The symplectic-topology tag has no wiki summary.

**6**

votes

**1**answer

118 views

### Homology class of a Lagrangian Klein bottle

Nemirovskii's 2008 paper, by the same title in this question, claims that any Lagrangian Klein bottle in a closed symplectic 4-manifold $M$ must realize a nontrivial homology class in $H_2(M; \mathbb ...

**3**

votes

**0**answers

45 views

### Flux group of surfaces with genus $g\ge2$

I have read in this paper by C. Campos-Apanco and Pedroza (at the end of page 2) that:
When $(M,\omega)$ is a closed surface of genus greater than one, the flux group is trivial.
Does anyone ...

**6**

votes

**1**answer

370 views

### Symplectic reversing diffeomorphisms on a compact symplectic manifold

I Ask this question in MSE and I received interesting comments and ideas. I repeat the question here for more discussion:
Let $(M,\omega)$ be a compact symplectic manifold.
Is there a ...

**1**

vote

**0**answers

40 views

### Perturbation of a Fredholm sections which preserves compactness of 0-set

I am learning Morse-Bott-Floer theory and found the following cool paper
http://de.arxiv.org/abs/1310.5080
by P. Albers and D Hein. In order to prove a cup-length estimate on the number of critical ...

**0**

votes

**1**answer

97 views

### Lagrangian flow preserves symplectic form

Let $X$ be a configuration space and $L: TX \rightarrow \mathbb{R}$ a Lagrangian. Then I want to show that the Lagrangian flow $F^t(x(0),x'(0)) = (x(t),x'(t))$ preserves the symplectic form just like ...

**0**

votes

**0**answers

71 views

### Is there an explicit way to glue a stable map in projective space by writing down the family of maps explicitly in terms of polynomials?

Let $v_1:\mathbb{P}^1 \longrightarrow \mathbb{P}^2$
and $v_2:\mathbb{P}^1 \longrightarrow \mathbb{P}^2$ be two holomorphic maps
of degree $d_1$ and $d_2$ respectively. Suppose they agree at some ...

**2**

votes

**0**answers

88 views

### Generation of compact Lagrangians over fields with characteristic 2

Let $\pi:X\rightarrow\mathbb{C}$ be a Lefschetz fibration, and assume that the fibers are exact or monotone. A classical result of Seidel says that all the closed weakly unobstructed Lagrangian ...

**2**

votes

**0**answers

102 views

### McDuff's classification of symplectic manifolds

According to a theorem of McDuff, if $(X,\omega)$ is a closed symplectic 4-manifold, which contains a symplectic sphere of self-intersection +1, then $X$ is symplectomorphic to $CP^2$ blown up a few ...

**2**

votes

**0**answers

73 views

### Is there an action functional for the s-dependent Floer equation?

The usual Floer equation (in local coordinates)
\begin{equation*}
\partial_su+J(t,u)(\partial_tu-X_{H_t}(u))=0
\end{equation*}
is derived as the gradient flow of the symplectic action functional ...

**1**

vote

**1**answer

141 views

### Show that the symplectic action 1-form on loop space is closed

I am struggling to see how the symplectic action 1-form $\alpha_H$ on the loop space of a symplectic manifold $(M,\omega)$ is closed. It is defined by
$\alpha_H(Y) = \int_0^1 ...

**0**

votes

**1**answer

228 views

### On Gromov's Theorem on Symplectic Homotopy

I want to understand the proof of the following theorem due to Gromov which I'll state in the context of Euclidean spaces. While I tried to read the proof from Macduff-Salamon, it turned out that my ...

**1**

vote

**0**answers

112 views

### Complex but not Symplectic

For every $n$ there exist a smooth manifold $M$ of $dim M = n$ that admits a complex structure but not a symplectic one?

**1**

vote

**0**answers

60 views

### why is there such a 1-form on a planar open book?

Suppose $B$ is the binding (with more than one component) of a planar open book on a 3-manifold $Y$ and let $L\subset B$ be the complement of a single component of the binding. Now we perform ...

**1**

vote

**0**answers

106 views

### Lagrangian fibrations with isolated singular fibers

Let $(M,\omega)$ be a symplectic manifold, and assume $\dim_\mathbb{R}(M)>4$. Suppose there exists a real Lagrangian fibration $\pi:M\rightarrow Q$ so that $Q$ becomes an integral affine manifold ...

**2**

votes

**0**answers

157 views

### Fully faithful embedding of the exact Fukaya category

Let $\mathscr{F}(X)$ be the exact Fukaya category of an exact symplectic manifold $(X^{2n},\omega)$, i.e. the objects in $\mathscr{F}(X)$ are all closed exact Lagrangian submanifolds with Maslov index ...

**2**

votes

**0**answers

106 views

### Examples of symplectic manifolds which are twisted $T^n$ bundles over $T^n$

I'm looking for certain (higher-dimensional) analogues of the Kodaira-Thurston manifolds, i.e. I want to know whether in $\dim_\mathbb{R}X\geq6$ we have examples of symplectic manifolds satisfying the ...

**1**

vote

**0**answers

82 views

### Does some square of the first Chern class preserved by conifold transition?

Let $X$ be a smooth projective 3-fold or a symplectic 6-manifold.
Suppose $Y$ is a conifold transition on a single nullhomologous
Lagrangian sphere $S^{3}$ in $X$. Then there is a exact sequence $0\to
...

**0**

votes

**2**answers

322 views

### When are Maslov $0$ disks non-trivial in $\pi_2(M,L)$?

My goal is to better understand the Maslov-index of pseudoholomorphic disks.
For a symplectic manifold $(M,\omega)$ and a Lagrangian submanifold $L\subset M$, the Maslov-index of a pseudoholomorphic ...

**0**

votes

**0**answers

94 views

### Connectedness of the symplectic automorphism of the 2-sphere $S^2$

The 2-sphere, endowed with the round Riemann metric with constant curvature 1, is a symplectic manifolds.
My question is:
Is the group of symplectic automorphisms
of $S^2$ with respect to this ...

**10**

votes

**6**answers

695 views

### When do you go hunting for Lagrangian submanifolds?

Similar to this question, I'm trying to figure out why one would be interested in Lagrangian submanifolds. But from a more geometric point of view. My best find so far is Exercise 12.4 in McDuff, ...

**2**

votes

**0**answers

145 views

### Finding a smooth 6-manifold with a closed 2-form which is degenerate only along some embedded 2-spheres

Given a symplectic 6-manifold $(M,\omega)$ and an embedded
symplectic 2-sphere $C\subset M$ whose normal bundle has the first
Chern class -2. How to find on $M$ another closed 2-form $\eta$ which
only ...

**1**

vote

**0**answers

69 views

### Does the twisted product $K^{\mathbb{C}}\times_{Z(k)^{\mathbb{C}}} X^k$ have a natural Kähler or sympletic structure?

Let $K$ be a connected compact Lie group, and suppose that $(X,J,\omega)$ is a compact Kähler manifold on which the group $K$
acts holomorphically such that the group $K$ preserves the
Kähler ...

**1**

vote

**0**answers

80 views

### Decomposition of Lefschetz fibrations into surface bundles and Lefschetz fibrations over spheres

Assume that $M$ is a Lefschetz fibration with fiber genus $g$ over a base of genus $h>0$ (with at least one singular fiber). What obstructions exist to decomposing $M$ into a surface bundle $S$ of ...

**3**

votes

**0**answers

100 views

### The relative h-principle and extension problems

As a beginner for h-principles, I want to know why the relative
h-principle cannot imply a positive solution to the problems for
extending symplectic structures.
The following is a relative ...

**3**

votes

**2**answers

249 views

### Reference Request: “Neck Stretching Procedure” (In Symplectic Field Theory)

I've been reading some papers in Symplectic Geometry which refer to something called "Stretching the neck", and give reference to Eliashberg, Givental and Hofer's SFT paper ...

**4**

votes

**0**answers

116 views

### Symplectic sum and Symplectic cut

The symplectic sum of Gompf and the symplectic cut of Lerman are known to be inverse of each other, in the sense that if you apply one of these first and the other one afterward, you obtain the ...

**2**

votes

**1**answer

144 views

### On Lerman's description of symplectic cut

Assume $(X,\omega)$ is a compact real $2n$-dimensional symplectic manifold with a Hamiltonian torus action corresponding to the moment map $\mu:X\to \mathfrak{t}^*\cong \mathbb{R}^k$.
In this ...

**8**

votes

**2**answers

264 views

### The importance of differentiable dynamics from outside dynamics? (mainly topology)

I'm looking for examples that highlight how dynamical systems (particularly, Hamiltonian and Reeb dynamics) can be used to shed light in other areas of mathematics. This could potentially include ...

**4**

votes

**0**answers

143 views

### What can be said about compact embedded exact Lagrangians in the generalized pair of pants?

What can be said about compact embedded exact Lagrangians in the $n$-dimensional generalized pair of pants e.g. the hypersurface in $(\mathbb{C}^*)^{n+1}$ defined by the equation:
$$ 1+\Sigma_i z_i = ...

**2**

votes

**0**answers

203 views

### $C^0$ estimates in wrapped Lagrangian Floer cohomology

Let $(M, d\theta, \theta, Z)$, be an exact Liouville domain, where $Z$ is the Liouville vector field and $\theta$ is the primitive of the symplectic form. Let $\bar{M}$, be the symplectic completion ...

**1**

vote

**1**answer

216 views

### A question on existence of $Spin^c$-structure $P\to M$

Let $(M,\omega)$ be a compact symplectic manifold and the cohomology class $$[\omega]+\frac{1} {2}c_1(\wedge_{\mathbb C}^{0,n}(TM, J))\in H_{dR}^2(M)$$
is integral, for some almost complex structure ...

**1**

vote

**1**answer

159 views

### Relation between volume of reduced space and phase space

Let $G$ ba a compat Lie group and $\frak g$ be its Lie algebra, then by Marsden-Weinstein reduction theory we know that if $J:M\to \frak g^*$ be its equivariant moment map then the reduced space is ...

**1**

vote

**1**answer

182 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 ...

**12**

votes

**3**answers

1k 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). ...

**7**

votes

**1**answer

271 views

### Does the smooth manifold $\#_{l}CP^{2}\#_{k}(-CP^{2})$ admit a symplectic structure?

Let $-CP^{2}$ denote the complex projective surface $CP^{2}$ with the reverse orientation. I have seen some results about the existence of symplectic structures on the connected sums ...

**8**

votes

**1**answer

415 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, ...

**3**

votes

**1**answer

178 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 ...

**3**

votes

**3**answers

521 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 ...

**7**

votes

**2**answers

525 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 $ \# ...

**1**

vote

**1**answer

233 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 ...

**1**

vote

**2**answers

433 views

### On the de Rham cohomology of 1-forms in cotangent bundle.

We know that a cotangent bundle $T^\star M$ has a canonical symplectic form and $M$ is a natural Lagrangian submanifold of it. A well known result is that any submanifold $X=\{(p,f(p)): p\in M\}$, ...

**0**

votes

**1**answer

242 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 ...

**6**

votes

**2**answers

487 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 ...

**6**

votes

**2**answers

349 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 = ...

**5**

votes

**4**answers

392 views

### Lagrangian Kleinian bottles

I remember some talks some time ago about proofs of nonexistence of Lagrangian Kleinian bottles in C^2 for the standard symplectic structure, mentioning that this were the only compact surface for ...

**8**

votes

**0**answers

491 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 ...

**8**

votes

**2**answers

724 views

### Relative version of Symplectic Thom conjecture.

Ozsváth and Szabó proved Symplectic Thom conjecture [Annals of Mathematics, 151(2000), 93-124]. It states: An embedded symplectic surface in a closed, symplectic 4-manifold is ...

**1**

vote

**1**answer

235 views

### codimension of bubbling of disk and sphere

I would like to understand why bubbling of disks are said to be co-dimension 1 phenomena and bubbling of spheres co-dimension 2 phenomena.

**4**

votes

**1**answer

313 views

### “Rounding the corners” to get contact boundary

Suppose we have symplectic manifolds $(M_1, \omega_1)$ and $(M_2, \omega_2)$ with non-empty boundary of contact . Often we need to deal with the product $M_1 \times M_2$ with the product symplectic ...

**3**

votes

**1**answer

632 views

### Are there any symplectic but not holomorphic Calabi-Yau manifolds in real dimensions 4 and 6?

Are there any symplectic but not complex Calabi-
Yau manifolds in real dimensions 4 and 6?