Fukaya \(A_\infty\)-structures associated to Lefschetz fibrations. I
Algebraic structures such as \(A_\infty\)-algebras, bimodules and morphisms between bimodules obtained from the fibre and total space with vanishing cycles and Lefschetz thimbles of an exact symplectic Lefschetz fibration \(\pi: E\to\mathbb{C}\) are compared (Theorem 7.1). The main result says that the associated directed \(A_\infty\)-algebra \({\mathcal A}\) has two maps from \({\mathcal A}^\vee\), the dual diagonal bimodule of \({\mathcal A}\) by working in terms of the fibre and total space of the fibration, that agree up to homotopy and composition with a quasi-morphism from \({\mathcal A}^\vee\) to itself (\S1. Theorem cf. Corollary 7.1). The author says that the theorem of \S1 corresponds to a weak version of conjecture 4 of \textit{P. Seidel} [Proc. Symp. Pure Math. 80, Pt. 1, 415--434 (2009; Zbl 1179.53085)] and is intended to be the start of a wider project concerning Fukaya categories of Lefschetz fibration [\textit{M. Maydanskiy},``Exotic symplectic manifolds from Lefschetz fibration'', (2009), \url{arXiv:0906.2224}; \textit{M. Maydanskiy} and \textit{P. Seidel}, J. Topol. 3, No. 1, 157--180 (2010; Zbl 1235.53088)]. The author also remarks that this paper has a significant overlap in the eventual goals, even though not necessary in intermediate steps, with [\textit{F. Bourgeois} et al., Geom. Topol. 16, No. 1, 301--389 (2012; Zbl 1322.53080)]. An exact symplectic manifold \(M\) has an exact symplectic form \(\omega= d\theta\) and a \(\omega\)-compatible almost complex structure \(I\) which is weakly convex (Definition 2.1). A Lefschetz fibration \(\pi: E\to\mathbb{C}\) is said to be an exact symplectic Lefschetz fibration if \(E\) is an exact symplectic manifold and \(\partial E\) is weakly convex (Definition 5.1). Moreover, the fibration is assumed to be trivial near infinity (as for detailed meaning of ``trivial near infinity'' cf. \S4.1). Vanishing cycles in the fibre \(M\) of \(\pi: E\to\mathbb{C}\), denoted by \((V_1,\dots, V_m)\), are assumed to be in general position. The corresponding Lefschetz thimbles in \(E\) are denoted by \((L_1,\dots, L_m)\) (cf. \S5.1). In general, let \(M\) be an exact symplectic manifold, \((V_1,\dots, V_m)\) be an ordered collection of Lagrangian submanifolds, such that the intersections \(V_i\cap V_j\) are transversal and \(V_i\cap V_j\cap V_k=\emptyset\). Then the Floer cochain group \(CF(V_i,V_j)\) is the vector space over \(\mathbb{K}= \mathbb{Z}/2\mathbb{Z}\) with the Floer differential \[ \mu^1(x_1)= \sum_{x_0} n^1(x_0, x_1)x_0, \] where \(n^1(x_0,x_1)\) is the (modulo 2) number of solutions of the pseudoholomorphic map equation (Floer equation) \[ (du)^{0,1}= \textstyle{{1\over 2}} (du(z)+ J_z\circ du(z)\circ j_S) \] joining \(x_0\) and \(x_1\) at \(s\to\pm\infty\) limits. Here \(J_z\) is an \(\omega\)-compatible almost complex structure that agrees with the given \(I\) in some neighborhood of \(\partial M\), i.e., on the strip-like ends, the family \((J_z)\), \(z\in{\mathcal S}\), \({\mathcal S}\) is a punctured Riemann surface, is translation invariant and \(j_{{\mathcal S}}\) is the complex structure of \({\mathcal S}\) (\S2.2 and 2.3). Then, the cohomology of \(\overline{\mathcal A}= \bigoplus_{i<j} CF(V_i, V_j)\) is the direct sum of Floer cohomology groups \(\bigoplus_{i<j} HF(V_i, V_j)\). Moreover, denoting \(n^d(x_0,\dots, X_d)\) to be the (modulo 2) number of isolated points of the moduli space of pseudo-holomorphic maps \({\mathcal M}^d_{{\mathcal A}}(x_0,\dots, x_d)\) [\textit{P. Seidel}, Fukaya categories and Picard-Lefschetz theory. Zürich: European Mathematical Society (EMS) (2008; Zbl 1159.53001), \S9], and defining \(\mu^d:\overline{\mathcal A}^{\otimes d}\to\overline{\mathcal A}\) by \[ \mu^d(x_d,\dots, x_1)= \sum_{x_0} n^d(x_0,\dots, x_d) x_0, \] \(\overline{\mathcal A}\) becomes a \(A^\infty\)-algebra (\S2.3). Let \(R\) be \(\mathbb{K}^m=\mathbb{K} e+1\oplus\cdots\oplus \mathbb{K} e_m\), \(e^2_i= e_i\), \(e_i e_j= 0\), \(i\neq j\). Then \({\mathcal B}=\bigoplus_{i,j} CF(V_i, V_j)\) is an \(R\)-bimodule. \({\mathcal B}\) is defined without considering any ordering. Reflecting this to the perturbed holomorphic strip equation and the Floer differential, \(H_{{\mathcal B}}\)-perturbed Floer cochain spaces \(CF_{{\mathcal B}}(V_i, V_j)\) are defined (\S2.4). Taking a cocycle \(u_i\in CF_{{\mathcal B}}(V_i, V_i)\) which represents the unit class \(1\in H^*(V_i,\mathbb{K})\) for each \(i\), the bimodule \[ {\mathcal B}^+= \bigoplus_i \mathbb{K} u_i\oplus\bigoplus_{i<j} CF_{{\mathcal B}}(V_i, V_j)\subset{\mathcal B} \] is defined. \({\mathcal B}^+\) is independent of the choice of \(u_i\) up to quasi-isomorphism (\S2.4). Let \({\mathcal B}^-={\mathcal B}/{\mathcal B}^+\) and \(\Delta:{\mathcal B}^-\to{\mathcal B}^+\) be the boundary map associated to the short exact sequence \(0\to{\mathcal B}^+\to{\mathcal B}\to{\mathcal B}^-\to 0\). The algebraic structures on the fibre of \(\pi: E\to\mathbb{C}\) considered in this paper are \(A_\infty\)-algebras \(\overline{\mathcal A}\), bimodules \({\mathcal B}^{\pm}\) and the morphism \(\Delta:{\mathcal B}^+\to{\mathcal B}^-\), which are defined taking \((V_1,\dots, V_m)\) to be the vanishing cycles. Replacing vanishing cycles by Lefschetz thimbles, the corresponding \(A_\infty\)-algebra \(\overline{\mathcal J}\), bimodules \({\mathcal U}^{\pm}\) and morphism \(\Gamma:{\mathcal U}^-\to{\mathcal U}^+\) on the total space \(E\) are also defined in \S6. Descriptions of these modules by using Hochschild cocycles are also given in \S2.5 and \S6.5 and also quasi-isomorphisms \(\Phi:{\mathcal A}\to{\mathcal B}^+\) and corresponding map \(\Pi\) in the total space, where \[ {\mathcal A}= R\oplus\overline{\mathcal A}. \] The author says that this shows existence of connections between \(\overline{\mathcal A}\) and \({\mathcal B}\) and corresponding bimodules on the total space, that go beyond the bimodule structure. In \S7.4 (Lemma 7.7), a sub-module \({\mathcal V}\subset{\mathcal U}^+\) is singled out, and the following assertions are proved: 1. The map \(\Gamma\) lands in \({\mathcal V}\), and defines a quasi-isomorphism \({\mathcal U}^-\to{\mathcal V}\). 2. \({\mathcal U}^+/{\mathcal V}\) is isomorphic to \({\mathcal B}\) as a bimodule over \(\overline{\mathcal A}=\overline{\mathcal J}\). 3. The image of the Hochschild cocycle \(\overline\Pi\) in \(CC(\overline{\mathcal J},{\mathcal U}^+/{\mathcal V})= CC(\overline{\mathcal A},{\mathcal B})\) is equal to \(\Phi\) (cf. \S2.5 and \S6.5). (\S7. Theorem 7.1). These are the main results of the paper. As a corollary, existence of a bimodule quasi-isomorophism \(\Theta\) from \({\mathcal A}^\vee\) to itself such that the bimodule homomorphisms \(\Pi^{-1}\circ \Gamma\circ \Xi^{-1}\) and \(\Phi^{-1}\circ \Delta\circ \Psi^{-1}\circ\Theta\) are homotopic, is proved. Introducing unital \(A_\infty\)-algebras \({\mathcal A}\) and \({\mathcal J}= R\oplus\overline{\mathcal J}\), and using Hochschild cocycle representations and Poincaré duality in Floer cohomology (\S3), existence of quasi-isomorphisms \[ {\mathcal B}^+\backsimeq{\mathcal A},\quad{\mathcal B}^-\backsimeq{\mathcal A}^\vee,\qquad{\mathcal U}^+\backsimeq{\mathcal J},\quad{\mathcal U}^-\backsimeq{\mathcal J}^\vee \] is shown (Proposition 2.1, 3.1 and Proposition 6.1, 6.2). Here \({\mathcal A}^\wedge\) and \({\mathcal J}^\wedge\) mean duals of \({\mathcal A}\) and \({\mathcal J}\).
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