2.1 Dehn twist

Let $S$ be a connected closed manifold equipped with a Riemannian metric $g(\cdot ,\cdot )$ such that every geodesic is closed of length $2\unicode[STIX]{x1D70B}$ (i.e., the shortest period of every unit-speed geodesic is $2\unicode[STIX]{x1D70B}$ ). We identify $T^{\ast }S$ with $TS$ by $g$ and switch freely between the two. The following lemma is well known.

Lemma 2.1. The Hamiltonian $\unicode[STIX]{x1D70E}:T^{\ast }S\rightarrow \mathbb{R}$ defined by

$$\begin{eqnarray}\unicode[STIX]{x1D70E}(\unicode[STIX]{x1D709})=\Vert \unicode[STIX]{x1D709}\Vert\end{eqnarray}$$

for all $q\in S$ and $\unicode[STIX]{x1D709}\in T_{q}^{\ast }S$ has the property that its Hamiltonian vector field $X_{\unicode[STIX]{x1D70E}}$ generates the normalized geodesic flow on $T^{\ast }S\backslash \{0_{\text{section}}\}$ (this is the flow that parallel transports every tangent vector at unit-speed along the geodesic starting from it).

To define the Dehn twist, we need to introduce an auxiliary function. We first consider the case when $S$ is not diffeomorphic to a sphere. For $\unicode[STIX]{x1D716}>0$ small, we define a Dehn twist profile to be a smooth function $\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D716}}^{\text{Dehn}}:\mathbb{R}^{+}\rightarrow \mathbb{R}$ such that (see Figure 2):

1. (1) $\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D716}}^{\text{Dehn}}(r)=2\unicode[STIX]{x1D70B}-r$ for $r\ll \unicode[STIX]{x1D716}$ ;

2. (2) $0<\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D716}}^{\text{Dehn}}(r)<2\unicode[STIX]{x1D70B}$ for all $r<\unicode[STIX]{x1D716}$ ; and

3. (3) $\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D716}}^{\text{Dehn}}(r)=0$ for $r\geqslant \unicode[STIX]{x1D716}$ .

Definition 2.2. If $S$ is not diffeomorphic to a sphere, the model Dehn twist $(\unicode[STIX]{x1D70F}_{S},\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D716}}^{\text{Dehn}})$ on $T^{\ast }S$ is given by

$$\begin{eqnarray}\unicode[STIX]{x1D70F}_{S}(\unicode[STIX]{x1D709})=\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D716}}^{\text{Dehn}}(\Vert \unicode[STIX]{x1D709}\Vert )}^{\unicode[STIX]{x1D70E}}(\unicode[STIX]{x1D709})\end{eqnarray}$$

on $T^{\ast }S-\{0_{\text{section}}\}$ and identity on the zero section.

Figure 2.

A Dehn twist profile $\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D716}}^{\text{Dehn}}$ when $S\neq S^{n}$ .

We will simply write $\unicode[STIX]{x1D70F}_{S}$ instead of $(\unicode[STIX]{x1D70F}_{S},\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D716}}^{\text{Dehn}})$ .

When $S$ is diffeomorphic to a sphere, the spherical Dehn twist profile $\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D716}}^{\text{Dehn},\text{sp}}$ is picked with (1), (2) above replaced by:

( $1^{\prime }$ )

$\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D716}}^{\text{Dehn},\text{sp}}(r)=\unicode[STIX]{x1D70B}-r$ for $r\ll \unicode[STIX]{x1D716}$ ; and

( $2^{\prime }$ )

$0<\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D716}}^{\text{Dehn},\text{sp}}(r)<\unicode[STIX]{x1D70B}$ for all $r<\unicode[STIX]{x1D716}$ .

In this case, the Dehn twist $(\unicode[STIX]{x1D70F}_{S},\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D716}}^{\text{Dehn},\text{sp}})$ is defined analogously but the antipodal map is used to extend smoothly along the zero section instead of the identity map.

Example 2.3. Let $T^{\ast }S^{1}=\mathbb{R}/2\unicode[STIX]{x1D70B}\mathbb{Z}\times \mathbb{R}$ with coordinates $(q,p)\in \mathbb{R}/2\unicode[STIX]{x1D70B}\mathbb{Z}\times \mathbb{R}$ be equipped with the standard symplectic form $\unicode[STIX]{x1D714}_{S^{1}}=dp\wedge dq$ . For a spherical profile $\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D716}}^{\text{Dehn},\text{sp}}$ , $(\unicode[STIX]{x1D70F}_{S^{1}},\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D716}}^{\text{Dehn},\text{sp}})$ is defined by

$$\begin{eqnarray}\unicode[STIX]{x1D70F}_{S^{1}}(q,p)=\left\{\begin{array}{@{}ll@{}}\displaystyle \bigg(q+\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D716}}^{\text{Dehn},\text{sp}}(\Vert p\Vert )\frac{p}{\Vert p\Vert },p\bigg) & \text{for }p\neq 0,\\ (q+\unicode[STIX]{x1D70B},0) & \text{for }p=0.\end{array}\right.\end{eqnarray}$$

Consider the double cover $\unicode[STIX]{x1D704}_{\text{double}}:T^{\ast }S^{1}\rightarrow T^{\ast }\mathbb{RP}^{1}=\mathbb{R}/2\unicode[STIX]{x1D70B}\mathbb{Z}\times \mathbb{R}$ given by

(2.1) $$\begin{eqnarray}\unicode[STIX]{x1D704}_{\text{double}}(q,p)=(2q,{\textstyle \frac{1}{2}}p)=(\widetilde{q},\widetilde{p}).\end{eqnarray}$$

For $(\widetilde{q},\widetilde{p})=\unicode[STIX]{x1D704}_{\text{double}}(q,p)\in T^{\ast }\mathbb{RP}^{1}$ , we define

$$\begin{eqnarray}T(\widetilde{q},\widetilde{p})=\unicode[STIX]{x1D704}_{\text{double}}\circ \unicode[STIX]{x1D70F}_{S^{1}}(q,p)\end{eqnarray}$$

which is independent of the choice of the preimage $(q,p)$ of $(\widetilde{q},\widetilde{p})$ . It is an easy exercise to show that $T$ is Hamiltonian isotopic to $\unicode[STIX]{x1D70F}_{\mathbb{RP}^{1}}$ for the new Dehn twist profile under the change of coordinates (2.1). Also, if we identify $T^{\ast }\mathbb{RP}^{1}$ with $T^{\ast }S^{1}$ so that $\unicode[STIX]{x1D70F}_{S^{1}}$ is well defined on $T^{\ast }\mathbb{RP}^{1}$ , then $T$ is also Hamiltonian isotopic to $\unicode[STIX]{x1D70F}_{S^{1}}^{2}$ for an appropriate choice of spherical profile.

This example has the following well-known immediate generalizations.

Lemma 2.4. Let $\unicode[STIX]{x1D704}_{\text{double}}:T^{\ast }S^{n}\rightarrow T^{\ast }\mathbb{RP}^{n}$ be the symplectic double cover obtained from the double cover of the zero section. For $(\widetilde{q},\widetilde{p})=\unicode[STIX]{x1D704}_{\text{double}}(q,p)\in T^{\ast }\mathbb{RP}^{n}$ ,

$$\begin{eqnarray}T(\widetilde{q},\widetilde{p})=\unicode[STIX]{x1D704}_{\text{double}}\circ \unicode[STIX]{x1D70F}_{S^{n}}(q,p)\end{eqnarray}$$

is well defined and $T$ is Hamiltonian isotopic to $\unicode[STIX]{x1D70F}_{\mathbb{RP}^{n}}$ for an appropriate choice of auxiliary function defining $\unicode[STIX]{x1D70F}_{\mathbb{RP}^{n}}$ .

If $n>1$ , the choice of auxiliary function defining $\unicode[STIX]{x1D70F}_{\mathbb{RP}^{n}}$ is irrelevant up to Hamiltonian isotopy.

Proof. Since the Hamiltonian function is radial and invariant under the antipodal map on $S^{n}$ , the time- $t$ flow on $T^{\ast }\mathbb{R}\mathbb{P}^{n}$ lifts to a time- $t$ flow on $T^{\ast }S^{n}$ . More precisely, the function $\tilde{\unicode[STIX]{x1D70E}}:=\unicode[STIX]{x1D70E}\,\circ \,\unicode[STIX]{x1D704}_{\text{double}}:T^{\ast }S^{n}\rightarrow \mathbb{R}$ is given by $\tilde{\unicode[STIX]{x1D70E}}(\unicode[STIX]{x1D709})=\Vert \unicode[STIX]{x1D709}\Vert$ with respect to the pull-back bundle metric from $T^{\ast }\mathbb{RP}^{n}$ by $\unicode[STIX]{x1D704}_{\text{double}}$ so we have $\unicode[STIX]{x1D704}_{\text{double}}\circ \unicode[STIX]{x1D719}_{t}^{\tilde{\unicode[STIX]{x1D70E}}}=\unicode[STIX]{x1D719}_{t}^{\unicode[STIX]{x1D70E}}\circ \unicode[STIX]{x1D704}_{\text{double}}$ . In particular, we have

(2.2) $$\begin{eqnarray}\unicode[STIX]{x1D704}_{\text{double}}\circ \unicode[STIX]{x1D719}_{\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D716}}^{\text{Dehn}}(\Vert \unicode[STIX]{x1D709}\Vert )}^{\tilde{\unicode[STIX]{x1D70E}}}(\unicode[STIX]{x1D709})=\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D716}}^{\text{Dehn}}(\Vert \unicode[STIX]{x1D704}_{\text{double}}(\unicode[STIX]{x1D709})\Vert )}^{\unicode[STIX]{x1D70E}}\circ \unicode[STIX]{x1D704}_{\text{double}}(\unicode[STIX]{x1D709})\end{eqnarray}$$

for any Dehn twist profile $\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D716}}^{\text{Dehn}}$ . Since the shortest period of geodesics of $S^{n}$ with respect to the pull-back Riemannian metric by $\unicode[STIX]{x1D704}_{\text{double}}$ is $4\unicode[STIX]{x1D70B}$ instead of $2\unicode[STIX]{x1D70B}$ , the left-hand side of (2.2) equals $\unicode[STIX]{x1D704}_{\text{double}}\circ \unicode[STIX]{x1D70F}_{S^{n}}(\unicode[STIX]{x1D709})$ when $\Vert \unicode[STIX]{x1D709}\Vert \neq 0$ . The right-hand side of (2.2) equals $\unicode[STIX]{x1D70F}_{\mathbb{RP}^{n}}\circ \unicode[STIX]{x1D704}_{\text{double}}(\unicode[STIX]{x1D709})$ when $\Vert \unicode[STIX]{x1D709}\Vert \neq 0$ , so the result follows.◻

As usual, one may globalize the model Dehn twist.

For more details and the dependence of choices used to define $\unicode[STIX]{x1D70F}_{S}$ , see [Reference SeidelSei99] and [Reference SeidelSei03].

2.2 Lagrangian surgery through flow handles

2.2.1 Surgery at a point

We first recall the definition of a Lagrangian surgery at a transversal intersection point from [Reference Lalonde and SikoravLS91], [Reference PolterovichPol91] and [Reference Biran and CorneaBC13].

Definition 2.7. Let $a(s),b(s)\in \mathbb{R}$ . A smooth curve $\unicode[STIX]{x1D6FE}(s)=a(s)\,+\,ib(s)\in \mathbb{C}$ is called $\unicode[STIX]{x1D706}$ -admissible if (see Figure 3):

• ∙ $(a(s),b(s))=(-s+\unicode[STIX]{x1D706},0)$ for $s\leqslant 0$ ;

• ∙ $a^{\prime }(s),b^{\prime }(s)<0$ for $s\in (0,\unicode[STIX]{x1D716})$ ; and

• ∙ $(a(s),b(s))=(0,-s)$ for $s\geqslant \unicode[STIX]{x1D716}$ (note that $b(\unicode[STIX]{x1D716})=-\unicode[STIX]{x1D716}$ ).

Figure 3.

Picture of an admissible curve.

The part of a $\unicode[STIX]{x1D706}$ -admissible curve with $s\in [0,\unicode[STIX]{x1D716}]$ can be uniquely captured by $\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D706}}(r):=a(b^{-1}(-r))\in [0,\unicode[STIX]{x1D706}]$ . The main property of an admissible curve can be translated into properties of $\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D706}}$ as follows:

1. (1) $\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D706}}(0)=\unicode[STIX]{x1D706}>0$ , and $\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D706}}^{\prime }(r)<0$ for $r\in (0,\unicode[STIX]{x1D716})$ ;

2. (2) $\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D706}}^{-1}(r)$ and $\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D706}}(r)$ have vanishing derivatives of all orders at $r=\unicode[STIX]{x1D706}$ and $r=\unicode[STIX]{x1D716}$ , respectively.

Figure 4.

Admissible and semi-admissible functions.

Such a function will also be called $\unicode[STIX]{x1D706}$ -admissible. We will frequently use the two equivalent descriptions of admissibility interchangeably.

We also define a class of semi-admissible functions, by relaxing (2) to:

$(2^{\prime })$ $\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D706}}^{\prime }(0)=-\unicode[STIX]{x1D6FC}\in [-\infty ,0]$ . Here $\unicode[STIX]{x1D6FC}=\infty$ if $\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D706}}$ is admissible.

See an example of admissible and semi-admissible function in Figure 4.

Note that in all definitions of (semi-)admissibility there is an extra variable $\unicode[STIX]{x1D716}$ . We will see that the dependence on $\unicode[STIX]{x1D716}$ is not significant in this paper: we fix $\unicode[STIX]{x1D716}$ for each pair of Lagrangian submanifolds $(L_{1},L_{2})$ once and for all. In any surgery constructions appearing later, the resulting surgery manifold yields a smooth family of Lagrangian isotopic submanifolds as $\unicode[STIX]{x1D716}$ varies. In the context of Dehn twists, this family becomes Hamiltonian isotopic. As a result, we will suppress the dependence of $\unicode[STIX]{x1D716}$ unless necessary.

Given a $\unicode[STIX]{x1D706}$ -admissible curve $\unicode[STIX]{x1D6FE}$ , define the handle

$$\begin{eqnarray}H_{\unicode[STIX]{x1D6FE}}=\Big\{(\unicode[STIX]{x1D6FE}(s)x_{1},\ldots ,\unicode[STIX]{x1D6FE}(s)x_{n})\mid s,x_{i}\in \mathbb{R},\sum \mathop{x}_{i}^{2}=1\Big\}\subset \mathbb{C}^{n}.\end{eqnarray}$$

Lemma 2.8. For a $\unicode[STIX]{x1D706}$ -admissible $\unicode[STIX]{x1D6FE}$ , $H_{\unicode[STIX]{x1D6FE}}$ is a Lagrangian submanifold of $(\mathbb{C}^{n},\sum dx_{i}\wedge dy_{i})$ .

Proof. Let $J$ be the standard complex structure on $\mathbb{C}^{n}$ and $g(\cdot ,\cdot ):=\unicode[STIX]{x1D714}(\cdot ,J\cdot )$ be the standard flat metric. Observe that $T_{\unicode[STIX]{x1D6FE}(s)x}H_{\unicode[STIX]{x1D6FE}}=\operatorname{Span}_{\mathbb{R}}\{\unicode[STIX]{x1D6FE}^{\prime }(s)x\}\,\oplus \,\unicode[STIX]{x1D6FE}(s)T_{x}S^{n-1}$ , for $x=(x_{1},\ldots ,x_{n})\in S^{n-1}\subset \mathbb{R}^{n}$ . A tangent vector in the second summand can be represented by $\unicode[STIX]{x1D6FE}(s)y=\unicode[STIX]{x1D6FE}(s)(y_{1},\ldots ,y_{n})$ , where $y\in \mathbb{R}^{n}$ and $g(x,y)=0$ . We have

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D714}(\unicode[STIX]{x1D6FE}^{\prime }(s)x,\unicode[STIX]{x1D6FE}(s)y) & = & \displaystyle -g(\unicode[STIX]{x1D6FE}(s)^{\prime }x,J(\unicode[STIX]{x1D6FE}(s)y))\nonumber\\ \displaystyle & = & \displaystyle -g(a^{\prime }(s)x+b^{\prime }(s)J(x),-b(s)y+a(s)J(y))\nonumber\\ \displaystyle & = & \displaystyle a^{\prime }(s)b(s)g(x,y)-b^{\prime }(s)a(s)g(J(x),J(y))=0,\nonumber\end{eqnarray}$$

where $\unicode[STIX]{x1D6FE}(s)=(a(s),b(s))$ . The fact that $\unicode[STIX]{x1D714}|_{\unicode[STIX]{x1D6FE}(s)T_{x}S^{n-1}}=0$ is obvious, so the result follows.◻

As a consequence, we have the following corollary.

Corollary 2.9. Let $L_{1},L_{2}\subset (M,\unicode[STIX]{x1D714})$ be two Lagrangians transversely intersecting at $p$ . Let $\unicode[STIX]{x1D704}:U\rightarrow M$ be the Darboux chart around $p$ with a standard complex structure so that $\unicode[STIX]{x1D704}^{-1}(L_{1})\subset \mathbb{R}^{n}$ and $\unicode[STIX]{x1D704}^{-1}(L_{2})\subset i\mathbb{R}^{n}$ are disks centered at the origin. Then one can obtain a Lagrangian $L_{1}\text{\#}_{p,st}L_{2}$ by attaching a Lagrangian handle $\unicode[STIX]{x1D704}(H_{\unicode[STIX]{x1D6FE}})$ to $(L_{1}\cup L_{2})\backslash \unicode[STIX]{x1D704}(U)$ .

The Lagrangian $L_{1}\text{\#}_{p,st}L_{2}$ is called a Lagrangian surgery from $L_{1}$ to $L_{2}$ following [Reference Lalonde and SikoravLS91, Reference PolterovichPol91]. Note that, the Lagrangian $L_{2}\text{\#}_{p,st}L_{1}$ obtained by performing Lagrangian surgery from $L_{2}$ to $L_{1}$ is in general not even smoothly isotopic to $L_{1}\text{\#}_{p,st}L_{2}$ .

Now, we present an new approach to performing Lagrangian surgery which also motivates the definition of Lagrangian surgery along clean intersections.

Definition 2.10. Given the zero section $L\subset T^{\ast }L$ , a Riemannian metric $g$ on $L$ (hence inducing a bundle metric on $T^{\ast }L$ ), a point $x\in L$ , and a choice of $\unicode[STIX]{x1D70E}$ as in Lemma 2.1, we define the flow handle $H_{\unicode[STIX]{x1D708}}$ with respect to a $\unicode[STIX]{x1D706}$ -admissible function $\unicode[STIX]{x1D708}$ to be (see Figure 5)

$$\begin{eqnarray}H_{\unicode[STIX]{x1D708}}=\{\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D708}(\Vert p\Vert )}^{\unicode[STIX]{x1D70E}}(p)\in T^{\ast }L:p\in (T_{x}^{\ast }L)_{\unicode[STIX]{x1D716}}\backslash \{x\}\},\end{eqnarray}$$

where $(T_{x}^{\ast }L)_{\unicode[STIX]{x1D716}}$ denotes the cotangent vectors at $x\in L$ with length ${\leqslant}\unicode[STIX]{x1D716}$ .

The following elementary fact about geodesics will be repeatedly used.

Lemma 2.11. Let $L$ , $x$ and $H_{\unicode[STIX]{x1D708}}$ be as above. Let $\unicode[STIX]{x1D6FE}:[0,c]\rightarrow L$ be a unit-speed geodesic starting at $\unicode[STIX]{x1D6FE}(0)=x$ . Then there is an induced symplectic embedding $T^{\ast }[0,c]\rightarrow T^{\ast }L$ and

$$\begin{eqnarray}T^{\ast }[0,c]\cap H_{\unicode[STIX]{x1D708}^{\unicode[STIX]{x1D6FC}}}=\bigg\{\bigg(\frac{\unicode[STIX]{x1D708}(\Vert p\Vert )p}{\Vert p\Vert },p\bigg)\in T^{\ast }[0,c]\,\biggr|\,p\in (0,\unicode[STIX]{x1D716}],\unicode[STIX]{x1D708}(p)\leqslant c\bigg\}.\end{eqnarray}$$

Proof. We have an embedding $\unicode[STIX]{x1D6FE}_{\ast }:T[0,c]\rightarrow TL$ . The tangent bundles can be dualized to an embedding $T^{\ast }[0,c]\rightarrow T^{\ast }L$ using the standard metric on $T[0,1]$ and $TL$ . To see that this embedding is symplectic when $\unicode[STIX]{x1D6FE}$ is a geodesic, we can pick a geodesic ball $U$ with a normal coordinate system $q=(q_{1},\ldots ,q_{n})$ centered at $x$ . We can assume $\unicode[STIX]{x1D6FE}(t)=(t,0,\ldots ,0)$ . Let $0\neq v\in T[0,c]$ and $\unicode[STIX]{x1D6FE}_{\ast }v\in TL$ . By definition, the dual of $\unicode[STIX]{x1D6FE}_{\ast }v\in TL$ , denoted by $(\unicode[STIX]{x1D6FE}_{\ast }v)^{\ast }:=\,\,g(-,\unicode[STIX]{x1D6FE}_{\ast }v)\in T^{\ast }L$ , vanishes when it is evaluated at vectors perpendicular to $\unicode[STIX]{x1D6FE}_{\ast }v$ . Since $U$ is a geodesic ball, vectors perpendicular to $\mathbb{R}\unicode[STIX]{x1D6FE}_{\ast }v=\mathbb{R}\unicode[STIX]{x2202}_{q_{1}}$ are spanned by vectors $\unicode[STIX]{x2202}_{q_{2}},\ldots ,\unicode[STIX]{x2202}_{q_{n}}$ . Therefore, $(\unicode[STIX]{x1D6FE}_{\ast }v)^{\ast }=|v|\,dq_{1}$ where $|v|$ is the norm of $v$ measured in standard metric in $[0,c]$ . As a result, the embedding $T^{\ast }[0,c]\rightarrow T^{\ast }L$ is given by $(q^{\prime },p^{\prime })\mapsto (q,p)=((q^{\prime },0,\ldots ,0),(p^{\prime },\ldots ,0))$ which is clearly symplectic. By definition, we have $T^{\ast }[0,c]\,\cap \,H_{\unicode[STIX]{x1D708}^{\unicode[STIX]{x1D6FC}}}=\{(\unicode[STIX]{x1D708}(\Vert p\Vert )p/\Vert p\Vert ,p)\in T^{\ast }[0,c]\mid p\in (0,\unicode[STIX]{x1D716}],\unicode[STIX]{x1D708}(p)\leqslant c\}$ , so the result follows.◻

Figure 5.

A flow handle.

Remark 2.12. The time-1 Hamiltonian flow of $\widetilde{\unicode[STIX]{x1D708}}(\Vert p\Vert )$ , where $\widetilde{\unicode[STIX]{x1D708}}^{\prime }(s)=\unicode[STIX]{x1D708}(s)$ , is $\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D708}(\Vert p\Vert )}^{\unicode[STIX]{x1D70E}}$ . For this reason, the reader should keep in mind that $H_{\unicode[STIX]{x1D708}}$ is automatically Lagrangian for any choice of admissible $\unicode[STIX]{x1D708}$ . For our purposes, the discussion on $\unicode[STIX]{x1D708}$ will be more flexible so we suppress the role of the actual Hamiltonian function $\widetilde{\unicode[STIX]{x1D708}}$ unless otherwise specified.

Lemma 2.13. Let $S_{\unicode[STIX]{x1D706}}(T_{x}^{\ast }L)$ be the radius $\unicode[STIX]{x1D706}$ -sphere in the cotangent plane of $x$ . We use the exponential map on the cotangent bundle via the identification of $T^{\ast }L$ and $TL$ . If $\exp :S_{\unicode[STIX]{x1D706}}(T_{x}^{\ast }L)\rightarrow L$ is an embedding, and $\unicode[STIX]{x2202}H_{\unicode[STIX]{x1D708}}\,\cap \,L=\exp (S_{\unicode[STIX]{x1D706}}(T_{x}^{\ast }L))\subset L$ divides $L$ into two components, then $H_{\unicode[STIX]{x1D708}}$ glues with exactly one of the components, as well as $T_{x}^{\ast }L\setminus (T_{x}^{\ast }L)_{\unicode[STIX]{x1D716}}$ , to form a smooth Lagrangian submanifold coinciding with $T_{x}^{\ast }L$ outside a compact set for a $\unicode[STIX]{x1D706}$ -admissible $\unicode[STIX]{x1D708}$ .

Proof. The only thing left to prove is the smoothness of gluing on $\unicode[STIX]{x2202}H_{\unicode[STIX]{x1D708}}=\exp (S_{\unicode[STIX]{x1D706}}(T_{x}^{\ast }L))$ . Note that from the assumptions, $\unicode[STIX]{x1D706}$ is not a critical radius of the exponential map, hence the exponential map is a diffeomorphism near $S_{\unicode[STIX]{x1D706}}(T_{x}^{\ast }L)$ . The flow $\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D708}(\Vert p\Vert )}^{\unicode[STIX]{x1D70E}}$ , followed by the projection to $L$ , is the same as $\exp (\unicode[STIX]{x1D708}(\Vert p\Vert )\cdot (p/\Vert p\Vert ))$ , which is a diffeomorphism. Therefore, near $\unicode[STIX]{x2202}H_{\unicode[STIX]{x1D708}}$ , the handle is a smooth Lagrangian section of $T^{\ast }L$ over the open shell $\exp (B_{\unicode[STIX]{x1D706}}(T_{x}^{\ast }L)\backslash \overline{B_{\unicode[STIX]{x1D706}-\unicode[STIX]{x1D6FF}}(T_{x}^{\ast }L)})$ , which is a smooth open submanifold of $L$ . Here, $B_{r}(T_{x}^{\ast }L)$ is the radius- $r$ ball of $T_{x}^{\ast }L$ centered at the origin. Moreover, the section has vanishing derivatives for all orders on the boundary component $S_{\unicode[STIX]{x1D706}}(T_{x}^{\ast }L)$ due to the assumption of admissibility on $\unicode[STIX]{x1D708}(r)$ near $r=0$ . The conclusion follows.◻

Example 2.14. One may match the Lagrangian handle $H_{\unicode[STIX]{x1D6FE}}$ and flow handle $H_{\unicode[STIX]{x1D708}}$ for an admissible $\unicode[STIX]{x1D6FE}$ and its corresponding admissible $\unicode[STIX]{x1D708}(r)=a(b^{-1}(-r))$ (see Definition 2.7 and the paragraph after it) via the identification between $T^{\ast }\mathbb{R}^{n}$ and $\mathbb{C}^{n}$ .

To see this, take an admissible curve $\unicode[STIX]{x1D6FE}(s)=(a(s),b(s))$ and its flow handle $H_{\unicode[STIX]{x1D6FE}}$ . We consider the flow handle

$$\begin{eqnarray}H_{\unicode[STIX]{x1D708}}=\bigg\{\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D708}(\Vert p\Vert )}^{\unicode[STIX]{x1D70E}}(0,p)=\bigg(0+\frac{p}{\Vert p\Vert }\cdot \unicode[STIX]{x1D708}(\Vert p\Vert ),p\bigg):p\in (T_{0}^{\ast }\mathbb{R}^{n})_{\unicode[STIX]{x1D716}}\bigg\}.\end{eqnarray}$$

We now identify $T^{\ast }\mathbb{R}^{n}$ with $\mathbb{C}^{n}$ by sending $(q,p)\mapsto q-ip$ , which matches the symplectic form $dp\wedge dq$ and $(1/-2i)\,dz\wedge d\bar{z}=dx\wedge dy$ . Then by definition

$$\begin{eqnarray}\displaystyle \bigg(\unicode[STIX]{x1D708}(\Vert p\Vert )\frac{p}{\Vert p\Vert },p\bigg) & \mapsto & \displaystyle \unicode[STIX]{x1D708}(\Vert p\Vert )\frac{p}{\Vert p\Vert }-ip\nonumber\\ \displaystyle & = & \displaystyle (\unicode[STIX]{x1D708}(\Vert p\Vert )-i\Vert p\Vert )\frac{p}{\Vert p\Vert }\nonumber\\ \displaystyle & = & \displaystyle (a(b^{-1}(-\Vert p\Vert ))+ib(b^{-1}(-\Vert p\Vert )))\frac{p}{\Vert p\Vert }\nonumber\\ \displaystyle & = & \displaystyle (a(s)+ib(s))x\nonumber\end{eqnarray}$$

by a change of variable $s=b^{-1}(-\Vert p\Vert )$ and $x=p/\Vert p\Vert$ . By this identification, we will simply use $H_{\unicode[STIX]{x1D708}}$ to denote both handles.

Corollary 2.15. Let $L_{1},L_{2}\subset (M,\unicode[STIX]{x1D714})$ be two Lagrangians transversely intersecting at $p$ . Under the assumption in Lemma 2.13, one can obtain a Lagrangian $L_{1}\text{\#}_{p}^{\unicode[STIX]{x1D708}}L_{2}$ by gluing: (1) $L_{2}\backslash U$ , for an $\unicode[STIX]{x1D716}$ -neighborhood $U$ of $p$ ; (2) the Lagrangian flow handle $H_{\unicode[STIX]{x1D708}}$ ; and (3) an open set in $L_{1}$ that glues with $H_{\unicode[STIX]{x1D708}}$ given by Lemma 2.13. For appropriately chosen $\unicode[STIX]{x1D708}$ , $L_{1}\text{\#}_{p}^{\unicode[STIX]{x1D708}}L_{2}$ coincides with $L_{1}\text{\#}_{p,st}L_{2}$ defined in Corollary 2.9.

Proof. Clearly $H_{\unicode[STIX]{x1D708}}$ glues smoothly with $L_{2}\backslash U$ and the open set in $L_{1}$ by Lemma 2.13, and the result is a smooth Lagrangian submanifold. Let $\unicode[STIX]{x1D708}(x)=a(b^{-1}(-x))$ be chosen as in Example 2.14 for an admissible curve. The flow handle is then identified with the standard Lagrangian handle defined by Lalonde and Sikorav, and Polterovich.◻

The following lemma addresses the independence of the surgery Lagrangians on the choice of profile functions.

Lemma 2.16. If $L_{1}$ is simply-connected, and the surgery profile and the injectivity radius of the exponential map at $x$ satisfies $\unicode[STIX]{x1D708}(0)<\operatorname{inj}_{g}(x)$ , then the Hamiltonian isotopy type of the surgery Lagrangian is independent of the choice of $\unicode[STIX]{x1D708}(r)$ .

Proof. For any two flow handles with different surgery profiles, there is a Lagrangian isotopy between them by a family of flow handles. We want to show that this Lagrangian isotopy can be extended to a symplectic isotopy using (a minor modification of) Banyaga’s isotopy extension theorem [Reference McDuff and SalamonMS98, Theorem 3.19].

Let $W$ be a Weinstein neighborhood of $L_{1}$ and ${\mathcal{N}}$ be a neighborhood of a flow handle in $W$ . Following the proof of [Reference McDuff and SalamonMS98, Theorem 3.19], it is sufficient to show that the relative cohomology $H^{2}(W,\unicode[STIX]{x2202}W\cup {\mathcal{N}};\mathbb{R})$ vanishes. But this cohomology is nothing but the second cohomology of the following space: take the Thom space of $T^{\ast }L_{1}$ (whose second cohomology vanishes when dim $L_{1}>2$ ), then collapse an additional cycle obtained from the connected sum of a fiber and the zero section. Therefore, the vanishing of $H^{2}(W,\unicode[STIX]{x2202}W\cup {\mathcal{N}};\mathbb{R})$ can be obtained by the long exact sequence of the relative cohomologies when $L_{1}$ is simply connected and dim $L_{1}>2$ . When $\operatorname{dim}L_{1}=2$ , the long exact sequence reads

$$\begin{eqnarray}\displaystyle H^{1}(L_{1};\mathbb{R})\cong H^{1}(D^{2}\text{\#}L_{1},S^{1};\mathbb{R}) & \rightarrow & \displaystyle H^{2}(W,\unicode[STIX]{x2202}W\cup D^{2}\text{\#}L_{1};\mathbb{R})\rightarrow H^{2}(W,\unicode[STIX]{x2202}W)\nonumber\\ \displaystyle & \rightarrow & \displaystyle H^{2}(D^{2}\text{\#}L_{1},S^{1};\mathbb{R}).\nonumber\end{eqnarray}$$

The last arrow is an isomorphism by restriction of the Thom form, hence the first arrow is again surjective. The rest of the argument will then go through. To argue the symplectic isotopy obtained by Banyaga’s argument is Hamiltonian, use the assumption of simply-connectedness again.◻

We give two examples of flow handles that are different from standard handles.

Example 2.17. Let $r(p)$ be the injectivity radius of $p$ . For different choices of $\unicode[STIX]{x1D708}(r)$ with $\unicode[STIX]{x1D708}(0)<r(p)$ , these handles will define a family of different Lagrangian surgeries which are all Lagrangian isotopic to each other.

The situation becomes more interesting when $\unicode[STIX]{x1D708}(0)>r(p)$ . Some simple instances are given by $S=\mathbb{R}\mathbb{P}^{n}$ , $\mathbb{C}\mathbb{P}^{n}$ or any finite cover of rank-one symmetric space. Take $\mathbb{C}\mathbb{P}^{n}$ and its standard Fubini–Study metric as an example, for any $p\in S$ , the flow surgery can be performed for $kr(p)<\unicode[STIX]{x1D708}(0)<(k+1)r(p)$ for any $k\in \mathbb{Z}$ . Later we will see that such surgeries are indeed iterated surgeries in the ordinary sense in Lemma 3.3 (although surgeries along clean intersections will be involved).

Example 2.18. A less standard example is essentially given by exotic spheres in [Reference SeidelSei14]. Given any $f\in \operatorname{Diff}^{+}(S^{n-1})$ , one may form an exotic sphere $S_{f}=B_{-}\bigcup _{f}B_{+}$ by gluing two copies $B_{\pm }$ of $B^{n}$ via $f$ . There is a Riemannian metric on $S_{f}$ so that all geodesics starting from the origins $0_{\pm }\in B_{\pm }$ are closed, passing through both $0_{\pm }$ , and of the same length [Reference SeidelSei14, Lemma 2.1]. Take $p=0_{-}\in B_{-}$ . When $\unicode[STIX]{x1D706}$ is below the injectivity radius, the corresponding flow handle surgery is the original one considered in [Reference PolterovichPol91]. When $\unicode[STIX]{x1D708}(0)>r(p)$ , the generalized surgery defined above is identified with an iterated surgery along $p$ and $q=0_{+}\in B_{+}$ in a successive order, which is exactly the family constructed in [Reference SeidelSei14] by the geodesic flow.

The following lemma can be found in [Reference SeidelSei99], but we feel that it is instructive to sketch its proof from the point of view of flow handles to make our discussion complete.

Lemma 2.19 [Reference SeidelSei99].

Let $x\in S^{n}$ be a point and consider $L=\unicode[STIX]{x1D70F}_{S^{n}}(T_{x}^{\ast }S^{n})\subset T^{\ast }S^{n}$ . Then $S^{n}\text{\#}_{x}^{\unicode[STIX]{x1D708}}T_{x}^{\ast }S^{n}$ is Hamiltonian isotopic to $L$ through a compactly supported Hamiltonian, where $\unicode[STIX]{x1D708}$ is an admissible function such that $\unicode[STIX]{x1D708}(0)<r(x)$ , the injectivity radius of $x$ under the round metric.

Proof. Let $S^{n}$ be equipped with the round metric such that every embedded closed geodesic has length $2\unicode[STIX]{x1D70B}$ . This induces a metric on $T^{\ast }S^{n}$ which we will use throughout. Denote by $A:S^{n}\rightarrow S^{n}$ the antipodal map.

We consider two embeddings of open geodesic balls $B_{\unicode[STIX]{x1D70B}}(x){\hookrightarrow}S^{n}$ and $B_{\unicode[STIX]{x1D70B}}(A(x)){\hookrightarrow}S^{n}$ into the zero section, which are of radius $\unicode[STIX]{x1D70B}$ , and centered at $x$ and $A(x)$ , respectively. Also fix a normal coordinate system on each of these balls, which induces a trivialization (and a chart) on $T^{\ast }B_{\unicode[STIX]{x1D70B}}(x)$ and $T^{\ast }B_{\unicode[STIX]{x1D70B}}(A(x))$ .

The two embeddings give two symplectomorphisms $f_{x}:T^{\ast }B_{\unicode[STIX]{x1D70B}}(x)\rightarrow T^{\ast }S^{n}\backslash T_{A(x)}^{\ast }S^{n}$ and $f_{A(x)}:T^{\ast }B_{\unicode[STIX]{x1D70B}}(A(x))\rightarrow T^{\ast }S^{n}\backslash T_{x}^{\ast }S^{n}$ . Indeed, these symplectomorphisms come from the definition of the canonical symplectic form on a cotangent bundle, which patches the forms $dp\wedge dq$ for the two local charts on the zero section as above. Recall the notation of spherical Dehn twist profile from the paragraph following Lemma 2.1, and $\unicode[STIX]{x1D716}$ is an arbitrary small positive constant. Under these symplectomorphisms we have by Lemma 2.11

$$\begin{eqnarray}f_{x}^{-1}(L)=\bigg\{\bigg(\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D716}}^{\text{Dehn},\text{sp}}(\Vert p\Vert )\frac{p}{\Vert p\Vert },p\bigg)\in T^{\ast }B_{\unicode[STIX]{x1D70B}}(x):p\in \mathbb{R}^{n}\backslash \{0\}\bigg\}\end{eqnarray}$$

and

$$\begin{eqnarray}f_{A(x)}^{-1}(L)=\bigg\{\bigg(\unicode[STIX]{x1D70B}-\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D716}}^{\text{Dehn},\text{sp}}(\Vert p\Vert ))\frac{p}{\Vert p\Vert },p\bigg)\in T^{\ast }B_{\unicode[STIX]{x1D70B}}(A(x)):p\in B_{\unicode[STIX]{x1D716}}(0)\bigg\}.\end{eqnarray}$$

Here, we use the properties that $B_{\unicode[STIX]{x1D70B}}(x){\hookrightarrow}S^{n}$ are geodesic ball embeddings.

On the other hand, suppose $\unicode[STIX]{x1D708}=\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D706}}$ is such that $\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D706}}(0)=\unicode[STIX]{x1D706}<\unicode[STIX]{x1D70B}=r(x)$ (see § 2.2.1). Then $f_{x}^{-1}(H_{\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D706}}})$ is given by

$$\begin{eqnarray}\displaystyle f_{x}^{-1}(H_{\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D706}}}) & = & \displaystyle \bigg\{\bigg(\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D706}}(\Vert p\Vert )\frac{p}{\Vert p\Vert },p\bigg)\in T^{\ast }B_{\unicode[STIX]{x1D70B}}(x):p\in \mathbb{R}^{n}\backslash \{0\}\bigg\}\nonumber\\ \displaystyle & & \displaystyle \cup \,\{(q,0)\in T^{\ast }B_{\unicode[STIX]{x1D70B}}(A(x)):q\in B_{\unicode[STIX]{x1D70B}}(0)\backslash B_{\unicode[STIX]{x1D706}}(0)\}.\nonumber\end{eqnarray}$$

Let $\unicode[STIX]{x1D6FF}>0$ be such that $\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D716}}^{\text{Dehn},\text{sp}}(r)=\unicode[STIX]{x1D70B}-r$ for $r<\unicode[STIX]{x1D6FF}$ . We can pick $\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D706}}$ such that $\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D706}}(r)=\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D716}}^{\text{Dehn},\text{sp}}(r)$ for $r\geqslant \unicode[STIX]{x1D6FF}$ . The resulting $S^{n}\text{\#}_{x}^{\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D706}}}T_{x}^{\ast }S^{n}$ hence coincides with $L$ outside $T^{\ast }B_{\unicode[STIX]{x1D6FF}}(A(x))$ . Inside $T^{\ast }B_{\unicode[STIX]{x1D6FF}}(A(x))$ , even though $\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D716}}^{\text{Dehn},\text{sp}}$ is not an admissible function, both $S^{n}\text{\#}_{x}^{\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D706}}}T_{x}^{\ast }S^{n}$ and $L$ are graphs of exact 1-forms (although the primitive function does not vanish near the boundary). Therefore, $S^{n}\text{\#}_{x}^{\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D706}}}T_{x}^{\ast }S^{n}$ is Lagrangian isotopic to $L$ by varying the primitive function in $T^{\ast }B_{\unicode[STIX]{x1D6FF}}(A(x))$ but fixing them near the boundary, and hence Hamiltonian isotopic to $L$ by a compactly supported Hamiltonian due to the simply-connectedness of $T^{\ast }S^{n}$ and $L$ . (Extending an exact Lagrangian isotopy to a Hamiltonian isotopy is well known when the Lagrangian is compact, but when the exact Lagrangian has cylindrical ends, then it requires a relative version of Banyaga extension and the simply-connectedness assumption. See a completely parallel argument in Lemma 2.16.)◻

Remark 2.20. Consider the setting as in Corollary 2.15. For semi-admissible $\unicode[STIX]{x1D708}^{\unicode[STIX]{x1D6FC}}$ that is not admissible, the gluing with $L_{1}$ cannot be smooth in general (for example, consider a semi-admissible profile where $\unicode[STIX]{x1D708}(0)<\unicode[STIX]{x1D70B}$ in the case when $L_{1}$ is a sphere with radius $1$ , then we will have a corner locus which is the circle that we glue along). Lemma 2.19 is an instance when a surgery using a semi-admissible profile $\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D716}}^{\text{Dehn}}$ yields a smooth Lagrangian submanifold. Intuitively, the lemma regards $\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D716}}^{\text{Dehn}}$ as a degenerate case of an admissible function. The point is that, when $\unicode[STIX]{x1D706}=r(p)$ , we only need to glue $\operatorname{Cl}(H_{\unicode[STIX]{x1D708}})$ with $L_{2}\backslash U$ (compare with Lemma 2.13, which exemplifies the more common case where $\operatorname{Cl}(H_{\unicode[STIX]{x1D708}})$ glues with an open subset of $L_{2}$ ), where $\operatorname{Cl}(\cdot )$ denotes the closure and $U$ is an $\unicode[STIX]{x1D716}$ -neighborhood of $p$ .

In the case when a semi-admissible function defines a smooth Lagrangian surgery manifold, we will continue to denote it as $L_{1}\text{\#}_{p}^{\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D706}}^{\unicode[STIX]{x1D6FC}}}L_{2}$ . This applies to other surgeries along clean intersections and will be used several more times in a parametrized version in the paper.

2.2.2 Surgery along clean intersection

Let $L_{1}$ and $L_{2}$ be two Lagrangians in $(M,\unicode[STIX]{x1D714})$ which intersect cleanly at a submanifold $D$ . In other words, we have $T_{p}D=T_{p}L_{1}\cap T_{p}L_{2}$ for all $p\in D$ . The following well-known local proposition due to Pozniak allows us to extend the definition of flow handles to this case.

Proposition 2.21 [Reference PoźniakPoź99].

Let $L_{1},L_{2}\subset (M,\unicode[STIX]{x1D714})$ be two closed embedded Lagrangians with clean intersection at $L_{1}\cap L_{2}=D$ . Then there is a symplectic embedding $\unicode[STIX]{x1D711}$ from a neighborhood $U$ of $0_{\text{section}}\subset T^{\ast }L_{1}$ to $M$ such that $\unicode[STIX]{x1D711}(0_{\text{section}})=L_{1}$ and $\unicode[STIX]{x1D711}^{-1}(L_{2})\subset N_{D}^{\ast }$ , where $0_{\text{section}}$ is the zero section and $N_{D}^{\ast }$ is the conormal bundle of $D$ in $L_{1}$ .

Definition 2.22. Fix a metric $g$ on $L$ . We define the flow handle for $D\subset L$ with respect to an admissible function $\unicode[STIX]{x1D708}$ to be

$$\begin{eqnarray}H_{\unicode[STIX]{x1D708}}^{D}=\{\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D708}(\Vert \unicode[STIX]{x1D709}\Vert )}^{\unicode[STIX]{x1D70E}}(\unicode[STIX]{x1D709})\in T^{\ast }L:\unicode[STIX]{x1D709}\in (N_{D}^{\ast })_{\unicode[STIX]{x1D716}}\backslash D\},\end{eqnarray}$$

where $(N_{D}^{\ast })_{\unicode[STIX]{x1D716}}$ consists of covectors in the conormal bundle of $D$ in $L$ with length ${\leqslant}\unicode[STIX]{x1D716}$ .

Lemma 2.23. Let $S_{\unicode[STIX]{x1D706}}(N_{D}^{\ast })$ be the radius- $\unicode[STIX]{x1D706}$ sphere bundle in the conormal bundle of $D$ . If $\exp :S_{\unicode[STIX]{x1D706}}(N_{D}^{\ast })\rightarrow L$ is an embedding, and $\unicode[STIX]{x2202}H_{\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D706}}}^{D}\cap L\subset L$ divides $L$ into two components, then $H_{\unicode[STIX]{x1D708}}^{D}$ glues with exactly one of the components to form a smooth Lagrangian submanifold coinciding with $N_{D}^{\ast }$ outside a compact set.

The proof is exactly the same as Lemma 2.13 and we omit it. Below, we call $r(D)>0$ the injectivity radius of $D$ , which is the supremum of $r>0$ such that the (dual) exponential map of $(N_{D}^{\ast })_{{\leqslant}r}$ is an embedding. It is positive since it is easy to check to be positive in a chart, and we only consider the case when $D$ is compact. As in the transversal intersection case, the surgery is always well defined when we choose $\unicode[STIX]{x1D708}(0)=\unicode[STIX]{x1D706}<r(D)$ . Using Proposition 2.21, we globalize the construction as follows.

Corollary 2.24. Let $L_{1},L_{2}\subset (M,\unicode[STIX]{x1D714})$ be two Lagrangians intersecting cleanly at $D$ . By choosing a metric on $L_{1}$ , a symplectic embedding $\unicode[STIX]{x1D704}:(T^{\ast }L_{1})_{\unicode[STIX]{x1D716}}\rightarrow M$ such that $\unicode[STIX]{x1D704}(0_{\text{section}})=L_{1}$ and $\unicode[STIX]{x1D704}^{-1}(L_{2})\subset N_{D}^{\ast }$ , one can obtain a Lagrangian $L_{1}\text{\#}_{D}^{\unicode[STIX]{x1D708}}L_{2}$ by attaching a Lagrangian flow handle $\unicode[STIX]{x1D704}(H_{\unicode[STIX]{x1D708}}^{D})$ to $(L_{1}\backslash U_{1})\cup (L_{2}\backslash U_{2})$ , with $U_{i}\subset L_{i}$ appropriate open neighborhoods of $D$ , and $\unicode[STIX]{x1D716}$ being sufficiently small.

As in Example 2.17, we denote $L_{1}\text{\#}_{D}^{\unicode[STIX]{x1D708}}L_{2}$ by $L_{1}\text{\#}_{D}L_{2}$ if $\unicode[STIX]{x1D706}<r(D)$ .

2.3 $E_{2}$ -flow surgery and its family version

We will introduce a generalization of flow handle which will be useful when performing surgeries on Lagrangian submanifolds in product symplectic manifolds later. Heuristically, our previous constructions have taken advantage of the fact that $\Vert p\Vert$ has a well-defined Hamiltonian flow on the whole cotangent bundle except for the zero section. More crucially, the resulting flow handle should have an embedded boundary in $L_{1}$ . Indeed, any Hamiltonian function with such properties will suffice for defining a meaningful Lagrangian handle.

A variant of the flow handle can therefore be defined as follows. Let $L=K_{1}^{n-m}\times K_{2}^{m}$ be a product manifold equipped with a product Riemannian metric. Then there is an orthogonal decomposition $T^{\ast }L=E_{1}\oplus E_{2}$ given by the two factors respectively. Let $D\subset L$ be of codimension $m$ and transverse to $\{p\}\times K_{2}$ for all $p\in K_{1}$ . Let $\unicode[STIX]{x1D70B}_{2}:T^{\ast }L\rightarrow E_{2}$ be the projection to $E_{2}$ . One may then use the function $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D70B}}(\cdot )=\Vert \unicode[STIX]{x1D70B}_{2}(\cdot )\Vert _{g}:T^{\ast }L\rightarrow \mathbb{R}$ to define a new flow handle. Note that $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D70B}}=\Vert \unicode[STIX]{x1D70B}_{2}(\cdot )\Vert _{g}$ is smooth on $T^{\ast }L\backslash E_{1}$ .

Figure 6.

An $E_{2}$ -flow handle.

Definition 2.25. In the situation above, we define the $E_{2}$ -flow handle for $D$ (or flow handle along the $E_{2}$ -direction) with respect to an $\unicode[STIX]{x1D706}$ -admissible $\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D706}}$ to be (see Figure 6)

$$\begin{eqnarray}H_{\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D706}}}^{D,E_{2}}=\{\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D706}}(\Vert \unicode[STIX]{x1D70B}_{2}(\unicode[STIX]{x1D709})\Vert )}^{\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D70B}}}(\unicode[STIX]{x1D709})\subset T^{\ast }L:\unicode[STIX]{x1D709}\in (N_{D}^{\ast })_{\unicode[STIX]{x1D716},E_{2}}\backslash D\},\end{eqnarray}$$

where $(N_{D}^{\ast })_{\unicode[STIX]{x1D716},E_{2}}$ consists of covectors $\unicode[STIX]{x1D709}$ in the conormal bundle of $D$ in $L$ such that $\Vert \unicode[STIX]{x1D70B}_{2}(\unicode[STIX]{x1D709})\Vert \leqslant \unicode[STIX]{x1D716}$ .

We note that for any point $\unicode[STIX]{x1D709}=(\unicode[STIX]{x1D709}_{1},\unicode[STIX]{x1D709}_{2})\in E_{1}\oplus E_{2}$ , $\unicode[STIX]{x1D719}_{t}^{\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D70B}}}(\unicode[STIX]{x1D709})=(\unicode[STIX]{x1D709}_{1},\unicode[STIX]{x1D719}_{t}^{\unicode[STIX]{x1D70E}}(\unicode[STIX]{x1D709}_{2}))$ so $E_{2}$ -flow is the normalized (co)geodesic flow on the second factor and is trivial on the first factor.

Let $S_{\unicode[STIX]{x1D706}}(E_{2}|_{D})$ be the radius- $\unicode[STIX]{x1D706}$ sphere bundle of $E_{2}$ over $D$ . We consider $\exp _{\unicode[STIX]{x1D706}}^{E_{2}}:S_{\unicode[STIX]{x1D706}}(E_{2}|_{D})\rightarrow L$ , which is the exponential map restricted on $S_{\unicode[STIX]{x1D706}}(E_{2}|_{D})$ along the leaves of the foliation given by second factor. We define the $E_{2}$ -injectivity radius $r^{E_{2}}(D)$ of $D$ as the supremum of $\unicode[STIX]{x1D706}$ such that $\exp _{s}^{E_{2}}$ is an embedding for all $s<\unicode[STIX]{x1D706}$ .

Similarly to the cases we considered before, if $L_{1}=K_{1}\times K_{2}$ and $L_{2}$ are Lagrangians cleanly intersecting at $D$ as above, we can add an $E_{2}$ -flow handle to $L_{1}\cup L_{2}$ outside a tubular neighborhood of $D$ to get a new Lagrangian submanifold for $\unicode[STIX]{x1D706}<r^{E_{2}}(D)$ . We will denote the resulting Lagrangian submanifold by $L_{1}\text{\#}_{D,E_{2}}L_{2}$ , called the $E_{2}$ -flow surgery from $L_{1}$ to $L_{2}$ along  $D$ .

A family version of $E_{2}$ -flow surgery

We now consider the $E_{2}$ -flow surgery for a family over a symplectic base. Assume that we have a smooth manifold pair $(L=K_{1}\times K_{2},D)$ , a decomposition $T^{\ast }L=E_{1}\oplus E_{2}$ and a Lagrangian handle $H_{\unicode[STIX]{x1D708}}^{D,E_{2}}$ as above.

Let $(\mathscr{P},\unicode[STIX]{x1D714}_{\mathscr{P}})$ be a symplectic manifold equipped with a symplectic fiber bundle structure $\unicode[STIX]{x1D70B}:\mathscr{P}\rightarrow \mathscr{B}$ , such that it has a symplectic base $(\mathscr{B},\unicode[STIX]{x1D714}_{\mathscr{B}})$ and fibers symplectomorphic to $T^{\ast }L$ . Let $i:B{\hookrightarrow}\mathscr{B}$ be a Lagrangian submanifold such that the structure group $G$ of $i^{\ast }\mathscr{P}\rightarrow B$ is a subgroup of $\text{Isom}(L){\hookrightarrow}\text{Symp}(T^{\ast }L)$ . In particular, $G$ preserves $L$ and we make the following further assumptions on  $G$ .

1. (i) $G$ preserves $D$ , the subbundles $E_{1}$ and $E_{2}$ and $H_{\unicode[STIX]{x1D708}}^{D,E_{2}}$ .

2. (ii) For any loop $\unicode[STIX]{x1D6FE}:\mathbb{R}/\mathbb{Z}\rightarrow B$ , the symplectic monodromy around $\unicode[STIX]{x1D6FE}$ for the fiber bundle $\unicode[STIX]{x1D70B}:\mathscr{P}\rightarrow \mathscr{B}$ lies in $G$ .

3. (iii) Let $x\in \mathscr{P}$ , $b:=\unicode[STIX]{x1D70B}(x)\in B$ and $\mathscr{P}_{b}:=\unicode[STIX]{x1D70B}^{-1}(b)$ . Assume $\operatorname{Ker}_{x}$ is the symplectic orthogonal complement of the fiber $T_{x}\mathscr{P}_{b}\subset T_{x}\mathscr{P}$ at $x$ , then we ask $\unicode[STIX]{x1D714}_{\mathscr{P}}(v_{1},v_{2})=0$ for any $v_{1},v_{2}\in \operatorname{Ker}_{x}$ such that $\unicode[STIX]{x1D70B}_{\ast }v_{i}\in TB\subset T\mathscr{B}$ for both $i$ .

Since $L$ and $D$ are invariant under $G$ , we can use symplectic parallel transport to obtain a smooth manifold pair $(\mathscr{L},\mathscr{D})$ in $\mathscr{P}$ . More precisely, let $\mathscr{P}_{b}=T^{\ast }L$ be a reference fiber with based point $b\in B$ . We define $\mathscr{L}:=\bigcup _{\unicode[STIX]{x1D6FE}\in I_{B}}\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D6FE}}(L)$ , $\mathscr{D}:=\bigcup _{\unicode[STIX]{x1D6FE}\in I_{B}}\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D6FE}}(D)$ , where $I_{B}$ consists of all paths $\unicode[STIX]{x1D6FE}$ from $b$ to another point $b^{\prime }\in B$ and $\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D6FE}}$ is the symplectic parallel transport along $\unicode[STIX]{x1D6FE}$ . As a result, $(\mathscr{L},\mathscr{D})$ has a compatible fiber bundle structure over the base $B$ , that is,

where the two bundle structures are compatible with the inclusions $\mathscr{D}{\hookrightarrow}\mathscr{L}{\hookrightarrow}i^{\ast }\mathscr{P}$ .

All previous symplectic constructions on $T^{\ast }L$ are $G$ -invariant, by assumption (i), hence can be glued over $B$ . For example, $N_{D}^{\ast }L$ glues into $N_{\mathscr{D}}^{\ast }\mathscr{L}:=\bigcup _{\unicode[STIX]{x1D6FE}\in I_{B}}\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D6FE}}(N_{D}^{\ast }L)$ . When $i^{\ast }\mathscr{P}$ is regarded as a vector bundle over $\mathscr{L}$ , it comes with a natural splitting $i^{\ast }\mathscr{P}=\mathscr{E}_{1}\,\oplus \,\mathscr{E}_{2}$ , where $\mathscr{E}_{i}:=\bigcup _{\unicode[STIX]{x1D6FE}\in I_{B}}(\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D6FE}})_{\ast }(E_{i})$ for $i=1,2$ . Moreover, the $E_{2}$ -handle $H_{\unicode[STIX]{x1D708}}^{D,E_{2}}$ on fibers can be glued together, which gives a smooth handle $\mathscr{H}_{\unicode[STIX]{x1D708}}:=\bigcup _{\unicode[STIX]{x1D6FE}\in I_{B}}\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D6FE}}(H_{\unicode[STIX]{x1D708}}^{D,E_{2}})\subset i^{\ast }\mathscr{P}$ . The fact that $\mathscr{H}_{\unicode[STIX]{x1D708}}{\hookrightarrow}\mathscr{P}\supset i^{\ast }\mathscr{P}$ is indeed a Lagrangian embedding follows from the assumptions (ii) and (iii) on $G$ .

Despite the fact that many assumptions are imposed on $(\mathscr{P},\unicode[STIX]{x1D714}_{\mathscr{P}})$ and $G$ , there are practical examples where Lemma 2.27 applies (see § 3.3).

3.1 Fiber version

In this section, we are interested in $L$ being $\mathbb{RP}^{n}$ , $\mathbb{CP}^{m/2}$ or $\mathbb{HP}^{n}$ equipped with the Riemannian metric such that every geodesic is closed of length $2\unicode[STIX]{x1D70B}$ . All actual proofs will be given only in the case of $\mathbb{C}\mathbb{P}^{n}$ but are easily generalized. Let $x\in L$ be a point and $F_{x}=T_{x}^{\ast }L$ . We also let $D_{x}=\{y\in L\mid \text{dist}(x,y)=\unicode[STIX]{x1D70B}\}$ be the submanifold opposite to $x$ . In the case of $\mathbb{CP}^{m/2}$ , $D_{x}\simeq \mathbb{CP}^{m/2-1}$ .

Proof of Lemma 3.1.

For the first statement, we observe that the space of $\unicode[STIX]{x1D706}$ -admissible functions for $(k-1)\unicode[STIX]{x1D70B}<\unicode[STIX]{x1D706}<k\unicode[STIX]{x1D70B}$ is connected. A smooth isotopy $\{\unicode[STIX]{x1D708}_{t}\}$ from $\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D706}_{1}}$ to $\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D706}_{2}}$ in this space results in a smooth Lagrangian isotopy from $L\text{\#}_{x}^{\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D706}_{1}}}F_{x}$ to $L\text{\#}_{x}^{\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D706}_{2}}}F_{x}$ since $\unicode[STIX]{x2202}H_{\unicode[STIX]{x1D708}^{t}}$ does not pass any critical locus. This is a Hamiltonian isotopy because $H^{1}(L\text{\#}_{x}^{\unicode[STIX]{x1D708}_{t}}F_{x},\unicode[STIX]{x2202}^{\infty }(L\text{\#}_{x}^{\unicode[STIX]{x1D708}_{t}}F_{x});\mathbb{R})=0$ (cf. Example 2.17). Here, $\unicode[STIX]{x2202}^{\infty }(L\text{\#}_{x}^{\unicode[STIX]{x1D708}_{t}}F_{x})$ is the infinite end of $L\text{\#}_{x}^{\unicode[STIX]{x1D708}_{t}}F_{x}$ .

Figure 7.

Isotopy from $\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D706}_{1}}$ to $\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D70B}}$ to $\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D70B}}^{\unicode[STIX]{x1D6FC}}$ (Dehn twist profile).

For the second statement, we only consider the case that $k=1$ and $L=\mathbb{CP}^{m/2}$ , and the remaining cases are similar. In this case, denote $\unicode[STIX]{x1D708}^{\unicode[STIX]{x1D6FC}}=\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D70B}}^{\unicode[STIX]{x1D6FC}}$ . Then for a handle $H_{\unicode[STIX]{x1D708}^{\unicode[STIX]{x1D6FC}}}$ at $x$ , $\operatorname{Cl}(H_{\unicode[STIX]{x1D708}^{\unicode[STIX]{x1D6FC}}})\backslash H_{\unicode[STIX]{x1D708}^{\unicode[STIX]{x1D6FC}}}=D_{x}=\mathbb{CP}^{m/2-1}$ . We pick a local chart $U\subset L$ with local coordinates $(q_{1},\ldots ,q_{m})$ adapted to $D_{x}$ in the sense that $U\cap D_{x}=\{q_{1}=q_{2}=0\}$ and $c(t)=(tq_{1},tq_{2},q_{3},\ldots ,q_{m})$ are unit-speed geodesics perpendicular to $D_{x}$ at $t=0$ , for any $(q_{1},\ldots ,q_{m})$ such that $|q_{1}|^{2}+|q_{2}|^{2}=1$ . It induces canonically a Darboux chart $T^{\ast }U$ in $T^{\ast }L$ . We write a point in $T^{\ast }U$ as $(q_{a},q_{b},p_{a},p_{b})$ , where $q_{a}=(q_{1},q_{2})$ , $q_{b}=(q_{3},\ldots ,q_{m})$ and similarly for $p_{a}$ and $p_{b}$ . Since $H_{\unicode[STIX]{x1D708}^{\unicode[STIX]{x1D6FC}}}$ is defined by the geodesic flow, we have

(3.1) $$\begin{eqnarray}\displaystyle T^{\ast }U\cap H_{\unicode[STIX]{x1D708}^{\unicode[STIX]{x1D6FC}}} & = & \displaystyle \{(q_{a},q_{b},p_{a},0)\mid q_{a}=-\unicode[STIX]{x1D6FC}p_{a}\neq 0\}\end{eqnarray}$$

(3.2) $$\begin{eqnarray}\displaystyle T^{\ast }U\cap D_{x} & = & \displaystyle \{(0,q_{b},0,0)\}\end{eqnarray}$$

when $U$ is sufficiently small. To see that it is true, we consider a unit-speed geodesic $\unicode[STIX]{x1D6FE}:[0,\unicode[STIX]{x1D70B}]\rightarrow L$ from $x$ to a point $x^{\prime }\in D_{x}$ . By Lemma 2.11, we have a symplectic embedding $T^{\ast }[0,\unicode[STIX]{x1D70B}]\rightarrow T^{\ast }L$ and

(3.3) $$\begin{eqnarray}\displaystyle T^{\ast }[0,\unicode[STIX]{x1D70B}]\cap H_{\unicode[STIX]{x1D708}^{\unicode[STIX]{x1D6FC}}}=\bigg\{\bigg(\frac{\unicode[STIX]{x1D708}^{\unicode[STIX]{x1D6FC}}(\Vert p\Vert )p}{\Vert p\Vert },p\bigg)\in T^{\ast }[0,\unicode[STIX]{x1D70B}]\,\biggr|\,p\in (0,\unicode[STIX]{x1D716}]\bigg\}. & & \displaystyle\end{eqnarray}$$

On the other hand, the symplectic embedding $T^{\ast }[0,\unicode[STIX]{x1D70B}]\cap T^{\ast }U\rightarrow T^{\ast }U$ is given by

(3.4) $$\begin{eqnarray}\displaystyle (q,p)\mapsto (q_{a},q_{b},p_{a},p_{b})=((\unicode[STIX]{x1D70B}-q)c_{1},c_{2},-pc_{1},0) & & \displaystyle\end{eqnarray}$$

for some $c_{1},c_{2}$ such that $|c_{1}|=1$ because $U$ is adapted to $D_{x}$ (by the same reasoning as in the proof of Lemma 2.11). When $p>0$ close to zero, $q=\unicode[STIX]{x1D708}^{\unicode[STIX]{x1D6FC}}(\Vert p\Vert )p/\Vert p\Vert =\unicode[STIX]{x1D70B}-\unicode[STIX]{x1D6FC}p$ so we have $(\unicode[STIX]{x1D70B}-q)=\unicode[STIX]{x1D6FC}p$ and hence $(\unicode[STIX]{x1D70B}-q)c_{1}=\unicode[STIX]{x1D6FC}pc_{1}$ . Therefore, (3.3) and (3.4) imply (3.1) when we consider all possible unit-speed geodesics from $x$ to points on $D_{x}$ .

From the local description, it is clear that $H_{\unicode[STIX]{x1D708}^{\unicode[STIX]{x1D6FC}}}$ and $D_{x}$ can be glued smoothly to become $\operatorname{Cl}(H_{\unicode[STIX]{x1D708}^{\unicode[STIX]{x1D6FC}}})$ . The gluing from $H_{\unicode[STIX]{x1D708}^{\unicode[STIX]{x1D6FC}}}$ to $F_{x}-B_{\unicode[STIX]{x1D716}}$ is the same as in the admissible case. It results in a smooth Lagrangian $L\text{\#}_{x}^{\unicode[STIX]{x1D708}^{\unicode[STIX]{x1D6FC}}}F_{x}$ .

Finally, we want to show that $L\text{\#}_{x}^{\unicode[STIX]{x1D708}^{\unicode[STIX]{x1D6FC}}}F_{x}$ is Hamiltonian isotopic to $L\text{\#}_{x}^{\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D706}_{i}}}F_{x}$ . We can assume $\unicode[STIX]{x1D6FC}\neq 0$ , by a Hamiltonian perturbation if necessary. Locally near $D_{x}$ , we have

(3.5) $$\begin{eqnarray}\displaystyle & \displaystyle T^{\ast }U\cap (H_{\unicode[STIX]{x1D708}^{\unicode[STIX]{x1D6FC}}}\cup D_{x})=\{(-\unicode[STIX]{x1D6FC}p_{a},q_{b},p_{a},0)\}=\bigg\{\bigg(q_{a},q_{b},-\frac{1}{\unicode[STIX]{x1D6FC}}q_{a},0\bigg)\bigg\} & \displaystyle\end{eqnarray}$$

(3.6) $$\begin{eqnarray}\displaystyle & \displaystyle T^{\ast }U\cap L=\{(q_{a},q_{b},0,0)\}. & \displaystyle\end{eqnarray}$$

Notice that, by the choice of $U$ , we have $\text{dist}^{2}(\cdot ,D_{x})=|q_{a}|^{2}:=|q_{1}|^{2}+|q_{2}|^{2}$ in $U$ . It implies that $H_{\unicode[STIX]{x1D708}^{\unicode[STIX]{x1D6FC}}}\cup D_{x}$ is the graph of $d(-(1/2\unicode[STIX]{x1D6FC})\text{dist}^{2}(\cdot ,D_{x}))$ in $T^{\ast }U$ . Since the definition of the graph of $d(-(1/2\unicode[STIX]{x1D6FC})\text{dist}^{2}(\cdot ,D_{x}))$ is coordinate-free, by gluing charts that are adapted to $D_{x}$ , we know that there is a small $\unicode[STIX]{x1D6FF}>0$ such that $(H_{\unicode[STIX]{x1D708}^{\unicode[STIX]{x1D6FC}}}\cup D_{x})\cap T^{\ast }B_{\unicode[STIX]{x1D6FF}}(D_{x})$ is the graph of $d(-(1/2\unicode[STIX]{x1D6FC})\text{dist}^{2}(\cdot ,D_{x}))$ over $B_{\unicode[STIX]{x1D6FF}}(D_{x})$ , where $B_{\unicode[STIX]{x1D6FF}}(D_{x})$ is the $\unicode[STIX]{x1D6FF}$ neighborhood of $D_{x}$ in $L$ . Take a smooth decreasing function $f(r):[0,\unicode[STIX]{x1D6FF}]\rightarrow \mathbb{R}$ so that $f=0$ near $r=0$ and $f(r)=-(1/2\unicode[STIX]{x1D6FC})r$ near $r=\unicode[STIX]{x1D6FF}$ . Denote $f_{t}(r)=tf(r)-(1-t)(1/2\unicode[STIX]{x1D6FC})r$ .

Then the graph of $d(f_{t}\circ \text{dist}^{2}(\cdot ,D_{x}))$ can be patched with $H_{\unicode[STIX]{x1D708}^{\unicode[STIX]{x1D6FC}}}\backslash T^{\ast }B_{\unicode[STIX]{x1D6FF}}(D_{x})$ to give a Hamiltonian isotopy from $L\text{\#}_{x}^{\unicode[STIX]{x1D708}^{\unicode[STIX]{x1D6FC}}}F_{x}$ to $L\text{\#}_{x}^{\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D706}}}F_{x}$ for some admissible $\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D706}}$ with $0<\unicode[STIX]{x1D706}<\unicode[STIX]{x1D70B}$ . We remark that the Hamiltonian isotopy is invariant under $\text{Isom}(L)_{x}$ , the isometry group of $L$ fixing  $x$ . This concludes the proof.

The following could be helpful to understand the construction. An alternative way to describe the isotopy $d(f_{t}\circ \text{dist}^{2}(\cdot ,D_{x}))$ is that we use a family of semi-admissible function $\{\unicode[STIX]{x1D708}^{\unicode[STIX]{x1D6FC}_{t}}\}_{t\in [0,1)}$ ( $\unicode[STIX]{x1D6FC}_{t}>0$ ) such that $\unicode[STIX]{x1D6FC}_{0}=\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D708}^{\unicode[STIX]{x1D6FC}_{t}}$ approaches to $\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D706}}$ when $t$ goes to $1$ . Then we will have $L\text{\#}_{x}^{\unicode[STIX]{x1D708}^{\unicode[STIX]{x1D6FC}}}F_{x}=d(f_{t}\circ \text{dist}^{2}(\cdot ,D_{x}))$ near $D_{x}$ .◻

Later we will see that, when the surgery profile $\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D706}}$ has $\unicode[STIX]{x1D706}$ exceeding the injectivity radius, there is no cobordism directly associated to such a surgery. To fit such a surgery into the cobordism framework, in general we need to decompose the surgery into several steps. The following lemma shows how this works in the case of $\mathbb{C}\mathbb{P}^{n}$ (which easily generalizes to $\mathbb{R}\mathbb{P}^{n}$ and $\mathbb{HP}^{n}$ ).

Figure 8.

Left and middle: identification of $\mathbb{C}\mathbb{P}^{m/2}\text{\#}_{D_{x}}^{\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D706}}}Q_{x}$ and $\mathbb{C}\mathbb{P}^{m/2}\text{\#}_{x}^{\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D706}+\unicode[STIX]{x1D70B}}}F_{x}$ . Right: isotopy from an admissible function to a Dehn twist profile.

3.2 Product version

In this section we prove Theorem 1.1(1), (3). The proofs here are similar to that in the last section, and should be considered as family versions of it. In this subsection, we use $S$ to denote $S^{n}$ , $\mathbb{RP}^{n}$ , $\mathbb{CP}^{m/2}$ or $\mathbb{HP}^{n}$ equipped with the Riemannian metric such that every geodesic is closed of length $2\unicode[STIX]{x1D70B}$ .

For a symplectomorphism $\unicode[STIX]{x1D70F}:(M,\unicode[STIX]{x1D714})\rightarrow (M,\unicode[STIX]{x1D714})$ , we define the graph of $\unicode[STIX]{x1D70F}$ as

$$\begin{eqnarray}\displaystyle \text{Graph}(\unicode[STIX]{x1D70F}):=\{(p,\unicode[STIX]{x1D70F}(p))\in M\times M\mid p\in M\}. & & \displaystyle \nonumber\end{eqnarray}$$

In particular, $\text{Graph}(\unicode[STIX]{x1D70F})$ is a Lagrangian submanifold if we equip $M\times M$ with the symplectic form $\unicode[STIX]{x1D714}\oplus -\unicode[STIX]{x1D714}$ .

For the moment, let $S\subset (M,\unicode[STIX]{x1D714})$ be a Lagrangian sphere and $S^{-}:=S\subset M^{-}$ . One may consider the clean surgery of $L_{1}=S\times S^{-}$ and $L_{2}=\unicode[STIX]{x1D6E5}$ in $M\times M^{-}$ . In this case, they cleanly intersect along $D=\unicode[STIX]{x1D6E5}_{S}\subset S\times S^{-}$ . In Definition 2.25, take $E_{2}=S\times (T^{\ast }S)^{-}\subset T^{\ast }S\times (T^{\ast }S)^{-}$ , $E_{1}=T^{\ast }S\times S^{-}\subset T^{\ast }S\times (T^{\ast }S)^{-}$ and a $\unicode[STIX]{x1D70B}$ -admissible function $\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D70B}}$ .

Let $U\subset M$ be a Weinstein neighborhood of $S$ which can be identified with $T_{\unicode[STIX]{x1D716}}^{\ast }S$ , the open set of $T^{\ast }S$ consisting of covectors with length less than $\unicode[STIX]{x1D716}$ . It induces an identification between $U\times U$ , a Weinstein neighborhood of $L_{1}$ , and $T_{\unicode[STIX]{x1D716}}^{\ast }S\times T_{\unicode[STIX]{x1D716}}^{\ast }S$ . Now consider a point $(p,p)\in \unicode[STIX]{x1D6E5}_{U}:=\unicode[STIX]{x1D6E5}\cap (U\times U)$ , where $p$ can be considered as a point in $T_{\unicode[STIX]{x1D716}}^{\ast }S$ . The flow in Definition 2.25 defines a symplectomorphism fixing the first coordinate in $(T^{\ast }S\times (T^{\ast }S)^{-})\backslash E_{1}$ ; when restricted to $\unicode[STIX]{x1D6E5}_{U}$ , the $E_{2}$ -flow sends $(p,p)\mapsto (p,\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D70B}}(\Vert p\Vert )}^{\unicode[STIX]{x1D70E}}(p))$ . Therefore, the image of $\unicode[STIX]{x1D6E5}_{U}\backslash \unicode[STIX]{x1D6E5}_{S}$ under the flow is an open subset of the graph of $\unicode[STIX]{x1D70F}_{S}^{-1}$ (the inverse owes to the negation of symplectic form on $M^{-}$ ), except that we have used an admissible profile for the handle which is not a Dehn twist profile. Lemma 3.4 below ensures that this could be compensated by a local Hamiltonian perturbation. Hence modulo Lemma 3.4, this shows that $(S\times S^{-})\text{\#}_{\unicode[STIX]{x1D6E5}_{S},E_{2}}^{\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D70B}}}\unicode[STIX]{x1D6E5}=\text{Graph}(\unicode[STIX]{x1D70F}_{S}^{-1})$ (see Figure 9). The whole construction applies when $S$ is $\mathbb{RP}^{n}$ , $\mathbb{CP}^{m/2}$ or $\mathbb{HP}^{n}$ , except that the admissible profile has $\unicode[STIX]{x1D708}(0)=2\unicode[STIX]{x1D70B}$ .

Figure 9.

Obtaining the graph of a Dehn twist by $E_{2}$ -flow surgery.

We have the following corollary whose proof is similar to Corollary 3.2

Proof of Lemma 3.4.

The proof of the first statement is exactly the same as Lemma 3.1. For the second statement, we again only consider the case that $k=1$ and $S=\mathbb{CP}^{m/2}$ and the remaining cases are similar.

Define $D^{\operatorname{op}}=\{(x,y)\in S\times S\mid \text{dist}(x,y)=\unicode[STIX]{x1D70B}\}$ . Projection to the first factor in $S\times S$ equips $D^{\operatorname{op}}$ with a $\mathbb{CP}^{m/2-1}$ -bundle structure over $S=\mathbb{CP}^{m/2}$ . Therefore, a tubular neighborhood of $D^{\operatorname{op}}$ in $S\times S^{-}$ is the total space $\widetilde{\mathscr{V}}$ of a fiber bundle $\unicode[STIX]{x1D70B}_{\widetilde{\mathscr{V}}}:\widetilde{\mathscr{V}}\rightarrow S$ , whose fibers $\mathscr{V}$ are total spaces of topological ${\mathcal{O}}(1)$ -bundles over $\mathbb{CP}^{m/2-1}$ . We pick an open subset $U^{B}\subset S$ and a local trivialization $U^{B}\times U^{F}\simeq \unicode[STIX]{x1D70B}_{\widetilde{\mathscr{V}}}^{-1}(U^{B})$ (so $U^{F}=\mathscr{V}$ ). Readers should note that the product structure of $U^{B}\times U^{F}$ is not compatible with the product structure of $L=S\times S^{-}$ , but $\{q\}\times U^{F}$ is an open set of $\{q\}\times S^{-}$ for any $q\in U^{B}$ . In other words, we should regard $U^{B}$ as a parameter space and points in $U^{B}$ are parameterizing some open subsets of $S^{-}$ .

Pick a chart $U^{B}\times U^{F^{\prime }}$ on $U^{B}\times U^{F}$ with local coordinates $(q^{B},q^{F})=(q_{1}^{B},\ldots ,q_{m}^{B},q_{1}^{F},\ldots ,q_{m}^{F})$ adapted to $D^{\operatorname{op}}$ in the sense that:

• ∙ $(U^{B}\times U^{F^{\prime }})\cap D^{\operatorname{op}}=\{q_{1}^{F}=q_{2}^{F}=0\}$ ;

• ∙ $c(t)=(q^{B},tq_{1}^{F},tq_{2}^{F},q_{3}^{F},\ldots ,q_{m}^{F})$ are unit-speed geodesics for all $(q^{B},q^{F})$ such that $|q_{1}^{F}|^{2}+|q_{2}^{F}|^{2}=1$ ; and

• ∙ $c(t)\in \{q^{B}\}\times S$ is perpendicular to $\{q^{B}\}\times D_{q^{B}}\subset \{q^{B}\}\times S$ at $t=0$ with respect to the metric on $\{q^{B}\}\times S=S$ , where $D_{q^{B}}$ is the submanifold opposite to $q^{B}\in S$ .

Note that $c(0)\in D^{\operatorname{op}}$ and the projection of $c(0)$ to the first factor is $q^{B}$ so $c(0)\in \{q^{B}\}\times D_{q^{B}}$ .

The inclusions $U^{B}\times U^{F^{\prime }}\subset U^{B}\times U^{F}\subset S\times S^{-}$ induces canonical inclusions $T^{\ast }(U^{B}\times U^{F^{\prime }})\subset T^{\ast }(U^{B}\times U^{F})\subset T^{\ast }(S\times S^{-})$ . We write a point in $T^{\ast }(U^{B}\times U^{F^{\prime }})$ as $(q^{B},p^{B},q_{a}^{F},q_{b}^{F},p_{a}^{F},p_{b}^{F})$ , where $q_{a}^{F}=(q_{1}^{F},q_{2}^{F})$ , $q_{b}^{F}=(q_{3}^{F},\ldots ,q_{m}^{F})$ and similarly for $p_{a}^{F}$ and $p_{b}^{F}$ . We consider points $(q^{B},p^{B})\in T^{\ast }U^{B}$ as points in $T^{\ast }S$ (because $U^{B}\subset S$ ).

For each point $(q^{B},p^{B})\in T^{\ast }S$ , there is a corresponding point $\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D70B}}(\Vert p^{B}\Vert )}^{\unicode[STIX]{x1D70E}}(q^{B},p^{B})\in T^{\ast }S^{-}$ such that $((q^{B},p^{B}),\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D70B}}(\Vert p^{B}\Vert )}^{\unicode[STIX]{x1D70E}}(q^{B},p^{B}))\in H_{\unicode[STIX]{x1D708}^{\unicode[STIX]{x1D6FC}}}\subset T^{\ast }S\times T^{\ast }S^{-}$ . There are also corresponding open sets $\{(q^{B},p^{B})\}\times T^{\ast }U^{F^{\prime }}\subset \{(q^{B},p^{B})\}\times T^{\ast }S^{-}$ and $\{q^{B}\}\times U^{F^{\prime }}\subset \{q^{B}\}\times S^{-}$ such that $\{q^{B}\}\times U^{F^{\prime }}$ is adapted to $\{q^{B}\}\times D_{q^{B}}$ in the sense that:

• ∙ $\{q^{B}\}\times U^{F^{\prime }}\cap \{q^{B}\}\times D_{q^{B}}=\{q^{B}\}\times \{q_{a}^{F}=0\}$ ; and

• ∙ $c(t)=(q^{B},tq_{1}^{F},tq_{2}^{F},q_{3}^{F},\ldots ,q_{m}^{F})$ are unit-speed geodesics and perpendicular to $D_{q^{B}}$ at $t=0$ , for any $q^{F}\in U^{F}$ such that $|q_{1}^{F}|^{2}+|q_{2}^{F}|^{2}=1$ .

Since $\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D70B}}(\Vert \cdot \Vert )}^{\unicode[STIX]{x1D70E}}(\cdot )$ is defined by the geodesic flow on $\{(q^{B},p^{B})\}\times T^{\ast }S^{-}=T^{\ast }S^{-}$ and $\{q^{B}\}\times U^{F^{\prime }}$ is adapted to $\{q^{B}\}\times D_{q^{B}}$ , we know that $\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D708}^{\unicode[STIX]{x1D6FC}}(\Vert p^{B}\Vert )}^{\unicode[STIX]{x1D70E}}(q^{B},p^{B})=(-\unicode[STIX]{x1D6FC}p_{a}^{F},q_{b}^{F},p_{a}^{F},0)$ when $p^{B}\neq 0$ small (as in the proof of Lemma 3.1). Now, we have a parametrized version of (3.5)

(3.7) $$\begin{eqnarray}\displaystyle & & \displaystyle T^{\ast }(U^{B}\times U^{F^{\prime }})\cap H_{\unicode[STIX]{x1D708}^{\unicode[STIX]{x1D6FC}}}\nonumber\\ \displaystyle & & \displaystyle \quad =\{(q^{B},p^{B},-\unicode[STIX]{x1D6FC}p_{a}^{F},q_{b}^{F},p_{a}^{F},0)\mid p^{B}\neq 0,\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D708}^{\unicode[STIX]{x1D6FC}}(\Vert p^{B}\Vert )}^{\unicode[STIX]{x1D70E}}(q^{B},p^{B})=(-\unicode[STIX]{x1D6FC}p_{a}^{F},q_{b}^{F},p_{a}^{F},0)\}\end{eqnarray}$$

when $U^{B}$ and $U^{F^{\prime }}$ are sufficiently small.

Here, both $\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D70B}}^{\unicode[STIX]{x1D6FC}}(\Vert p^{B}\Vert )}^{\unicode[STIX]{x1D70E}}(q^{B},p^{B})$ and $(-\unicode[STIX]{x1D6FC}p_{a}^{F},q_{b}^{F},p_{a}^{F},0)$ are considered as points in $T_{\unicode[STIX]{x1D716}}^{\ast }S$ although they belong to different factors of $T^{\ast }(S\times S^{-})$ . Therefore, in $T^{\ast }(U^{B}\times U^{F^{\prime }})\cap H_{\unicode[STIX]{x1D708}^{\unicode[STIX]{x1D6FC}}}$ , fixing $q^{B}$ and letting $p^{B}$ go to $0$ linearly leads to fixing $q_{b}^{F}$ and letting $p_{a}^{F}$ go to zero linearly.

Since $H_{\unicode[STIX]{x1D708}^{\unicode[STIX]{x1D6FC}}}$ is globally defined, the above discussion is true for any charts $U^{B}\times U^{F^{\prime }}$ on $U^{B}\times U^{F}$ adapted to $D^{\operatorname{op}}$ . From the discussion using local charts on $U^{B}\times U^{F}$ and the fact that $p_{a}^{F}$ goes to zero linearly as $p^{B}$ goes to $0$ linearly, we can see that $H_{\unicode[STIX]{x1D708}^{\unicode[STIX]{x1D6FC}}}$ and $D^{\operatorname{op}}$ can be glued smoothly (linearity is not necessary for $H_{\unicode[STIX]{x1D708}^{\unicode[STIX]{x1D6FC}}}$ and $D^{\operatorname{op}}$ to be glued smoothly but it is sufficient). The fact that $H_{\unicode[STIX]{x1D708}^{\unicode[STIX]{x1D6FC}}}$ can be glued smoothly with $\unicode[STIX]{x1D6E5}$ is because all orders of derivatives of $\unicode[STIX]{x1D708}^{\unicode[STIX]{x1D6FC}}$ vanish at $r=\unicode[STIX]{x1D716}$ . It results in a smooth Lagrangian, which we denote by $(S\times S^{-})\text{\#}_{\unicode[STIX]{x1D6E5}_{S},E_{2}}^{\unicode[STIX]{x1D708}^{\unicode[STIX]{x1D6FC}}}\unicode[STIX]{x1D6E5}$ .

Finally, we show that $(S\times S^{-})\text{\#}_{\unicode[STIX]{x1D6E5}_{S},E_{2}}^{\unicode[STIX]{x1D708}^{\unicode[STIX]{x1D6FC}}}\unicode[STIX]{x1D6E5}$ is Hamiltonian isotopic to $(S\times S^{-})\text{\#}_{\unicode[STIX]{x1D6E5}_{S},E_{2}}^{\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D706}_{1}}}\unicode[STIX]{x1D6E5}$ . We can choose $\{\unicode[STIX]{x1D708}^{\unicode[STIX]{x1D6FC}_{t}}\}_{t\in [0,1)}$ interpolating $\unicode[STIX]{x1D708}^{\unicode[STIX]{x1D6FC}}$ and $\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D706}_{1}}$ as in the proof of Lemma 3.1. This is a smooth Lagrangian isotopy which is invariant under the diagonal $\text{Isom}(S)$ action.◻

In parallel to Lemma 3.3, we have the following.

Lemma 3.6 (Theorem 1.1(3)).

Let $S$ be $\mathbb{RP}^{n}$ , $\mathbb{CP}^{m/2}$ or $\mathbb{HP}^{n}$ and

$$\begin{eqnarray}\displaystyle D^{\operatorname{op}}=\{(x,y)\in S\times S^{-}\mid \text{dist}(x,y)=\unicode[STIX]{x1D70B}\}. & & \displaystyle \nonumber\end{eqnarray}$$

Up to Hamiltonian isotopy in $T^{\ast }S\times (T^{\ast }S)^{-}$ , we have

$$\begin{eqnarray}\displaystyle (S\times S^{-})\text{\#}_{D^{\operatorname{op}},E_{2}}((S\times S^{-})\text{\#}_{\unicode[STIX]{x1D6E5}_{S},E_{2}}\unicode[STIX]{x1D6E5})=\text{Graph}(\unicode[STIX]{x1D70F}_{S}^{-1}). & & \displaystyle \nonumber\end{eqnarray}$$

3.3 Family versions

One may also generalize the above example to the case of family Dehn twists [Reference Wehrheim and WoodwardWW16] which we now recall. Let $G$ be $\operatorname{SO}(l+1)$ with Lie algebra $\mathfrak{g}$ . A connection one form on a principal $G$ bundle $\unicode[STIX]{x1D70B}:P\rightarrow B$ is a one form $\unicode[STIX]{x1D6FC}\in \unicode[STIX]{x1D6FA}^{1}(P,\mathfrak{g})$ satisfying the following two conditions:

• ∙ $\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D709}_{P})=\unicode[STIX]{x1D709}$ for any $\unicode[STIX]{x1D709}\in \mathfrak{g}$ , where $\unicode[STIX]{x1D709}_{P}$ is the vector field generating the action of $\unicode[STIX]{x1D709}$ ;

• ∙ $g^{\ast }\unicode[STIX]{x1D6FC}=\operatorname{Ad}(g)^{-1}\unicode[STIX]{x1D6FC}$ for any $g\in G$ , where the adjoint action is on the values of $\unicode[STIX]{x1D6FC}$ .

We have a splitting

$$\begin{eqnarray}TP=\ker (\unicode[STIX]{x1D6FC})\oplus \ker (D\unicode[STIX]{x1D70B})\end{eqnarray}$$

which is invariant under the group action.

Suppose $B$ admits a symplectic structure $\unicode[STIX]{x1D714}_{B}$ . We equip $(T^{\ast }S^{l},\unicode[STIX]{x1D714}_{T^{\ast }S^{l}})$ with the Hamiltonian $G$ -action induced by the isometry of the zero section $G=\text{Isom}(S^{l})\subset \text{Symp}(T^{\ast }S^{l})$ , where $\unicode[STIX]{x1D714}_{T^{\ast }S^{l}}$ is the canonical symplectic structure on $T^{\ast }S^{l}$ . We denote the moment map by $\unicode[STIX]{x1D6F7}:T^{\ast }S^{l}\rightarrow \mathfrak{g}^{\vee }$ . The minimally coupling form on $P\times T^{\ast }S^{l}$ is defined by

(3.8) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D714}_{P\times T^{\ast }S^{l},\unicode[STIX]{x1D6FC}}:=\unicode[STIX]{x1D70B}_{P}^{\ast }\unicode[STIX]{x1D70B}^{\ast }\unicode[STIX]{x1D714}_{B}+\unicode[STIX]{x1D70B}_{S}^{\ast }\unicode[STIX]{x1D714}_{T^{\ast }S^{l}}+d\langle \unicode[STIX]{x1D70B}_{P}^{\ast }\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D70B}_{S}^{\ast }\unicode[STIX]{x1D6F7}\rangle \in \unicode[STIX]{x1D6FA}^{2}(P\times T^{\ast }S^{l}), & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D70B}_{P}:P\times T^{\ast }S^{l}\rightarrow P$ and $\unicode[STIX]{x1D70B}_{S}:P\times T^{\ast }S^{l}\rightarrow T^{\ast }S^{l}$ are projections to the first and second factors, respectively. The two form $\unicode[STIX]{x1D714}_{P\times T^{\ast }S^{l},\unicode[STIX]{x1D6FC}}$ has the property that $\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D709}_{P\times F}}\unicode[STIX]{x1D714}_{P\times F,\unicode[STIX]{x1D6FC}}=0$ and it descends to a symplectic form $\unicode[STIX]{x1D714}_{P(T^{\ast }S^{l})}$ on $P\times _{G}T_{\unicode[STIX]{x1D716}}^{\ast }S^{l}$ for some $\unicode[STIX]{x1D716}>0$ , where $\unicode[STIX]{x1D709}_{P\times F}$ is the vector field in $P\times F$ generating the diagonal action of $\unicode[STIX]{x1D709}$ and $T_{\unicode[STIX]{x1D716}}^{\ast }S^{l}$ consists of the cotangent vectors with norm less than $\unicode[STIX]{x1D716}$ . Moreover, $\unicode[STIX]{x1D70B}\circ \unicode[STIX]{x1D70B}_{P}:P\times T^{\ast }S^{l}\rightarrow B$ descends to a symplectic fiber bundle map $\unicode[STIX]{x1D70B}_{P(T^{\ast }S^{l})}:P\times _{G}T_{\unicode[STIX]{x1D716}}^{\ast }S^{l}\rightarrow B$ . An important feature of $\unicode[STIX]{x1D714}_{P(T^{\ast }S^{l})}$ is the following.

Recall that a spherically fibered coisotropic manifold $i:C^{2n-l}{\hookrightarrow}M^{2n}$ is a coisotropic submanifold so that there is a fibration $\unicode[STIX]{x1D70C}:C\rightarrow B^{2n-2l}$ over a symplectic base, whose fibers are leaves of the characteristic foliation (also called null-leaves) and are diffeomorphic to $S^{l}$ . In other words, $\unicode[STIX]{x1D70C}^{\ast }\unicode[STIX]{x1D714}_{B}=i^{\ast }\unicode[STIX]{x1D714}_{M}$ . Moreover, we equip the fibers with round metrics such that all their geodesics are closed of length $2\unicode[STIX]{x1D70B}$ and ask that the structure group of $\unicode[STIX]{x1D70C}$ lie in $\operatorname{SO}(l+1)$ .

A neighborhood $U$ of $C$ can be symplectically identified with $P\times _{\operatorname{SO}(l+1)}T_{\unicode[STIX]{x1D716}}^{\ast }S^{l}$ , where $P$ is the principal $\operatorname{SO}(l+1)$ -bundle associated to $C$ . The family Dehn twist $\unicode[STIX]{x1D70F}_{C}$ can then be defined fiberwise as the fiberwise Hamiltonian function $\widetilde{\unicode[STIX]{x1D708}}_{\unicode[STIX]{x1D716}}^{\text{Dehn}}(\Vert p\Vert )$ (see Remark 2.12) is preserved by the structure group. With respect to the fiberwise metric $g^{v}$ , the function $h(\cdot )=\widetilde{\unicode[STIX]{x1D708}}(\Vert \cdot \Vert _{g^{v}})$ defines a flow along fibers whose time- $1$ map is the desired Dehn twist (with a continuation over $C$ defined by the fiberwise antipodal map on $C$ ).

Equivalently, we can choose a reference fiber $F=T^{\ast }S^{l}$ of the symplectic fibration $\unicode[STIX]{x1D70B}_{P(T^{\ast }S^{l})}$ and the family Dehn twist restricted to any fiber $F^{\prime }$ is defined to be $\unicode[STIX]{x1D70F}_{C}|_{F^{\prime }}:=\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D6FE}^{-1}}\circ \unicode[STIX]{x1D70F}_{S^{l}}\circ \unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D6FE}}$ , where $\unicode[STIX]{x1D6FE}$ is a path from the base point of $F^{\prime }$ to the base point of $F$ , $\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D6FE}}$ is symplectic parallel transport along $\unicode[STIX]{x1D6FE}$ and $\unicode[STIX]{x1D70F}_{S^{l}}$ is the Dehn twist on $F$ . By Corollary 3.8, $\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D6FE}^{-1}}\,\circ \,\unicode[STIX]{x1D70F}_{S^{l}}\,\circ \,\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D6FE}}$ is independent of $\unicode[STIX]{x1D6FE}$ and gives a symplectomorphism of $F^{\prime }$ . The fiberwise symplectomorphisms patch together to a diffeomorphism $\unicode[STIX]{x1D70F}_{C}$ . To see that $\unicode[STIX]{x1D70F}_{C}$ is a symplectomorphism of $P\times _{\operatorname{SO}(l+1)}T_{\unicode[STIX]{x1D716}}^{\ast }S^{l}$ , it suffices to check that

(3.9) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D714}_{P(T^{\ast }S^{l})}((\unicode[STIX]{x1D70F}_{C})_{\ast }v_{1},(\unicode[STIX]{x1D70F}_{C})_{\ast }v_{2})=\unicode[STIX]{x1D714}_{P(T^{\ast }S^{l})}(v_{1},v_{2}) & & \displaystyle\end{eqnarray}$$

for any $v_{1},v_{2}\in \ker (\unicode[STIX]{x1D6FC})$ because $T(P\times _{\operatorname{SO}(l+1)}T_{\unicode[STIX]{x1D716}}^{\ast }S^{l})=\ker (\unicode[STIX]{x1D6FC})\oplus T(T_{\unicode[STIX]{x1D716}}^{\ast }S^{l})$ , $(\ker (\unicode[STIX]{x1D6FC}))^{\unicode[STIX]{x1D714}_{P(T^{\ast }S^{l})}}=T(T_{\unicode[STIX]{x1D716}}^{\ast }S^{l})$ by Lemma 3.7 and we already know that $\unicode[STIX]{x1D70F}_{C}$ is fiberwise symplectic. By construction, $(\unicode[STIX]{x1D70F}_{C})_{\ast }v_{i}\in \ker (\unicode[STIX]{x1D6FC})$ and $(\unicode[STIX]{x1D70B}_{P(T^{\ast }S^{l})}\circ \unicode[STIX]{x1D70F}_{C})_{\ast }v_{i}=(\unicode[STIX]{x1D70B}_{P(T^{\ast }S^{l})})_{\ast }v_{i}$ for $i=1,2$ so (3.9) follows from (3.8).

Now consider the natural Lagrangian embedding $\widetilde{C}:=C\times _{B}C{\hookrightarrow}M\times M$ , where $M$ is a symplectic manifold such that we have an inclusion $i:C^{2n-l}{\hookrightarrow}M^{2n}$ making $C$ a spherically fibered coisotropic manifold. Explicitly, the image of this map is

$$\begin{eqnarray}\widetilde{C}=\{(x,y)\in C\times C\subset M\times M:\unicode[STIX]{x1D70B}(x)=\unicode[STIX]{x1D70B}(y)\},\end{eqnarray}$$

where $\unicode[STIX]{x1D70B}:C\rightarrow B$ is the $S^{l}$ -bundle projection. Indeed, $\widetilde{C}=C^{t}\circ C$ is a composition Lagrangian in the sense of (5.2). Here we have abused the notation by identifying $C$ with its Lagrangian image in $B\times M$ defined by

$$\begin{eqnarray}\{(x,y)\in B\times C\subset B\times M:\unicode[STIX]{x1D70B}(y)=x\}.\end{eqnarray}$$

Note that $\widetilde{C}$ is a fiber bundle over $B$ with fiber $S^{l}\times S^{l}$ and structure group the diagonal $\operatorname{SO}(l+1)$ .

We continue to use $U$ to denote a neighborhood of $C$ in $M$ which can be identified with $P\times _{\operatorname{SO}(l+1)}T_{\unicode[STIX]{x1D716}}^{\ast }S^{l}$ . Consider a symplectic trivialization $U_{0}:=B_{0}\times T_{\unicode[STIX]{x1D716}}^{\ast }S^{l}$ of the symplectic fiber bundle $\unicode[STIX]{x1D70B}_{P(T^{\ast }S^{l})}:U=P\times _{\operatorname{SO}(l+1)}T_{\unicode[STIX]{x1D716}}^{\ast }S^{l}\rightarrow B$ for some contractible open set $B_{0}\subset B$ . Any point contained in $\unicode[STIX]{x1D6E5}\cap (U_{0}\times U_{0}^{-})\subset M\times M^{-}$ thus takes the form $((x,p),(x,p))$ , where $x\in B_{0}$ and $p\in T_{\unicode[STIX]{x1D716}}^{\ast }S^{l}$ . In this setting, the graph of $\unicode[STIX]{x1D70F}_{C}^{-1}$ in $U_{0}\times U_{0}^{-}$ consist of points

$$\begin{eqnarray}\text{Graph}(\unicode[STIX]{x1D70F}_{C}^{-1})=\{((x,p),(x,\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D708}^{\text{Dehn}}(\Vert p\Vert )}^{\unicode[STIX]{x1D70E}}(p)))\mid x\in B_{0},p\in T_{\unicode[STIX]{x1D716}}^{\ast }S^{l}\}.\end{eqnarray}$$

This is true for any contractible open subset $B_{0}\subset B$ and $\text{Graph}(\unicode[STIX]{x1D70F}_{C}^{-1})$ coincides with $\unicode[STIX]{x1D6E5}$ outside $U\times U^{-}$ .

As before, we want to realize $\text{Graph}(\unicode[STIX]{x1D70F}_{C}^{-1})$ as a surgery from $\widetilde{C}$ to $\unicode[STIX]{x1D6E5}$ . In this case, we want to perform a family $E_{2}$ -surgery.

In the notation of § 2.3, let $L=S^{l}\times (S^{l})^{-}$ , $D=\unicode[STIX]{x1D6E5}_{S^{l}}\subset L$ , $\mathscr{P}=U\times U^{-}$ , $\mathscr{B}=B\times B^{-}$ , $i:B\rightarrow \unicode[STIX]{x1D6E5}_{B}\subset \mathscr{B}$ be the diagonal embedding, $\mathscr{L}=\widetilde{C}$ , $N_{\mathscr{D}}^{\ast }\mathscr{L}=\unicode[STIX]{x1D6E5}_{U}$ , $\mathscr{D}=\unicode[STIX]{x1D6E5}_{U}\cap \widetilde{C}$ , and $G=\operatorname{SO}(l+1)=\text{Isom}_{\unicode[STIX]{x1D6E5}}(L)\subset \operatorname{SO}(l+1)\times \operatorname{SO}(l+1)$ is the diagonal isometry group. It is clear how to define the fiber bundle structures

compatible with the inclusions $\mathscr{D}{\hookrightarrow}\mathscr{L}{\hookrightarrow}i^{\ast }\mathscr{P}$ . As in the calculation for Lemma 3.7 and Corollary 3.8, we know that the symplectic orthogonal complement of the fibers of $\unicode[STIX]{x1D70B}:\mathscr{P}\rightarrow \mathscr{B}$ is given by the product horizontal distribution $\ker (\unicode[STIX]{x1D6FC})\oplus (\ker (\unicode[STIX]{x1D6FC}))^{-}$ and the symplectic monodromy for loops in $B$ lies in the diagonal isometry group $G$ . We leave it as an exercise for readers to check that $G$ satisfies all the other assumptions in § 2.3. As a result, we can define a global Lagrangian handle $\mathscr{H}_{\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D70B}}}\subset \mathscr{P}$ .

By the same token, we can define a projectively fibered coisotropic manifold which is a coisotropic manifold with null-leaves complex (or real, quaternionic) projective spaces. Family Dehn twists for these spaces are defined similarly.