# Questions tagged [symplectic-topology]

The symplectic-topology tag has no usage guidance.

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### A clarification in the definition of Seidel's absolute Maslov index for a pair of transverse Lagrangians

I'm reading Seidel's paper Graded Lagrangian submanifolds where he introduces the absolute Maslov index of a pair of graded lagrangians as follows:
Let $\mathcal{L}(V,\beta)$ be the Lagrangian ...

**3**

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52 views

### Symplectic vector fields everywhere transverse to a co-dimension one hypersurface

Usually when speaking about vector fields transverse to a hypersurface in a symplectic manifold, we talk about Liouville vector fields, i.e. vector fields $X$ with the property that $\mathcal{L}_X\...

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**1**answer

159 views

### Floer equation and Cauchy Riemann equation

Consider a symplectic manifold $(M,\omega)$ with the property that $\pi_2(M) = 0$. Given a time dependent hamiltonian $H_t$ on $M$, and a $\omega$-compatible almost complex structure J on M, we may ...

**2**

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86 views

### Limitations of the splitting construction and SFT

I am trying to understand the so-called symplectic field theory (SFT) machinery used in symplectic topology. As I understand it, one of the applications of SFT (or rather, of the splitting ...

**2**

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61 views

### Transitivity of Diff on the space of embeddings of balls

Given 2 symplectic embeddings $g_0$ and $g_1$ of a 4-ball of radius $r \leq 1$ into the 4-ball of radius 1 (all equipped with the standard symplectic form coming from $\mathbb{R}^4$), does there exist ...

**4**

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66 views

### Pairs of J-holomorphic curves

Let $(M, \omega)$ be a symplectic 4-manifold and let $A$ and $B$ be symplectic submanifolds on M such that $A \cap B = p \in M$.Under what conditions can I find a $\omega$-compatible almost complex ...

**5**

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**1**answer

192 views

### Intuition about bubbling off a ghost bubble

I'm trying to improve my intuition about the bubbling phenomenon for $J$-holomorphic curves $\Sigma \to (M,\omega)$, where $\Sigma$ is a compact Riemann surface with possibly boundary. I assume that ...

**4**

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106 views

### Diffeomorphism of $ \mathbb{C}P^2 \# ~\overline{\mathbb{C}P^2}$

I am currently reading Dusa McDuff's paper "Blow ups and symplectic embedding in dimension 4" and had a few questions regarding the paper.
In the paper McDuff uses the following notation. $X = \...

**4**

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**1**answer

499 views

### The singular cohomology embeds into the symplectic cohomology

Viterbo's theorem on cotangent bundles $M=T^*N$ tells you in particular that singular cohomology $H^*(M)$ gets embedded in $SH^*(M)$ via the $c^*$ map. Having a Weinstein manifold (or more generally ...

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97 views

### Closed symplectic manifold with Euler characteristic 2

I am working about an article. In this article, author said that if close symplectic manifold $M$ has two fixed points implies that either $M$ is 2-sphere or $\dim M=6$.
The closed symplectic manifold ...

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126 views

### Cohomological indices VS cup-length, Lusternik-Schnirelmann category, betti sum in critical point set theory

Let $M$ be a (closed) manifold acted on by a compact Lie group $G$. For any characteristic class $\alpha \in H^*(BG)$, one can define a so-called cohomological index of $M$ as follows:
$$\text{ind}_{\...

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**1**answer

126 views

### Normal coordinates for isotropic submanifolds

Let $(M,\omega)$ be a symplectic manifold and $N$ an isotropic submanifold. For a point $p\in N$, can we always find coordinates $(x_1,\ldots,x_n,y_1,\ldots,y_n)$ in a neighbourhood $U$ of $p$ such ...

**10**

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378 views

### Moduli space of curves

Let $(M,\omega)$ be a symplectic manifold, and let $\mathscr{J}$ be the set of compatible almost complex structures on $M$.Finally let $A \in H^2(M,\mathbb{Z})$. Then we can consider the moduli space ...

**4**

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114 views

### Complement of Donaldson divisors in dimension 4

Let $(X,\omega)$ be a symplectic 4-manifold such that $\omega$ has a rational cohomology class. I am interested in Donaldson divisors (surfaces) $D$ in $(X,\omega)$ whose complement is a 1-handle body....

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**1**answer

184 views

### Compact simply-connected homogeneous symplectic manifold

I was reading a paper in which the authors use the fact that any compact simply-connected homogeneous symplectic manifold has non-zero Euler characteristic. They prove it by quoting a theorem by ...

**9**

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**1**answer

994 views

### What is the current status of the Arnold conjecture?

Let $(M, \omega)$ be a symplectic manifold. V.Arnold conjectured that the number of fixed points of a Hamiltonian symplectomorphism is bounded below by the number of critical points of a smooth ...

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149 views

### Where can I find good surveys on Symplectic and Contact geometry

Are there any good survey articles in symplectic and contact geometry, which focus on the "big picture", i.e how this discipline fits into the mathematical world ?
In the symplectic case : I am ...

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107 views

### Unobstructedness of nodal holomorphic curve in symplectic manifold

Suppose $(X,\omega)$ is a compact symplectic manifold and $J$ is an $\omega$-compatible almost complex structure on $X$ (the symplectic structure seems to be irrelevant for this question actually). ...

**9**

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393 views

### DGLA controlling deformation of holomorphic curves

Suppose $C$ is a compact Riemann surface and $X$ is a compact Kähler manifold. Suppose $f:C\to X$ is a stable holomorphic map. Then, the deformations of $f$ are controlled by the complex $L^\bullet = ...

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120 views

### space of $\omega$-tame almost complex structures and $\mathrm{Diff}(M)$

Let $(M,\omega)$ be a symplectic manifold, and $J$ is an almost complex structure on $M$. $J$ is said to be $\omega$-tame if
$$
\omega(v, Jv) >0
$$
for all non-zero $v\in TM$.
It is commonly said ...

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75 views

### On the measure of regular and chaotic regions in a phase space

Consider a Hamiltonian, non-linear, dynamical system associated to $H(\vec{q},\vec{p})$. Assume that the number of effective degrees of freedom is relatively small, say $D=3,4,5$. Now choose a certain ...

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**2**answers

192 views

### Non-trivial examples of overtwisted contact structures

Are there any non-trivial examples of overtwisted contact structures on closed contact $3$-manifolds? By non-trivial I mean any examples besides the trivial one $\xi = \ker (\cos(r)dz - r \sin(r)d\...

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159 views

### Recovering topological invariants of symplectic manifold from the group of Hamiltonian diffeomorphisms?

There is a famous result of Banyaga stating that if two closed symplectic manifolds $(M_1, \omega_1)$ and $(M_2, \omega_2)$ have isomorphic groups of Hamiltonian diffeomorphisms $\mathrm{Ham}(M_1, \...

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334 views

### Question about Obstruction Bundle Gluing paper of Hutchings-Taubes

I'm trying to learn about Embedded Contact Homology. To understand the proof of $d^2=0$, I started by watching Hutchings' lectures on Obstruction Bundle Gluing on YouTube (1, 2, 3) and have now ...

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81 views

### Is there a simply connected contact manifold, “non-exactly” fillable, cappable, such that the whole thing is symplectically aspherical?

Is there an example of a simply connected contact manifold W, with a non exact symplectic filling $M_1$, (that is, $M_1$ is a symplectic manifold, with contact boundary $W$ and a Liouville vector ...

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111 views

### Lagrangian foliation for a holomorphic symplectic manifold

I am interested in gathering as many examples as possible for Lagrangian foliations of holomorphically symplectic manifolds $(X, \omega)$, where $X$ is a $2n$-dimensional complex manifold equipped ...

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**1**answer

170 views

### symplectic topology of (perturbed) KAM tori

Consider a real analytic $H_0:\mathbb{R}^n\to \mathbb{R}$ whose Hessian is everywhere non-degenerate as well as a real analytic $F:\mathbb{T}^n\times \mathbb{R}^n\to \mathbb{R}$. KAM theory studies ...

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125 views

### Existence of isotopy preserving the action

Let $\gamma_1$ and $\gamma_2$ be simple closed curves in $R^4.$ Let $\lambda= x_1 dy_1+ x_2dy_2.$ Suppose that $\int_{\gamma_1} \lambda= \int_{\gamma_2} \lambda.$
I am looking for a reference for ...

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250 views

### Symplectic reduction of 4-manifolds with circle actions

Let $(M,\omega)$ be a $4$-dimensional closed symplectic manifold. Assume there exists a Hamiltonian $S^1$-action on $M$, let $\mu:M \to \mathbb{R}^*$ be its moment map and let $M_{\text{red}}=\mu^{-1}(...

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53 views

### Symplectic displacement energy for several intersection points?

Let $(X, \omega)$ be a symplectic manifold. For any non-empty subset $Y \subset X$ we may define the displacement energy as
$$
e(Y)=\mathrm{inf}\{||\phi||_H \: | \phi \in Ham(X, \omega), \phi(Y) \cap ...

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294 views

### Two embedded symplectic spheres with zero square in a symplectic $4$-manifold

I am aware that the following result is a classical one (by now). But I am not able to understand who proved it. What should be a proper reference to this statement?
Theorem. Let $M^4$ be a compact ...

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371 views

### Chart for Deligne-Mumford Compactification

Suppose $g\ge 0$ and $n\ge 0$ are integers. We have the space $\overline{\mathcal M}_{g,n}$ of stable curves of arithmetic genus $g$ with $n$ marked points. The topology on this space is the one ...

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321 views

### Finding basis of cohomology of a symplectic manifold by using Symplectic Minimal Model Program

My question is about Floer theory via symplectic surgery of Minimal Model program for finding basis of cohomology.
Motivation: Perelman for solving Thurston's Geometrization Conjecture used some sort ...

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**1**answer

408 views

### Describing a Lefschetz fibration whose fiber is plumbing of $T^*S^n$

I am studying an example of symplectic Lefschetz fibrations. As far as I know, given a Weinstein manifold $F$ and a collection $V_1,\ldots,V_k$ of exact framed Lagrangian spheres of $F$, there exists ...

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131 views

### Existence of $K$-invariant complex structure

Let $K$ be a compact connected real Lie group, let $M$ be a symplectic manifold with a symplectic left action of $K$, and let $ω$
be the symplectic structure of $M$.
Does there always exist a $K$-...

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481 views

### $(M,\omega)$ not symplectomorphic to $(M,-\omega)$

Looking for an example of a symplectic manifold $(M,\omega)$ that is not symplectomorphic to $(M,-\omega)$.
In particular this means that $M$ must be chiral (i.e. doesn't admit an orientation-...

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174 views

### The structure of Banach manifolds in symplectic geometry

Let $M$ be a symplectic manifold, and let $L_0$ and $L_1$ be Lagrangian submanifolds which transverse to each other. In Floer theory, we need to consider a Banach manifold $\mathcal B$ of maps $u:\...

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69 views

### Is every contractible open bounded domain in $\mathbb R^{2n}$ symplecomorphic to a star-shaped domain?

In Hofer & Zehnder's book "Symplectic Invariants and Hamiltonian Dynamics" (Page 99) they present an example of a star shaped domain (bounded, with smooth boundary) in the shape of a "Bordeaux ...

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**1**answer

122 views

### Contradiction between fixed points of a hamiltonian diffeomorphism of a torus and quasi-periodic motion on a torus

Again a very simple question. I currently hold two contradictory ideas in my head
1) A hamiltonian diffeomorphism of a torus necessarily has fixed points
2) most hamiltonian actions on a torus in an ...

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98 views

### Embeddings of the configuration space into the phase space of integrable systems

As always, I'm not sure if I'm about to ask a very stupid question, and I apologise if that is the case.
Most systems from physics come from classical Hamiltonians, defined on the phase space of ...

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**1**answer

186 views

### How to understand geometrically, the count of pseudoholomorphic discs by (multi)section perturbation of the kuranish structure on the moduli space?

When defining the $A_\infty$ algebra of a Lagrangian (as done in the book by FOOO) it is done by "counting" (integrating over the moduli space or over the fiber of evaluation map) pseudoholomorphic ...

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100 views

### The isotopy class of a Boothby-Wang contact structure

Let $p:\Sigma\to M$ be a non-trivial principal $S^1$-bundle over a closed orientable surface $M$. Let's call $V$ the vector field generating the $S^1$-action. It is known that there exists a Boothby-...

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### Can one choose a sufficiently generic path of a.c.s such that only “codimension 1” bubbling occurs?

Consider a symplectic manifold $(M,\omega)$ of dimension $2n$ (closed or open with bounded geometry). Let $L\subset M$ be a compact Lagrangian submanifold (not necessarily connected).
Consider two ...

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146 views

### what are advantages of considering $\{J_z\}$-holomorphic curves for a parametrized family $\{J_z\}$ of almost complex structure?

Reference: McDuff-Salamon's book J-holomorphic curves and Symplectic Topology, second edition.
In section 6.7, the book introduces the moduli space of $\{J_z\}$-holomorphic curves (see page 184 for ...

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476 views

### Categorical mapping class group action

Let $\Sigma$ be a closed genus $g$ surface. Assume that $\mathcal{C}$ is a smooth and proper dg- (or $A_\infty$-) category which admits a faithful action
$$ MCG(\Sigma) \to Auteq(\mathcal{C})$$
by ...

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**1**answer

450 views

### Almost complex structures on $\mathbb CP^2$ that are not tamed

Recall that an almost complex structure $J$ on a manifold $M^{2n}$ is called tamed if there exists a symplectic form $\omega$ on $M^{2n}$ such that $\omega(v,Jv)>0$ for any non-zero tangent vector $...

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628 views

### Gluing symplectic manifolds

The condition that allows gluing of symplectic manifolds, is the existence of a fixed point free $S^1$ action on the boundary, such that the orbits of the $S^1$ action are tangent to the kernel of the ...

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105 views

### Reference request: Liouville integrability of a torus action of small dimension on a symplectic manifold

Consider a hamiltonian toric acion on a connected real symplectic manifold of dimension 2n. The dimension of the torus, which we denote by $k$, may be less than $n$. The generators of the action will ...

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**1**answer

351 views

### Generating Fukaya category vs split-generating Fukaya category

I just started learning about Fukaya categories and got slightly confused by the following question. It looks like the statement that a collection of objects generate Fukaya category is stronger than ...

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**1**answer

194 views

### the tangent space $T_J\mathcal J^k$ of the space of $\omega$-compatible almost complex structure

Let $(M,\omega)$ is a symplectic manifold, and you can assume it is compact if necessary.
Denote by $\mathcal J^k = \mathcal J^k(M, \omega)$ the set of all $\omega$-compatible almost complex ...