Questions tagged [symplectic-topology]

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Associativity of orientations of determinant bundles in Floer homology

I have been reading the paper "Coherent orientations for periodic orbits problems in symplectic geometry" by Floer and Hofer, trying to understand how we can orient the moduli spaces that ...
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Compactness of Moduli spaces in Lagrangian Floer Cohomology

I have been reading Denis Aurox lecture notes on Fukaya Categories https://arxiv.org/pdf/1301.7056.pdf , and in page $9$ he starts to discuss the compactness properties of the moduli spaces and how we ...
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3 votes
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96 views

Removal of singularities theorem for J-holomorphic curves in non-compact manifolds

Following the book by Mcduff and Salamon, J-holomorphic curves and Symplectic Topology, we know that every $J-$holomorphic curve on the punctured disk with values in a compact symplectic manifold ...
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6 votes
0 answers
260 views

Computation of the Fredhom index in Floer theory

I have been reading Salamon's lecture notes on Floer homology, and to compute the Fredholm index of operators they use the fact that the spectral flow of $A(s)$ is the Fredholm index. Now in the proof ...
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Compactness properties in floer homology of cotangent bundles in the non-periodic case

Following the paper https://arxiv.org/pdf/math/0408280.pdf I have been interested in studying the case of solutions $x:[0,1] \rightarrow T^*M$ such that $x(0)\in T_{q_0}^*M$ and $x(1)\in T_{q_1}^*M$ ...
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Linking number of specific Reeb orbits in a toric domain ($S^3$ diffeomorphic)

Consider a toric domain defined by the region bounded on the first quadrant by a function $f:[0,a]\mapsto [0,b]$ with $a,b>0;f(0)=b,f(a)=0,f(x)>0 \hspace{2mm} \forall x\in [0,a)$. We know that $\...
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Using the removal of singularities theorem in $\mathbb{C}\mathbb{P}^1-\{0,\infty\}$ with lagrangian boundary conditions

Reading the paper "Floer Cohomology of Lagrangian intersections" the authors construct a map $f: \mathbb{R}^n \times [0,2^N]\rightarrow \mathbb{C}\mathbb{P}^n$ such that $f(\tau,0)=f(\tau,2^...
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7 votes
1 answer
273 views

Cotangent bundles of surfaces as varieties

As far as I understand, it is easy to see (and find in the literature) that the affine variety $$z_1^2+z_2^2+z_3^2=1$$ with the restriction of the standard $\omega_{std}$ of $\mathbb{C}^3$ is ...
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Are there paths of non-degenerate $2$-forms joining two symplectic structures on open $4$-manifolds?

There is a Theorem of Conolly, L$\hat{\text{e}}$ and Ono which states that on a closed simply-connected $4$-manifold two cohomologous symplectic forms can be joined by a path of non-degenerate $2$-...
Atilda the Hun's user avatar
3 votes
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Properties of $I_{\mu}$ for Lagrangian Floer Homology in the Cotangent bundle

Following the notation of the book "Lagrangian intersection Floer theory anomaly and obstruction" suppose we have that our symplectic manifold is a cotangent bundle $T^*M$ with the canonical ...
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Higher genus (Hamiltonian perturbed) holomorphic curves in cotangent bundle of S^1

Consider $T^*S^1$ as symplectic manifold, with hamiltonian function $H(x,y) = y^2$ (y is the fiber direction, I know this is morse bott but it can be perturbed). consider the set of maps $u: \Sigma \...
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11 votes
1 answer
402 views

A theorem about the symplectic geometry of projective bundles

I am trying to understand the following theorem about symplectomorphisms of projective bundles. Theorem 1.5 of "Characteristic Classes in Symplectic Topology" A.G. Reznikov. Selecta ...
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How does the Maslov index of a loop `project’ to the rotation number?

I’m trying to learn some Legendrian contact homology and the grading of the generators of the DGA are given by computing a fractional rotation number. In the symplectisation, this number is the Conley-...
no_idea's user avatar
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1 answer
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(Contradiction) All symplectic manifolds are holomorphic

I’m studying symplectic manifolds and almost complex structures. This lead to two propositions: Proposition 1 (from da Silva’s Lectures on Symplectic Geometry): If $J_0$ and $J_1$ are almost complex ...
Anthony D'Arienzo's user avatar
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38 views

Is the forgetful map a submersion away from the nodal points?

Let $\overline{\mathcal{M}}_{0,n}$ be the moduli space of stable curves of genus zero with $n$ marked points. For $n \geq 4$ we have a forgetful map $\pi \colon \overline{\mathcal{M}}_{0,n}\rightarrow ...
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0 answers
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Displacing a conormal Lagrangian from the zero section

I was told that the conormal bundle $\nu^*K$ of a knot $K\subset S^3$ can be displaced from the zero section $0_{S^3}$ in $T^*S^3.$ Having no intuition about whether/how often this happens in general, ...
Filip's user avatar
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1 vote
0 answers
517 views

On prequantization bundles over integral symplectic manifolds

I am trying to clarify certain subtleties regarding prequantization bundles over symplectic manifolds, for which I haven't found any clear explanation so far. Let me fix some definitions first. ...
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3 votes
0 answers
99 views

Continuation map interpolating two quadratic Hamiltonians with respect to different contact boundaries

Let $(M,\lambda)$ be a Liouville manifold. Consider two different contact boundaries $\partial_{\infty}^1M$ and $\partial_{\infty}^2M$ with respect to the same Liouville flow $Z$. Each of them ...
ChiHong Chow's user avatar
6 votes
0 answers
490 views

Hamiltonian dynamics on cotangent bundle

I'm stuck with the following claim made in Section 13.1 of Y-G. Oh's book "Symplectic topology and Floer homology". Assume that $N$ is a differential manifold and $S_0 ,S_1\subseteq N$ two ...
TheWildCat's user avatar
3 votes
1 answer
391 views

On some prerequisites for J-holomorphic curves and Gromov-Witten invariants

I'm currently reading through J-holomorphic curves and Quantum cohomology by McDuff and Salamon, and I've been facing some unfamiliarity issues with respect to the PDE and functional analysis tools ...
manav gaddam's user avatar
2 votes
0 answers
83 views

Two closed lagrangians embedded in $\mathbb{R}^{2n}$ can be shifted and perturbed to intersect at two points

In Polterovich paper "The surgery of Lagrange manifolds". It is stated in page 204 - in proof of Theorem 7a - the following, If $L_1$ and $L_2$ are two Lagrangian submaifolds of $\mathbb{C}^...
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1 vote
0 answers
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Definition of signs of isomorphisms $c_u : o(x_1) \to o(x_0)$ in the definition of Floer cohomology via Seidel's book

I'm reading Paul Seidel's book "Fukaya Categories and Picard-Lefschetz Theory", chapter 12, and I'm currently trying to understand the differential on Floer cohomology in terms of ...
Matija Sreckovic's user avatar
3 votes
0 answers
127 views

Moment map of $\mathrm{O}(n)$-action on $\mathbb{C}^n$

Let $(\mathbb{C}^n, \omega_0)$ be the complex Euclidean space of dimension $n$ with the standard Kähler structure $\omega_0$. I am looking for a Hamiltonian $\mathrm{O}(n)$-action on $(\mathbb{C}^n, \...
Math1016's user avatar
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8 votes
2 answers
558 views

Which curves are boundary of pseudoholomorphic curves?

I have posted it on Mathstackexchange but nobody replied. Consider a loop $\gamma:\mathbb{S}^1\to M^{2n}$ in a symplectic manifold $(M^{2n},\omega)$. Let $J$ be an $\omega$-compatible almost complex ...
Overflowian's user avatar
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10 votes
0 answers
764 views

Roadmap to Floer homotopy theory?

I am a young postdoc working in symplectic topology. Recently I became intrigued by Floer homotopy, especially after seeing it had been applied to classical questions in symplectic topology. (e.g. ...
1 vote
1 answer
339 views

Terminology for exact symplectomorphism

Let $(M,\omega = d\alpha)$ be an exact symplectic manifold. Then a symplectomorphism $\varphi \colon M \to M$ is said to be exact, iff $\varphi^*\alpha - \alpha$ is exact. Is there a terminology for ...
TheGeekGreek's user avatar
13 votes
1 answer
423 views

How not to use J-holomorphic curves [closed]

The field of symplectic topology is filled with subtle traps for the unwary, particularly when it comes to the analysis of $J$-holomorphic curves. So that the next generation of symplectic topologists ...
Anonymous Geometer's user avatar
2 votes
0 answers
92 views

Local contractibility of group of symplectomorphisms for open manifolds

It is well know that for a closed symplectic manifold $(M, \omega)$ the group of symplectomorphisms in locally contractible. The gist of this proof goes as follows. Given a $\psi \in \operatorname{...
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6 votes
1 answer
299 views

Is the symplectic quotient $\mu^{-1}(0)/G$ unique up to something?

Given a Hamiltonian action of a compact Lie group $G$ on a symplectic manifold $(M,\omega)$, we may choose a moment map $\mu \colon M\to \mathfrak{g}^* $ and obtain the symplectic reduction $M/\!\!/G =...
Gaussler's user avatar
  • 295
2 votes
0 answers
101 views

Augmentations of wrapped Floer cochains

Let $M$ be a closed, simply-connected spin manifold and let $F_b \subset T^*M$ be the cotangent fiber over a point $b \in M$. Let $CW^*(L,L)$ be the $A_{\infty}$-algebra of wrapped Floer cochains over ...
user142700's user avatar
3 votes
1 answer
360 views

Viterbo restriction map surjective on Weinstein neighbourhood

In a Liouville manifold $M$ having a Liouville subdomain $i: N \hookrightarrow M$, there is the so-called Viterbo restriction map in symplectic cohomology $$SH^*(i): SH^*(M)\rightarrow SH^*(N).$$ In ...
Filip's user avatar
  • 1,617
2 votes
1 answer
188 views

Superlevel sets of a parametrized quadratic forms

Let $N$ be an odd integer, $n\in\mathbb{N}$, and $-\frac{2T}{NR^2}\leq a\leq0$ for given $R,T\in\mathbb{R}$ with $\frac{T}{NR^2}\leq\frac{\pi}{2}$. Now consider the quadratic form $\Omega(a)=\sum_{l\...
SoYu's user avatar
  • 203
4 votes
2 answers
396 views

Contactomorphisms have in general no fixed points

Let $(V, \xi = \ker \alpha)$ be a cooriented contact manifold. A contactomorphism of $(V, \xi)$ is a diffeomorphism $\phi$ which preserves the contact structure $\xi$ and its coorientation. In other ...
BrianT's user avatar
  • 1,197
2 votes
1 answer
178 views

Comparing the minimal Chern number and the cup-length of a symplectic manifold

Let $(M,\omega)$ be a symplectic manifold. One can define its minimal Chern number $N_M$ as: $$ N_M := \text{inf} \lbrace k > 0 \ |\ \exists A \in H_2(M; \mathbb{Z}), \langle c_1, A \rangle = k \...
BrianT's user avatar
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7 votes
1 answer
645 views

Lagrangian intersection Floer homology: understanding some assumptions

Let $(X,\omega)$ be a symplectic manifold and $L\subset X$ be a Lagrangian subspace. Let $\mu_L:H_2(X,L;\mathbb{Z})\to \mathbb{Z}$ be the Maslov index homomorphism. Usual hypothesis Recall that $L$...
Overflowian's user avatar
  • 2,523
2 votes
1 answer
482 views

Two Lagrangian submanifolds with clean intersections

Having two closed exact Lagrangian submanifolds $L_1$ and $L_2$ that intersect cleanly inside a Liouville manifold $M$ with $c_1(M)=0,$ is there (with possibly some other conditions) any relation ...
Filip's user avatar
  • 1,617
10 votes
0 answers
1k views

Roadmap to understanding Gromov's Non-squeezing theorem

I'm a graduate student starting out to venture into the areas of Symplectic Geometry/Topology, and was somewhat motivated by the essence of Gromov's non-squeezing theorem which in a sense made me feel ...
manav gaddam's user avatar
10 votes
0 answers
394 views

Words and ranks

Let me state two problems that look very much alike. The first one can be solved putting together answers that different people have given to some questions I asked here a few weeks ago. The second ...
H A Helfgott's user avatar
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5 votes
1 answer
448 views

Diffeomorphic but not isotopic symplectic forms

Do we know of any closed symplectic manifold $M$ with 2 cohomologous symplectic forms $\omega_1$ and $\omega_2$ such that there exist $\psi \in \text{Diff}(M)$ and $\psi^* \omega_1 = \omega_2$ but $\...
cr1t1cal's user avatar
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3 votes
1 answer
185 views

Star-shaped domain in $\mathbb{C}P^2$

Consider $(\mathbb{C}P^2,\omega_{FS})$ where $\omega_{FS}$ is the standard Fubini-Study form. Let $L$ denote a sphere in $\mathbb{C}P^2$ in the class $\mathbb{C}P^1$. Further let $\int_{L} \omega_{FS} ...
cr1t1cal's user avatar
  • 755
5 votes
1 answer
480 views

Lagrangian torus fibrations and Arnol'd-Liouville theorem

Let $(X, \omega)$ be a closed symplectic manifold of dimension $2n$ and $\pi: X \rightarrow Q$ a Lagrangian torus fibration. Let $F_q$ denote the fiber at $q \in Q$. It is claimed in a paper of ...
John Rached's user avatar
4 votes
1 answer
133 views

Orientable surface bundle

Is it true that every orientable surface bundle can be made into a symplectic fibration?If yes, why? What about the particular case that $M$ is a connected compact 4-manifold?
Ramtin.VA's user avatar
  • 207
3 votes
1 answer
227 views

Compactly supported symplectomorphisms of $D^2$

I'm trying to understand a more basic version about Gromov's theorem about the compactly supported symplectomorphisms of $D^2 \times D^2$ being contractible. Consider the dimensional disk $D^2 \...
cr1t1cal's user avatar
  • 755
2 votes
0 answers
359 views

The norm squared of a moment map

I am studying the paper by E. Lerman: https://arxiv.org/abs/math/0410568 Let $(M,\sigma)$ be a connected symplectic manifold with a Hamiltonian action of a compact Lie group $G$, so that there exist ...
user56980's user avatar
  • 432
4 votes
1 answer
324 views

Every symplectic submanifold is J-holomorphic

I am trying to show that every symplectic submanifold $N$ of a 2n- dimensional symplectic manifold $(M,\omega)$ is J-holomorphic for some compatible almost complex structure $J$. The way I am ...
cr1t1cal's user avatar
  • 755
1 vote
0 answers
71 views

Invariance under tame almost complex structure of the fibre tangent space of the symplectic normal bundle

I am trying to understand the construction of symplectic inflation and I am stuck in the following point. Suppose we have a 4 dimensional symplectic manifold $(M, \omega)$. Also suppose that $N \...
cr1t1cal's user avatar
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5 votes
0 answers
241 views

Legendrian surgery and invertible elements in zeroth degree symplectic cohomology

Is there anything known about the relation between Legendrian handle attachment and invertible elements in $\mathit{SH}^0(M)$? As the simplest interesting case, take $M_0$ to be the cotangent bundle $...
YHBKJ's user avatar
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2 votes
0 answers
187 views

What is the relation between the different generating functions thought as finite approximations of action functionals

In the book Introduction to symplectic topology by MC Duff and Salamon, a discrete analogue of the action functional is defined on $\mathbb{R}^{2n}$. The idea is that a Hamiltonian isotopy can be ...
BrianT's user avatar
  • 1,197
4 votes
0 answers
143 views

Mixed characteristic in symplectic geometry

Are there any mixed-characteristic phenomena in symplectic geometry/mirror symmetry? There are papers on symplectic geometry by Abouzaid (inspired by Kontsevich--Soibelman, I believe) in which there ...
user avatar
6 votes
1 answer
580 views

Relationship between Gromov-Witten and Taubes' Gromov invariant

Fix a compact, symplectic four-manifold ($X$, $\omega$). Recall Taubes' Gromov invariant is a certain integer-valued function on $H^2(X; \mathbb{Z})$ defined by weighted counts of pseudoholomorphic ...
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