# Tagged Questions

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**1**answer

124 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 ...

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**3**answers

893 views

### Is there a physical intuition for Darboux's theorem?

We know that there is a physical interpretation for symplectic manifolds (briefly, the fact that a sympletic form assigns to any Hamiltoninan a vector field which describes the motion of particles). ...

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**1**answer

312 views

### Does there always exist a line bundle whose Chern class represents an integer symplectic form?

Let $(M, \omega, J)$ be a compact symplectic manifold with a
compatible almost complex structure $J$, such that the symplectic
form determines an integer cohomology class, ie
$$ [\omega] \in H^2(M, ...

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**1**answer

142 views

### Can symplectic blow up increase symplectic capacities?

Let $N$ be a symplectic submanifold of $M$. Symplectic blow up of $M$ along $N$ is an operation replacing a tubular neighborhood of $N$ with the projectivization of that neighborhood. So it decreases ...

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347 views

### Symplectic blow-up

Blow-ups of points can also be performed in the symplectic category; for a given point $p\in (X,\omega)$ we choose a Darboux chart around $p$ and then use the symplectic cut corresponding to the ...

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**2**answers

418 views

### How to prove Arnold Conjecture without using S^1 localization?

By the Arnold Conjecture, I mean the following statement:
Let $M$ be a closed symplectic manifold, and $\phi:M\to M$ a Hamiltonian symplectomorphism with nondegenerate fixed points. Then $ \# ...

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**1**answer

166 views

### A basic question related to Hamiltonian isotopy in symplectic geometry

In any standard symplectic geometry/topology textbook, the concept of Hamiltonian isotopy was introduced:
$(M, \omega)$ is a sympplectic manifold. Given a symplectic isotopy
$\phi_t : M \rightarrow ...

**0**

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**1**answer

234 views

### $q_{S^*\omega}(X)=S^{\ast}q_{\omega}(X)$ ?

Definition: Let $(V,\Omega)$ be a symplectic vector space, we define
$\perp:\Lambda ^k(V^*)\to\Lambda ^{k-2}(V^{\ast})$
by $\perp(\omega)=i_{X_{\Omega}}(\omega)$
here if ...

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votes

**2**answers

455 views

### A Question on Exterior Forms

For the last few days, I have been trying to answer the following algebraic question in exterior algebra. The following question appears as an algebraic step in the context of existence of solutions ...

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327 views

### Square root for Hamiltonian diffeomorphisms

Let $\psi_t: X\to X$, $t \in [0,1]$, be a path Hamiltonian diffeomorphism on a symplectic manifold $X$, given by functions $H_t$. If $H_t \equiv H$ is independent of $t$ then
$$ \psi_1 = ...

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**0**answers

431 views

### Pseudocycle definition of open Gromov-Witten invariants

I decided my original question was unnecessarily long, and have edited to simply ask the desired question directly (the below is much more direct than my original post, though it may seem just as ...

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**0**answers

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 ...

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**3**answers

769 views

### How transitive are the actions of symplectomorphism groups ?

This question is motivated by the classical fact from differential geometry :
Let $M$ be a smooth manifold of dimension at least $2$. Then for any $n$ the diffeomorphism group $\textrm{Diff}(M)$ ...

**2**

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**1**answer

429 views

### The Hard Lefschetz property on Almost-Kahler manifolds

In the realm of almost-Kahler geometry , to what extent , the hard Lefschetz property is still holds?

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463 views

### Reference for Almost-Kahler geometry

Is there any comprehensive reference for Almos-Kahler geometry or more generally to Almost- Hermitian geometry ?

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**0**answers

404 views

### Almost-Kahler Einstein four manifolds

Are the odd Betti numbers of an Almost-Kahler Einstein four manifolds necessarily even ?

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**1**answer

353 views

### Is every contractible loop contained in a Darboux chart?

Let $(M,\omega)$ be a symplectic manifold and $\gamma:S^1 \rightarrow M$ be a contractible smooth loop.
Is it possible to find an open set $U \subset M$ such that $\gamma(S^1) \subset U$ and such ...

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**4**answers

1k views

### When are two symplectic forms “isotopic”?

I've been skulking around MathOverflow for about a month, reading questions and answers and comments, and I guess it's about time I asked a question myself, so here is one has interested me for a long ...

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**1**answer

412 views

### Real interpretations of Discontinuities in Floer homology

This question is motivated by the answer in this question (you dont have to read it to understand the following).
I am not that proficient in calculating Floer homology, and I held back on answering ...

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**1**answer

440 views

### Hamiltonian displaceability of tori in symplectic balls

Here is my first try at a question, which is a really easy to state question
about displaceability:
Let $D$ be the unit disk in the complex plane $D = \{ |z| \leq 1 \}$ equipped
with its standard ...

**9**

votes

**5**answers

2k views

### Understanding moment maps and lie brackets

I'm trying to learn about moment maps in symplectic topology (suppose our Lie group is G with lie algebra g, acting on the symplectic manifold (M,w) by symplectomorphisms). I'm having a hard time, and ...

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827 views

### Why is every symplectomorphism of the unit disk Hamiltonian isotopic to the identity?

That is, for any symplectomorphism $\psi: D^2 \to D^2$, there should be a time-dependent Hamiltonian Ht on D2 such that the corresponding flow at time 1 is equal to $\psi$.
I found this in claim a ...

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**2**answers

512 views

### Why (and whether) is any smooth embedded torus in R^4 isotopic to an embedded Lagrangian torus?

The question is pretty self-explanatory; we are dealing with the standard symplectic structure on ℝ4.
Some background: I'm reading the thesis "Lagrangian Unknottedness of Tori in Certain ...

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1k views

### Periodic orbits of Hamiltonian systems

Given a Hamiltonian $H$ on $\mathbb{R}^{2n}$ and a periodic orbit $\gamma$, what in general can one say about the existence of periodic orbits near $\gamma$?
I'm almost embarrassed to ask this ...

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3k views

### Is the Fukaya category “defined”?

Sometimes people say that the Fukaya category is "not yet defined" in general.
What is meant by such a statement? (If it simplifies things, let's just stick with Fukaya categories of compact ...