Morse函数的复化与有向Donaldson-Killaya范畴
Complexifications of Morse functions and the directed Donaldson-Fukaya category
摘要：
设N是一个闭的四维流形，其中自索引Morse函数f只有3个临界值0，2，4和唯一的最大值和最小值.设g是N上的黎曼度量，使得（f，g）是Morse-Smale的.本文从（N，f，g）出发构造了一个六维精确辛流形M，以及M中的一些精确拉格朗日球面V_4，V_2^j，V_0，j=1，. K.这些球对应于f的临界点x_4、x_2 ^j、x_0，其中下标表示Morse指数。(In在另一篇文章中，我们解释了（M，V_4，{V_2 ^j}，V_0）是f的复化的正则纤维和消失球的模型，f被看作是圆盘余切丛D（T^*N）上的莱夫谢茨纤维化。本文的主要结果是计算Lagrange弗洛尔同调群HF（V_4，V_2^j），HF（V_2^j，V_0），HF（V_4，V_0）和三角积μ_2：HF（V_4，V_2^j）\otimes HF（V_2^j，V_0）--> HF（V_4，V_0）.结果是（M，V_4，{V_2^j}，V_0）的有向Donaldson-Fukaya范畴与（N，f，g）的流范畴同构。
英文摘要：
Let N be a closed four dimensional manifold which admits a self-indexing Morse function f with only 3 critical values 0,2,4, and a unique maximum and minimum. Let g be a Riemannian metric on N such that (f,g) is Morse-Smale. We construct from (N,f,g) a certain six dimensional exact symplectic manifold M, together with some exact Lagrangian spheres V_4, V_2^j, V_0 in M, j=1,...,k. These spheres correspond to the critical points x_4, x_2^j, x_0 of f, where the subscript indicates the Morse index. (In a companion paper we explain how (M, V_4,{V_2^j},V_0) is a model for the regular fiber and vanishing spheres of the complexification of f, viewed as a Lefschetz fibration on the disk cotangent bundle D(T^*N).) Our main result is a computation of the Lagrangian Floer homology groups HF(V_4,V_2^j), HF(V_2^j,V_0), HF(V_4,V_0) and the triangle product mu_2: HF(V_4,V_2^j) \otimes HF(V_2^j,V_0) --> HF(V_4,V_0). The outcome is that the directed Donaldson-Fukaya category of (M,V_4,{V_2^j},V_0) is isomorphic to the flow category of (N,f,g).
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