times\\{p\\})=N_{\Delta_{S}}^{*}\times\\{p\\}$.

  * •

$\pi_{\mathbb{C}}\circ\Psi_{t}=\pi_{\mathbb{C}}$ is clear.

Thus the surgery model $(S\times
S\times\mathbb{R})\\#_{\Delta_{S}}N^{*}_{\Delta_{S}\times\\{0\\}}$ is
Hamiltonian isotopic to the image of itself under $\Phi$. The smoothing and
the extension to a cobordism with ends $S\times S$, $\Delta$ and $(S\times
S)\\#_{\Delta_{S}}\Delta$ can be done while keeping the Hamiltonian isotopy
type. Hence the surgery part of the cobordism is Hamiltonian isotopic to the
image of itself under $\Phi$.

###  6.3. The symmetry of the suspension part.

This is very similar to the preceeding surgery part. Again, it is enough to
show the statement for the surgery model. Let $(S\times
S)\\#_{\Delta_{S}}^{\nu_{t}}\Delta$, $t\in[0,1]$ be a Hamiltonian isotopy,
where all $\nu_{t}$ are admissible, except for $\nu_{1}$ which coincides with
$\nu_{\epsilon}^{\mathrm{Dehn}}$. The Hamiltonian $K_{t}\colon T^{*}(S\times
S)\to T^{*}(S\times S)$ generating the isotopy can be chosen to be of the form
$K_{t}(\xi_{1},\xi_{2})=K_{t}(\norm{\xi_{1}},\norm{\xi_{2}})$, see [MW18,
Lemma 3.6]. Moreover, $K_{t}$ can be chosen to be zero near $0$ and near $1$.
The suspension cobordism is the cylindrical extension of the Lagrangian

$\mathcal{S}:=\left\\{(\psi_{t}^{K}(x),t-iK_{t}(\psi_{t}^{K}(x)))\in M\times
M^{-}\times\mathbb{C}\,\big{|}\,x\in H^{\nu_{0}},t\in[0,1]\right\\}.$

Consider the Hamiltonian isotopy $\Psi_{t}$ from before. We claim that

  * •

$\Psi_{1}(\mathcal{S})=\Phi(\mathcal{S})$

  * •

$\Psi_{t}\left({\left((S\times
S)\\#_{\Delta_{S}}^{\nu_{0}}\Delta\right)\times\\{p\\}}\right)={\left((S\times
S)\\#_{\Delta_{S}}^{\nu_{0}}\Delta\right)\times\\{p\\}}$ for
$p\in\mathbb{R}_{<0}$

  * •

$\Psi_{t}\left({\left((S\times
S)\\#_{\Delta_{S}}^{\nu_{0}}\Delta\right)\times\\{p\\}}\right)={\left((S\times
S)\\#_{\Delta_{S}}^{\nu_{1}}\Delta\right)\times\\{p\\}}$ for
$p\in\mathbb{R}_{>1}$

  * •

$\pi_{\mathbb{C}}\circ\Psi_{t}=\pi_{\mathbb{C}}$

Let us check these properties.

  * •

Elements of “the handle part” of $\mathcal{S}$ can be written as

$\alpha=(\xi,\psi^{\sigma}_{\nu(||\xi||)}(-\xi),t-iK_{t}(\norm{\xi}))$

for some $\xi\in T_{\epsilon}^{*}S$. Hence elements of the corresponding part
of $\Psi_{1}(\mathcal{S})$ are of the form

$\Psi_{1}(\alpha)=(cc_{0}^{*}(\xi),-cc_{0}^{*}(-\psi^{\sigma}_{\nu(\norm{\xi})}(-\xi)),t-iK_{t}(\norm{\xi}).$

Elements of the corresponding part of $\Phi(\mathcal{S})$ are of the form

$(c(-\psi^{\sigma}_{\nu(\norm{\xi})}(-\zeta)),-c(\zeta),t-iK(\norm{\eta})$

for $\zeta\in T_{\epsilon}^{*}S$. As before, using Lemma 6.1, the elements are
in $1:1$-correspondence via $\zeta=c_{0}^{*}(-\phi_{\nu(\norm{\xi})}(-\xi))$.

  * •

It is very similar, but simpler to see that $\Psi_{t}$ preserves
$H^{\nu_{0}}\times\\{t\\}$ for $t\in\mathbb{R}_{<0}$, and also
$H^{\nu_{1}}\times\\{t\\}$ for $t\in\mathbb{R}_{>1}$.

  * •

The last item is obvious.

Therefore, the suspension part $\mathcal{S}$ of the cobordism is Hamiltonian
isotopic to $\Phi(\mathcal{S})$.

The symmetry of the surgery part shown in section 6.2 and the symmetry of the
suspension part shown above together prove Theorem B.

##  7\. Background on Floer cohomology

###  7.1. Floer cohomology for symplectomorphisms.

For convenience of the reader we briefly collect the basic ideas and notation
for Floer cohomology of a symplectomorphism following [DS94]. For more
detailed expositions, we refer the reader to [DS94] for the monotone case, and
to [Sei97b] and [Lee05] for $W^{+}$-symplectic manifolds.

Let $(M,\omega)$ be a closed symplectically aspherical symplectic manifold.
Let $f\in\mathrm{Symp}(M)$ be a symplectomorphism. We first need to choose a
Hamiltonian perturbation, namely a family of Hamiltonian functions
$\\{H_{s}\colon X\longrightarrow\mathbb{R}\\}_{s\in\mathbb{R}}$. It should be
$f$-periodic, in the sense that

$H_{s}=H_{s+1}\circ f.$

Roughly speaking, Floer cohomology of $f$ is Morse cohomology on the twisted
loop space

$\Omega_{f}:=\left\\{x\in
C^{\infty}(\mathbb{R},X)\;\mid\;x(s+1)=f(x(s))\right\\}$

with the closed $1$-form

$\lambda_{H}(x)(\xi)=\int_{0}^{1}\omega\left(\dot{x}(s)-X^{H}_{s}(x(s)),\xi(s)\right)\,ds.$

Here, $X^{H}_{s}$ denotes the Hamiltonian vector field of $H_{s}$. We write
$P_{f}(H)$ for the set of $x\in\Omega_{f}$ satisfying
$\dot{x}(s)=X_{s}^{H}(x(s))$. For a generic choice of $H$, $P_{f}(H)$ is a
finite set. The vector space underlying the Floer complex is the
$\Lambda$-vector space generated by $P_{f}(H)$:

$\displaystyle\mathrm{CF}^{*}(f;H)=\bigoplus_{x\in P_{f}(H)}\Lambda x.$

$\mathrm{CF}^{*}(f)$ is $\mathbb{Z}/2$-graded as follows. A generator $x\in
P_{f}(H)$ corresponds to a fixed point $x(0)$ of
${f}_{H}:=(\psi_{1}^{H})^{-1}f$. The degree $\mathrm{deg}(x)\in\mathbb{Z}/2$
of $x$ is related to the index of $x(0)$ by

$(-1)^{\mathrm{deg}(x)}=\mathrm{sign}\left(\det(\mathrm{id}-(Df_{H})_{x(0)})\right).$

To define the differential, we need to choose a family of almost complex
structures $\mathcal{J}=\\{J_{s}\\}_{s\in\mathbb{R}}$ on $M$, compatible with
$\omega$ and $f$-periodic, meaning $J_{s}={f}^{*}(J_{s+1})$. One considers
finite-energy solutions

$u\colon\mathbb{R}\times\mathbb{R}\longrightarrow X,(s,t)\longmapsto u(s,t)$

of Floer’s equation

$\frac{\partial u}{\partial t}+J_{s}(u)\left(\frac{\partial u}{\partial
s}-X^{H}_{s}(u)\right)=0,$

which are $f$-periodic in $s$, $u(s+1,t)=f(u(s,t))$, and satisfy the
asymptotic conditions

$\lim_{t\to-\infty}u(s,t)=x(s)\text{ and }\lim_{t\to\infty}u(s,t)=y(s)$

for some Hamiltonian chords $x,y\in\Omega_{f}$. Consider the moduli space
$\mathcal{M}(x,y;\mathcal{J},H)$ of all such solutions $u$. For regular
$(\mathcal{J},H)$, the moduli space is a smooth manifold. $\mathbb{R}$ acts on
the one-dimensional component $\mathcal{M}^{1}(x,y;\mathcal{J},H)$ by
translation, and the quotient set
$\widehat{\mathcal{M}}^{1}(x,y;\mathcal{J},H)=\mathcal{M}^{1}(x,y;\mathcal{J},H)/\mathbb{R}$
is discrete.

The Floer differential
$\partial\colon\mathrm{CF}^{*}(f;\mathcal{J},H)\longrightarrow\mathrm{CF}^{*}(f;\mathcal{J},H)$
is defined by

$\displaystyle\partial(x)=\sum\limits_{y\in
P_{\varphi}(H)}\sum\limits_{u\in\widehat{\mathcal{M}}^{1}(x,y;\mathcal{J},H)}y.$

The homology of the chain complex $CF^{*}(f)$ is called the Floer cohomology
of $f$ with Floer data $(\mathcal{J},H)$ and denoted by
$HF^{*}(f;\mathcal{J},H)$.

There are graded continuation maps for different choices of Floer datum:
Suppose $(H,\mathcal{J})$ and $(H^{\prime},\mathcal{J}^{\prime})$ are regular
Floer data as above. Choose a family $(H_{s,t},J_{s,t})$ that satisfies the
periodicity assumptions

$J_{s}=f^{*}(J_{s+1})\text{ and }J_{s}^{\prime}=f^{*}(J_{s+1}^{\prime})$

and interpolate between $(H_{s},J_{s})$ and $(H_{s}^{\prime},J_{s}^{\prime})$,
i.e.

$\displaystyle H_{s,t}=H_{s}^{\prime},J_{s,t}=J_{t}^{\prime}\qquad$
$\displaystyle\text{for $t$ near $-\infty$},$ $\displaystyle
H_{s,t}=H_{s},J_{s,t}=J_{t}\qquad$ $\displaystyle\text{for $t$ near
$\infty$}.$

We denote by $\mathcal{M}(x,y;J_{s,t},H_{s,t})$ the moduli space of solutions
to the $1$-parametric Floer equation

$\frac{\partial u}{\partial t}+J_{s,t}(u)\left(\frac{\partial u}{\partial
s}-X^{H}_{s,t}(u)\right)=0$

that are $f$-periodic in $s$ and tend to $x$ and $y$ as $t\to\pm\infty$. For
generic choice of $(H_{s,t},J_{s,t})$ the moduli space is a manifold and its
zero-dimensional component $\mathcal{M}^{0}(x,y;J_{s,t},H_{s,t})$ is discrete.
The chain-level continuation map is the chain map

$\displaystyle C_{H_{s,t},J_{s,t}}\colon\mathrm{CF}^{*}(f;\mathcal{J},H)$
$\displaystyle\longrightarrow\mathrm{CF}^{*}(f,\mathcal{J}^{\prime},H)$
$\displaystyle x$ $\displaystyle\longmapsto\sum\limits_{y\in
P_{\varphi}(H)}\sum\limits_{u\in\mathcal{M}^{0}(x,y;J_{s,t},H_{s,t})}y.$

The map induced in cohomology is independent of the choice of homotopy
$(H_{s,t},J_{s,t})$. This allows us to identify the cohomology groups
$\mathrm{HF}^{*}(f,\mathcal{J},H)$ and
$\mathrm{HF}^{*}(f,\mathcal{J}^{\prime},H^{\prime})$ and simply write
$\mathrm{HF}^{*}(f)$ for the cohomology group of $f$.

###  7.2. Lagrangian Floer cohomology.

We recall here Lagrangian Floer cohomology for relatively aspherical
Lagrangians. Given two closed Lagrangians $L_{0},L_{1}\subset M$, choose $H$
so that $\psi_{1}^{H}(L_{0})\cap L_{1}$ is a transverse intersection at
finitely many points. Then the underlying $\Lambda$-vectorspace of
$CF(L_{0},L_{1};H,J)$ is generated by those points. The differential is
defined by counting $J$-holomorphic strips, using a $w$-compatible almost
complex structure $J$ on $M$. Floer’s equation reads:

$\begin{cases}\frac{\partial u}{\partial t}+J_{s}(u)\left(\frac{\partial
u}{\partial s}-X_{s}^{H}(u)\right)=0\\\ u(0,t)\in L_{0},\qquad u(1,t)\in
L_{1}\\\ \lim_{t\to-\infty}u(s,t)=\psi_{s}^{H}(z)\text{ for some }z\in
L_{0}\\\ \lim_{t\to\infty}u(s,t)=\psi_{s}^{H}(w)\text{ for some }w\in L_{0}\\\
\end{cases}\,.$

If $L_{0}$ and $L_{1}$ are oriented, we define the degree of $x$ as follows:

$(-1)^{\mathrm{deg}(x)}=(-1)^{\frac{n(n+1)}{2}}\nu(x;L_{0},L_{1}),$

where $\nu(x;L_{0},L_{1})\in\\{\pm 1\\}$ denotes the intersection index of
$L_{0}$ and $L_{1}$ at $x$. This number is defined to be $+1$ if
$v_{1},\dots,v_{2n}$ is a positive basis for $T_{x}M$ whenever
$v_{1},\dots,v_{n}$ is a positive basis for $T_{x}L_{0}$ and
$v_{n+1},\dots,v_{2n}$ is a positive basis for $T_{x}L_{1}$. See [Sei00,
Section 2d] for the grading, and [RS22] for the intersection index.

###  7.3. Proof of Proposition 3.2.

Choose Floer datum $H_{s}$ and $J_{s}$ as in Section 7.1. The generators of
$\mathrm{CF}(\phi;H_{s},J_{s})$ are points $x\in M$ such that
$\phi(x)=\phi_{1}^{H}(x)$. For the Lagrangian Floer complex, we choose the
following Floer data:

$\displaystyle
K_{s}(x,y)=-\frac{1}{2}H_{\frac{1-s}{2}}(x)-\frac{1}{2}H_{\frac{s+1}{2}}(y).$

and

$\displaystyle\tilde{J}_{s}:=\tilde{J}_{s}^{(1)}\oplus\tilde{J}_{s}^{(2)}:=J_{\frac{1-s}{2}}\oplus(-J_{\frac{s+1}{2}}).$

Generators of $\mathrm{CF}(\Delta,\Gamma_{\phi};K_{s},\tilde{J}_{s})$ are of
the form $(x,\phi(x))\in\psi_{1}^{K}(\Gamma_{\mathrm{id}})$. We show that the
map

$\displaystyle\mathrm{CF}(\phi;H_{s},J_{s})$
$\displaystyle\longrightarrow\mathrm{CF}(\Delta,\Gamma_{\phi};K_{s},\tilde{J}_{s})$
$\displaystyle x$ $\displaystyle\longmapsto(x,\phi(x))$

is a chain isomorphism. This follows from checking that generators get mapped
to generators, and solutions to

$\begin{cases}\frac{\partial v}{\partial
t}+\tilde{J}_{s}(v)\left(\frac{\partial v}{\partial
s}-X_{s}^{K}(v)\right)=0\\\ v(0,t)\in\Delta\text{ and
}v(1,t)\in\Gamma_{\phi}\\\ \lim_{t\to-\infty}v(s,t)=\psi_{s}^{K}(z)\text{ for
some }z\in\Delta\\\ \lim_{t\to\infty}v(s,t)=\psi_{s}^{K}(w)\text{ for some
}w\in\Delta\\\ \end{cases}$

are in one to one correspondence to solutions of

$\begin{cases}\frac{\partial u}{\partial t}+J_{s}(u)\left(\frac{\partial
u}{\partial s}-X_{s}^{H}(u)\right)=0\\\ u(1,t)=\phi(u(0,t)\\\
\lim_{t\to-\infty}u(s,t)=\psi_{s}^{K}(x)\\\
\lim_{t\to\infty}u(s,t)=\psi_{s}^{K}(y).\\\ \end{cases}$

The correspondence is given by

$\displaystyle v(s,t)=(v_{1}(s,t),v_{2}(s,t))\longleftrightarrow
u(s,t)=\begin{cases}v_{1}(1-2s),-2t)\qquad s\in[0,\frac{1}{2}]\\\
v_{2}(2s-1,-2t)\qquad s\in[\frac{1}{2},1].\end{cases}$

For the grading: Let $(x,x)\in\Delta\cap\Gamma_{\phi}$. Let $\mathcal{B}^{M}$
be a basis of $T_{x}M$ and consider the bases $\mathcal{B}^{\Delta}$ and
$\mathcal{B}^{\Gamma_{\phi}}$ of $T_{(x,x)}\Delta$ and
$T_{(x,x)}\Gamma_{\phi}$ associated to $\mathcal{B}^{M}$. Note that
$\mathcal{B}^{\Delta}$ and $\mathcal{B}^{\Gamma_{\phi}}$ are either both
positive or both negative. Hence $\nu(x,x)=1$ if and only if the basis
$\mathcal{B}=\left(\mathcal{B}^{\Delta},\mathcal{B}^{\Gamma_{\phi}}\right)$ is
a positive of $T_{(x,x)}M\times M^{-}$ One computes

$\mathcal{B}=\begin{pmatrix}\mathrm{Id}&\mathrm{Id}\\\
\mathrm{Id}&D\phi\end{pmatrix}\mathcal{B}_{0},$

where $\mathcal{B}_{0}=\left((\mathcal{B}^{M},0),(0,\mathcal{B}^{M})\right)$.
$\mathcal{B}_{0}$ is positively oriented if and only if $n$ is even. The
determinant of the matrix is
$\det(D\phi-\mathrm{Id})=\det(\mathrm{Id}-D\phi)$. Hence

$\nu(x,x)=(-1)^{n}\mathrm{sign}\det(\mathrm{Id}-D\phi)$

and

$\displaystyle(-1)^{\mathrm{deg}(x,x)}$
$\displaystyle=(-1)^{n}(-1)^{\frac{2n(2n+1)}{2}}\nu(x,x)$
$\displaystyle=(-1)^{n}(-1)^{\frac{2n(2n+1)}{2}}(-1)^{n}\mathrm{sign}\det(\mathrm{Id}-D\phi)$
$\displaystyle=(-1)^{n}(-1)^{\frac{2n(2n+1)}{2}}(-1)^{n}(-1)^{\mathrm{deg}(x)}$
$\displaystyle=(-1)^{\mathrm{deg}(x)}.$

This shows that the isomorphism above indeed preserves the grading.

## Appendix A. Algebraic background.

We briefly explain the algebraic background relevant for the definition of the
the main character of this paper: the element $A\in\mathrm{HF}(\tau^{-1})$. We
follow the conventions for $A_{\infty}$-machinery from [Sei08].

Suppose $\mathcal{A}$ is a homologically unital $A_{\infty}$-category. The
Yoneda embedding is a functor

$\mathcal{Y}\colon\mathcal{A}\rightarrow mod_{\mathcal{A}}$

taking an object $L$ to the $\mathcal{A}$-module $\mathcal{Y}(L)$ defined by

$\mathcal{Y}(L)(K):=Mor_{\mathcal{A}}(K,L).$

and

$\mu^{d}_{\mathcal{Y}(L)}(b,a_{d-1},\dots,a_{1}):=\mu^{d}(b,a_{d-1},\dots,a_{1})$

for $a_{i}\in Mor_{\mathcal{A}}(K_{i-1},K_{i})$, $i\in\\{1,\dots,d-1\\}$ and
$b\in\mathcal{Y}(L)(K_{d-1})=Mor_{\mathcal{A}}(K_{d-1},L)$.

By [Sei08, Section 2g] the Yoneda embedding induces a unital, full and
faithfull embedding

$\mathrm{H}(\mathcal{Y})\colon\mathrm{H}(\mathcal{A})\to\mathrm{H}(mod_{\mathcal{A}}).$

The derived cateogory $\mathcal{DA}$ of $\mathcal{A}$ can be constructed as
follows: Take a triangulated completion of the image of $\mathcal{Y}$ in
$mod_{\mathcal{A}}$ and take its homology category.

The following is an immediate consequence of the properties of the Yoneda
embedding.

######  Corollary 7.1.

Each $f\in Mor_{D\mathcal{A}}(\mathcal{Y}(L_{1}),\mathcal{Y}(L_{2}))$ can be
represented by $\mathcal{Y}(\alpha)$ for some $\alpha\in
Mor_{\mathcal{A}}(L_{1},L_{2})$. Moreover, $[\alpha]\in
Mor_{H(\mathcal{A})}(L_{1},L_{2})$ is uniquely defined.

###### Proof.

First, note that

$Mor_{D\mathcal{A}}(\mathcal{Y}(L_{1}),\mathcal{Y}(L_{2}))\cong\mathrm{H}(Mor_{mod_{\mathcal{A}}}(\mathcal{Y}(L_{1}),\mathcal{Y}(L_{2}))).$

For any object $K$, $\mathcal{Y}(\alpha)$ determines the map

$\mathcal{Y}(L_{1})(K)\cong
Mor(K,L_{1})\xrightarrow{\mu^{2}(\alpha,-)}Mor(K,L_{2})\cong\mathcal{Y}(L_{2})$

The existence and uniqueness of $\alpha$ follow immediately from
$\mathrm{H}(\mathcal{Y})$ being full and faithful. ∎

These notions are applied in this paper to the $A_{\infty}$-category
$\mathcal{F}uk(M)$.

## Acknowledgements

This work is part of my doctoral studies at ETH under the supervision of Paul
Biran. I would like to express my deep gratitude to Paul Biran for his
guidance, many patient explanations and for sharing his insights with me. I’m
grateful to Jonny Evans for our conversation about examples. I would also like
to thank Alessio Pellegrini for reading this work and helping to improve the
paper. The author was partially supported by the Swiss National Science
Foundation (grant number 200021 204107).

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