Let

Y

be a symplectic manifold, and let

S^{n} \subset Y

be a Lagrangian sphere. We now describe a symplectomorphism called symplectic Dehn twist:

τ_{S^{n}} : Y \to Y,

described in [Sei03].

1: construction of the symplectic dehn twist

Fix the standard metric

g

S^{n}

, and let

B_{r}^{*} S^{n}

be the radius

r

conormal ball of

S^{n}

. We first describe a symplectomorphism of

B_{r}^{*} S^{n}

. Let

π : B_{r}^{*} S^{n} \to S^{n}

be projection to the base. Consider the function

\begin{aligned} f : B_{r}^{*} S^{n} \to & R \\ (q, p) \mapsto & | p |_{g}^{2} . \end{aligned}

The function

f

is a smooth map on

B_{r}^{*} S^{n}

, and the Hamiltonian flow of

f

is the geodesic flow. This is a smooth function on

B_{r}^{*} S^{n} ∖ S^{n}

. On the symplectic manifold

B_{r}^{*} S^{n} ∖ S^{n}

the time

π

flow of

\sqrt{f}

is the antipodal map on the

S^{n}

base ((Dehn twist as surgery)). We take a smooth function

ρ : R \to R

with the property that

ρ \circ f = f

when

f < ϵ

ρ \circ f = \sqrt{f}

when

f > r - ϵ

, and

ρ

is increasing.

Let

H = ρ \circ f : B_{r}^{*} S^{n} \to R

, and let

ϕ_{H} : B_{r}^{*} S^{n} \to B_{r}^{*} S^{n}

be the time-one Hamiltonian isotopy of

H

. Finally, let

- id : S^{n} \to S^{n}

the antipodal map, which extends to a symplectomorphism

- id : B_{r}^{*} S^{n} \to B_{r}^{*} S^{n}

. Define

- ϕ_{H} := - id \circ ϕ_{H}

. Observe that the map

- ϕ_{H} : S^{n} \to S^{n}

is a symplectomorphism of

B_{r}^{*} S^{n}

, which acts by the identity in a neighborhood of

\partial B_{r}^{*} S^{n}

. It acts by the antipodal map on the zero section.

definition 1.0.1

Given a Lagrangian sphere

S^{n} \subset Y

, pick

r

small enough to identify a Weinstein neighborhood

S^{n} \subset B_{r}^{*} S^{n} \subset X

. We define symplectic Dehn twist as the symplectomorphism:

τ_{S^{n}} (x) := {\begin{array}{cc} x & for x \notin B_{r}^{*} S^{n} \\ - ϕ_{H} (x) & for x \in B_{r}^{*} S^{n} \end{array}

figure 1.0.2:Performing a Dehn twist on the zero section of $T^{*} S^{1}$

theorem 1.0.3

Let

X

be a symplectic manifold, and

S \subset X

a Lagrangian sphere, and

L \subset X

another Lagrangian submanifold. There is an exact triangle in the Fukaya category

\dots \to {CF}^{∙} (S, L) \otimes S \to L \overset{ev}{\to} τ_{S} (L) \overset{[1]}{\to} \dots .

Here,

{CF}^{∙} (S, L) \otimes S

is a twisted complex. Recall that (as a vector space)

{CF}^{∙} (S, L) = ⨁_{x \in S \cap L} Λ ⟨ x ⟩

. The twisted complex

{CF}^{∙} (S, L) \otimes S

is given by

⨁_{x \in S \cap L} S ⟨ x ⟩

, which is to say that formal direct sum of copies of

S

whose grading is determined by the intersection points

x

. The differential on a twisted complex is a collection of maps

δ_{x y}^{E} \in {CF}^{∙} (S ⟨ x ⟩, S ⟨ y ⟩)

. The morphism we take is

δ_{x y}^{E} = ⟨ m^{1} (x), y ⟩ id .

We now describe the map

ev : {CF}^{∙} (S, L) \otimes S \to L

. Recall that a morphism of twisted complexes is a collection of maps. We must pick for each

S ⟨ x ⟩

a morphism in

hom (S ⟨ x ⟩, L)

. Fortunately, there is a canonical choice (which is

x

itself).

References

[Sei03]

Paul Seidel. A long exact sequence for symplectic Floer cohomology. Topology, 42:1003--1063, 2003.