I’d like to eventually explain Fukaya’s result that the Morse cochain complex admits an A- structure because I think it will help in understanding what an A- algebra/module/category is. But first, I need to talk a little more about compactification of moduli spaces. As you may recall, compactification of 1-dimensional moduli spaces by broken flow lines is the reason in Morse homology and any Floer homology setting.
Here I’ll restrict to talking about moduli of gradient trajectories of a Morse function. Let (f,g) be a Morse-Smale pair of a function and Riemannian metric . If are critical points of , then the space
is the moduli of gradient flow lines from to , modulo translation. Its expected dimension is where denotes the Morse index of a critical point.
Suppose , then is a 1-dimensional manifold. In other words, a union of open line segments:
In fact, it’s possible to compactify these space by adding limit points at the end corresponding to broken flows. Given any sequence of flow lines , a subsequence will converge to a pair of flow lines, where and for some critical point of index . Thus, the open line segments can be though of a closed segments:
Now, let’s go up a dimension and suppose that . Then is an open polygon:
This too can be compactified by adding in broken flow lines. However, a flow line can “break” at a critical point of index or a critical point of index or both. The points in the open line segments of the boundary are given by flow lines in or . Thus, each line segment of the boundary corresponds to a pair or of a rigid flow line and a connected component of a 1-dimensional moduli space. Each vertex is given by a triple of flow lines , where is the broken trajectory compactifying at that end and is the broken trajectory compactifying at that end.
The key insight here is that all moduli spaces of Morse trajectories can be compactified by tuples of lower-dimensional moduli. In other words, flow lines degenerate but only to flow lines you already know about, in some sense. Then, based upon this insight, it’s possible to establish algebraic relations in cochain complexes because these compactifications describe how all the moduli spaces are related to one another.
Now, an example. Let be a 3-manifold and choose to be a self-indexing Morse function so that if is a critical point of index , then and furthermore assume there is a unique critical point of index 3 and a unique critical point of index 0. Using this function, it’s possible to find a Heegaard diagram encoding the 3-manifold as follows. First, the Heegaard surface is given as the level set . Label the index 1 critical points and the index 2 critical points . Then the ascending/stable manifold of an index 1 critical point is a 2-dimensional disk, which intersects the Heegaard surface transversely in a closed loop, denoted . Similarly, the descending/unstable manifold of an index 2 critical point is a 2-dimensional disk, which intersects the Heegaard surface transversely in a closed loop, denoted .
Every noncritical point lies on a unique flow line and contains no critical points. By the above discussion, it’s clear that , since corresponds to the flow lines descending to and corresponds to flow lines descending from . Furthermore, and . Finally, .
Thus, you can “see” the moduli spaces from a Heegaard diagram and understand how they can be compactified. As a concrete example, lets take the genus 1 Heegaard diagram for .
Then, the manifold is a solid torus and let be a meridian and longitude, respectively. This solid torus contains an index 0 critical point and an index 1 critical point . Flow lines descending to fill the space, connecting to the Heegaard surface. However, there are also two flow lines connecting to . The moduli of flow lines from to is parametrized by and as such a flow line approaches in the moduli space, it degenerates to the union of some and a flow line from to . Finally, there exists a single flow line from to , given by the intersection . We can assume that there exist exactly 2 flow lines connecting to and so the compactified moduli space of flows from to looks like:
(Remember, this was a discussion of Morse theory in the setting of a Heegaard diagrams, not a discussion of moduli of holomorphic curves in Heegaard Floer theory)