# Morse Homology and Lagrangian Floer Homology (Thursday, January 28) ## Morse Homology :::{.remark} Last time: defined the Morse complex. Assumed $(f, g)$ was a Morse-Smale pair, where $f$ is a Morse function and $g$ is a Riemannian metric, and this guarantees that if $p, q\in \crit(f)$ with $\ind(p) - \ind(q) = 1$, then (among other things) there are finitely many gradient trajectories $p\leadsto q$. We denoted this \( \mathcal{M}(p, q) \). The chain complex was defined by $C_i(f, g) \da \bigoplus_{\ind(p) = i} \ZZ_2 \gens{ p }$ with differential $\bd_i: C_i \to C_{i-1}$ was defined by sending an index $i$ critical point $p$ to $\sum_{\ind(q) = i-1} \# \mathcal{M}(p, q) q \mod 2$. ::: :::{.theorem title="The Morse Complex is a Chain Complex"} $\bd_{i} \circ \bd_{i+1} = 0$. ::: :::{.proof title="?"} Idea of the proof: we can directly compute \[ \bd(\bd p) &= \bd \qty{ \sum_{\ind(q) = i-1} \# \mathcal{M}(p, q) q } \\ &= \sum_{\ind(q) = i-1} \# \mathcal{M}(p, q) \bd q \\ &= \sum_{\ind(q) = i-1} \# \mathcal{M}(p, q) \qty{ \sum_{\ind(r) = i-2 \# \mathcal{M}(q, r) r }} \\ &= \sum_{\ind(r) = i-2} \qty{\sum_{\ind(q) = i-1} \# \mathcal{M}(p, q) \# \mathcal{M}(q, r) } r \\ &= \sum_{\ind(r) = i-2} c_{p,q,r} r \\ &= 0 && \text{(claim)} .\] This happens if and only if $c_{p, q, r} = 0 \mod 2$ for all $r$ with $\ind(r) = i-2$. This is multiplication of the number of trajectories: ![image_2021-01-28-11-23-19](figures/image_2021-01-28-11-23-19.png) In other words, this is the total number of trajectories $p\leadsto r$ that pass through $q$. These trajectories "break" at $q$, and so we refer to these as **broken trajectories**. ::: :::{.definition title="Broken Trajectories"} Suppose $\ind(r) = \ind(p) - 2$, then a **broken trajectory** from $p$ to $r$ is a trajectory from $p$ to $q$ followed by a trajectory $q$ to $r$ where $\ind(q) = \ind(p)-1 = \ind(r) + 1$. ![image_2021-01-28-11-26-25](figures/image_2021-01-28-11-26-25.png) ::: :::{.question} Why is the number of broken trajectories even? ::: :::{.answer} We can check that $\dim \mathcal{M}(p, r) = \dim \qty{ W^u(p) \transverse W^s(r)}/\RR = (\ind(p) - \ind(r)) - 1 = 2-1 = 1$. We can compactify \( \mathcal{M}(p, r) \) by adding in all of the broken trajectories to define \[ \overline{\mathcal{M}(p, r)} \union \qty{ \Union_{\ind(q) = i-1} \mathcal{M}(p, q) \cross \mathcal{M}(q, r) } .\] This is useful here because we can appeal to the classification of smooth compact 1-dimensional manifolds, which are unions of copies of $S^1$ and $D_1 = I$. In particular, the number of boundary points \[ \bd \overline{\mathcal{M}(p, r)} = \Union_{\ind(q) = i-1} \mathcal{M}(p, q) \cross \mathcal{M}(q, r) \] is even: ![image_2021-01-28-11-32-34](figures/image_2021-01-28-11-32-34.png) ::: :::{.example title="Morse Homology of the Torus"} Suppose you have two critical points of the same index. The Morse-Smale condition implies that there's no trajectory between them. A counterexample would be $p_3 \leadsto p_2$ on the torus with the height function: ![image_2021-01-28-11-45-16](figures/image_2021-01-28-11-45-16.png) However, if you perturb this slightly, the trajectories can be made to miss $p_2$ and end at $p_1$ instead. All of the trajectories are disjoint, so we end up with a situation like the following after perturbing the metric: ![image_2021-01-28-11-48-06](figures/image_2021-01-28-11-48-06.png) We can cut along a curve on the bottom to better analyze these trajectories: ![image_2021-01-28-11-48-57](figures/image_2021-01-28-11-48-57.png) ![image_2021-01-28-11-50-27](figures/image_2021-01-28-11-50-27.png) Now cut this cylinder along the trajectories $p_1\leadsto p_3 \leadsto p_1$, i.e. the green trajectories here: ![image_2021-01-28-11-51-31](figures/image_2021-01-28-11-51-31.png) ![image_2021-01-28-11-53-32](figures/image_2021-01-28-11-53-32.png) Here we can see that as the trajectories approach the corners, they limit to broken trjacetories: ![image_2021-01-28-11-54-44](figures/image_2021-01-28-11-54-44.png) We can compute - $C_0 = \ZZ/2\ZZ \gens{ p_1 }$ - $C_1 = \ZZ/2\ZZ \gens{ p_2, 3 }$ - $C_2 = \ZZ/2\ZZ \gens{ p_4 }$ Since there are exactly two trajectories $p_4$ to $p_2$ or $p_3$, we get $\bd_2 = 0$. Similarly $\bd_1 = 0$, and we get $HM_i(T) = [\ZZ/2\ZZ, \ZZ/2\ZZ^2, \ZZ/2\ZZ, 0, \cdots]$, which is the same as its singular homology. ::: :::{.theorem title="?"} \[ HM_i(f, g) \cong H_i^{\sing}(M; \ZZ/2\ZZ) .\] In particular, it doesn't depend on the choice of Morse-Smale pair $(f, g)$. See proof in references, e.g. Audin. ::: :::{.proof title="?"} By definition, $\# \crit_i(f) = \rank C_i(f, g) = \rank HM_i(f, g)$, and in any chain complex the rank of the chain groups are always at least the rank of the homology. ::: ## Lagrangian Floer Homology :::{.remark} Suppose $L_0^n, L_1^n \subset M^{2n}$ are compact with $L_0 \transverse L_1$, so the intersection is finitely many points. ![image_2021-01-28-12-16-27](figures/image_2021-01-28-12-16-27.png) We can do Morse theory on the space of paths between them: \[ \mathcal{P}(L_0, L_1) \da \ts{ \gamma: I\to M \st \gamma(0) \in L_0, \gamma(1) \in L_1} .\] We'll find analogs of Morse functions on $P(L_0, L_1)$ such that the critical points are constant paths, i.e. $L_0 \intersect L_1$. The Morse inequalities then gives bounds on the number of intersection points between $L_0$ and $L_1$. ::: :::{.definition title="Symplectic Manifolds"} A **symplectic manifold** is a pair $(M^{2n}, \omega)$ with \( \omega \) a 2-form which is - Closed, i.e. $d \omega = 0$, and - Nondegenerate, i.e. $\Wedge^n \omega \neq 0$. ::: :::{.definition title="Lagrangian Submanifolds"} A half-dimensional submanifold $L^n \subset M^{2n}$ is called **Lagrangian** if \( \ro{ \omega}{L^n} = 0 \). ::: :::{.example title="?"} The pair $(\RR^{2n}, \sum_{i=1}^n dx_i \wedge dy_i$ is a symplectic manifold (and also a symplectic vector space). Note that this 2-form is also a bilinear form of the following shape: \[ \begin{bmatrix} 0 & \id_n \\ -\id_n & 0 \end{bmatrix} .\] This has a Lagrangian submanifold $\RR^n \da \ts{y_1 = \cdots = y_n = 0}$. > Note: See Darboux theorem. ::: :::{.remark} The general setup for next time: we'll have $(M^{2n}, \omega)$ a symplectic manifold, a pair $L_0, L_1 \subset M$ such that $L_0 \transverse L_1$, and we want to do Morse Homology on \( \mathcal{P}(L_0, L_1) \). :::