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)