---
title: "Chapter 9"
---
Tags:
#geomtop/symplectic-topology #projects/my-talks
Refs:
[Projects/0000 Reading Notes/Audin-Damian Morse Theory and Floer Homology Reading Notes](Projects/0000%20Reading%20Notes/Audin-Damian%20Morse%20Theory%20and%20Floer%20Homology%20Reading%20Notes.md)
# Background, Notation, Setup
**Goals**
:::{.theorem title="Arnold Conjecture (Symplectic Morse Inequalities?)"}
Let $(W, \omega)$ be a compact symplectic manifold and
\[
H: W\to \RR
\]
a time-dependent Hamiltonian with nondegenerate 1-periodic solutions.
Then
\[
\size \ts{\text{1-Periodic trajectories of }X_H} \geq \sum_{k\in \ZZ} \dim_{?} HM_k(W; \, \ZZ/2\ZZ)
.\]
> Here $HM_*(W)$ is the Morse homology, and *nondegenerate* means the differential of the flow at time 1 has no fixed vectors.
:::
**Important Ideas for This Chapter**:
:::{.theorem title="Use Broken Trajectories to Compactify"}
$\mathcal{L}(x, y)$ is compact, where the compactification is given by adding in
\[
\bd \mathcal{L}(x, y) = \ts{\text{"Broken Trajectories"}}
\]
:::
:::{.theorem title="Gluing Yields a Chain Complex"}
\[
\del^2 = 0
\]
:::
\newpage
**Strategy**:
> In the background, have a Hamiltonian $H: W\to \RR$.
> Basic idea: cook up a gradient flow.
1. Define the action functional $\mathcal{A}_H$
> On an infinite-dimensional space, critical points are periodic solutions of $H$
2. Construct the chain complex (graded vector space) $CF_*$.
> Uses analog of the *index* of a critical point.
3. Define the vector field $X_H$ using $-\grad \mathcal{A}_H$.
> This will be used to define $\del$ later.
4. Count the trajectories of $X_H$
5. Show finite-energy trajectories connect critical points of $\mathcal{A}_H$.
6. Show *Gromov Compactness* for space of trajectories of finite energy
7. Define $\del$
> Uses another compactness property
8. Show space of trajectories is a manifold, plus analog of "Smale property"
9. \textbf{Show that} $\del^2 = 0$ using a gluing property
10. Show that $HF_*$ doesn't depend on $\mathcal{A}_H$ or $X_H$
11. Show $HF_* \cong HM_*$, and compare dimensions of the vector spaces $CM_*$ and $CF_*$.
\newpage
**Ingredients**:
- $(W, \omega, J)$ with $\omega \in \Omega^2(W)$ is a symplectic manifold
- With $J: T_p W \to T_p W$ an almost complex structure, so $J^2 = -\id$.
- $H \in C^\infty(W; \RR)$ a Hamiltonian
- $X_H$ the corresponding symplectic gradient.
- Defined by how it acts on tangent vectors in $T_x M$:
\[
\omega_x(\wait , X_H(x)) = (dH)_x(\wait)
.\]
- Zeros of vector field $X_H$ correspond to critical points of $H$:
\[
X_H(x) = 0 \iff (dH)_x = 0
.\]
- Take the associated flow, assumed 1-periodic:
\[
\psi^t \in C^\infty(W, W) \qquad \psi^1 = \id
,\]
- Critical points of $H$ are periodic trajectories.
- $u \in C^\infty(\RR\cross S^1; W)$ is a solution to the Floer equation.
- The Floer equation and its linearization:
\[
\mcf(u) &= \dd{u}{s} + J \dd{u}{t} + \grad_u(H) = 0 \\
\qty{d\mcf}_u(Y) &= \dd{Y}{s} + J_0 \dd{Y}{t} + S\cdot Y \\ \\ \\
&Y\in u^* TW,~ S \in C^\infty(\RR\cross S^1; \endo(\RR^{2n}))
.\]
\newpage
- $\mcl W$ is the *free loop space* on $W$, i.e. space of contractible loops on $W$, i.e. $C^\infty(S^1; W)$ with the $C^\infty$ topology
- Elements $x\in \mcl W$ can be viewed as maps $S^1\to W$.
- Can extend to maps from a closed disc, $u: \bar \DD^2 \to M$.
- Loops in $\mcl W$ can be viewed as maps $S^2\to W$, since they're maps $I\cross S^1\to W$ with the boundaries pinched:
- The action functional is given by
\[
\mca_H: \mcl W &\to \RR\\
x &\mapsto -\int_{\DD} u^* \omega + \int_0^1 H_t(x(t)) ~dt
\]
- Example: $W = \RR^{2n} \implies A_H(x) = \int_0^1 \qty{H_t ~dt - p~dq}$.
- A correspondence
\[
\correspond{\text{Solutions to the } \\\text{Floer equation}}
\iff
\correspond{\text{Trajectories} \\ \text{of } \grad \mathcal{A}_H}
.\]
- $x, y$ periodic orbits of $H$ (nondegenerate, contractible), equivalently critical points of $\mathcal{A}_H$.
\newpage
- Assumption of *symplectic asphericity*, i.e. the symplectic form is zero on spheres.
Statement: for every $u\in C^\infty(S^2, W)$,
\[
\int_{S^2} u^* \omega = 0 \qtext{or equivalently} \inner{\omega}{\pi_2 W} = 0
.\]
- Assumption of *symplectic trivialization*: for every $u\in C^\infty(S^2; M)$ there exists a symplectic trivialization of the fiber bundle $u^* TM$, equivalently
\[
\inner{c_1 TW}{\pi_2 W} = 0
.\]
> Locally a product of base and fiber, transition functions are symplectomorphisms.
- Maslov index: used the fact that
- Every path in $\gamma: I\to \Sp(2n, \RR)$ can be assigned an integer coming from a map $\tilde \gamma: I \to S^1$ and taking (approximately) its winding number.
- $\mathcal{M}(x, y)$, the moduli space of contractible finite-energy solutions to the Floer equation connecting $x, y$.
- After perturbing $H$ to get transversality, get a manifold
- Dimension:
\[
\dim \mcm(x, y) = \mu(x) - \mu(y)
.\]
- How we did it:
- Describe as zeros of a section of a vector bundle over $\mathcal{P}^{1, p}(x, y)$
(Banach manifold modeled on the Sobolev spaces $W^{1, p}$),
- Apply Sard-Smale to show $\mathcal{M}(x, y)$ is the inverse image of a regular value of some map.
- Needed tangent maps to be Fredholm operators, proved in Ch. 8 and used to show transversality.
- Showed $(d\mathcal{F})_u$ is a Fredholm operator of index $\mu(x) - \mu(y)$.
$\qed$
\newpage
# Reminder of Goals
**Overall Goal**:
:::{.theorem title="Symplectic Morse Inequalities"}
\[
\size \ts{\text{1-Periodic trajectories of }X_H} \geq \sum_{k\in \ZZ} HM_k(W; \, \ZZ/2\ZZ)
.\]
:::
---
**Important Ideas for This Chapter**:
:::{.theorem title="Using Broken Trajectories to Compactify"}
$\mathcal{L}(x, y)$ is compact,
\[
\bd \mathcal{L}(x, y) = \ts{\text{"Broken Trajectories"}}
\]
:::
:::{.theorem title="Using Gluing to Get a Chain Complex"}
\[
\del^2 = 0
\]
:::
\newpage
# 9.1 and Review
- Defined moduli space of (parameterized) **solutions**:
\[
\mathcal{M}(x, y) &= \ts{\text{Contractible finite-energy solutions connecting }x, y }
\\ \\
\mcm &= \ts{\text{\textbf{All} contractible finite-energy solutions to the Floer equation}} \\
&= \bigcup_{x, y} \mathcal{M}(x, y)
.\]
- The moduli space of (unparameterized) **trajectories** connecting $x, y$:
\[
\mathcal{L}(x, y) \da \mathcal{M}(x, y) / \RR
.\]
- Use the quotient topology, define sequentially:
\[
\tilde u_n \converges{n\to\infty}\too \tilde u
\quad\iff\quad
\exists \ts{s_n}\subset \RR \text{ such that } u_n(s_n + s, \wait) \converges{n\to\infty}\too u(s, \wait)
.\]
- When $\abs{\mu(x) - \mu(y)} = 1$, get a compact 0-manifold, so the number of trajectories $$n(x, y) \da \size \mathcal{L}(x, y)$$ is well-defined.
- $C_k(H) \da \ZZ/2\ZZ[\ts{\text{Periodic orbits of } X_H \text{ of Maslov index } k }]$.
- Finitely many since they are nondegeneracy implies they are isolated.
:::{.remark}
Some notation:
\begin{center}
\begin{tikzcd}
\RR \ar[r] & \mathcal{M}(x, z)\ar[d, "\pi"] \\
& \mathcal{L}(x, z) \\
\end{tikzcd}
\end{center}
Hats will generally denote maps induced on quotient.
:::
- Defined a differential
\[
\del: C_k(H) &\to C_{k-1}(H) \\
x &\mapsto \sum_{\mu(y) = k-1} n(x, y) y \\ \\
n(x, y) &\da \size \ts{\text{Trajectories of } \grad \mathcal{A}_H \text{ connecting } x, y} \mod 2 \\
&= \size \mathcal{L}(x, y) \mod 2
.\]
- Examined $\del^2$:
\[
\del^2: C_{k}(H) &\to C_{k-2}(H) \\
x &\mapsto \del(\del(x)) \\
&= \del \qty{\sum_{\mu(y) = \mu(x)-1} n(x, y) y} \\ \\
&= \sum_{\mu(y) = \mu(x) - 1} n(x, y) \del(y) \\ \\
&= \sum_{\mu(y) = \mu(x) - 1} n(x, y) \qty{\sum_{\mu(z) = \mu(y)-1} n(y, z) z} \\ \\
&= \sum_{\mu(y) = \mu(x) - 1} \,\,\sum_{\mu(z) = \mu(y)-1} n(x, y) n(y, z) \,z \\
&= \sum_{\mu(z) = \mu(y) - 1} \qty{\sum_{\mu(y) = \mu(x)-1} n(x, y) n(y, z)}\,z \hspace{4em} \text{(finite sums, swap order)}
,\]
so it suffices to show
\[
\sum_{\mu(y) = \mu(x)-1} n(x, y) n(y, z) = 0 \qtext{when} \mu(z) = \mu(x) - 2
.\]
> Easier to examine parity, so we'll show it's zero mod 2.
\newpage
- When $\mu(z) = \mu(x) - 2$, $\mathcal{L}(x, z)$ is a non-compact 1-manifold, so we compactify by adding in *broken trajectories* to get $\bar{\mathcal{L}}(x, y)$.
- We'll then have
\[
\bar{\mathcal{L}}(x, z) = \mathcal{L}(x, z) \union \bd \bar{\mathcal{L}}(x, z), \qquad \bd \bar{L}(x, z) = \bigcup_{\mu(y) = \mu(x) - 1} \mathcal{L}(x, y) \cross \mathcal{L}(y, z)
,\]
which "space-ifies" the equation we want.
- We'll show $\bd \bar{\mathcal{L}}(x, z)$ is a 1-manifold, which must have an even number of points, and thus
\[
\sum_{\mu(y) = \mu(x)-1} n(x, y) n(y, z) =
\size \qty{\bd \bar{\mathcal{L}}(x, z)} \equiv 0 \mod 2
.\]
> Image here of relations between spaces!
$\qed$
\newpage
# Three Important Theorems
## First Theorem: Convergence to Broken Trajectories
- Recall: *broken trajectories* are unions of intermediate trajectories connecting intermediate critical points.
- Shown last time: a sequence of trajectories can converge to a broken trajectory, i.e. there are broken trajectories in the closure of $\mathcal{L}(x, z)$.
- This theorem describes their behavior:
:::{.theorem title="9.1.7: Convergence to Broken Trajectories"}
Let $\ts{u_n}$ be a sequence in $\mathcal{M}(x, z)$, then there exist
\
- A subsequence $\ts{u_{n_j}}$
\
- Critical points $\ts{x_0, x_1, \cdots, x_{\ell+1}}$ with $x_0=x$ and $x_{\ell+1} = z$
\
- Sequences $\ts{s_n^1}, \ts{s_n^2}, \cdots, \ts{s_n^\ell}$.
\
- Elements $u^k \in \mathcal{M}(x_k, x_{k+1})$ such that for every $0\leq k \leq \ell$,
\[
u_{n_j} \cdot s_n^k \converges{n\to\infty}\too u^k
.\]
:::
- Upshots:
- Every sequence upstairs has a subsequence which (after reparameterizing) converges
- This descends to actual convergence after quotienting by $\RR$?
- Yields uniqueness of limits in $\mathcal{L}(x, z)$, thus a separated topology
- Sequentially compact $\iff$ compact since $\mathcal{L}(x, z)$ is a metric space?
:::{.corollary title="Compactness"}
$\bar{\mathcal{L}}(x, z)$ is compact.
:::
\newpage
## Second Theorem: Compactness of $\bar{\mathcal{L}}(x, z)$
:::{.definition title="Regular Pair"}
For an almost complex structure $J$ and a Hamiltonian $H$, the pair $(H, J)$ is **regular** if the Floer map $\mathcal{F}$ is transverse to the zero section in the following vector bundle:
\
\begin{center}
\begin{tikzcd}
E_u \da \ts{\text{Vector fields tangent to $M$ along $u$}} \ar[r] & C^\infty(\RR\cross S^1; TM)\ar[dd] \\
& \\
& C^\infty(\RR\cross S^1; M) \ar[uu, bend left, "\mathcal{F}", dotted] \ar[uu, bend right, "\mathbf{0}"', dotted]\\
\end{tikzcd}
\end{center}
:::
Most of chapter 9 is spent proving this theorem:
:::{.theorem title="9.2.1"}
Let $(H, J)$ be a regular pair with $H$ nondegenerate and $x, z$ be two periodic trajectories of $H$ such that
\[
\mu(x) = \mu(z) + 2
.\]
Then $\bar{\mathcal{L}}(x, z)$ is a compact 1-manifold with boundary with
\[
\bd \bar{\mathcal{L}}(x, z) = \bigcup_{y\in \mathcal{I}(x, z)} \mathcal{L}(x ,y) \cross \mathcal{L}(y, z) \\
\text{where}\qquad \mathcal{I}(x, z) =
\ts{
y \st \mu(x) < \mu(y) < \mu(z)
}
.\]
> Note: possibly a typo in the book? Has $x, y$ on the LHS.
:::
:::{.corollary}
\[\bd^2 = 0.\]
:::
\newpage
## Third Theorem: Gluing
:::{.theorem title="9.2.3: Gluing"}
Let $x,y,z$ be three critical points of $\mathcal{A}_H$ with three consecutive indices
\[
\mu(x) = \mu(y)+1 = \mu(z) + 2
.\]
and let
\[
(u, v) \in \mathcal{M}(x, y) \cross \mathcal{M}(y, z) \quad\leadsto\quad (\hat u, \hat v)\in \mathcal{L}(x, y) \cross \mathcal{L}(y, z)
.\]
Then
1. There exists a $\rho_0 > 0$ and a differentiable map
\[
\psi: [\rho_0, \infty) &\to \mathcal{M}(x, z)
\]
such that $\hat \psi$, the induced map on the quotient
\begin{center}
\begin{tikzcd}
{[\rho_0, \infty)} \ar[r, "\psi"] \ar[rd, "\hat\psi"', dotted] & \mathcal{M}(x, z)\ar[d, "\pi"] \\
& \mathcal{L}(x, z) \\
\end{tikzcd}
\end{center}
is an embedding that satisfies
\[
\hat\psi(\rho) \converges{\rho\to\infty}\too (\hat u, \hat v) \in \bar{\mathcal{L}}(x, z)
.\]
\newpage
2. ("Uniqueness") For any sequence $\ts{\ell_n}\subseteq \mathcal{L}(x, z)$,
\[
\ell_n \converges{n\to\infty}\too (\hat u, \hat v) \quad\implies\quad \ell_n \in \im(\hat \psi) \text{ for } n \gg 0
.\]
:::
- We already know that $\bar{\mathcal{L}}(x, z)$ is compact and $\mathcal{L}(x, z)$ is a 1-manifold, so we look at neighborhoods of boundary points.
- Why unique: will show that the broken trajectory $(\hat u, \hat v)$ is the endpoint of an embedded interval in $\bar{\mathcal{L}}(x, z)$.
- Then show that any other sequence converging to $(\hat u, \hat v)$ must approach via this interval, otherwise could have cuspidal points:
$\qed$
\newpage
# Gluing Theorem
Broken into three steps:
1. **Pre-gluing**:
- Get a function $w_\rho$ which interpolates between $u$ and $v$ in the parameter $\rho$.
- Not exactly a solution itself, just an "approximation".
2. **Newton's Method**:
- Apply the Newton-Picard method to $w_p$ to construct a true solution
\[
\psi: [-\rho, \infty) &\to \mathcal{M}(x, z) \\
\rho &\mapsto \oldexp_{w_p}\qty{\gamma(p)} \\ \\
\text{for some } \gamma(p) &\in W^{1, p}(w_p^*\, TW) = T_{w_p} \mathcal{P}(x, z)
\]
- [GIF of Newton's Method](https://www.maplesoft.com/support/help/content/4702/plot552.gif)
3. **Project and Verify Properties**:
- Check that the projection $\hat \psi = \pi \circ \psi$ satisfies the conditions from the theorem.
$\qed$
\newpage
# 9.3: Pre-gluing, Construction of $w_\rho$
- Choose (once and for all) a bump function $\beta$ on $B_{\eps}(0)^c \subset \RR \to [0, 1]$ which is 1 on $\abs{x} \geq 1$ and $0$ on $\abs{x} < \eps$
- Split into positive and negative parts $\beta^\pm(s)$:
- Define an interpolation $w_\rho$ from $u$ to $v$ in the following way: let
- $\oldexp \left[ \wait \right] \da \oldexp_{y(t)}(\wait)$ and
- $\ln(\wait) \da \oldexp\inv_{y(t)}(\wait)$,
then
\[
w_\rho: x &\to z \\
w_\rho(s,t) &\da
\begin{cases}
{\color{blue} u(s+\rho, t)} & s\in (-\infty, -1] \\ \\
\oldexp\left[ \beta^-(s)\ln({\color{blue} u(s+\rho, t) }) + \beta^+(s)\ln({\color{purple} u(s-\rho, t)} )\right] & s\in [-1, 1] \\ \\
{\color{purple} u(s-\rho, t)} & s\in [1, \infty)
\end{cases}
.\]
- Why does this make sense?
\[
\abs{s}\leq 1 \implies u(s\pm \rho, t) \in
\ts{\oldexp_{y(t)} Y(t) \st \sup_{t\in S^1} \norm{Y(t)}\leq r_0 } \subseteq \im \oldexp_{y(t)} (\wait)
,\]
so we can apply $\oldexp_{y(t)}\inv(\wait)$.
- Can make $\abs{s}\leq 1$ for large $\rho$, since
\[
u(s, t) \converges{s\to\infty}\too \quad &y(t) \\
v(s, t) \converges{s\to-\infty}\too \quad &y(t)
.\]
- So pick a $\rho_0$ such that this holds for $\rho > \rho_0$.
- Might have to increase $\rho_0$ later in the proof, so $\rho > \rho_0$ just means $\rho \gg 0$.
- Some properties:
- $w_\rho \in C^\infty(x, z)$ and is differentiable in $\rho$.
- $s\in [-\eps, \eps] \implies w_\rho(s, t) = y(t)$.
\[
w_\rho(s-\rho, t) &\converges{\rho\to\infty}\too u(s, t)\qtext{in} C^\infty_\loc \\ \\
w_\rho(s, t) &\converges{\rho\to\infty}\too y(t) \qtext{in} C^\infty_\loc
.\]
- Now carry out the linearized version on tangent vectors, to which we will apply Newton-Picard:
- Let $Y\in T_u \mathcal{P}(x, y)$
- Let $Z\in T_v \mathcal{P}(x, y)$
- Replace $w_\rho$ with the interpolation
\[
Y\size_\rho Z \in T_{w_\rho} \mathcal{P}(x, y) = W^{1, p}(w_\rho^* TW)
.\]
defined by
\[
(Y\size_\rho Z) (s, t) =
\begin{cases}
{\color{blue} Y(s+\rho, t)} & s\in (-\infty, -1] \\ \\
\oldexp_T\left[ \beta^-(s) \ln_T({\color{blue} Y(s+\rho, t) }) \, + \beta^+(s) \ln_T({\color{purple} Z(s-\rho, t)} )\right] & s\in [-1, 1] \\ \\
{\color{purple} Z(s-\rho, t)} & s\in [1, \infty)
\end{cases}
,\]
where the subscript $T$ indicates taking tangents of the exponential maps at appropriate points.
$\qed$
\newpage
# 9.4: Construction of $\psi$.
## Summary
- Newton-Picard method, general idea:
- Allows finding zeros of $f$ given an approximate zero $x_0$, using the extra information of the 1st derivative $f'$.
- Original method and variant: find the limit of a sequence
\[
x_{n+1} = x_n - {f(x_n) \over f'(x_n)}
,\qquad x_{n+1} = x_n - {f(x_n) \over f'({\color{red} x_0} )}
.\]
- Second variant more useful: only need derivative at one point:
\newpage
- Pregluing function $w_\rho \in C^\infty_{\searrow}(x, z)$ from previous section
- Exponential decay
- Want to construct true solution $\psi_\rho \in \mathcal{M}(x, z)$, so $\mathcal{F}(\psi_p) = 0$.
- Suffices to get a weak solution
- Automatic continuity + elliptic regularity $\implies$ strong solution
- Define $\mathcal{F}_\rho$ as $\mathcal{F} \circ \oldexp_{w_\rho}$ expanded bases $Z_i$ from trivialization of $TW$.
- $L_\rho = (d\mathcal{F}_\rho)_0$ will be the linearization of the Floer operator at zero.
\vspace{5em}
- Adapting Newton-Picard to operators:
- $L_\rho$ won't be invertible on entire space, but
\[ {1\over f'(x_0)} \iff L_\rho^{-1}, \]
- Decompose
\[
T_{w_\rho} \mathcal{P}(x, z)
= W^{1, p}(w_\rho^* TW)
= W^{1, p}(\RR\cross S^1; \RR^{2n})
= \ker(L_{\rho}) \oplus W_{\rho}\perp,
\]
where $L_\rho$ will have a right inverse on $W_\rho\perp$.
- Where does $W_\rho\perp$ come from?
Essentially the kernel of some linear functional given by an integral:
\[
W_\rho\perp \da \ts{Y\in W^{1, p} \st \int_{\RR\cross S^1} \inner{Y}{\cdots}\,ds\,dt = 0,\, \text{ plus conditions}}
.\]
- Run Newton-Picard in $W_{\rho}\perp$
- Will obtain for every $\rho \geq \rho_0$ an element $\gamma(\rho) \in W_\rho\perp$ with
\[
\mathcal{F}_\rho(\gamma(\rho)) = 0
.\]
\newpage
- Where does $\gamma$ come from?
Intersection-theoretic interpretation on page 320:
\[
\qty{\oldexp_{w_\rho}}\inv \mathcal{M}(x, z) \intersect W_{\rho}\perp &\subseteq T_{w_\rho}\mathcal{P}(x, z) &\leadsto \gamma
\\
\mathcal{M}(x, z) \cap \ts{\oldexp_{w_\rho} W_\rho\perp \st \rho\geq \rho_0}&\subseteq \mathcal{P}(x, z) &\leadsto \psi(\rho)
,\]
which we get by exponentiating.
- This gives a codimension 1 subspace in $\mathcal{M}(x, z)$, which we take to be $\psi(\rho)$:
\newpage
> Schematic picture here for $\gamma, \psi(\rho)$.
\newpage
- Apply the implicit function theorem to show differentiability of $\gamma$ in $\rho$.
- Use a trivialization $Z_i^\rho$ of $TW$ to get a vector field along $w_\rho$
- This is exactly an element of $T_{w_\rho}\mathcal{P}(x, z)$
- Exponentiate to get an element of $\mathcal{M}(x, z)$:
\[
\psi(\rho) \da \oldexp_{w_\rho}\qty{\gamma(\rho)}
.\]
- **Final Result**: project onto $\mathcal{L}(x, z)$ to get $\hat \psi$.
\vspace{4em}
**Checking Properties**:
- Existence: show $\hat \psi$ is a proper injective immersion $\implies$ embedding.
- Uniqueness: show the broken trajectory $(\hat u, \hat v)$ is the endpoint of an embedded interval in $\bar{\mathcal{L}}(x, z)$.
- Show that any other sequence converging to $(\hat u, \hat v)$ must approach via this interval, otherwise could have cuspidal points:
> Probably not worth going farther than this! Extremely detailed analysis.
$\qed$
\newpage