Complexifications of Morse functions and the directed Donaldson-Fukaya category

TL;DR

This paper constructs a symplectic manifold from a 4D manifold with a Morse function and computes its Floer homology, revealing an isomorphism between the Donaldson-Fukaya category and the flow category of the original manifold.

Contribution

It introduces a new symplectic construction linking Morse theory on 4-manifolds to Floer homology and the Donaldson-Fukaya category, establishing a categorical isomorphism.

Findings

Computed Floer homology groups for specific Lagrangian spheres.

Established the isomorphism between the directed Donaldson-Fukaya category and the flow category.

Demonstrated the symplectic model corresponds to the complexification of the Morse function.

Abstract

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).