Floer cohomology of Dehn twists
along real Lagrangian spheres

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

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

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:

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.

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

Floer-theoretic considerations yield a homomorphism

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 $D\mathcal{F}uk(M\times M^{-})$:

The associated long exact sequence is

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

Consider the symplectomorphism

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

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

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

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).

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.

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].

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

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

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 $\epsilon>0$.

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

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

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

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.

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

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.

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.

Let

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.

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.

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

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

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

$\mathrm{HF}^{*}(f)$ 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 $\mathrm{HF}^{*}(L,K)$ .

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.

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

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.

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)

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

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

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

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 $\varphi=\tilde{c}$.

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