IMPA conference: Mihai Damien

Mihai discussed some results about the topology of Lagrangian submanifolds and pointed out the difference between monotone (rigid) and the flexible results of Yasha, Tobias, Ivan and Emmy. His key method: Lifted Floer homology.

Morse Theory

Want to do Morse theory on the universal cover of a manifold. Have an infinite dimensional complex. Define $d \tilde x = \sum \tilde y$, where the sum is over lifts of gradient lines. If you fix one lift $\tilde x$ in each class, then you can reinterpret this as a complex generated by critical points but now over $R[ \pi_1(L) ]$ ($R$ is the base ring.)

Floer theory

Look at image of $u(s,0)$ in differential to tell us how to interpret going between two different lifts of a given intersection point of $L, L'$. Problem: if $N_L =2$, $d^2 \ne 0$ in this case. Can bubble disks. The issue is then that
(Thm, Oh) $$ d^2 x = \sum (disks) x$$, where we consider the count of Maslov 2 disks passing through $x$.

*Prop* (Oh): L monotone, $N_L = 2$, and $L, L'$ Hamiltonian isotopic. Then, the number passing through $x$ on $L$ and the number on $L'$ passing through $x$ have the same parity.

Therefore, at least $mod 2$, we have $d^2 = 0$, at least in classical Floer homology.

 In this lifted picture on the universal cover, we have some trouble:
If $g \in \pi_1(L, x)$ has $g = [w( \partial D]$, then this disk on $L$ counts as $g$ and a similar disk on $L'$ counts as 1. Thus, $$d^2(\tilde x) = \sum_{g \in \pi_1(L) } (1+g) \#_2^g(x).$$
 And where $\#_2^g(x)$ is the [mod 2] number of Maslov 2 disks whose boundaries are in homotopy class $g$.
 In particular then, have well-defined if $N_L \ge 3$.

 If not defined: Fix some $g$ so $\#_2^g = 1$. Now, take $\gamma \in \pi_1(L, x)$. We now construct some kind of cobordism to go from $g$ to $\gamma g \gamma^{-1}$. (Idea is to move the marked point in the evaluation map around the loop $\gamma$. It is important that we consider the based fundamental group and not the free one.)
 Thus, the set $\{ \gamma g \gamma^{-1} \, | \, \gamma \in \pi_1(L, x) \}$ is finite. (Gromov compactness + rigidity)
Thus, $Z(g)$ is of finite index.

UPSHOT: defined/not-defined is a key tool in Mihai's work. This is a really cool variation of the usual pain we have to say something is well-defined.