2
votes
1answer
81 views

Computation of symplectic quasi-state

A subset of a symplectic manifold is called strongly non-displaceable if it cannot be displaced by symplectomorphisms. A meridian in a $2$-torus is displaceable by a symplectomorphism, but not by a ...
3
votes
1answer
100 views

A question about solutions to Floer's equation which are asymptotic to a stationary point

Let $M$ be a compact symplectic manifold and $H$ a time independent Hamiltonian on $M$. Let $\alpha$ be a solution to Floer's equation $$ u(t,s): S^1 \times \mathbb{R} \to M$$ $$(du+X_H\otimes ...
1
vote
1answer
123 views

Question about transversality for PSS map in Hamiltonian Floer cohomology

Let X be a compact symplectic manifold and $H_t,J_t$ a Floer regular pair of $\mathbb{S}^1$ dependent Hamiltonians and complex structures. The PSS maps are defined by considering $\mathbb{C}$ with a ...
5
votes
0answers
109 views

Unobstructed Lagrangian tori

Let $X$ be a compact symplectic manifold, $U$ a Darboux chart, and $L$ a standard Lagrangian torus in $U$. Is $H^1(L)$ weakly unobstructed in the sense of Fukaya-Oh-Ohta-Ono (that is, does $b \in ...
7
votes
2answers
423 views

Orientations for pseudoholomorphic curves with totally real boundary condition

I am trying to understand what the obstructions are to orienting moduli spaces of pseudoholomorphic curves with totally real boundary condition. I believe that Fukaya-Oh-Ohta-Ono have shown that if ...
7
votes
1answer
400 views

Why is Heegaard Floer Homology defined in terms of Sym$^g\Sigma_g$ instead of Pic$^g\Sigma_g$?

Recall the definition of Heegaard Floer homology: $\Sigma_g$ is a closed surface, and $\{\alpha_1,\ldots,\alpha_g\}$ and $\{\beta_1,\ldots,\beta_g\}$ are sets of attaching circles. Then Heegaard ...
4
votes
0answers
204 views

Lie-infinity structure in Lagrangian Floer theory ?

Is there (besides the A-infinity structure) also a L-infinity structure in Lagrangian Floer theory (forming together a G-infinity structure) - like in Hochschild cohomology ?
9
votes
6answers
2k views

Introduction to Floer Theory?

Can anyone suggest a good overview/introduction of the Floer machine for a beginner? (Someone pointed out some intriguing connections to surface mapping class groups, which might be enough incentive ...
5
votes
2answers
444 views

Maslov index and heegard floer homology

I am an undergraduate who wants to learn Knot Floer homology. I was told to start with this expository paper which was working quite well until I reached the actual definition of the differential. ...
4
votes
1answer
283 views

path of almost complex structure in the definition of heegaard floer homology

In order to define Heegaard Floer Homology for a connected, closed, oriented 3 manifold, we fix a generic path of nearly symmetric almost complex strucutre $J_s$ over $Sym^g(\Sigma)$. By ...
3
votes
2answers
441 views

Dimension of moduli space in Lagrangian Floer homology

Let $(M,\omega)$ be symplectic manifold with $\omega=c_{1}=0$ on $\pi_{2}M$. Let $\Lambda\subseteq M$ be Lagrangian submanifold. Let $H:M\times S^{1}\rightarrow\mathbf{R}$ be Hamiltonian and $J$ be ...
10
votes
1answer
571 views

Where are $+$, $-$ and $\infty$ in bordered Heegaard-Floer theory?

Here goes my first MO-question. I've just read Lipshitz, Ozsváth and Thurston's recently updated "A tour of bordered Floer theory". To set the stage let me give two quotes from this paper. ...
8
votes
0answers
405 views

SFT gluing on chain level in Floer homology?

I came across a new type of gluing in the FOOO paper http://arxiv.org/abs/1002.1660 (maybe not so new to experts). The situation is as follows: one considers a relative (to lagrangian) class and ...
3
votes
0answers
177 views

Regular points for solutions of Floer's equation

Let $(W,\omega)$ be a symplectic manifold and $H\in C^{\infty}(W\times S^{1};\mathbb{R})$. Let $(J_{t})_{t\in S^{1}}$ denote a family of $\omega$-compatible almost complex structures. Fix a map $u\in ...
7
votes
0answers
352 views

Degenerate moduli spaces in Floer homology

Let $(W,\omega)$ be a closed symplectially aspherical symplectic manifold, and fix a Hamiltonian $H\in C^{\infty}(W\times S^{1};\mathbb{R})$ and a compatible almost complex structure $J$ on $W$. Given ...
8
votes
3answers
760 views

Index theorem interpretation of the spectral flow for a pseudo holomorphic curve

Let $(M , \omega)$ be a symplectic manifold, $J$ a compatible almost complex structure. We call pseudo holomorphic strip a solution $u : \mathbb R \times I \to M$ of the equation $\partial_s u + J ...
2
votes
1answer
403 views

Length of Floer flow lines

Suppose $(X,\omega)$ is a closed symplectic manifold. Let $H$ denote a time-dependent Hamiltonian, all of whose critical points are non-degenerate, and fix an $\omega$-compatible time dependent family ...
6
votes
1answer
479 views

Almost complex structures in Floer theory

When defining the Floer cohomology $HF(L_0,L_1)$ of 2 Lagrangians in a symplectic manifold $(M,\omega)$, one first has to choose some extra data such a 1-parameter family of almost complex structures ...
3
votes
1answer
1k views

What is Floer homology of a knot?

I've heard that there are different theories providing knot invariants in form of homologies. My understanding is that if you embed knot in a special way into a space, there is a special homology ...