December | 2012 | The Electric Handle Slide

I’d like to eventually explain Fukaya’s result that the Morse cochain complex admits an A- $\infty$ structure because I think it will help in understanding what an A- $\infty$ 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 $\partial^2 = 0$ 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 $f:M \rightarrow \mathbb{R}$ and Riemannian metric $g$ . If $p,q$ are critical points of $f$ , then the space

$\widetilde{\mathcal{M}}(p,q) :=$

$\{\gamma:\mathbb{R} \rightarrow M;$

$\lim_{t \rightarrow -\infty} = p,$

$\lim_{t \rightarrow \infty} = q,$

$\nabla (f) (\gamma(t)) = \gamma ' (t)\} \forall t \in \mathbb{R}$

is the moduli of gradient flow lines from $p$ to $q$ , modulo translation. Its expected dimension is $I(p) - I(q) - 1$ where $I$ denotes the Morse index of a critical point.

Suppose $I(p) - I(q) - 1 = 1$ , then $\widetilde{\mathcal{M}}(p,q)$ 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 $\{\gamma_{v}\} \in \widetilde{\mathcal{M}}(p,q)$ , a subsequence will converge to a pair $(\alpha_1, \alpha_2)$ of flow lines, where $\alpha_1 \in \widetilde{\mathcal{M}}(p,r)$ and $\alpha_2 \in \widetilde{\mathcal{M}}(r,q)$ for some critical point $r$ of index $I(r) = I(p) - 1 = I(q) + 1$ . Thus, the open line segments can be though of a closed segments:

Now, let’s go up a dimension and suppose that $I(p)-I(q)-1 = 2$ . Then $\widetilde{\mathcal{M}}(p,q)$ 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 $r$ of index $I(r) = I(p) - 1 = I(q)+2$ or a critical point $s$ of index $I(s) = I(p) - 2 = I(q)+1$ or both. The points in the open line segments of the boundary are given by flow lines in $\widetilde{\mathcal{M}}(p,r) \times \widetilde{\mathcal{M}}(r,q)$ or $\widetilde{\mathcal{M}}(p,s) \times \widetilde{\mathcal{M}}(s,q)$ . Thus, each line segment of the boundary corresponds to a pair $(\alpha, \widetilde{\mathcal{M}}(r,q))$ or $(\widetilde{\mathcal{M}}(p,s), \beta)$ 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 $(\alpha, \gamma, \beta)$ , where $(\alpha,\gamma)$ is the broken trajectory compactifying $\widetilde{\mathcal{M}}(p,s)$ at that end and $(\gamma,\beta)$ is the broken trajectory compactifying $\widetilde{\mathcal{M}}(r,q))$ 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 $M$ be a 3-manifold and choose $f$ to be a self-indexing Morse function so that if $c$ is a critical point of index $i$ , then $f(c) = i$ and furthermore assume there is a unique critical point $c$ of index 3 and a unique critical point $d$ 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 $f^{-1}(3/2)$ . Label the index 1 critical points $a_1,\dots,a_k$ and the index 2 critical points $b_1,\dots,b_k$ . Then the ascending/stable manifold $W_s(a_i)$ of an index 1 critical point is a 2-dimensional disk, which intersects the Heegaard surface transversely in a closed loop, denoted $\alpha_i$ . Similarly, the descending/unstable manifold $W_u(b_i)$ of an index 2 critical point is a 2-dimensional disk, which intersects the Heegaard surface transversely in a closed loop, denoted $\beta_i$ .

Every noncritical point lies on a unique flow line and $H$ contains no critical points. By the above discussion, it’s clear that $\widetilde{\mathcal{M}}(b_i,a_j) = \beta_i \cap \alpha_j$ , since $\alpha_j$ corresponds to the flow lines descending to $a_j$ and $\beta_i$ corresponds to flow lines descending from $b_i$ . Furthermore, $\widetilde{\mathcal{M}}(c,a_i) = \alpha_i - \beta$ and $\widetilde{\mathcal{M}}(b_i,d) = \beta_i - \alpha$ . Finally, $\widetilde{\mathcal{M}}(c,d) = H - \{\alpha \cup \beta\}$ .

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 $S^3$ .

Then, the manifold $f^{-1}([0,3/2])$ is a solid torus and let $\alpha, \beta$ be a meridian and longitude, respectively. This solid torus contains an index 0 critical point $d$ and an index 1 critical point $a$ . Flow lines descending to $d$ fill the space, connecting $d$ to the Heegaard surface. However, there are also two flow lines $\gamma_1, \gamma_2$ connecting $d$ to $a$ . The moduli of flow lines from $c$ to $d$ is parametrized by $T^2 - \{\alpha \cup \beta\}$ and as such a flow line approaches $\alpha$ in the moduli space, it degenerates to the union of some $\gamma_i$ and a flow line from $c$ to $a$ . Finally, there exists a single flow line $\epsilon$ from $b$ to $a$ , given by the intersection $\alpha \cap \beta$ . We can assume that there exist exactly 2 flow lines $\delta_1,\delta_2$ connecting $c$ to $b$ and so the compactified moduli space of flows from $c$ to $d$ 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)

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