What does a moduli space look like?

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:
Moduli_open
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:
Moduli_compactified

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:
Moduli_dimension2_Labeled
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.

Morse_Heegaard

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.

Heegaard_torusS3

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:
Moduli_torusS3
(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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