# Tagged Questions

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### Connectedness of the space of symplectic embeddings into a higher dimensional manifold

Suppose $M$ and $N$ are symplectic manifolds, $N$ is compact, $\dim_{\mathbb R}(N) \leq \dim_{\mathbb R}(M) -4$. Suppose there are embeddings $f_i:N \to M$, $i=0,1$ such that $f_i^*\omega_M$ is non-...
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### Triangulation of S^2xS^2

Could someone tell me or give a reference for the minimal triangulation of $S^2\times S^2$ and $S^2\times S^1\times S^1$ ? Thanks,
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### Open Hamiltonian Gromov-Witten Invariants

Both open Gromov-Witten invariants and Hamiltonian Gromov-Witten invariants have been studied. I am interested in knowing whether anyone has considered open Hamiltonian Gromov-Witten invariants before....
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### Shape of the bubbling limit of holomorphic discs

I will present my question in the specifics I encountered it, so maybe some of the details are irrelevant for the desired conclusion. Consider $(S^2\times S^2,\omega_{std})$ the product of two ...
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### holomorphic curves in almost toric fibration and their relation to tropical curves

My goal is to get better understanding how the projection of holomorphic curves converge to tropical disks. We are given an almost toric fibration $X\rightarrow B$ with special Lagrangian fibers with ...
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### Handle attachment in symplectic category

It is known that for an exact symplectic manifold $(M,\omega_M)$ with a convex boundary $(\partial M,\theta_M)$, where $d\theta_M=\omega_M$ (usually called a Liouville domain), one can attachment to ...
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### What is the formula for the homology class represented by the diagonal?

Let $M$ be a compact oriented manifold and $\{\mu_{i}\}$ a basis for the homology $H_*(M, \mathbb{Z})$ (we are ignoring any torsion). Now consider the diagonal $\Delta_{M}$ inside $M\times M$. ...
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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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### 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 ...
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### 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 ...
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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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### 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 ...
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### 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 ...
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### 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 ...
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### 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 h-...
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### 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 (http://arxiv.org/abs/math/...
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### 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 ...
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### 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 ...
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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 ...
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 = ...
### $C^0$ 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 ...