Questions tagged [symplectic-topology]
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244
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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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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 ...
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2
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$(M,\omega)$ 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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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 $\mathfrak g$, acting on the symplectic manifold $(M,\omega)$ by symplectomorphisms). I'm ...
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Does Fukaya see all symplectic topology?
I recently had a debate with my friend about how much of symplectic topology is about Fukaya category. I thought that for the most part, symplectic topology is not about Fukaya category. Now, to prove ...
user137767
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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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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 $\textrm{Diff}(M)$ ...
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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 ...
- 15.3k
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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, \...
user74900
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Floer homology and status of the Arnold conjecture
The Arnold conjecture on a closed symplectic manifold $(M,\omega)$ says in the weakest version that for a non-degenerate Hamiltonian there are at least $k$ 1-periodic orbits where $k$ is the sum of ...
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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 ...
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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, ...
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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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Are the stiefel-Whitney classes of the tangent bundle determined by the mod 2 cohomology?
Let $G=\mathbb{Z}/2\mathbb{Z}$. Let $f\colon L \to N$ be a smooth map of connected smooth closed $n$-dimensional manifolds such that the induced map
$$f^* \colon H^*(N,G) \to H^*(L,G)$$
is an ...
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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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How are invariants represented in category theory?
I'm trying to better understand how to think about invariance in the setting of category theory.
In some cases it seems there's an obvious interpretation: for instance, the fundamental group $\pi_1$ ...
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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 ...
- 1,618
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2
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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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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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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 ...
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Current status of the linearity of mapping class group
In the paper A faithful linear-categorical action of the mapping class group of a surface with boundary it is claimed that it was (as of 2014) still unknown that mapping class group is linear, but the ...
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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 genus-...
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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 ...
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Real interpretations of Discontinuities in Floer homology
This question is motivated by the answer in this question (you don't 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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Maslov class as a relative cohomology class in $H^2(M, L)$
Let $(M, \omega)$ be a symplectic manifold and $L \subseteq M$ - a Lagrangian submanifold. I am trying to understand under what circumstances the Maslov homomorphism $I_{\mu, L} \colon \pi_2(M, L) \to ...
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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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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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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. ...
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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 ...
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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 ...
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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 $\#_{l}CP^{2}\#_{...
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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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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 $ \# \...
- 18.3k
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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 ...
- 2,008
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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 long!...
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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 ...
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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
$$ [\omega] \in H^2(M, ...
- 3,235
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Symplectic reduction of 4-manifolds with circle actions
Let $(M,\omega)$ be a $4$-dimensional closed symplectic manifold. Assume there exists a Hamiltonian $S^1$-action on $M$, let $\mu:M \to \mathbb{R}^*$ be its moment map and let $M_{\text{red}}=\mu^{-1}(...
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Compact simply-connected homogeneous symplectic manifold
I was reading a paper in which the authors use the fact that any compact simply-connected homogeneous symplectic manifold has non-zero Euler characteristic. They prove it by quoting a theorem by ...
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Symplectic reversing diffeomorphisms on a compact symplectic manifold
I Ask this question in MSE and I received interesting comments and ideas. I repeat the question here for more discussion:
Let $(M,\omega)$ be a compact symplectic manifold.
Is there always a ...
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Square root for Hamiltonian diffeomorphisms
Let $\psi_t: X\to X$, $t \in [0,1]$, be a path Hamiltonian diffeomorphism on a symplectic manifold $X$, given by functions $H_t$. If $H_t \equiv H$ is independent of $t$ then
$$ \psi_1 = \psi_{\frac{...
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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 ...
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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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Lagrangian Kleinian bottles
I remember some talks some time ago about proofs of nonexistence of Lagrangian Kleinian bottles in ${\mathbb C}^2$ for the standard symplectic structure, mentioning that this were the only compact ...
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Definition of "Lagrangian skeleton"
I'm looking for the precise definition of "Lagrangian skeleton", as I'm eventually going to give a talk on this topic. As I asked a professor in my university about references on Lagrangian skeleta, ...
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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 $L$...
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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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Example where Calabi invariant is nontrivial?
Let $D^2$ denote the closed unit disk in $\mathbb{R}^2$. Let $\omega := dx \wedge dy$ denote the standard area form on $\mathbb{R}^2$ (and on $D^2$ by restriction). Let $\phi$ be a diffeomorphism of $...
user74319
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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 ...
- 14k
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On exotic symplectic structures of smooth closed 4-manifolds
What are some known techniques and examples of exotic symplectic structures on a fixed smooth closed 4-manifolds [by exotic I mean two symplectic structures that are not symplectomorphic]. This sounds ...
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