Exact triangle for fibered Dehn twists
 Katrin Wehrheim^{1} and
 Chris T. Woodward^{2}Email author
DOI: 10.1186/s406870160065x
© The Author(s) 2016
Received: 20 April 2015
Accepted: 26 April 2016
Published: 15 August 2016
Abstract
We use quilted Floer theory to generalize Seidel’s long exact sequence in symplectic Floer theory to fibered Dehn twists. We then apply the sequence to construct versions of the Floer and Khovanov–Rozansky exact triangles in Lagrangian Floer theory of moduli spaces of bundles.
1 Introduction
Seidel’s long exact sequence [41, Theorem 1] describes the effect of a symplectic Dehn twist on Lagrangian Floer cohomology. In many examples (moduli spaces of bundles, nilpotent slices, etc.), the relevant fibrations have Morse–Bott rather than Morse singularities and the associated monodromy maps are fibered Dehn twists. Many years ago, Seidel suggested that this sequence should generalize to the fibered case. In this paper, we show how to carry out this suggestion using quilted Floer theory developed in Wehrheim–Woodward [44, 45, 50]. Quilted Floer theory gives an expression of the third term in the exact triangle as a push–pull functor, similar to the situation in the analogous triangle in algebraic geometry developed by Horja [16].
To state the main result, suppose that M is an exact or monotone symplectic manifold. If M has boundary \(\partial M\), then we assume that the boundary \(\partial M\) is convex so that our spaces of pseudoholomorphic curves satisfy good compactness properties. A Lagrangian brane in M is a compact, oriented Lagrangian submanifold L equipped with a grading in the sense of [45] and relative spin structure in the sense of [48]. We say that a Lagrangian brane L is admissible if L is monotone in the sense of [45], the image of the fundamental group \(\pi _1(L)\) of L in \(\pi _1(M)\) is torsion for any choice of base point, and L has minimal Maslov number at least three. This notion of admissibility is chosen so that any pair \(L^0,L^1\) of admissible Lagrangian branes in M is monotone as a pair, which implies an energyindex relation for pseudoholomorphic strips. This relation in turn implies that disk bubbles cannot obstruct the proof of \(\partial ^2=0\) in the construction of the Lagrangian Floer homology group \(HF(L^0,L^1)\). The underlying complex for this group is generated by perturbed intersection points of \(L^0,L^1\). The differential counts finite energy holomorphic strips with boundary in \(L^0,L^1\). Taking the Floer groups as morphism spaces, one obtains the cohomology of the Fukaya category of Lagrangian branes in M. For the moment, we work with \(\mathbb {Z}_2\) coefficients, although the main result will be stated with \(\mathbb {Z}\) coefficients.
Theorem 1.1
Seidel’s result is often referred to as a categorification of Picard–Lefschetz since by taking the Euler characteristics one essentially recovers the Picard–Lefschetz formula as in Arnold [4, Chapter I]. The Fukayacategorical version (conjectured by Kontsevich) is developed in Seidel’s book [42].
Definition 1.2
 (a)
(Fibrating) the null foliation of C is fibrating over a symplectic base B with fiber \(S^c\) a sphere of dimension c and
 (b)
(Orthogonal structure group) the structure group of \(p: C \rightarrow B\) is equipped with a reduction to \(SO(c+1)\), that is, a principal \(SO(c+1)\)bundle \(P\rightarrow B\) and a bundle isomorphism \(P\times _{SO(c+1)} S^c \cong C\).
Theorem 1.3
A similar triangle was developed by Perutz, as part of the program described in [33]. A different approach to exact triangles via Lagrangian cobordism is given in the recent work of Mak and Wu [25] who also treated the codimension one case with \(\mathbb {Z}_2\) coefficients for the first time. We also treat the codimension one case below (see Remark 5.10) although the monotonicity assumptions required in this case are more complicated.
As in Seidel’s work, there is a connection with the mapping cone construction in the derived Fukaya category, which we establish in Sect. 7.1 as follows.
Theorem 1.4
We briefly outline the contents of the paper. Section 2 contains background results on fibered Dehn twists and Lefschetz–Bott fibrations. Section 3 describes various situations in which surface Dehn twists induce generalized Dehn twists on moduli spaces of flat bundles; these are mostly minor improvements of results of Seidel and Callahan. Sections 4 and 5 contain the proof of the exact triangle. Section 6 applies the triangle to moduli spaces of flat bundles to obtain generalizations of Floer’s exact triangle for surgery along a knot, as well as surgery exact triangles for crossing changes in knots which have the same form as the surgery exact triangles as Khovanov [17] and Khovanov and Rozansky [19]. Finally, Sect. 7 describes generalizations to the \(A_\infty \) setting. These are limited to the case of minimal Maslov number greater than two.
We thank Mohammed Abouzaid, Tim Perutz, and Reza Rezazadegan for helpful comments, and especially Paul Seidel who got us involved in this area.
The present paper is an updated and more detailed version of a paper the authors have circulated since 2007. The authors have unreconciled differences over the exposition in the paper, and explain their points of view at https://math.berkeley.edu/~katrin/wwpapers/ resp. http://christwoodwardmath.blogspot.com/. The publication in the current form is the result of a mediation.
2 Lefschetz–Bott fibrations and fibered Dehn twists
This section covers the generalization of the theory of symplectic Lefschetz fibrations to the Lefschetz–Bott case, that is, to the case that the singularities of the fibration are not isolated but still nondegenerate in the normal directions. Most of this material is covered in an unpublished manuscript of Seidel [39] and in the works of Perutz [33, 34]. For more recent appearance of fibered Dehn twists, see Chiang et al. [8].
2.1 Symplectic Lefschetz–Bott fibrations
In our examples, we will not have global complex structures on E and S (at least no canonical ones). Instead, we work with symplectic versions of Lefschetz–Bott fibrations. The definition of the symplectic version uses the following condition introduced in Perutz [33]. Let \((M,\omega )\) be a symplectic manifold equipped with an almost complex structure J and \(M' \subset M\) an almost complex submanifold. The submanifold \(M'\) is said to be normally Kähler if a tubular neighborhood N of \(M'\) in M is foliated by Jcomplex normal slices \(\{ N_e \subset N\}, e \in M'\), such that \(J  N_e\) is integrable and \(\omega  N_e\) is Kähler for each e.
Definition 2.1
 (a)(Symplectic fibrations) A symplectic fibration is a manifold E equipped with a closed twoform \(\omega _E \in \Omega ^2(E)\) and a fibration \(\pi : E \rightarrow S\) over a smooth surface S, such that the restriction of \(\omega _E\) to any fiber of \(\pi \) is symplectic:$$\begin{aligned} (\omega _E(e) _{D_e \pi ^{1}(0)} )^{\dim (E)1} \ne 0 \quad \forall e \in E . \end{aligned}$$
 (b)(Symplectic Lefschetz–Bott fibrations) A symplectic Lefschetz–Bott fibration consists of
 (i)
a smooth manifold E equipped with a closed twoform \(\omega _E\);
 (ii)
a smooth, oriented surface S;
 (iii)a smooth proper map \(\pi : E \rightarrow S\) with critical set and values$$\begin{aligned} E^{{{\text {crit}}}} := \{e \in E \  \ {\text {rank}}(D_e \pi ) < 2 \}, \quad S^{{{\text {crit}}}}=\pi (E^{{{\text {crit}}}})\subset S ; \end{aligned}$$
 (iv)
a positively oriented complex structure \(j_0\in \mathrm{End}(TS_{\mathcal {U}})\) defined in a neighborhood \({\mathcal {U}}\subset S\) of the critical values \(S^{{{\text {crit}}}}\); and
 (v)
an almost complex structure \(J_0\in \mathrm{End}(TE_{\mathcal {V}})\) defined in a neighborhood \({\mathcal {V}}\subset E\) of the critical set \(E^{{{\text {crit}}}}\)
 (i)
\(E^{{{\text {crit}}}}\subset E\) is a smooth submanifold with finitely many components;
 (ii)
\(E^{{{\text {crit}}}}\) is normally Kähler;
 (iii)
\(\pi _{\mathcal {V}}: {\mathcal {V}}\rightarrow {\mathcal {U}}\) is \((J_0,j_0)\) holomorphic;
 (iv)
the normal Hessian \(D^2\pi _e  T^{\otimes 2}N_e\) at any critical point is nondegenerate;
 (v)
\(\omega _E\) is nondegenerate on \( {\text {ker}}(D\pi )\subset TE\);
 (vi)
\(\omega _E_{\mathcal {V}}\) is nondegenerate and compatible with \(J_0\).
 (i)
Remark 2.2
Proposition 2.3
Each vanishing cycle \(C_t\) from (6) is a coisotropic submanifold of the fiber \(E_{\gamma (t)}\). The map \(\rho _{t,1}:C_t \rightarrow B\) is smooth and gives \(C_t\) the structure of a spherically fibered coisotropic submanifold in the sense of Definition 1.2.
Proof
2.2 Fibered Dehn twists
The symplectic Dehn twist along a Lagrangian sphere in [41] can be generalized to spherically fibered coisotropics using the associated symplectic fiber bundle construction. This construction associates with any principal bundle and Hamiltonian action with small moment map a symplectic fiber bundle.
 (a)
the evaluation \(\alpha (\xi _P)\) is the constant function equal to \(\xi \) for any \(\xi \in \mathfrak {g}\), where \(\xi _P \in {\text {Vect}}(P)\) is the vector field generating the action of \(\xi \), and
 (b)
the pullback \( g^* \alpha \) is equal to \( {\text {Ad}} (g)^{1} \alpha \) for any \(g \in G\), where the adjoint action is on the values of \(\alpha \).

a base symplectic manifold \((B,\omega _B)\);

a principal bundle \(\pi : P \rightarrow B\) with structure group G;

a fiber symplectic manifold \((F,\omega _F)\) equipped with a Hamiltonian Gaction with moment map \(\varPhi _F: F \rightarrow \mathfrak {g}^\vee := {\text {Hom}}(\mathfrak {g},\mathbb {R})\); and

a connection oneform \(\alpha \in \Omega ^1(P,\mathfrak {g})^{G}\).
Theorem 2.4
 (a)The minimally coupled form \(\omega _{P \times F,\alpha }\) descends to a closed twoformThe form \(\omega _{P(F),\alpha }\) is nondegenerate on the fibers in a neighborhood$$\begin{aligned} \omega _{P(F),\alpha } \in \Omega ^2(P(F)), \quad P(F) := P \times _G F. \end{aligned}$$of the associated bundle to the zerolevel set \(P(\varPhi ^{1}_F(0)) \subset P(F)\). Here \(\vert \varPhi _F \vert \) denotes the norm of \(\varPhi _F\) with respect to an invariant metric on \(\mathfrak {g}\cong \mathfrak {g}^\vee \). Hence \(P(F)_\epsilon \) is a symplectic fiber bundle over B for sufficiently small \(\epsilon > 0 \).$$\begin{aligned} P(F)_\epsilon := P( \{ \vert \varPhi _F \vert < \epsilon \}) \end{aligned}$$
 (b)
If \(\varPhi ^{1}_F(0)\) is smooth, then \(P(\varPhi ^{1}_F(0))\) is a smooth submanifold of \(P(F)_\epsilon \) with coisotropic fibers.
 (c)
Given two choices of connection \(\alpha _j, j = 0,1\), there exists an isomorphism of symplectic fiber bundles from \((P(F)_\epsilon ,\omega _{P(F),\alpha _0})\) to \((P(F)_\epsilon , \omega _{P(F),\alpha _1})\) for sufficiently small \(\epsilon > 0 \).
 (d)
The association \((F,\omega _F,\varPhi _F) \rightarrow (P(F)_\epsilon ,\omega _{P(F)})\) is functorial in the sense that any isomorphism of Hamiltonian Gmanifolds \((F_0,\omega _{F_0},\varPhi _{F_0})\) to \((F_1,\omega _{F_1}, \varPhi _{F_1})\) induces an isomorphism of symplectic fiber bundles \(P(F_0)_\epsilon \rightarrow P(F_1)_\epsilon \).
Example 2.5
(Associated bundles with cotangentsphere fibers) We are mainly interested in the following special case of the general construction. For any integer \(c \ge 1\), let \(S^c\) denote a sphere of dimension c and \(T^\vee S^c\) its cotangent bundle. Consider \(T^\vee S^c\) with canonical symplectic form \(\omega _{T^\vee S^c}\) and the canonical \(SO(c+1)\)action. The action is Hamiltonian with a moment map \( \varPhi _{T^\vee S^c}\) whose zerolevel set is \(S^c\). Thus for any principal \(SO(c+1)\)bundle \(\pi : P \rightarrow B\) the associated fiber bundle construction yields a symplectic fiber bundle \(P(T^\vee S^c)_\epsilon \) over B. By functoriality, any automorphism \(\tau \) of \((T^\vee S^c,\omega _{T^\vee S^c},\varPhi _{T^\vee S^c})\) induces a bundle isomorphism \(\tau _{P(T^\vee S^c)}: P(T^\vee S^c) \rightarrow P(T^\vee S^c)\). The latter is an isomorphism of symplectic fiber bundles on \(P(T^\vee S^c)_\epsilon \).
The notion of Dehn twist is most familiar from Riemann surface theory, where a Dehn twist denotes a diffeomorphism obtained by twisting around a circle on a handle. In [41], Seidel introduces a generalized notion of Dehn twist which is a symplectomorphism around a Lagrangian sphere, called a generalized Dehn twist. The symplectomorphisms we consider here are further generalized by allowing the twists to be fibered, so that the vanishing cycles are fibered coisotropics. To save space, we call these simply fibered Dehn twists. We begin with the local model introduced by Seidel [41].
Definition 2.6
We construct fibered Dehn twists along spherically fibered coisotropics as follows.
Definition 2.7
 (a)
(Coisotropic embedding) Recall that C is diffeomorphic to an associated fiber bundle \(P(S^c) := P \times _{SO(c+1)} S^c ,\) for some principal \(SO(c+1)\)bundle \(\pi : P \rightarrow B\). By the coisotropic embedding theorem [14, p. 315], a neighborhood of C in M is symplectomorphic to a neighborhood of the zero section in \(P(T^\vee S^c)_\epsilon \) as in Theorem 2.4.
 (b)(Model fibered Dehn twists) Any \(SO(c+1)\)equivariant model Dehn twist \(\tau _{T^\vee S^c}: T^\vee S^c \rightarrow T^\vee S^c\) induces a symplectomorphismby functoriality of the associated symplectic fiber bundle construction as in Example 2.5. Given a symplectomorphism \(\phi \) of a neighborhood U of \(C\subset M\) with \(P(T^\vee S^c)_\epsilon \), we define a symplectomorphism \( \tau _C : M \rightarrow M \) by \(\tau _{ P(T^\vee S^c)}\) on the neighborhood of C and the identity outside:$$\begin{aligned} \tau _{P(T^\vee S^c)} : P(T^\vee S^c)_\epsilon \rightarrow P(T^\vee S^c)_\epsilon \end{aligned}$$We call \(\tau _C\) a model fibered Dehn twist along C.$$\begin{aligned} \tau _C_U = \phi ^{1} \tau _{T^\vee P(S^c)}_{\phi (U)} \phi , \quad \tau _C _{M  U}= {\text {Id}}_M . \end{aligned}$$
 (c)
(Fibered Dehn twists) A symplectomorphism \(\tau _C\) of M is called a fibered Dehn twist along C if \(\tau _C\) is Hamiltonian isotopic to a model Dehn twist.
Remark 2.8
The Hamiltonian isotopy class of a fibered Dehn twist is independent of the choice of local model and fibered Dehn twist used in its construction: Any two local models for a fibered coisotropic are isotopic, by a family version of Moser’s construction. This fact implies that any two fibered Dehn twists \(\tau _{C,0}, \tau _{C,1}\) defined using different local models and model twists may be connected by a family \(\tau _{C,t}\). The vector field \( v_t : = (\tau _{C,t}^{1} )_* \frac{d}{dt}\tau _{C,t} \in {\text {Vect}}(M)\) vanishes on C and is necessarily Hamiltonian in a neighborhood of C.
2.3 Equivariant fibered Dehn twists

a principal \(SO(c+1)\)bundle \(\pi : P\rightarrow B\) equipped with an action of G by bundle automorphisms (i.e., \(SO(c+1)\)equivariant diffeomorphisms) and

a Gequivariant bundle isomorphism \(P\times _{SO(c+1)} S^c \cong C\), where the Gaction is induced by the action on the first factor.
Lemma 2.9
Proof
Lemma 2.10
Suppose that \(C \subset M\) is a spherically fibered Gcoisotropic over a base B where M is a Hamiltonian Gmanifold with moment map \(\varPhi \). Then \(\varPhi \) is constant on the fibers of C and the induced action of G on B is Hamiltonian with moment map \(\varPhi _B:B \rightarrow \mathfrak {g}^\vee \) the unique map satisfying \(p^* \varPhi _B = \varPhi  C\).
Proof
By assumption, the action of G on P is \(SO(c+1)\)equivariant and so induces an action on B. For any \(\xi \in \mathfrak {g}\), the infinitesimal action \(\xi _M\in {\text {Vect}}(M)\) is tangent to C. Hence \(L_v \langle {\varPhi },\xi \rangle = {\omega }(\xi _M,v) = 0\) for all fiber tangent vectors \(v \in T^{{\text {vert}}} {C}=TC^{\omega }\). It follows that \(\varPhi \) is constant on the fibers of \(p: C \rightarrow B\). So \(\varPhi \) induces a map \(\varPhi _B: B \rightarrow \mathfrak {g}^\vee \), satisfying \({\text{ d }}\langle \varPhi _B, \xi \rangle = \iota (\xi _B) \omega _B\) for all Lie algebra vectors \(\xi \in \mathfrak {g}\) as claimed. \(\square \)
Remark 2.11
In this setting, every Gequivariant fibered Dehn twist along C descends to a fibered Dehn twist of \(M{//}G\) along \(C{//}G\):
Theorem 2.12
Proof
2.4 Lefschetz–Bott fibrations associated with fibered Dehn twists
In this section, we explain that any fibered Dehn twist appears as the monodromy of a symplectic Lefschetz–Bott fibration. Conversely, the monodromy of a symplectic Lefschetz–Bott fibration is given by a fibered Dehn twist up to isotopy by Theorem 2.14 of Perutz [33] recalled below. (Theorem 2.14 is not used in this paper; we mention it only for its conceptual importance linking Lefschetz–Bott fibrations and fibered Dehn twists.)
Proposition 2.13
Let M be a symplectic manifold, \(C \subset M\) a spherically fibered coisotropic, and \(\tau _C: M \rightarrow M \) a fibered Dehn twist around C. There is a standard Lefschetz–Bott fibration \(E_C\) with generic fiber M and symplectic monodromy \(\tau _C\).
Proof
Theorem 2.14
[33, Theorem 2.19] Suppose that \(\pi : E \rightarrow S\) is a symplectic Lefschetz–Bott fibration, and \(s_0 \in S^{{{\text {crit}}}}\) is such that \(\pi ^{1}(s_0) \cap E^{{{\text {crit}}}}\) has a unique connected component. Then the symplectic monodromy around \(s_0\) is a fibered Dehn twist.
2.5 Further examples of fibered Dehn twists
Fibered Dehn twists are often induced by flows of components of moment maps. First let \(U(1) = \{ z \in \mathbb {C}\  \ z = 1 \}\) denote the circle group. We identify the Lie algebra of U(1) with \(\mathbb {R}\) via division by \(2 \pi i\). The integers \(\mathbb {Z}= \exp ^{1}(1)\) are then the coweight lattice. Let \((M,\omega )\) be a symplectic manifold, and \(M_0 \subset M\) an open subset equipped with a free Hamiltonian action of U(1) with moment map \(\varPhi : M_0 \rightarrow (c_,c_+)\).
Proposition 2.15
(Fibered Dehn twists via Hamiltonian circle actions) Let \(\psi \in C^\infty [c_,c_+]\) be a function such that \(\psi ' = 1\) on a neighborhood of \(c_+\), and \(\psi ' = 0 \) on a neighborhood of \(c_\). Then the timeone flow of \( \psi \circ \varPhi \) on \(M_0\) extends to a smooth flow on M equal to the identity on the complement of \(M_0\) and the extension is a fibered Dehn twist along \(\varPhi ^{1}(c)\) for any \(c \in (c_,c_+)\). If \(M_0\) is a Hamiltonian Gmanifold for a compact Lie group G and \(\psi \) is Ginvariant, then this fibered Dehn twist is equivariant.
Proof
Remark 2.16
(Standard Dehn twists as symplectic Dehn twists) Standard Dehn twists of complex curves arise from the construction in 2.15 as follows. Suppose that M is a complex curve and \(C \rightarrow M\) an embedded circle equipped with an identification \(C \cong S^1\). Choose an area form \(\omega _M\) on M. Since \(C \subset M\) is Lagrangian, by the Lagrangian embedding theorem there exists a tubular neighborhood \(M_0 = C \times (c_,c_+)\) on which the symplectic form is standard. Then the U(1) action by rotation on the left factor of \(M_0\) is free, and the projection \(\varPhi \) on the second factor is a moment map. For any \(\psi \) with the properties in Proposition 2.15, the flow of \(\psi \circ \varPhi \) is a standard Dehn twist.
Next we consider Dehn twists induced by flows of moment maps of SU(2)actions. We fix a metric on the Lie algebra \(\mathfrak {su}(2)\) so that nonzero elements \(\xi \) with \(\exp (\xi ) = 1\) have minimal length 1.
Proposition 2.17
(Fibered Dehn twists via Hamiltonian SU(2)actions) Suppose that \((M,\omega ,\varPhi )\) is a Hamiltonian SU(2)manifold with moment map \(\varPhi : M \rightarrow \mathfrak {su}(2)^\vee \) and the stabilizer H of the action of SU(2) on any point in \(\varPhi ^{1}(0)\) is trivial resp. U(1). Let \(\psi \in C^\infty _c[0,\infty )\) be a compactly supported function such that \(\psi ' = 1/2\) in a neighborhood of 0. Then \(\varPhi ^{1}(0)\) is a spherically fibered coisotropic of codimension 3 resp. 2 and the flow of \(\psi \circ \vert \varPhi \vert \) is a Dehn twist along \(\varPhi ^{1}(0)\).
Proof
The zerolevel set \(P := \varPhi ^{1}(0)\) is a Gequivariant G / Hbundle over the symplectic quotient \(M {//}G\), by the assumption on stabilizers and existence of local slices. We identify \(G/H \cong S^c\) with \(c = 3\) resp. \(c = 2\), in the trivial stabilizer resp. U(1)stabilizer case. We show that P is induced from an \(SO(c+1)\) principal bundle and that the flow of \(\psi \circ \vert \varPhi \vert \) is obtained by a Dehn twist by the symplectic fiber bundle construction.
\(\square \)
3 Fibered Dehn twists on moduli spaces of flat bundles
This section describes a natural collection of fibered Dehn twists on moduli spaces of flat bundles, which are our motivating examples.
3.1 Moduli spaces of flat bundles
We first recall the construction of symplectic structures on moduli spaces of flat bundles on surfaces with markings, that is, flat bundles on the complement of the markings with holonomies around them in fixed conjugacy classes.
Definition 3.1
 (a)(Conjugacy classes in compact 1connected Lie groups) Let G be a simple compact, 1connected Lie group, with maximal torus T and Weyl group \(W = N(T)/T\). Let \(\mathfrak {g},\mathfrak {t}\) denote the Lie algebras of G and T. We choose a highest root \(\alpha _0 \in \mathfrak {t}^\vee \) and positive closed Weyl chamber \(\mathfrak {t}_+ \subset \mathfrak {t}\). Conjugacy classes in G are parametrized by the Weyl alcovesee [36]. For any \(\mu \in \mathfrak {A}\), we denote by$$\begin{aligned} \mathfrak {A}:= \{ \xi \in \mathfrak {t}_+, \ \alpha _0(\xi ) \le 1 \} , \end{aligned}$$the corresponding conjugacy class. Inverting each conjugacy class defines an involution$$\begin{aligned} {\mathcal {C}}_\mu = \{ g \exp (\mu ) g^{1}, \ g \in G \} \end{aligned}$$In the case \(G = SU(2)\), we identify \( \mathfrak {t}\cong \mathbb {R}\) and \( \mathfrak {A}\cong [0,1/2] \) so that$$\begin{aligned} *: \mathfrak {A}\rightarrow \mathfrak {A}, \quad {\mathcal {C}}_{*\mu } = {\mathcal {C}}_\mu ^{1} . \end{aligned}$$(11)In particular,$$\begin{aligned} {\mathcal {C}}_\mu = \{ {\text {Ad}} (g) {\text {diag}}( \exp ( 2\pi i \mu ), \exp ( 2\pi i \mu )) \} . \end{aligned}$$(12)is the conjugacy class of traceless elements of SU(2).$$\begin{aligned}{\mathcal {C}}_{1/4} = \{ A \in SU(2) \  \ {\text {Tr}}(A) = 0 \} \end{aligned}$$
 (b)(Marked surfaces) By a marked surface we mean a compact oriented connected surface X equipped with a collection of distinct pointsand a collection of labels$$\begin{aligned} \underline{x} = (x_1,\ldots , x_n) \in X^n \end{aligned}$$For simplicity, we denote such a surface \((X,\underline{\mu })\).$$\begin{aligned} \underline{\mu } = (\mu _1,\ldots , \mu _n) \in \mathfrak {A}^n . \end{aligned}$$
 (c)(Holonomies) Let \(P \rightarrow X\) be a Gbundle equipped with a flat connection \(A \in {\mathcal {A}}(P)\). Parallel transport around loops in X gives rise to a holonomy representationIn particular, for any point \(x \in X\) a small loop around x defines a conjugacy class in \(\pi _1(X)\), obtained by joining the loop to a base point, and the holonomy around x is well defined up to conjugacy.$$\begin{aligned} \pi _1(X) \rightarrow G . \end{aligned}$$
 (d)
(Moduli spaces of bundles on marked surfaces) Let \((X,\underline{\mu })\) be a marked surface. Let \(M(X,\underline{\mu })\) denote the moduli space of isomorphism classes of flat Gbundles on \(X  \{ x_1,\ldots , x_n \}\) with holonomy around \(x_i\) conjugate to \(\exp (\mu _i)\), for each \(i = 1,\ldots ,n\), see, e.g., Meinrenken and Woodward [31]. If \(M(X,\underline{\mu })\) contains no reducible bundles (bundles with noncentral automorphisms), then \(M(X,\underline{\mu })\) is a compact symplectic orbifold.
Remark 3.2
 (a)(Action of central bundles) Let Z denote the center of G. Let \(M_Z(X)\) denote the moduli space of Zbundles on X with trivial holonomy around the markings. The group multiplication on Z induces a group structure on \(M_Z(X)\), isomorphic to \(Z^{2g}\) where g is the genus of X. The action of Z on G induces a symplectic action of \(M_Z(X)\) on \(M(X,\underline{\mu })\)corresponding to twisting the holonomies by elements of Z.$$\begin{aligned} M_Z(X) \times M(X,\underline{\mu }) \rightarrow M(X,\underline{\mu }) \end{aligned}$$
 (b)(Combining central markings) A label \(\mu \in \mathfrak {A}\) is central if \(\exp (\mu )\) lies in the center Z of G. In the case \(G = SU(r)\), the central labels are the vertices of the alcove \(\mathfrak {A}\). Several central labels may be combined into a single central label as follows: Suppose that \(\lambda _1 \in \mathfrak {A}\) resp. \(\lambda _2 \in \mathfrak {A}\) are labels corresponding to \(z_1,z_2 \in Z\), and \(\lambda _{12} \in \mathfrak {A}\) is the label that corresponds to \(z_1z_2 \in Z\). Then there is a symplectomorphismThis follows immediately from the description of the moduli space as representations of the fundamental group as in (17).$$\begin{aligned} M(X,\lambda _1,\lambda _2,\lambda _3, \ldots ,\lambda _n) \rightarrow M(X,\lambda _{12},\lambda _3,\ldots ,\lambda _n) . \end{aligned}$$
Remark 3.3
(Moduli of flat bundles as a symplectic quotient by the loop group) The moduli space of flat bundles may be realized as the symplectic quotient of the moduli space of framed bundles on a cut surface described in [31].
First we introduce notation for the cut surface. Let \(Y \subset X\) be an embedded circle, that is, a compact, oriented, connected onemanifold, disjoint from the markings \(\underline{x}\). Let \(X_{{\text {cut}}}\) denote the surface obtained from X by cutting along Y as in Fig. 1, with boundary components \((\partial X_{{\text {cut}}})_j \cong S^1, j =1,2\). The cut surface \(X_{{\text {cut}}}\) may be disconnected or connected depending on whether Y is separating.
3.2 Symplectomorphisms induced by Dehn twists
Any Dehn twist on a marked surface induces a symplectomorphism of the moduli space of flat bundles. In this section, we explicitly describe this symplectomorphism as the Hamiltonian flow of a nonsmooth function.
Proposition 3.4
Proposition 3.5
(The action of a Dehn twist is a Hamiltonian flow, c.f. [2, Theorem 4.5]) Let \((X,\underline{\mu })\) be a marked surface such that \(M(X,\underline{\mu })\) contains no reducibles, \(Y \subset X\) an embedded circle and \(\tau _Y: X \rightarrow X\) a Dehn twist around Y. Then \(\tau _Y\) acts on \(M(X,\underline{\mu })\) by the timeone Hamiltonian flow of \(h_Y\) on \(\rho _Y^{1}(\mathfrak {A}^\circ )\). In particular, the timeone Hamiltonian flow of \(h_Y\) extends smoothly to all of \(M(X,\underline{\mu })\).
Lemma 3.6
Proof
Since a Dehn twist along a circle in a Riemann surface is only defined up to isotopy, we may assume that \(\tau _Y\) is a Dehn twist along a small translation of the boundary component \((\partial X_{{\text {cut}}})_2\). This twist induces a Dehn twist on \(X_{{\text {cut}}}\), also denoted \(\tau _Y\).
Proof of Proposition 3.5
Lemma 3.7
3.3 Full twists for rank two bundles
In this section, we show that Dehn twists on a surface induce fibered Dehn twists of the moduli space of flat rank two bundles with trivial determinant. The following is a slight generalization of a result of Callahan (unpublished) resp. Seidel [39, Section 1.7] in the case of a separating resp. nonseparating curve on a surface.
Theorem 3.8
 (a)(Separating case gives a codimension one Dehn twist) If Y is separating then \(\tau _Y\) acts on \(M(X,\underline{\mu })\) by a fibered Dehn twist along the fibered coisotropicfor any \(\lambda \in (0,{\frac{1}{2}})\) such that \(M(X,\underline{\mu },\lambda ,*\lambda )\) contains no reducibles.$$\begin{aligned} C_\lambda := \rho _Y^{1}(\lambda ) \end{aligned}$$
 (b)(Nonseparating case gives a codimension three Dehn twist) If Y is nonseparating and \(M(X_{{{\text {cap}}}},\underline{\mu })\) contains no reducibles, then \(\tau _Y\) acts by a fibered Dehn twist along the fibered coisotropicof bundles with holonomy along Y equal to \(\exp (1/2)= I \in SU(2)\).$$\begin{aligned} C_{1/2} := \rho _Y^{1}(1/2) \end{aligned}$$
Proof
We have already expressed the action of the Dehn twist as the action of holonomy in Proposition 3.5. It remains to identify these flows as Dehn twists in the rank two case. For this, it suffices as in Theorem 2.12 to show that the corresponding Dehn twists induce equivariant Dehn twists on the moduli space of the cut surface.
It now follows from the results on equivariant Dehn twists that the Dehn twist on the surface acts by a fibered Dehn twist on the moduli space. That is, \((\tau _Y^{1})^*\) is a Dehn twist along any \(\varPhi _2^{1}(\lambda ) \subset M(X_{{\text {cut}}},\underline{\mu })^\circ \) for \(\lambda \) generic. By Proposition 2.15, \((\tau _Y^{1})^*\) acts as an equivariant Dehn twist on \(M(X_{{\text {cut}}},\underline{\mu })^\circ \). By Theorem 2.12, the action of \((\tau _Y^{1})^*\) descends to a fibered Dehn twist on \(M(X,\underline{\mu })\).
3.4 Half twists for rank two bundles with fixed holonomies
In this section, we show that a half twist on a marked surface \((X,\underline{\mu })\) induces a fibered Dehn twist on the moduli space \(M(X,\underline{\mu })\) of flat \(G = SU(2)\) bundles with fixed holonomy.
Definition 3.9
Example 3.10
The following is slight generalization of a result of Seidel [39].
Theorem 3.11
(For \(G = SU(2)\) half twists around pairs of markings act by codimension two fibered Dehn twists on the moduli space) Let \(G = SU(2)\). Let \((X,\underline{\mu })\) be a marked surface such that \(M(X,\underline{\mu })\) contains no reducibles and \(\mu _i = \mu _j = 1/4\). Let \(Y \subset X\) be an embedded circle disjoint from the markings that is the boundary of a disk containing \(x_i,x_j\). Then the set \(C_Y\) from (3.9) is a spherically fibered coisotropic submanifold of codimension 2, and the action of \({\tau _Y}\) on \(M(X,\underline{\mu })\) is a fibered Dehn twist along \(C_Y\).
Proof
3.5 Half twists for higherrank bundles
For special labels, a half twist on the marked surface X induces a fibered Dehn twist on the moduli space of flat SU(r) bundles with fixed holonomy.
Definition 3.12
Theorem 3.13
(Codimension two fibered Dehn twists via half twists) Suppose that \(G = SU(r)\) and \(\mu _i,\mu _j, \nu _k^1,\nu _k^2\) are as in Definition 3.12 such that the moduli spaces \(M(X,\underline{\mu })\) and \(M(X,\underline{\mu },(\mu _i,\mu _j) \mapsto \nu _k^2 )\) contain no reducibles. Let \(Y \subset X\) denote an embedded circle enclosing only the ith and jth markings. Then the subset \(C_Y\) from (21) is a spherically fibered coisotropic submanifold of codimension 2, fibered over the moduli space \(M(X,\underline{\mu }, (\mu _i,\mu _j) \mapsto \nu _k^2)\). The action of \({\tau _Y}\) on \(M(X,\underline{\mu })\) is a fibered Dehn twist along \(C_Y\).
Proof
4 Pseudoholomorphic sections of Lefschetz–Bott fibrations
In this section, we describe the relative invariants associated with Lefschetz–Bott fibrations. These are maps between Lagrangian Floer cohomology groups obtained by counting pseudoholomorphic sections.
4.1 Monotone Lagrangian Floer cohomology
Novikov rings are not needed to define Lagrangian Floer cohomology for a monotone pair of Lagrangian submanifolds. However, we will need our cochain complexes to admit “action filtrations.” For this, we find it convenient to use the version incorporating a formal variable keeping track of area, as in the construction of the spectral sequence in Fukaya et al. [12].
Definition 4.1
 (a)(Symplectic backgrounds) Fix a monotonicity constant \(\lambda \ge 0\) and an even integer \(N > 0\). A symplectic background is a tuple \((M,\omega ,b,{\text {Lag}}^N(M))\) as follows.
 (i)
(Bounded geometry) \((M,\omega )\) is a compact smooth symplectic manifold with either empty or convex boundary;
 (ii)
(Monotonicity) \(\omega \) is \(\lambda \)monotone, i.e., \([\omega ] = \lambda c_1(TM)\);
 (iii)
(Background class) \(b \in H^2(M,\mathbb {Z}_2)\) is a background class, which will be used for the construction of orientations; and
 (iv)
(Maslov cover) \({\text {Lag}}^N(M) \rightarrow {\text {Lag}}(M)\) is an Nfold Maslov cover such that the induced twofold Maslov covering \({\text {Lag}}^2(M)\) is the oriented double cover.
 (i)
 (b)(Monotone Lagrangians) A Lagrangian \(L \subset M \backslash \partial M\) is monotone if there is an areaindex relation for disks with boundary in L, that is,where \(A(u) = \langle [\omega ], [u] \rangle \) (the pairing of \([\omega ] \in H^2(M,L)\) with \([u] \in H_2(M,L)\)) is again the symplectic area and I(u) is the Maslov index of u as in [29, Appendix].$$\begin{aligned} A(u) = \frac{\lambda }{2} I(u) , \quad \forall u: (D,\partial D) \rightarrow (M,L), \end{aligned}$$
 (c)(Monotone tuples of Lagrangians) A tuple \((L_b)_{b\in {\mathcal {B}}}\) of Lagrangians in M is monotone if the following holds: Let S be any compact surface with boundary given as a disjoint union of onemanifolds \(C_b\)(with \(C_b\) possibly empty or disconnected). Then for some constant \(c(S,M, (L_b)_{b \in {\mathcal {B}}})\) independent of u,$$\begin{aligned} \partial S=\sqcup _{b\in {\mathcal {B}}} C_b \end{aligned}$$where I(u) is the sum of the Maslov indices of the totally real subbundles \((u_{C_b})^*TL_b\) in some fixed trivialization of \(u^*TM\). There is a similar definition of monotonicity for tuples of Lagrangian correspondences [45].$$\begin{aligned} A(u) = \frac{\lambda }{2} \cdot I(u) + c(S,M,(L_b)_{b \in {\mathcal {B}}}), \quad \forall u:(S, ( C_b)_{b \in {\mathcal {B}}}) \rightarrow (M, (L_b)_{b \in {\mathcal {B}}}) \end{aligned}$$(24)
 (d)
(Admissible Lagrangians) As mentioned in the Introduction, we say that a compact monotone Lagrangian submanifold L is admissible if the image of the fundamental group of L in M is torsion and L has minimal Maslov number at least 3. One may also allow the case that L has minimal Maslov number 2 and disk invariant (number of disks passing through a generic point) 0. However, the Maslov index 2 case is discussed separately in 5.3. Any tuple of admissible, monotone Lagrangians is automatically monotone, by the argument in Oh [32]. This argument involves completing each boundary component of the surface by adding a disk obtained by contracting a loop, possibly after passing to a finite cover. Products of admissible Lagrangians are also automatically admissible. Thus if \(L^0,L^1\) resp C. are admissible Lagrangians in M resp. \(M^ \times B\) then the tuples \((L^0,L^1)\), \((L^0 \times C, C^t \times L^1)\) are monotone.
 (e)(Generalized Lagrangian correspondences) Let \(M,M'\) be symplectic manifolds. A generalized Lagrangian correspondence \(\underline{L}\) from M to \(M'\) consists of
 (i)
a sequence \(N_0,\ldots ,N_r\) of any length \(r+1\ge 2\) of symplectic manifolds with \(N_0 = M\) and \(N_r = M'\) ,
 (ii)
a sequence \(L_{01},\ldots , L_{(r1)r}\) of compact Lagrangian correspondences with \(L_{(j1)j} \subset N_{j1}^\times N_{j}\) for \(j=1,\ldots ,r\).
as a generalized Lagrangian correspondence from \(M'\) to M. A generalized Lagrangian is called admissible if each component is.$$\begin{aligned} \underline{L}^t := \bigl (L_{(r1)r}^t,\ldots ,L_{01}^t\bigr ) . \end{aligned}$$  (i)
 (f)
(Lagrangian branes) Generally speaking, a Lagrangian brane means a Lagrangian with extra structure sufficient for the definition of Floer cohomology. In particular, a grading of a Lagrangian submanifold \(L \subset M\) is a lift of the canonical section \(L \rightarrow {\text {Lag}}(M)\) to \({\text {Lag}}^N(M)\), as in Seidel [40]. A brane structure on a connected Lagrangian L consists of a grading and a relative spin structure for the embedding \(L \rightarrow M\) (equivalent to a trivialization of \(w_2(M)\) in the relative chain complex for (M, L), see, e.g., [48]). A Lagrangian brane is an oriented Lagrangian submanifold equipped with a brane structure. A generalized Lagrangian brane is a generalized Lagrangian correspondence equipped with a brane structure.
Proposition 4.2
Remark 4.3
For later use, we recall a basic fact about approximate intersection points.
Lemma 4.4
Suppose that M is a compact Riemannian manifold and \(L^0,L^1 \subset M\) are compact submanifolds. For any \(\epsilon > 0\), there exists an \(\delta > 0\) such that if \(m \in M\) is a point with \(d(m,L^0) < \delta \) and \(d(m,L^1) < \delta \), then \(d(m, L^0 \cap L^1) < \epsilon \).
Proof
It follows that any map that is sufficiently close to both \(L^0\) and \(L^1\) at every point in the domain is in fact contained in a small neighborhood of \(L^0 \cap L^1\).
4.2 Relative invariants for Lefschetz–Bott fibrations
We may now associate with Lefschetz–Bott fibrations over surfaces with striplike ends relative invariants that are morphisms of Floer cohomology groups associated with the ends, given by counting pseudoholomorphic sections. The following material can also be found, in a slightly different form, in Perutz [34].
Definition 4.5
 (a)Given \(u \in \Gamma (E)\) with image disjoint from the critical set define its index and symplectic areaNote that the form \(\omega _E\) is only fiberwise symplectic. Thus the area A(u) may be negative, in the general case.$$\begin{aligned} I(u) = 2(c_1( u^* T^{{\text {vert}}} E),[S]), \ \ \ A(u) = \int _S u^* \omega _E . \end{aligned}$$
 (b)A symplectic Lefschetz–Bott fibration E is monotone with monotonicity constant \(\lambda \ge 0\) if there exists a constant c(E) such that$$\begin{aligned} \lambda I(u) = 2 A(u) + c(E) \quad \forall u \in \Gamma (E) . \end{aligned}$$
Remark 4.6
Proposition 4.7
Let \(\pi : E \rightarrow S\) be a symplectic Lefschetz–Bott fibration with S, E compact. If the generic fiber M of E is monotone, \(H_1(M) = 0 \) and all vanishing cycles in E have codimension at least 2 then E is monotone.
Proof
We now wish to allow our Lefschetz–Bott fibrations to have striplike ends. The monotonicity conditions in this case will be similar, but with additional constants depending on the limits.
Definition 4.8
 (a)Let S be a complex curve with boundary obtained from a compact curve with boundary \(\overline{S}\) be removing points on the boundary \(z_1,\ldots ,z_n\). A striplike end for the jth puncture of S is a holomorphic mapsuch that \(\exp (2 \pi ( ( \epsilon _j^{1})_{1} + i (\epsilon _{j}^{1})_{2} ))\) is a local holomorphic coordinate on the closure of the image of \(\epsilon _j\); the end is called incoming resp. outgoing if the sign is negative resp. positive. A collection of striplike ends \({\mathcal {E}}\) is a set of striplike ends, one for each \(j = 1,\ldots , n\). We write \({\mathcal {E}}= {\mathcal {E}}_ \cup {\mathcal {E}}_+\) the union of the incoming and outgoing ends.$$\begin{aligned}\epsilon _j: [0,\pm \infty ) \times [0,1] \rightarrow S \end{aligned}$$
 (b)A symplectic Lefschetz–Bott fibration over a surface with striplike ends S with fiber given by a symplectic manifold M is a Lefschetz–Bott fibration \(E \rightarrow S\) with \(S^{{\text {crit}}}\) contained in the interior of S, together with a trivializationfor each end \(e \in {\mathcal {E}}\), such that \( \varphi _{s,e}^* \omega _E = \pi _M^* \omega _M \) where$$\begin{aligned} \varphi _{S,e}: \epsilon _{S,e}^* E \rightarrow (0,\pm \infty ) \times [0,1] \times M \end{aligned}$$is projection on the last factor.$$\begin{aligned} \pi _M: (0,\pm \infty ) \times [0,1] \times M \rightarrow M \end{aligned}$$
 (c)Let S be a surface with striplike ends and \(\pi : E \rightarrow S\) a symplectic Lefschetz–Bott fibration with fiber M. A Lagrangian boundary condition for E is a submanifold \(F \subset \partial E\) such that
 (i)
\(\pi F \) is a fiber bundle over \(\partial S\);
 (ii)
each fiber \(F_z \subset E_z, z \in \partial S\) is a Lagrangian submanifold;
 (iii)for each \(e \in {\mathcal {E}}\) there exist Lagrangian submanifolds \(L^{0,e},L^{1,e} \subset M\) such that F is constant sufficiently close to \(z_e\), that is,and$$\begin{aligned} \varphi _{S,e}(F_{\epsilon _{S,e}(s,j)}) = L^{j,e}, \ \ \ \pm s \gg 0 ; \end{aligned}$$
 (iv)
for each \(e \in {\mathcal {E}}\), the intersection \(L^{0,e} \cap L^{1,e}\) is transversal.
 (i)
Definition 4.9
 (a)(Monotonicity condition for Lefschetz–Bott fibrations with boundary) Let S be a compact surface with boundary, (E, F) a bundle with boundary condition, and \(\Gamma (E,F)\) the set of smooth sections \(u: (S,\partial S) \rightarrow (E,F)\). Each \(u \in \Gamma (E,F)\) takes values in the smooth locus of E. By pullback one obtains bundlesTaking the Maslov index of this pair gives rise to a Maslov index map$$\begin{aligned} u^* T^{{\text {vert}}}E \rightarrow S, \quad (u  \partial S)^* T^{{\text {vert}}} F \rightarrow \partial S . \end{aligned}$$We also define the symplectic area$$\begin{aligned} I: \Gamma (E,F) \rightarrow \mathbb {Z}. \end{aligned}$$keeping in mind that the form \(\omega _E\) is only symplectic on the fibers. A pair (E, F) monotone with monotonicity constant \(\lambda \) if the index depends linearly on the area; that is,$$\begin{aligned} A: \Gamma (E,F) \rightarrow \mathbb {R}, \ \ u \mapsto \int _S u^* \omega _E, \end{aligned}$$for some constant c(E, F).$$\begin{aligned} \lambda I(u) = 2 A(u) + c(E,F) \quad \forall u \in \Gamma (E,F) \end{aligned}$$(26)
 (b)(Linearized operator) Suppose that S is a surface with boundary and striplike ends \({\mathcal {E}}= {\mathcal {E}}_ \cup {\mathcal {E}}_+\), and (E, F) a bundle with boundary condition. For any collectionlet$$\begin{aligned} (x_e \in {\mathcal {I}}(L^{0,e},L^{1,e}))_{e \in {\mathcal {E}}} , \end{aligned}$$denote the space of sections with boundary values in F and asymptotic limits \(x_e\). For sections \(u \in \Gamma (E,F)\) let$$\begin{aligned} \Gamma (E,F;(x_e)_{e \in {\mathcal {E}}}) = \left\{ u \in \Gamma (E) \, \left \, \quad \begin{array}{c} \lim _{s \rightarrow \pm \infty } u(\epsilon _e(s,t)) = x_e \quad \forall e \in {\mathcal {E}}\\ u _{\partial S} \in \Gamma (F) \end{array} \right. \right\} \end{aligned}$$denote the Cauchy–Riemann equation associated with the pair (j, J). The linearized operator$$\begin{aligned} \overline{\partial }u = {\frac{1}{2}}( {\text{ d }}u + J(u) \circ {\text{ d }}u \circ j) = 0 \end{aligned}$$(27)is given by differentiating the Cauchy–Riemann operator along a path \(\exp _u(t\xi )\) of geodesic exponentials and using parallel transport \(\Pi _{t\xi }^{1}\) back to u. The operator \(D_u\) is Fredholm since the boundary conditions at infinity are assumed transversal.$$\begin{aligned} D_u:\,&\Omega ^0( u^* T^{{\text {vert}}} E,\; u^* T^{{\text {vert}}} F) \rightarrow \Omega ^{0,1}(u^* T^{{\text {vert}}} E), \nonumber \\&\xi \mapsto \frac{d}{dt}_{t =0} \Pi _{t\xi }^{1} \overline{\partial }\exp _u(t\xi ) \end{aligned}$$(28)
 (c)(Monotonicity condition for Lefschetz–Bott fibrations with striplike ends) The pair (E, F) is monotone with monotonicity constant \(\lambda \ge 0\) if for any \((x_e)_{e \in {\mathcal {E}}}\) there exists a constant \(c(E,F;(x_e)_{e \in {\mathcal {E}}})\) such that$$\begin{aligned} \lambda {\text {Ind}}(D_u) = 2 A(u) + c(E,F;(x_e)_{e \in {\mathcal {E}}}) \quad \forall u \in \Gamma (E,F;(x_e)_{e \in {\mathcal {E}}}) . \end{aligned}$$(29)
Proposition 4.10
(Condition for monotonicity of the fiber to imply monotonicity of a fibration with striplike ends) Let \(E \rightarrow S\) be a symplectic Lefschetz–Bott fibration over a surface with boundary S with connected and simply connected fibers. Let \(F \subset E  \partial S\) be a Lagrangian boundary condition with connected and simply connected fibers. Suppose that the generic fiber of E is monotone and the vanishing cycles of E have codimension at least 2. Then (E, F) is monotone.
Proof
We now turn to the construction of relative invariants associated with symplectic Lefschetz–Bott fibrations. Let \(\pi : E \rightarrow S\) be such a fibration, equipped as in Definition 2.1 with a complex structure \(j_0\) on a neighborhood of the critical values in S and \(J_0\) on a neighborhood of the critical points in E. In the case that S has ends, we assume furthermore that almost complex structures \(J_e \in {\mathcal {J}}(M)\) are fixed making the moduli spaces of Floer trajectories for that end regular. We wish to extend these to almost complex structures on the entire fibration and base, so that we may define moduli spaces of pseudoholomorphic sections.
Definition 4.11
 (a)(Compatible almost complex structures) Let \(\pi : E \rightarrow S\) be a Lefschetz–Bott fibration over a surface with striplike ends S. A complex structure j on S is compatible with E if \(j = j_0\) in an neighborhood of \(S^{{\text {crit}}}\). An almost complex structure J on E is compatible with \(\pi ,j\) iff
 (i)
\(J = J_0\) in a neighborhood of \(E^{{\text {crit}}}\);
 (ii)
\(\pi \) is (J, j)holomorphic in a neighborhood of \(E^{{\text {crit}}}\), that is, \( J \circ {\text{ d }}\pi = {\text{ d }}\pi \circ j\); and
 (iii)
\(\omega _E(\cdot ,J \cdot )\) is symmetric and positive definite on \(TE^v_x\), for any \(x \in E\).
 (iv)
J is equal to fixed almost complex structure \(j \times J_e, e \in {\mathcal {E}}(S)\) on the ends of S in the given trivializations on a neighborhood \(U_e \subset S\) of each end.
 (i)
 (b)
(Moduli space of pseudoholomorphic sections) Let \({\mathcal {M}}(E,F;(x_e)_{e \in {\mathcal {E}}})\) denote the set of finite area sections \(u \in \Gamma (E,F,(x_e)_{e \in {\mathcal {E}}})\) such that u is (j, J)holomorphic, with limits \((x_e)_{e \in {\mathcal {E}}}\) along the ends.
Theorem 4.12
 (a)
\({\mathcal {M}}(E,F;(x_e)_{e \in {\mathcal {E}}})\) is a smooth manifold with tangent space at u given by \( {\text {ker}}(D_u)\);
 (b)
the zerodimensional component \({\mathcal {M}}(E,F;(x_e)_{e \in {\mathcal {E}}})_0 \subset {\mathcal {M}}(E,F;(x_e)_{e \in {\mathcal {E}}})\) is finite;
 (c)the onedimensional component \({\mathcal {M}}(E,F;(x_e)_{e \in {\mathcal {E}}})_1 \subset {\mathcal {M}}(E,F;(x_e)_{e \in {\mathcal {E}}})\) has a compactification with boundaryconsisting of pairs of a section with bubbledoff trajectory; and$$\begin{aligned} \bigcup _{(x_e)_{e \in {\mathcal {E}}},f,x_f'} {\mathcal {M}}(E,F;(x_e)_{e \in {\mathcal {E}}}, x_f \mapsto x_f')_0 \times {\mathcal {M}}(x_f,x_f')_0 \end{aligned}$$
 (d)
any relative spin structure on (E, F) induces a set of orientations on the manifolds \({\mathcal {M}}(E,F;(x_e)_{e \in {\mathcal {E}}})_0\) that are coherent in the sense that they are compatible with the gluing maps from (c) in the sense that the inclusion of the boundary in (c) has the signs \((1)^{\sum _{e<f} x_{e}^}\) (for incoming trajectories) and \((1)^{\sum _{e<f} x_{e}^+}\) (for outgoing trajectories.)
The proof is similar to that of [41] in the exact case. Bubbling for sections can occur only in the fiber. So sphere and disk bubbling on the zero and onedimensional moduli spaces is ruled out by the monotonicity condition. The construction of coherent orientations is given in [48].
Remark 4.13
(Sketch of construction of orientations from [48]) We briefly recall the construction of orientations: On any disk with Lagrangian boundary conditions where the Lagrangian is equipped with a relative spin structure, the linearized operator \(D_u\) over the surface S may be deformed via nodal degeneration to a boundary value problem \(D_u'\) on the sphere \(S_s\) and a constant boundary value problem on the disk \(S_d\). On the sphere, the linearized Cauchy–Riemann operator \(D_u'  S_s\) is homotopic to a complex linear operator \(D_u''\). The determinant line \(\det (D_u')\) inherits an orientation via the complex structure on the kernel \( {\text {ker}}(D_u'')\) and cokernel \( {\text {coker}}(D_u'')\). On the other hand, on the disk the boundary condition \((u _{\partial S_d})^* TL\) admits a canonical stable trivialization determined by the relative spin structure. The determinant line on the disk admits an orientation induced from a trivialization of \((u _{\partial S_d})^* TL\) and an isomorphism of the kernel \( {\text {ker}}(D_u'  S_d)\) with the tangent space to the Lagrangian \(T_{u(z)}L\) at any point on the boundary. These combine to an orientation on the determinant line \(\det (D_u) \cong \det (D_u'  S_s) \otimes \det (D_u'  S_d)\).
Remark 4.14
For later use, we recall the basic fact that maps with sufficiently small energy must also have small diameter:
Lemma 4.15
Let \(\underline{S}\) be a holomorphic quilt with striplike ends, with symplectic labels \(\underline{M}\) and Lagrangian boundary and seam condition \(\underline{L}\). For any \(\delta > 0, \ell _0\), there exists \(\epsilon > 0\) such that if \(u: \underline{S} \rightarrow \underline{M}\) is a pseudoholomorphic quilt with energy \(E(u) < \epsilon \), and \(\gamma : [0,1] \rightarrow \underline{S}\) is a path connecting boundary components of length less than \(\ell _0\), then the length of \(u \circ \gamma \) is less than \(\delta \).
Proof
Definition 4.16
Remark 4.17
(Independence from almost complex structure and fiberwise symplectic form) The cohomologylevel invariants \(\varPhi (E,F;\Lambda )\) are independent of the choice of compatible almost complex structure J on E, by an argument using parametrized moduli spaces similar to that of Theorem 4.12.
Definition 4.18
A Lefschetz–Bott fibration E with twoform \(\omega _E\) has nonnegative curvature if \(\omega _E(v,jv) \ge 0 \) for all v in the horizontal subspace \(TE^h\), that is, \(f(e) \ge 0\) for all \(e \in E\).
Remark 4.19
Nonnegative curvature implies that a small perturbation of the twoform is symplectic: Recall that the total space of any Lefschetz–Bott fibration \(\pi :E \rightarrow S\) admits a canonical isotopy class of symplectic structures given as follows. If \(\omega _S \in \Omega ^2(S)\) is a sufficiently positive twoform, then \(\omega _E + \pi ^* \omega _S\) is a symplectic form on \(TE_e\) for any \(e \in E\). If E is compact, then \(\omega _E + \pi ^* \omega _S\) is symplectic on E for \(\omega _S\) sufficiently positive. If E is nonnegative, then \(\omega _E + \pi ^* \omega _S\) is symplectic for any positive form \(\omega _S \in \Omega ^2(S)\).
Proposition 4.20
(Nonnegative curvature of standard Lefschetz–Bott fibrations) If \(C \subset M\) is a spherically fibered coisotropic, then the standard Lefschetz–Bott fibration \(E_C\) of 2.13 has nonnegative curvature.
Proof
Let \(v \in V\), the standard representation of \(SO(c+1)\), and \((p,v) \in P \times V\). The horizontal subspace \(H_v \subset T_v V\) pairs trivially with \( {\text {ker}}(\alpha ) \times T(\pi ^{1}_V(\pi _V(v)))\) under the pairing given by the twoform (7), where \(\pi _V\) is the projection (9). It follows that the image \([H_v]\) of \(H_v\) in P(V) is the horizontal subspace at \([p,v] \in P(V) := (P \times V)/G\). Let \(J_V\) denote the standard complex structure on V, and \(J_0\) the induced complex structure on E. Since \(J_{V}\) is nonnegative on \(H_v\), \(J_0\) is nonnegative on \([H_v]\). \(\square \)
Proposition 4.21
Let E be a Lefschetz–Bott fibration with Lagrangian boundary condition F and relative spin structure. If E has nonnegative curvature, then the exponents of q in the formula (31) are all nonnegative.
Proof
Since the form \(\omega _E(\cdot , J \cdot )\) is nonnegative for any \(J \in {\mathcal {J}}(E)\), any pseudoholomorphic section has nonnegative area. The qexponents in (31) are the areas. So these are nonnegative as well. \(\square \)
We do not give formula for the degree of the relative invariant, see [44] for a formula for the degree in the case without singularities.
4.3 Invariants for quilted Lefschetz–Bott fibrations
The main difference between the triangle for the fibered case and the original Seidel triangle [41] is the appearance of invariants associated with quilted surfaces. The following definitions are taken from [44].
Definition 4.22
 (a)(Quilted surfaces with striplike ends) A quilted surface with striplike ends \(\underline{S}\) consists of the following data:
 (i)a collection \(\underline{S} = (S_k)_{k=1,\ldots ,m}\) of surfaces with striplike ends, see [41, 44], called patches. Each \(S_k\) carries a complex structure \(j_k\) and has striplike ends \((\epsilon _{k,e})_{e\in {\mathcal {E}}(S_k)}\). Denote the limits of these endsand denote the boundary components$$\begin{aligned} \lim _{s \rightarrow \pm \infty } \epsilon _{k,e}(s,t) =: z_{k,e} \in \partial {\overline{S}}_k \end{aligned}$$$$\begin{aligned} \partial S_k = (I_{k,b})_{b\in {\mathcal {B}}(S_k)} ; \end{aligned}$$
 (ii)a collection \({\mathcal {S}}\) of seams: pairwise disjoint twoelement subsetsand for each \(\sigma = \{ I_{k,b}, I_{k',b'} \}\) a real analytic isomorphism$$\begin{aligned} \sigma \subset \bigcup _{k=1}^m \bigcup _{b \in {\mathcal {B}}(S_k)} I_{k,b} \end{aligned}$$where the isomorphisms \(\varphi _\sigma \) should be compatible with the striplike ends in the sense that on each end \(\varphi _\sigma \) should be a translation;$$\begin{aligned} \varphi _\sigma : I_{k,b} \rightarrow I_{k',b'} , \end{aligned}$$
 (iii)
Orderings of the boundary components of each set of patches \({\mathcal {B}}(S_k), k = 1,\ldots , m\); and
 (iv)Orderings of the incoming and outgoing ends of \(\underline{S}\)$$\begin{aligned} {\mathcal {E}}_(\underline{S})=\left( \underline{e}^_1,\ldots ,\underline{e}^_{N_(\underline{S})}\right) , \quad {\mathcal {E}}_+(\underline{S})=\left( \underline{e}^+_1,\ldots ,\underline{e}^+_{N_+(\underline{S})}\right) . \end{aligned}$$
 (i)
 (b)(Quilted symplectic Lefschetz–Bott fibrations) A (quilted) symplectic Lefschetz–Bott fibration \(\underline{E}\) over a quilted surface \(\underline{S}\) with striplike ends consists of a collection of Lefschetz–Bott fibrations \(E_k \rightarrow S_k, k = 0,\ldots , m\). A Lagrangian boundary/seam condition for E consists of a collection \(\underline{F}\) of submanifolds of the boundaries and seamswhere \(I_{k,b}\) ranges over true boundary components resp. \(I_{k_0,b_0}, I_{k_1,b_1}\) range over identified boundary components, such that$$\begin{aligned} F_{k,b} \subset E_{k,b}  I_{k,b}, \quad F_{(k_0,b_0),(k_1,b_1)} \subset E_{k_0} I_{k_0,b_0} \times \varphi _\sigma ^* \left( E_{k_1}  I_{k_1,b_1}\right) , \end{aligned}$$
 (i)each fiberover \(z \in \underline{S}\) is a Lagrangian submanifold; and$$\begin{aligned} F_{(k_0,b_0),(k_1,b_1),z} \subset E_{k_0,z} \times \left( \varphi _{(k_0,b_0),(k_1,b_1)}^* E_{k_1}\right) _z \end{aligned}$$
 (ii)over the striplike ends the fibers \(F_{(k_0,b_0),(k_1,b_1),z}\) over \(z \in \underline{S}\) are given by fixed Lagrangians \(L^{k_{e,i,}b_{e,i}}\) on the striplike ends, with the property that the compositionis transversal, where l(e) is the number of patches on the end e.$$\begin{aligned} L^{(k_{e,0},b_{e,0})}\circ L^{(k_{e,1},b_{e,1})} \ldots \circ L^{(k_{e,l(e)},b_{e,l(e)})} \end{aligned}$$
 (i)
We say that a quilted Lefschetz–Bott fibration is monotone if sections satisfy an areaindex relation similar that for pseudoholomorphic maps. In the case without singularities (i.e., fibrations) admissibility for the Lagrangians guarantees monotonicity in the quilted setting, see [44, Remark 3.7]. For Lefschetz–Bott fibrations, admissibility together with the codimension conditions in Lemma 4.10 guarantees monotonicity, by the same arguments.
Definition 4.23
As in the unquilted case in Remark 4.17, these invariants are independent of the choice of almost complex structure and deformation of the twoform on the total space.
4.4 Vanishing theorem
In this section, we use gluing along a seam to obtain a vanishing theorem analogous to [41, Section 2.3] for the invariants associated with standard fibrations associated with a fibered Dehn twist.
Remark 4.24
 (a)
(Glued surface) For \(k =0,1\) let \(\underline{S}_k\) be quilted surfaces with \(d_k + 1\) striplike ends, and \(z_k\) a seam point in \(\underline{S}_k\). Let \(\rho > 0\) be a gluing parameter. Let \(\underline{S}^\rho \) be the quilted surface with \(d_0 + d_1 + 2\) striplike ends formed by gluing together quilted disks \(D_0,D_1\) around \(z_0,z_1\) using the map \(z \mapsto \rho /z\). See Fig. 5.
 (b)
(Glued bundles) Let \((\underline{E}_k,\underline{F}_k)\) be Lefschetz–Bott fibrations over \(\underline{S}_k\), equipped with a trivialization of \(\underline{E}_k,\underline{F}_k\) in a neighborhood of \(z_k\), and a symplectomorphism of \((\underline{E}_{k,z_k},\underline{F}_{k,z_k})\) for \(k = 0,1\). The seam connect sum \(\underline{E}^\rho \rightarrow \underline{S}^\rho \) is formed by patching \(\underline{E}_0\) and \(\underline{E}_1\), and similarly for the boundary and seam conditions \(\underline{F}^\rho \).
 (c)
(Glued complex structures) Suppose that the following are given: a \((\pi _k,j_k)\)compatible almost complex structure \(\underline{J}_k\) for \(\pi _k:\underline{E}_k \rightarrow \underline{S}_k\) that is constant in a neighborhood of \(z_k\) (with respect to the given trivialization) for \(k \in \{ 0,1 \}\) and such that \(\underline{J}_0\) agrees with \(\underline{J}_1\) on the glued fiber. One can patch together these almost complex structures to obtain a compatible almost complex structure \(\underline{J}\) for \(\underline{E}\rightarrow S\).
Theorem 4.25
 (a)
the evaluation map \({\text {ev}}_0 \times {\text {ev}}_1\) is transverse to the diagonal;
 (b)for any pair \((u_0,u_1)\), there exists a gluing map on a neighborhood \(U(u_0,u_1)\) of \((u_0,u_1)\) given by$$\begin{aligned} \Theta ^\rho : {\mathcal {M}}(\underline{E}_0,\underline{F}_0) \times _{{\text {ev}}_0,{\text {ev}}_1} {\mathcal {M}}(\underline{E}_1,\underline{F}_1) \supseteq U(u_0,u_1) \rightarrow {\mathcal {M}}(\underline{E}^\rho ,\underline{F}^\rho ) ; \end{aligned}$$
 (c)
as \((u_0,u_1)\) varies over points in the zerodimensional component of the lefthandside, \(\Theta ^\rho \) is surjective onto the zerodimensional component of \({\mathcal {M}}(\underline{E}^\rho ,\underline{F}^\rho )\); and
 (d)
for any \(u \in {\mathcal {M}}(\underline{E}_0,\underline{F}_0) \times _{{\text {ev}}_0,{\text {ev}}_1} {\mathcal {M}}(\underline{E}_1,\underline{F}_1)\), the sequence \(\Theta ^\rho (u)\) Gromov converges to u as \(\rho \rightarrow 0\).
See McDuff and Salamon [29, Chapter 10] for the case of gluing at an interior point and Abouzaid [1] for the details of gluing along a point in the boundary.
Next we give a formula for the dimension for the pseudoholomorphic sections of the standard fibration studied in Propositions 2.13 and 4.20.
Definition 4.26
Lemma 4.27
Proof
We claim that the given sections are the sections of lowest index for sufficiently small radius. Indeed, the area \(A(w_{r,a,b})\) approaches zero as \(r \rightarrow 0\), for all a; this fact holds even after the adjustment [41, (1.17)] since the adjustment is by the pullback of the differential of a bounded oneform. Choose r sufficiently small so that \(A(w_{r,a,b}) \le 1/2\lambda \). The areaindex monotonicity relation and nonnegativity of the curvature in Proposition 4.20 imply that any other section u has positive area. So the index of this section I(u) is at least the index \(I(w_{r,a,b})\) of \(w_{r,a,b}\). \(\square \)
In order to define relative invariants for Lefschetz–Bott fibrations with codimension one vanishing cycles, a stronger monotonicity assumption must be assumed.
Definition 4.28
(Strong monotonicity) In the case that \(C \subset M\) is a fibered coisotropic of codimension one, we denote by \(M_C\) the cut space as in Lerman [23]. The cut space \(M_C\) is the space obtained by cutting M along C and collapsing the resulting manifold (whose boundary is two copies of C) by the circle action on the boundary. Thus \(M_C\) contains two copies \(B_\pm \) of B. Denote by \([B_\pm ] \in H^2(M_C)\) the dual classes of \(B_\pm \) and by \([\omega _C] \in H^2(M_C)\) the symplectic class. A monotone fibered coisotropic C is strongly monotone if either \({\text {codim}}(C) \ge 2 \) or both \({\text {codim}}(C) = 1\) and \(c_1(M_C)[B_+][B_]\) is a positive multiple of \([\omega _C]\).
Proposition 4.29
Suppose that the vanishing cycles of E are strongly monotone. Then the sections of (E, F) satisfy a monotonicity relation, and counting pseudoholomorphic sections defines a relative invariant \(\varPhi (E,F;\Lambda )\) as in (32).
Proof
Remark 4.30
Corollary 4.31
(Vanishing of the relative invariant associated with a standard fibration) Suppose that \(E \rightarrow S\) is a Lefschetz fibration with the Lagrangian boundary condition F obtained by a seam connect sum from a LefschetzBott fibration \((E_{C,r},F_{C,r})\) over the disk Dr corresponding to a spherically fibered coisotropic submanifold C, with an arbitrary quilted Lefschetz–Bott fibration \(E_0 \rightarrow S_0\) with boundary condition \(F_0\). Suppose that all these fibrations with boundary conditions are monotone and equipped with relative spin structures, so that in particular the relative invariant \(\varPhi (E,F)\) is defined. Then \(\varPhi (E,F)= 0 \). More precisely, there exists a null homotopy of the chainlevel relative invariant \(C\varPhi (E,F)\) that, if r is sufficiently small, has positive qexponents.
Proof
Remark 4.32
(Independence of choices) Let \(S_0,E_0,F_0,E_C,F_C,D_r\) be as above. Define an invariant associated with the nodal surface \(S_r\) by identifying a point on the seam \(S_0\) and disk D. Define a nodal Lefschetz fibration \(E_r\) by identifying a fiber of \(E_0\) with a fiber of \(E_C\) with boundary condition \(F_r \subset E_r\). By counting pseudoholomorphic sections with matching condition at the node one obtains an invariant \(\varPhi (E_r,F_r)\) for the nodal fibration, equal to the invariant for the glued fibration \(\varPhi (E,F)\). In particular, vanishing of \(\varPhi (E_r,F_r)\) for r small implies vanishing for any r.
4.5 Horizontal invariants
As in Seidel [41], the computation of the relative invariants in the special cases needed for the exact triangle uses only horizontal sections, defined as follows.
Definition 4.33
 (a)(Horizontal sections) A section \({u}: \underline{S} \rightarrow \underline{E}\) (i.e., a collection of sections \(u_k: S_k \rightarrow E_k,\ k = 1,\ldots , m\)) is horizontal iffor all \(s \in S_k, k = 1,\ldots , m\). Let \( {\mathcal {M}}^h(\underline{E},\underline{F}) \) denote the space of horizontal sections.$$\begin{aligned} {\text {Im}}Du_k(s) = TE^h_{k,u_k(s)} \end{aligned}$$
 (b)(Horizontal almost complex structures) A collection of compatible almost complex structures \(\underline{J} \in {\mathcal {J}}(\underline{E})\) is horizontal ifor equivalently if \(\omega _{E_k}(J \cdot , \cdot )\) is symmetric for \(k = 1,\ldots , m\). Let \({\mathcal {J}}^h(\underline{E})\) denote the set of horizontal compatible almost complex structures. If \(J \in {\mathcal {J}}^h(\underline{E})\), then any horizontal section is Jholomorphic, that is,$$\begin{aligned} J_k(e) T_e^hE_k = T^h_e E_k, \quad \forall e \in E_k E_k^{{\text {crit}}}, \quad k = 1,\ldots , m, \end{aligned}$$$$\begin{aligned} (J \in {\mathcal {J}}^h(\underline{E})) \implies ({\mathcal {M}}^h(\underline{E},\underline{F}) \subset {\mathcal {M}}(\underline{E},\underline{F})). \end{aligned}$$
 (c)(Clean moduli spaces) \({\mathcal {M}}^h(\underline{E},\underline{F})\) is clean if \({\mathcal {M}}^h(\underline{E},\underline{F})\) is a smooth submanifold of the Banach manifold of sections of a suitable Sobolev class of \(\underline{E}\) with tangent spacethe set of horizontal sections of \(u^* T^{{\text {vert}}}\underline{E}\) with boundary/seams in \(( u  \partial \underline{S})^* T^{{\text {vert}}} \underline{F}\).$$\begin{aligned} T_{u} {\mathcal {M}}^h(\underline{E},\underline{F}) = \{ \xi \in \Omega ^0(u^* T^{{\text {vert}}}\underline{E}, ( u  \partial \underline{S})^* T^{{\text {vert}}} \underline{F}) \  \ \nabla \xi = 0 \} \end{aligned}$$
Remark 4.34
Proposition 4.35
Proof
Proposition 4.36
(Criterion for the existence of only horizontal sections) Suppose that \(\underline{E}\) is a quilted Lefschetz–Bott fibration with nonnegative curvature and is equipped with a horizontal almost complex structure. If \(c((x_e)_{e \in {\mathcal {E}}}) = 0 \) then \({\mathcal {M}}(\underline{E},\underline{F}; (x_e)_{e \in {\mathcal {E}}} )_0\) consists of horizontal sections.
Proof
If u is a pseudoholomorphic section with index 0, then the monotonicity relation (29) implies that u has nonnegative symplectic area equal to \(c((x_e)_{e \in {\mathcal {E}}}) \). If this constant vanishes, then all such sections must have vanishing symplectic area. By the nonnegativity of the curvature and (40), any such section has vanishing vertical energy and so is horizontal. \(\square \)
5 Floer versions of the exact triangle
The proofs of the exact triangles described in the introduction follow the lines of the proof of Floer’s exact triangle [6], in Seidel [41] and Perutz [35]. Namely, one first constructs a short sequence of cochain groups that is exact up to leading order, and then uses a spectral sequence argument to deduce the existence of a long exact sequence of cohomology groups. In this section, we also describe various extensions, to the case of minimal Maslov number two and the case of periodic Floer cohomology.
5.1 Fibered Picard–Lefschetz formula
In this section, we prove the exact triangle on the level of vector spaces; this is essentially equivalent to the fibered Picard–Lefschetz formula in Theorem 1.
Definition 5.1
Lemma 5.2
Proof
5.2 Lagrangian Floer version
5.2.1 Definition of the maps
Let M be a monotone symplectic background and \(C \subset M\) a spherically fibered coisotropic submanifold of codimension \(c \ge 2\). Let \(L^0,L^1 \subset M\) be admissible Lagrangian branes, and suppose that C is equipped with an admissible brane structure as a Lagrangian submanifold of \(M^ \times B\). These conditions imply that all Lefschetz–Bott fibrations discussed below are monotone as in Lemma 4.10. We may assume, after Hamiltonian perturbation, that \(C,L^0,L^1\) all intersect transversally.
Definition 5.3
 (a)(Chaps map) The first map in the exact sequence is defined as the relative invariant associated with a “quilted pair of pants,” or more accurately, “quilted chaps” in American dialect. Let \(\underline{S}_1\) denote the quilted surface shown in Fig. 6: Let \((S_B,S_M)\) denote the patches of \(\underline{S}_1\), and \(\underline{E} = (S_M \times M,S_B \times B)\), where B is the base of the fibration \(p: C \rightarrow B\). We identify C with its image in \(M \times B\). Let \(\underline{F}\) denote the Lagrangian seam/boundary condition for \(\underline{E}\) given by \(L^0,L^1,C\) and consider the relative invariantThis invariant was defined in Definition 4.23 by counting points in the zerodimensional component of the moduli space \({\mathcal {M}}_1\) of pseudoholomorphic quilts on \(\underline{S}_1\).$$\begin{aligned} \varPhi _1: HF(L^0 ,C, C^t, \tau _C^{1} L^1;\Lambda )[\dim (B)] \rightarrow HF(L^0, \tau _C^{1} L^1;\Lambda ) . \end{aligned}$$(44)
 (b)(Lefschetz–Bott map) The second map in the exact sequence is a relative invariant associated with a Lefschetz–Bott fibration with monodromy given by the Dehn twist. Namely, let \(E_C \rightarrow D\) denote the standard Lefschetz–Bott fibration with monodromy \(\tau _C\) from Lemma 2.13 and Definition 4.26. Gluing in \(E_C\) with the trivial fibration over a strip (using the identity as transition map to the left of the disk, and \(\tau _C\) as transition map to the right) as in Seidel [41, p. 7] gives a Lefschetz–Bott fibration \(({E}_2,{F}_2)\) over the infinite strip shown in Fig. 7. Letdenote the associated relative invariant. Relative invariants were defined in (32) by counting points in the zerodimensional component of the moduli space \({\mathcal {M}}_2\) of pseudoholomorphic sections of \(E_2 \rightarrow S_2\) with boundary in \(F_2\). (It follows from Theorem 5.5 below that \(\varPhi _2\) has degree zero.)$$\begin{aligned} \varPhi _2: \ HF(L^0, \tau _C^{1} L^1;\Lambda ) \rightarrow HF(L^0,L^1;\Lambda ) \end{aligned}$$(45)
The first step in the proof of the exact sequence is to show that the composition of the chaps and Lefschetz–Bott maps vanishes:
Lemma 5.4
(Exactness at the middle term) The composition \(\varPhi = \varPhi _2 \circ \varPhi _1\) (the relative invariant associated with picture on the left in Fig. 8) vanishes; more precisely, there exists a null homotopy of the chainlevel maps \(C \varPhi _2 \circ C \varPhi _1\) whose terms have positive qexponent for r sufficiently small.
Proof
The composition of the two relative invariants is the relative invariant associated with a Lefschetz–Bott fibration over the glued surface by (34). Consider the deformation \(\underline{S}_t\) of \(\underline{S}\) obtained by moving the critical value of the Lefschetz–Bott fibration toward the boundary marked C and pinching off a disk in \(M \times B\) with boundary values in C. This process is shown in the rightmost two pictures in Fig. 8. The bundles \(\underline{E}\) and Lagrangian boundary/seam conditions \(\underline{F}\) naturally extend to families \(\underline{E}_t,\underline{F}_t\) that are obtained from gluing for \(t \gg 0\). It follows from Corollary 4.31 that the relative invariant \(C\varPhi \) is null homotopic. \(\square \)
5.2.2 Exactness to leading order
The proof that the maps \(\varPhi _1,\varPhi _2\) of (44), (45) fit into a long exact sequence follows a standard argument, familiar from Floer’s exact triangle [6]. In this argument, one first proves that the “leadingorder terms” in the cochainlevel map define a short exact sequence and then applies a spectral sequence to deduce the triangle. Recall that \( L^0 \cap \tau _C^{1} L^1\) is the disjoint union of the images of \(i_1\) and \(i_2\) of the map in Proposition 5.2.
Theorem 5.5
 (a)
(Small triangles as leadingorder contributions to \(C\varPhi _1\)) If \(x \in ((L^0 \times C) \cap (C^t \times \tau _C^{1} L^1))\) then \(C\varPhi _1( \langle {x} \rangle )\) is equal to \(q^\nu \langle {i_1(x)} \rangle \) for some \(\nu < \epsilon /2 \), plus terms of the form \(q^\mu \langle {z} \rangle \) with \(\mu > \epsilon \) and \(z \in L^0 \cap \tau _C^{1} L^1\).
 (b)(Horizontal sections as leadingorder contributions to \(C\varPhi _2\))
 (i)
For any \(x \in L^0 \cap L^1\), if \(y=i_2(x)\) then \(C\varPhi _2( \langle {y} \rangle )\) is equal to \( \langle {x} \rangle \) plus terms of the form \(q^\nu \langle {z} \rangle \) with \(z \in {\mathcal {I}}(L^0, L^1)\) and \(\nu > \epsilon \).
 (ii)
If \(y \ne i_2(x)\) for any x, then \(C\varPhi _2( \langle {y} \rangle )\) is a sum of terms of the form \(q^\nu \langle {z} \rangle \) for \(z \in {\mathcal {I}}(L^0, L^1)\) and \(\nu > \epsilon \).
 (i)
Proof
(a) We aim to reduce to the exact, unfibered case considered by Seidel [41].
(b) The second part of the Lemma is somewhat easier, since the leadingorder terms have order exactly zero arising from the horizontal sections.
Step 1: We show that the degree zero terms arise from horizontal sections. Let u be the horizontal section of \(E_2\) on the infinite strip with value x. Then \( {\text {Ind}}(D_{u,J}) = 0\), since the boundary conditions are constant. Hence u is regular for horizontal J, and the count for \(y = i_1(x)\) follows by Proposition 4.36. The map \(\varPhi _2\) has degree 0, since the horizontal sections have zero index. Because of the nonnegative curvature of the standard Lefschetz–Bott fibration in 4.20, any nonhorizontal section has positive area. Thus the qexponentzero terms arise only from horizontal sections.
Remark 5.6
(Energy gap for Floer trajectories) Let \(C,L^0,L^1\) be as in the previous two Lemmas. We claim that there exists \(\lambda \) sufficiently large such that any nonconstant Floer trajectory for \((L^0,C^t,C,\tau _C^{1}L^1)\) or \((L^0,\tau _C^{1}L^1)\) has symplectic area at least \(\epsilon \). We consider only the case of trajectories \(u: \mathbb {R}\times [0,1] \rightarrow M\) for \((L^0,\tau _C^{1}L^1)\); the case of trajectories for \((L^0,C^t,C,\tau _C^{1}L^1)\) is similar. Choose an open neighborhood U of m disjoint from \(L^1\). Each component of \(\tau _C^{1}L^1 \cap U \) converges to \(p^{1}(p(C \cap L^1))\cap U\) as \(\lambda \rightarrow \infty \) as smooth submanifolds. The discussion above, again using Lemma 4.15, shows that if u has sufficiently small area A(u) then the image \(u(\mathbb {R}\times [0,1])\) is contained in a small neighborhood of either some intersection point \(m \in L^0 \cap L^1\) or an intersection point in \(m \in L^0 \cap \tau _C^{1}L^1\). Since each component of \(\tau _C^{1}L^1 \cap U \) contains at most one intersection point with m and the image of the \(\mathbb {R}\times \{ 1 \}\) under u is connected, this implies that any index one trajectory u connects an intersection point m to itself and is homotopic to the trivial trajectory. Hence u has vanishing area.
5.2.3 Isomorphism with the mapping cone
Every mapping cone of cochain complexes gives rise to an exact triangle. To construct an exact triangle, it suffices to prove an isomorphism of a third complex with a mapping cone. So to prove Theorem 1.3 in the unquilted case (of simple Lagrangians \(L^0,L^1\)), it suffices to show the following:
Theorem 5.7
(Isomorphism with the mapping cone) Let \(L^0,L^1,\epsilon ,\tau _C\) as in Theorem 1.3. Then the map \(C\varPhi _{2}\) induces an isomorphism of \(CF(L^0,\tau _C^{1} L^1)\) with the mapping cone on \(C\varPhi _{1}\).
Before we give the proof of the theorem, we recall a bit of homological algebra, explained, for example, in [13].
Remark 5.8
 (a)(Mapping cone) If \(C_j= (C_j,\partial _j), j = 0,1\) are cochain complexes and \(f:C_0 \rightarrow C_1\) is a cochain map, then the mapping cone on f is the complex$$\begin{aligned} {\text {Cone}}(f) := C_0[1] \oplus C_1, \ \ \partial (c_0,c_1) = (\partial _0 c_0,\partial _1 c_1 + f(c_0)) . \end{aligned}$$
 (b)(Quasiisomorphisms from mapping cones) A cochain map from \({\text {Cone}}(f)\) to a complex \(C_2\) is equivalent to pair (k, h) consisting of a cochain map \(k:C_1 \rightarrow C_2\) together with a cochain homotopySuch a map induces a quasiisomorphism if and only if \({\text {Cone}}(k[1]\oplus h)\) is acyclic.$$\begin{aligned} h: C_0 \rightarrow C_2, \quad k \circ f = h \partial _0 + \partial _2 h . \end{aligned}$$
We will need the following criterion for a cochain map (k, h) as in Remark 5.8 (b) to induce a quasiisomorphism, similar to that in Seidel [41] and Perutz [35, Lemma 5.4]. By an \(\mathbb {R}\)graded \({\Lambda }\)cochain complex, we mean a \(\Lambda \)linear cochain complex equipped with an \(\mathbb {R}\)grading so that multiplication by \(q^\lambda \) shifts the grading by \(\lambda \).
Lemma 5.9
 (a)
The differentials \(\delta _0, \delta _1, \delta _2\) each have positive order while h has nonnegative order.
 (b)
We have \(f = f_0 + f_1\), \(k = k_0 +k_1\), where \(f_0 , k_0\) have order zero while \(f_1, k_1\) have positive order.
 (c)The leadingorder terms \(f_0,k_0\) give a short exact sequence of abelian groups (not necessarily cochain complexes)$$\begin{aligned} 0 \rightarrow C_0 \mathop {\rightarrow }\limits ^{f_0} C_1 \mathop {\rightarrow }\limits ^{k_0} C_2 \rightarrow 0 \end{aligned}$$
Proof
The proof is similar to that in Perutz [35, Lemma 5.4]. The leadingorder differential in \(C := {\text {Cone}}({\text {Cone}}(f),C_2)\) is acyclic by standard homological algebra: Given \((c_0,c_1,c_2)\) with leadingorder coboundary \(( 0, f_0(c_0), k_0(c_1) + h_0(c_0)) = 0\), we have \(c_0 =0 \) since \(f_0\) is injective, and so \(k_0(c_1) = 0\). Hence \(c_1 = f_0(b_0)\) for some \(b_0 \in C_0\). Now \(h_0(b_0) =  b_2\) for some \(b_2 \in C_2\) and \(c_2 + b_2= k_0(b_1)\) for some \(b_1 \in C_1\). So the coboundary of \((b_0,b_1,b_2)\) is \((0,c_1,c_2+ b_2  b_2) = (c_0,c_1,c_2)\) as desired. Since all maps f, h, k have finitely many terms, there exists an \(\epsilon > 0\) such that, if \(C^{\ge n} \subset C\) is the subcomplex of terms with order in \([n \epsilon ,\infty )\), the first page of the associated spectral sequence has vanishing cohomology. It follows that C is itself acyclic. \(\square \)
Proof of Theorem 5.7

the first union consists of pairs of pseudoholomorphic sections of \(\underline{E}_1\) and \(\underline{E}_2\), and

the second two unions correspond to bubbling off Floer trajectories \([u] \in {\mathcal {M}}(y,z)_0\) in M or a Floer trajectory \( [u] \in {\mathcal {M}}(x,y)_0\) in \(M \times B \times M\).
The quilted version of Theorem 1.3, where \(\underline{L}^0,\underline{L}^1\) are generalized Lagrangian branes, is proved similarly, but replacing the boundary labeled \(L^0,L^1\) with collections of strips corresponding to the symplectic manifolds appearing in the generalized Lagrangian branes \(\underline{L}^0,\underline{L}^1\). The details are left to the reader.
5.3 Minimal Maslov two case
In general, Lagrangian Floer cohomology is defined only the case that certain holomorphic disk counts vanish. In the case that one of the Lagrangians has minimal Maslov number two, the relevant disk count is that of Maslov index two holomorphic disks. First we recall some basics of the derived category of matrix factorizations from [47].
Definition 5.11
 (a)(Category of matrix factorizations) For any \(w \in \mathbb {Z}\), let \({\text {Fact}}(w)\) denote the category of factorizations of \(w{\text {Id}}\).
 (i)The objects of \({\text {Fact}}(w)\) consist of pairs \((C,\partial )\), whereis a \(\mathbb {Z}_2\)graded free abelian group and \(\partial \) is a group homomorphism squaring to a multiple of the identity:$$\begin{aligned} C = C = C^0 \oplus C^1 \end{aligned}$$$$\begin{aligned} \partial : C^\bullet \rightarrow C^{\bullet + 1} , \quad \partial ^2 = w{\text {Id}}. \end{aligned}$$
 (ii)For objects \((C,\partial ),(C',\partial ')\), the space of morphismsis the space of grading preserving maps intertwining the “differentials”$$\begin{aligned} {\text {Hom}}_{{\text {Fact}}}((C,\partial ),(C',\partial ')) \end{aligned}$$$$\begin{aligned} f: C^\bullet \rightarrow (C')^\bullet , \quad f \partial = \partial ' f . \end{aligned}$$
 (i)
 (b)(Cohomology) For any matrix factorization \((C,\partial )\), let \(H((C,\partial )\otimes _\mathbb {Z}\mathbb {Z}_w)\) denote the cohomology of the differential \(\partial \otimes _\mathbb {Z}\mathrm{Id} : C \otimes _\mathbb {Z}\mathbb {Z}_w \rightarrow C\otimes _\mathbb {Z}\mathbb {Z}_w \) obtained from \(\partial \) by tensoring with \(\mathbb {Z}_w\). Any morphism in \({\text {Fact}}(w)\) defines a homomorphism of the corresponding cohomology groups. The cohomology with coefficients functor has target the category \({\text {Ab}}\) of \(\mathbb {Z}_2\)graded abelian groups,$$\begin{aligned} {\text {Fact}}(w) \rightarrow {\text {Ab}}, \ \ (C,\partial ) \mapsto H( (C,\partial ) \otimes _\mathbb {Z}\mathbb {Z}_w) . \end{aligned}$$
Definition 5.12
Denote by \({\mathcal {J}}_t(M,L^0,L^1) \subset {\mathcal {J}}_t(M)\) the subset of tdependent almost complex structures whose restriction to a fixed small neighborhood of \( t = 0\) resp. \(t = 1\) lies in \({\mathcal {J}}^{{\text {reg}}}(M,L^0)\) resp. \({\mathcal {J}}^{{\text {reg}}}(M,L^1)\).
Proposition 5.13
 (a)
\(\partial ^2 = (w(L^0)  w(L^1)){\text {Id}}\).
 (b)
\((CF(L^0,L^1;\Lambda ), \partial )\) is independent of the choice of J, H up to cochain homotopy.
If \(L^k \subset M, k= 0,1\) are monotone Lagrangian branes with the same disk invariant, then the exact triangle in Theorem 1.3 holds, with the same proof. More generally, the disk invariant for a generalized Lagrangian brane \(\underline{L} = (L_1,\ldots , L_k)\) is the sum of the disk invariants for the components \(L_1, \ldots , L_k\). Furthermore, the disk invariant for a correspondences \(L_{01} \subset M_0^ \times M_1\) is the opposite of the disk invariant for its transpose \(L_{01}^t \subset M_1^ \times M_0\), because the change in complex structure reverses orientations on Maslov index two disks [48]. With these conventions, the quilted version of Theorem 1.3 also holds provided that the disk invariants of the generalized correspondences \(\underline{L}^0,\underline{L}^1\) are equal.
5.4 Periodic Floer version
One can also formulate a version of the exact triangle for symplectomorphisms, that is, in periodic Floer theory. In this formulation, the exact triangle relates the symplectic Floer cohomology of the Dehn twist with the Lagrangian Floer cohomology of the vanishing cycle and the identity:
Theorem 5.14
6 Applications to surgery exact triangles
In this section, we apply the exact triangle to obtain versions of the Floer [10], Khovanov [17], and Khovanov–Rozansky [19] exact triangles. We have already established in Sect. 3 that Dehn twists of surfaces induce fibered Dehn twists of moduli spaces.
6.1 Exact triangle for three bordisms
In [46], we introduced a categoryvalued field theory associated with certain connected, decorated surfaces and bordisms. We use freely the notation and terminology from [46]:
Definition 6.1
 (a)
(Decorated surface) A decorated surface is a compact connected oriented surface X equipped with a line bundle with connection \(D \rightarrow X\). The degree of D is the integer \(d = (c_1(D),X)\).
 (b)(Decorated bordisms) A bordism from \(X_\) to \(X_+\) is a compact, oriented threemanifold Y with boundary equipped with an orientationpreserving diffeomorphismHere \(\overline{X}_\) denotes the manifold \(X_\) equipped with the opposite orientation. We often omit \(\phi \) from the notation and assume that \(X_,X_+\) are boundary components of Y. A decorated bordism of degree d is a compact connected oriented bordism Y between decorated surfaces \((X_\pm ,D_\pm )\) equipped with a line bundle with connection \(D \rightarrow Y\) such that \(D  X_\pm = D_\pm \). Given a decorated bordism (Y, D) between \((X_\pm ,D_\pm )\), one can obtain another decorated bordism by tensoring D with any line bundle that is trivial on the boundary of Y.$$\begin{aligned}\phi : \partial Y \rightarrow \overline{X}_ \sqcup X_+ . \end{aligned}$$
 (c)
(Moduli spaces for decorated surfaces) Suppose that X is a decorated surface with line bundle D. Let M(X, D) denote the moduli space of constant curvature rank 2 bundles on X with fixed determinant. The space M(X, D) is a compact, monotone symplectic manifold with monotonicity constant 1 / 4 with a unique Maslov cover of any even order. In [46], we describe how to equip M(X, D) with the background classes so that M(X, D) is equipped with the structure of a symplectic background.
 (d)(Correspondences for decorated bordisms) Suppose that Y is a decorated surface with line bundle D between decorated surfaces \((X_,D_)\) and \((X_+,D_+)\). Letdenote the set of isomorphism classes of constant central curvature bundles with fixed determinant that extend over the interior of Y. If Y is an elementary bordism (i.e., admits a Morse function with a single critical point), then L(Y, D) is a smooth Lagrangian correspondence. In [46] we describe how to equip L(Y, D) with relative spin structures and gradings, so that L(Y, D) has the structure of an admissible Lagrangian brane.$$\begin{aligned} L(Y,D) \subset M(X_,D_)^ \times M(X_+,D_+) \end{aligned}$$
For the purposes of the exact triangle, we will need an alternative description as flat bundles with fixed holonomy around an additional marking.
Definition 6.2
 (a)
(Marked surfaces) For any integer \(n \ge 0\), an n marked surface is a compact oriented surface X equipped with a tuple \(\underline{x} = (x_1,\ldots , x_n)\) of distinct points on X and a labeling \(\underline{\mu } = (\mu _1,\ldots ,\mu _n)\). In this section, we take labels all equal to 1 / 2 corresponding to the central element \(I\) of \(\mathfrak {A}\).
 (b)
(Moduli spaces for marked surfaces) Denote by \(M(X,\underline{\mu })\) the moduli space of flat bundles on \(X  \underline{x}\) with holonomy \(I\) around each marking \(x_i\); we consider more general holonomies in the next subsection. The moduli space \(M(X,\underline{\mu })\) may be identified with the moduli space M(X, D) where D is a line bundle of degree n, by a construction described in Atiyah and Bott [5] given by twisting by a fixed central connection.
 (c)
(Marked bordisms) A marked bordism from \((X_,\underline{x}_)\) to \((X_+,\underline{x}_+)\) is a compact, oriented bordism Y equipped with a tangle (compact oriented onedimensional submanifold transverse to the boundary) \(K \subset Y\) such that \(K \cap X_\pm = \underline{x}_\pm \).
 (d)(Moduli spaces for marked bordisms) Denote bythe moduli space of bundles that extend over \(Y  K\) with holonomy \(I\) around K. The identification \(M(X_\pm ,\underline{\mu }_\pm ) \rightarrow M(X_\pm ,D_\pm )\) induces a homeomorphism \(M(Y,K) \rightarrow M(Y,D)\) where \(D \rightarrow Y\) is a line bundle whose first Chern class is dual to the tangle K. Thus, in particular, the addition to K of a circle component \(K'\) corresponds to twisting the determinant line bundle D by a line bundle whose first Chern class is dual to the homology class of \(K'\).$$\begin{aligned} L(Y,K) \subset M(X_,\underline{\mu }_)^ \times M(X_+,\underline{\mu }_+) \end{aligned}$$
The following proposition relates the moduli spaces of bundles with fixed holonomy around an embedded circle with the Lagrangian correspondences associated with elementary bordisms.
Proposition 6.3
 (a)
Let Y be a bordism from \(X_\) to \(X_+\) containing a single critical point of index 1 and a trivial tangle K (i.e., a union of intervals connecting \(\underline{x}_\) to \(\underline{x}_+\)) and \(C \subset X_+\) is the attaching cycle. The Lagrangian L(Y, K) is diffeomorphic via the projection to \(M(X_+,\underline{\mu }_+)\) to the subset of connections on \(X_+  \underline{x}_+\) with holonomy along C equal to I.
 (b)
Let Y be a decorated bordism from \(X_\) to \(X_+\) containing a single critical point of index 1, \(C \subset X_+\) the attaching cycle, \(K_0\) a trivial bordism connecting \(\underline{x}_\) to \(\underline{x}_+\) and \(K_1 \subset Y\) the unstable manifold of the critical point. The Lagrangian \(L(Y,K_0 \cup K_1)\) is diffeomorphic to the subset of flat bundles on \(X_+  \underline{x}_+\) with holonomy along C equal to \(I\).
Proof
By Seifert–van Kampen, \(\pi _1(Y  K)\) is the quotient of \(\pi _1(X_+  K)\) by the ideal generated by the element [C] obtained from C by joining by a path to the base point. Hence in the first case, L(Y, K) is diffeomorphic to the submanifold of \(M(X,\underline{\mu }_+)\) obtained by setting the holonomy along C equal to the identity. For the second case, the gradient flow the Morse function defines a deformation retract of \(Y  K_0  K_1\) to \(X_+  \underline{x}_+\). Homotopy invariance implies that \(\pi _1(Y  K_0  K_1 )\) is isomorphic to \(\pi _1(X_+  \underline{x}_+)\). Since C is a loop around \(K_1\), the holonomy around \(K_1\) is equal to the holonomy along C, hence the claim. \(\square \)
In order to obtain smooth Lagrangian correspondences, we break the given bordism into elementary bordisms.
Definition 6.4
 (a)(Cerf decompositions) A Cerf decomposition of a bordism Y is a decomposition of Y into elementary bordisms \(Y_1,\ldots ,Y_K\), that is, bordisms admitting a Morse function with at most one critical point. Associated with any Cerf decompositionand a decoration on Y is a generalized Lagrangian correspondence$$\begin{aligned} Y = Y_1 \cup _{X_1} \ldots \cup _{X_{k1}} Y_k \end{aligned}$$$$\begin{aligned} \underline{L}(Y) = (L(Y_1),\ldots , L(Y_k)) . \end{aligned}$$
 (b)The generalized correspondence \(\underline{L}(Y)\) may be equipped with a relative spin structure via its structure as a fibration over the moduli space of the incoming or outgoing surface. Thus \(\underline{L}(Y)\) gives rise to a functor of generalized Fukaya categories This functor is independent of the choice of Cerf decomposition [46].
 (c)
We prove the following surgery exact triangle for the invariants \(HF(Y;L^,L^+)\).
Definition 6.5
 (a)
(Knots) A knot in a bordism Y is an embedded, connected onemanifold \(K \subset Y\) disjoint from the boundary.
 (b)
(Knot framings) A framing of a knot \(K \subset Y\) is a nonvanishing section of its normal bundle up to homotopy. Given a framed knot, the other framings are obtained by twisting by representations of \(\pi _1(K) \cong \mathbb {Z}\) into SO(2) and so are indexed by \(\mathbb {Z}\).
 (c)
(Knot surgeries) For \(\lambda \in \mathbb {Z}\) the \(\lambda \)surgery \(Y_{\lambda ,K}\) of Y along K is obtained by removing a tubular neighborhood of K and gluing in a solid torus \(D^2 \times S^1\) so that the meridian \(\partial D^2 \times \{ {\text {pt}}\}\) is glued along the curve given by the framing of the knot corresponding to \(\lambda \). Denote by \(K_\lambda \) the knot in \(Y_{\lambda ,K}\) corresponding to a longitude in \(\partial D^2 \times S^1\). Thus \(K_\lambda \) intersects the meridian transversally once.
Remark 6.6
 (a)
The zerosurgery \(Y_{0,K}\) has a decomposition into elementary bordisms with two additional pieces, \(Y_\cup , Y_\cap \) inserted between \(Y_i\) and \(Y_{i+1}\). The knot \(K_0 \subset Y_{0,K}\) is divided into the two additional pieces \(K_0 \cap Y_\cup \) and \(K_0 \cap Y_\cap \). The correspondence \(L(Y_\cup , Y_\cup \cap K_0) \circ L( Y_\cap ,Y_\cap \cap K_0)\) is the correspondence associated with the moduli space of bundles \(L(Y_\cup ) \circ L(Y_\cap )\) on the decorated surface \(Y_\cup \cup Y_\cap \) with the shifted line bundle as in Proposition 6.3.
 (b)
The \(1\) surgery \(Y_{1,K}\) has decomposition into simple bordisms \(Y_1,\ldots , Y_l\) but where the identification \((\partial Y_i)_+ \rightarrow (\partial Y_{i+1})_\) is the Dehn twist along K.
Lemma 6.7
(Existence of Cerf decompositions compatible with a knot) For any framed knot \(K \subset Y\), there exists a Cerf decomposition \(Y= Y_1 \cup \ldots \cup Y_l\) so that K is contained in the boundary \((\partial Y_i)_+ = (\partial Y_{i+1})_\) for some \(i = 1,\ldots , l\) and the framing is the direction normal to the boundary.
Proof
Choose a Morse function \(f: Y \rightarrow \mathbb {R}\) such that f is constant on K and the framing is given by the gradient of f at K. The level set \(f^{1}(\lambda )\) containing K can be made connected by adding 1handles in Y, so that \(f^{1}(\lambda )\) becomes a connected surface containing K separating the boundary components of Y. By taking a selfindexing Morse function on \(f^{1}(\pm \infty ,\,\lambda ], f\) can be deformed away from \(f^{1}(\lambda )\) so that f is Morse and has connected fibers. \(\square \)
Theorem 6.8
Proof
Remark 6.9
(Dehn twists for separating curves) We do not discuss here the exact triangle for a Dehn twist around a separating curve, because the moduli space M does not satisfy the “strong monotonicity” condition of Definition 4.28. Instead, one needs to establish positivity properties of the form \(c_1(M_C)  [B_]  [B_+]\) with respect to the canonical complex structure on the moduli space of parabolic bundles \(M_C\). This seems like to hold but would take us outside the framework of this paper.
6.2 Exact triangles for tangles
We obtain Floertheoretic invariants of tangles constructed in [47] exact triangles that are the same as those obtained by Khovanov [17, 18] and Khovanov–Rozansky [19]. We assume freely the terminology from [47], in particular the terminology for moduli spaces of bundles with fixed holonomy for marked surfaces. In this subsection, we take \(G = SU(2)\) and \(\mathfrak {A}\cong [0,1/2]\) via the identification (12).
Definition 6.10
 (a)
(Correspondences for elementary tangles) Let \(X_\pm \) be a compact, oriented surface with odd numbers of markings \(\underline{x}_\pm \) admissible labels \(\underline{\mu }_\pm \) all equal to \(1/4 \in \mathfrak {A}\). Let \(K \subset Y: =X \times [1,1]\) be a tangle, that is, a bordism between marked surfaces \((X_,\underline{\mu }_)\) and \((X_+,\underline{\mu }_+)\). Let \(M(X_\pm ,\underline{\mu }_\pm )\) denote the moduli space of SU(2) bundles in Definition 3.1. Let L(K) denote the subset of \(M(X_,\underline{\mu }_)^ \times M(X_+,\underline{\mu }_+)\) of bundles that extend over the interior of the bordism. Assuming that K is elementary (admits a Morse function with at most one critical point on K, and none on Y), the Lagrangian L(K) has an admissible brane structure [47].
 (b)(Functors for tangles) More generally, let \(K \subset Y = X \times [1,1]\) be an arbitrary tangle, and \(K = K_1 \cup \ldots \cup K_r\) a decomposition into elementary tangles. Associated with each elementary tangle \(K_i\) is a Lagrangian correspondence \(L_i\) and so a generalized Lagrangian correspondence associated with KComposing the functors \(\varPhi (L_i)\) for these correspondences gives a functor In [47], we proved that \(\varPhi (K)\) is independent, up to isomorphism, of the choice of decomposition \(K_1 \cup \ldots \cup K_r\).$$\begin{aligned} \underline{L}(K) = (L_1,\ldots , L_r ) . \end{aligned}$$
 (c)
We prove the following surgery exact triangle for these invariants.
Definition 6.11
Given a tangle \(K \subset Y\), a separating embedded surface \(\Sigma \subset Y\), and a disk \(D \subset \Sigma \) meeting K in two points. let be the tangles obtained by modifying K by a half twist, respectively, adding a cup and cap as in Fig. 11.
Theorem 6.12
Proof
More generally, in higherrank invariants we obtain an exact triangle for the Khovanov–Rozansky modification.
Definition 6.13
 (a)
(Admissible labels) An admissible label is a projection of the barycenter of \(\mathfrak {A}\) onto some face. In the absence of reducibles, given a marked surface \((X,\underline{x})\) with admissible labels \(\underline{\mu }\) the moduli space \(M(X,\underline{\mu })\) of bundles with fixed holonomy around the markings \(\underline{x}\) in the conjugacy classes associated with \(\underline{\mu }\) is a smooth, compact, monotone symplectic manifold.
 (b)(Correspondences for admissible graphs) Letbe a trivalent graph with admissible labels, where each trivalent vertex is of the form described in Definition 3.12. This means that K is made up of a finite number of vertices that are points in Y, and edges that are embedded, compact onemanifold with boundary. The endpoints of the edges are either vertices or points on the boundary of Y. Trivalence means that each vertex is required to lie in the boundary of exactly three edges. Each edge is labeled by either \(\nu _k^1\) or \(\nu _k^2\), so that at any vertex the labels are \(\nu _k^1,\nu _k^1,\nu _k^2\), see Fig. 12 where the squiggly edge represents a vertex labeled \(\nu _k^2\).$$\begin{aligned} K \subset Y := X \times [1,1] \end{aligned}$$
 (c)(Decomposition into elementary graphs) Letbe a decomposition into elementary graphs admitting cylindrical Morse functions with at most one critical point or vertex. Each elementary graph \(K_i\) defines a smooth Lagrangian correspondence with admissible brane structure \(L_i\) consisting of bundles that extend over the interior, see [47], and so a generalized Lagrangian correspondence \(\underline{L} = (L_1,\ldots , L_e)\). The functor obtained by composing the functors \(\varPhi (L(K_i))\) is independent of the choice of decomposition into elementary graphs [47], up to \(A_\infty \) homotopy.$$\begin{aligned} K = K_1 \cup \ldots \cup K_e \end{aligned}$$
 (d)
(Khovanov–Rozansky modification of a graph) Suppose \(K \subset X \times [1,1]\) is a trivalent graph with admissible labels. Let \((K_1,\ldots ,K_e)\) be a cylindrical Cerf decomposition of K and \( K \cap (X \times \{ b_i \}) \) a slice such that two points have the same label \(\nu _k^1\) from 3.12. We obtain a new trivalent graphs by inserting a half twist, respectively, inserting two new vertices as shown in Fig. 13. Here the intermediate edge represented by a squiggle is labeled \(\nu _k^2\) from 3.12.
Theorem 6.14
Note that this generalizes the SU(2) exact triangle since if \(k = 2\) then \(\omega _0 = \omega _2 = 0\). The exact triangle of Khovanov–Rozansky [19] has a similar form. The theories for the other standard markings will not in general have surgery exact triangles of this form, since the corresponding symplectomorphisms are not, in general, Dehn twists. It would be interesting to understand whether there is a replacement for the surgery exact triangle in these more general cases.
7 Fukaya versions of the exact triangle
In [27], the authors constructed \(A_\infty \) functors for Lagrangian correspondences between Fukaya categories. The gluing results necessary for the construction of the \(A_\infty \) functors for Lagrangian correspondences are proved in [28]. Applied to the Lagrangian correspondences arising from moduli spaces of flat bundles one obtains a (partial) \(A_\infty \) categoryvalued field theory. We now explain the Fukayacategorical versions of the exact triangle for fibered Dehn twist.
7.1 Open Fukayacategorical version
Theorem 7.1
Proposition 7.2
The morphism \(k \in {\text {Hom}}(L, \tau _C L)\) of (56) induces an isomorphism of the mapping cone \({\text {Cone}}(f: \varPhi (C) \varPhi (C^t) L \rightarrow L)\) with \(\tau _C L\).
The proof of Proposition 7.2 depends on the following lemma, whose proof is left as an exercise (c.f. [37, Lemma 2.6]).
Lemma 7.3
Proof of Proposition 7.2
Now let \(L^1\) be another object in , for simplicity unquilted. Acyclicity of the differential (58) is shown as follows. It suffices to prove acyclicity with L replaced with \(\tau _C^{1}L\). The terms of lowest order in q are \(\mu ^2(f,a)\) and \(\mu ^2(k,b)\). As in Sect. 5.2, the leading term of \(\mu ^2(f,a)\) corresponds to the canonical injection \( {\mathcal {I}}(L^1, C^t,C,\tau _C^{1} L) \rightarrow {\mathcal {I}}(L^1,\tau _C^{1} L) \). On the other hand, the leading term of \(\mu ^2(k,b)\) corresponds to the canonical injection \( {\mathcal {I}}(L^1,L) \rightarrow {\mathcal {I}}(L^1,\tau _C^{1} L) .\) As before, the lowest order terms in complex are acyclic, after a small shift in the \(\mathbb {R}\)degrees of the generators. Filtering the complex by energy shows that entire complex is acyclic. An application of Lemma 7.3 completes the proof of Proposition 7.2. \(\square \)
Theorem 7.1 follows by taking the long exact sequence associated with the mapping cone in Proposition 7.2.
7.2 Periodic Fukayacategorical version
Theorem 7.4
Sketch of proof

the outer circle represents a quilted cylindrical end with seams \(C^t,C\),

the lightly shaded patch \(S_M\) maps to M and

the darkly shaded patch \(S_B\) maps to B.
Remark 7.5
(\(A_\infty \) results for minimal Maslov two) Similar results hold in the case of minimal Maslov number two for the \(A_\infty \) categories whose objects L have disk invariant \(w = w(\underline{L})\), counting the number of Maslov index disks passing through a generic point in \(\underline{L}\). See [27, Section 4.4] for more on the disk invariant and Fukaya category .
Declarations
Acknowledgements
Partially supported by NSF Grants CAREER 0844188 and DMS 0904358.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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