Complexifications of Morse functions and the directed Donaldson-Fukaya category
Floer theory of symplectic Lefschetz fibrations derived from a Morse function on a four manifold is studied. Let \(f: N\to\mathbb{R}\) be a real analytic Morse function. Then, in favorable circumstances, the complexification \(f_{\mathbb{C}}\) of \(f\) can be regarded as a symplectic Lefschetz fibration [cf. \textit{B. Hall} and \textit{W. Kirwin}, Adapted complex structure and the geodesic flow, \url{arXiv:0811.3083}, to appear in Math. Ann.]. For smooth \(f\) (not necessary real analytic), taking a Riemannian metric \(g\) such that \((f, g)\) is Morse-Smale, and constructing an exact symplectic manifold \(M\), \(\dim M= 2\dim N- 2\), with some exact Lagrangian spheres \(V_1,\dots, V_m\subset M\), one for each critical point of \(f\), for a given handle decomposition of \(N\), we can derive a symplectic Lefschetz fibration from \(f\) [cf. \textit{P. Seidel}, Fukaya categories and Picard-Lefschetz theory. Zürich Lectures in Advanced Mathematics. Zürich: European Mathematical Society (EMS). vi, 326 p. (2008; Zbl 1159.53001); hereafter referred to as [1]]. Assuming \(N\) is compact and \(f\) is a self-indexing Morse function with either two, three, or four critical values: \(\{0, n\}\), \(\{0, n,2n\}\) or \(\{0,n,n+ 1, 2n+ 1\}\), the author carried out this procedure in [The Picard-Lefschetz theory of complexified Morse functions, \url{arXiv:0906.1218}, to appear in Geometry and Topology, hereafter referred to as [2]]. In 82), Let \(\pi: E\to\mathbb{C}\) be the corresponding symplectic Lefschetz fibration, then 1. \(N\) embeds in \(E\) as an exact Lagrangean submanifold; 2. All the critiacal points of \(\pi\) lie in \(N\) and \(\text{Crit}(T)= \text{Crit}(f)\); 3. \(\pi|N= f: N\to\mathbb{R}\) (referred Theorem A in this paper). Adopting a version of plumbing [\textit{R. E. Gompf} and \textit{A. I. Stipsicz}, 4-manifolds and Kirby calculus, Graduate Studies in Mathematics. 20. Providence, RI: American Mathematical Society (AMS). xv, 558 p. (1999; Zbl 0933.57020)], explicit constructions of \(M\) and \(V_i\) in the case treated in this paper (\(\dim N= 4\), and critical values \(\{0,2,4\}\)), are precisely described in \S5 and \S6. Assuming \(\dim N= 4\) and critical values \(\{0,2,4\}\), this paper is a sequel of [2]. Let \(V_4\), \(V^j_2\), \(V_0\) be the Lagrangian spheres corresponding to the critical points \(x_4\), \(x^j_2\), \(x_0\), and let \(\text{Flow}(x, y)\) be the compactification of the space of unparameterized \((f,g)\)-gradient trajectories from \(x\) to \(y\). Then, denoting by \(\text{HF}(V_4, V^j_2)\), \(\text{HF}(V^j_2, V_0)\), \(\text{HF}(V_4,V_0)\) and \(\mu_2: \text{HF}(V_4,V^j_2)\otimes\text{HF}(V^j_2, V_0)\to\text{HF}(V_4, V_2)\), the Lagrangian Floer homology groups and the triangle product, the following is shown. Theorem B. We have \[ \text{HF}(V_4, V_2)\cong H_*(\text{Flow}(x_4, x^j_2)),\;\text{HF}(V^j_2, V_0)\cong H_*(\text{Flow}(x^j_2,x_0)), \] \[ \text{HF}(V_4, V_0)\cong H_*(\text{Flow}(x_4, x_0)), \] and by this correspondence, the triangle product is transferred to the composition in the Flow category; \[ \mu^{\text{Flow}}: H_*(\text{Flow}(x_4, x^j_2))\otimes H_*(\text{Flow}(x^j_2, x_0))\to H_*(\text{Flow}(x_4, x_0)). \] Theorem B is the main result of this paper and proved in \S7 (Computation of Floer homologies; Prop. 7. 1. Prop.7.1 and 7.2 in its proof are misprints in Lemma 7.1 and 7.2) and \S10 (Computation of triangle product; Theorem 10.1). Since signs and gradings for Morse-Bott Floer homology are not treated in this paper, this theorem is limited to \(\mathbb{Z}/2\) coefficients and ungraded Floer groups (see, Lemma 7.1). The author remarks that the \(V_i\)'s have a certain rotational symmetry, which allows to compute the moduli space of holomorphic triangles explicitly, but cannot perturb the \(V_i\)'s to make them transverse. This is the reason why the whole calculation in this paper is done in the context of Morse-Bott Floer homology (A review of Morse-Bott Floer homology is given in \S13.) In a certain sense, for any Lefschetz fibration \(\pi: E\to\mathbb{C}\), the Lefschetz thimbles \(\Delta_1,\dots, \Delta_m\subset E\) generate the whole Fukaya category \(\text{Fuk}(E)\) [1]. To perform computations in \(M\) (not in \(E\)), replacing \(\Delta_i\) by \(V_i\), the directed Fukaya category \(\text{Fuk}(M, V_1,\dots, V_m)\) is introduced. At the level of homology, it is called the directed Donaldson-Fukaya category and denoted by \(H(\text{Fuk}^\to(M, V_4,\{V^j_2\}, V_0))\). Then Theorem B is reformulated as follows: Theorem B\('\). The directed Donaldson-Fukaya category of \((M,\{V^j_2\},V_0)\) is isomorphic to the flow category of \((N,f, g)\). This answers a conjecture of \textit{P. Seidel} [More about vanishing cycles and mutation. Fukaya, K. (ed.) et al., Symplectic geometry and mirror symmetry. Proceedings of the 4th KIAS annual international conference, Seoul, South Korea, 2000. Singapore: World Scientific. 429--465 (2001; Zbl 1079.14529)]. Explanations of the flow category are given in \S3. The author says that Theorem B\('\) leads to the following conjecture: Conjecture 1.1. Let \(\pi\) be the projection of Lefschetz fibration obtained in Theorem A from \(f\), \(\Lambda\) the set of conormal bundles of the unstable manifolds of \(f\), then the two versions \(\text{Fuk}_S(T^*N,\pi)\) and \(\text{Fuk}_{NZ}(T^*N,\Lambda)\) of the Fukaya category of \(T^*N\), due to \textit{P. Seidel} ([1]) and \textit{D. Nadler} and \textit{E. Zaslow} [J. Am. Math. Soc. 22, No.~1, 233--286 (2009; Zbl 1227.32019)] are equivalent. This paper splits into three parts (\S2--\S4, \S5--\S10 and \S11--\S13). The first part outlines the main argument of this paper taking \(N=\mathbb{R} \mathbb{P}^2\) (\S2) and \(N= \mathbb{C} \mathbb{P}^2\) (\S4) as examples. \S2 is a simple case, while \S4 outlines the paper in detail. The main problem is to describe vanishing spheres (\(V_i\), etc.), an appropriate way to prove Theorem B (\S4.2--\S4.6). To study the triangle product, the \(V_i\)'s are modified to new exact versions \(\widetilde V_i\), and Theorem B (for \(\mathbb{C} \mathbb{P}^2\)) is proved in \S4.7 and \S4.8. The second part (\S5--\S10) is the core of this paper. It extends arguments presented in \S4 for general four manifolds. \S5--\S7 construct \(M\) and the \(V_j\)'s and compute Bott-Morse Floer homologies related to the \(V_i\)'s. Then \(\widetilde V_j\)'s are constructed and study Bott-Morse Floer homologies related to \(\widetilde V_j\)'s in \S8--\S9. After these preparations, the directed Donaldson-Fukaya category is described and Theorem B is proved in \S10. Some details of proofs in this part (e.g. Lemma 7.3, Prop.8.2) are postponed to the last part. The third part deals with technical details used in previous parts. \S11 proves that relevant moduli spaces of holomorphic strips are in 1-1 correspondence with the gradient flow lines of certain functions. This is an extension to manifolds with boundaries of the standard correspondence defined for closed manifolds [\textit{A. Floer}, J. Differ. Geom. 30, No.~1, 207--221 (1989; Zbl 0678.58012) and used in the proof of Lemma 7.3. \S12 proves Prop. 8.2 by describing explicitly the moduli space of holomorphic triangles (inside a Weinstein neighborhood \(D(T^*V^j_2)\subset M\) of each \(V^j_2\)), which allow to use \(\widetilde V_j\)'s in the computations of such moduli spaces. \S13, the last section, reviews Morse-Bott-Lagrangian Floer homology, including explanations of applications of Fredholm theory to this homology (\S13.10).
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