# Questions tagged [symplectic-topology]

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### 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 ...

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### 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 \...

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### 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 $...

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### 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 ...

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### 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 ...

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### 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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### 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 ...

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### Maslov index of pair of paths in $\mathcal{L}(2n)$ and its relation with the Maslov index of a loop in $\mathcal{L}(2n)$

I'm reading [RS] and I was wondering what kind of connection there is between the Maslov index for a pair of paths $\lambda_0,\lambda_1 \colon [a,b] \to \mathcal{L}(2n)$ as defined in [RS] and the ...

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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 ...

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### 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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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...