3
votes
0answers
150 views

Transversality in Bourgeois Oancea's non-equivariant contact homology

In their paper, "AN EXACT SEQUENCE FOR CONTACT- AND SYMPLECTIC HOMOLOGY", Bourgeois and Oancea defined, whenever there is sufficient transversality, a "non-equivariant contact homology". Essentially, ...
2
votes
2answers
534 views

Kenji Fukaya's Lecture series at Simons center

In the past decade, theory of Kuranishi structures on moduli space of pseudo-holomorphic curves has been in the center of debates between some mathematicians in the field of symplectic geometry. ...
1
vote
1answer
120 views

Each coadjoint orbit of a compact connected Lie group $G$ admits a $G$-invariant generalized complex structure

I am looking for a proof or counterexample for following assertion Each coadjoint orbit of a compact connected Lie group $G$ admits a $G$-invariant generalized complex structure (In sense of ...
0
votes
0answers
105 views

When the leaves of a distribution are compact

Let $(M,\omega)$, be a symplectic manifolds. A subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex polarization if $P$ is Lagrangian P involutive ...
0
votes
1answer
153 views

An example for the case tht if the leaves of polarization be non-compact then the polarized sections are not square integrable

Let $(M,\omega)$, be a symplectic manifolds. A subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex polarization if $P$ is Lagrangian P involutive ...
-1
votes
1answer
137 views
3
votes
0answers
197 views

Looking for a necessary and sufficient condition for the polarization $\mathbb{P}$ being positive

My question is about positivity of polarization in Geometric quantization theory. Let $\mathbb{P}$, be a complex polarization on symplectic manifold $(M,\Omega)$. For every $m\in M$, we can define a ...
2
votes
1answer
320 views

looking for an identity for higher jet bundle $J^kM$?

We know this fact that the first jet Bundle $J^1M$ is diffeomorphic with $T^*M×\mathbb{R}$.i.e, ($J^1M=T^*M×\mathbb{R}$) Is there something like this identity for higher jet bundle $J^kM$? I editted ...
0
votes
1answer
163 views

Coadjoint orbits and homogeneous symplectic $G$-manifolds

We know this important fact from A.A.Kirillov that : Every homogeneous symplectic $G$-manifold is locally isomorphic to an orbit in the coadjoint representation of the group $G$ or a central ...
1
vote
0answers
227 views

contact structure on 3 manifolds

every orientable closed 3-manifold admits a contact structure. how we can construct this contact structure? if the manifold is prime what will happen?
10
votes
2answers
1k views

What is Kirillov's method good for?

I am planing to study Kirillov's orbit method. I have seen Kirillov's method in several branch of mathematics, for instance, functional analysis, geometry, .... Why is this theory important for ...
7
votes
2answers
418 views

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 $ \# ...
7
votes
1answer
228 views

quasi conformal, area preserving homomorphisms of the disc

Restricting a quasi-conformal homeomorphism of the disc to the boundary gives a surjective homomorphism from $QC(D^2)$ (quasi-conformal homeos of $D^2$) to $QS(S^1)$ (quasi-symmetric homeos of the ...
1
vote
1answer
90 views

Is the space of half-dimensional symplectic linear subspaces of $\mathbb{R}^{4n}$ which trivially intersect $\mathbb{R}^{2n}\times\{0\}$ contractible?

I would like to know whether the subspace of the symplectic Grassmanian $Gr_{2n}^{Sp}(\mathbb{R}^{4n})$ consisting of symplectic linear subspaces in $\mathbb{R}^{4n}$ which have dimension $2n$ and ...
4
votes
1answer
206 views

Smooth projective toric varieties which are quotients of product of spheres and torii by a free torus action?

My question is just as in the box. Is every smooth projective toric variety diffeomorphic to a quotient of $\prod_i S^{n_i} \times T^k$ (I know torus is a one-sphere but I just wanted to make clear I ...
2
votes
1answer
189 views

Hamiltonian actions and contractible loops

Let $(M, \omega)$ be a symplectic manifold and $G$ be a compact Lie group. Suppose we have a Hamiltonian $G$-action on $M$, with moment map $\mu: M \to {\mathfrak g}^*$. We assume that the moment map ...
4
votes
1answer
583 views

an extended question of Gromov: Every **generalized open almost complex manifold** admits a **generalized symplectic structure**? [closed]

Definition (Open Manifolds):An open manifold is a manifold without boundary with no compact component. For a connected manifold, "open" is equivalent to "without boundary and non-compact. we know that ...
1
vote
2answers
225 views

Delauney triangulation in high (>20) dimensions

Hi all, I know that its very hard to find the Delauney triangulation of high dimensional spaces, especially if there are several thousand points that need to be triangulated. So I was wondering . . ...
10
votes
4answers
2k views

Geometric invariant theory for geometers

I am trying to learn "Geometric invariant theory" like it was introduced by Mumford. But I do not have a strong background in algebraic geometry since I work in geometric topology and geometry. So ...
1
vote
1answer
463 views

Problem:Gromov-Witten;Moduli space

Let us consider a map from a $\Sigma_g \longrightarrow N$, where $N$ is a symplectic manifold. Then we define the moduli space as $M= \{ f | f \mbox{ is a pseudoholomorphic map } \Sigma_g \to N, ...
1
vote
0answers
246 views

Transversality and isolated degenerate critical points

Maybe some of the following statements are not precise. Please correct them. Let $M$ be a compact smooth manifold. Let $f: M \to {\mathbb R}$ be a Morse function. Then a generic Riemannian metric $g$ ...
7
votes
1answer
400 views

Why is Heegaard Floer Homology defined in terms of Sym$^g\Sigma_g$ instead of Pic$^g\Sigma_g$?

Recall the definition of Heegaard Floer homology: $\Sigma_g$ is a closed surface, and $\{\alpha_1,\ldots,\alpha_g\}$ and $\{\beta_1,\ldots,\beta_g\}$ are sets of attaching circles. Then Heegaard ...
-4
votes
1answer
367 views

Symplectic forms and 1-forms

Suppose we have a real symplectic manifold $(M,\omega)$. Under what conditions can we find a global 1-form $\alpha$ such that $\omega = \alpha \wedge\alpha$? Obviously there are some simple ...
6
votes
4answers
595 views

Symplectic structures from Lagrangians?

In Witten's paper 'Quantization of Chern-Simons Gauge Theory with Complex Gauge Group,' he makes the statement (p. 35) that the symplectic form on the moduli space of flat $G_\mathbb{C}$ connections ...
9
votes
6answers
2k views

Introduction to Floer Theory?

Can anyone suggest a good overview/introduction of the Floer machine for a beginner? (Someone pointed out some intriguing connections to surface mapping class groups, which might be enough incentive ...
5
votes
2answers
444 views

Maslov index and heegard floer homology

I am an undergraduate who wants to learn Knot Floer homology. I was told to start with this expository paper which was working quite well until I reached the actual definition of the differential. ...
3
votes
1answer
371 views

Wanted: differential coming from higher genus surface in Heegaard Floer Homology

I am interested in studying moduli of complex surfaces which arise in computing the differential on the Heegaard Floer Homology chain complex. In particular, I am interested in the generic case, when ...
4
votes
1answer
284 views

path of almost complex structure in the definition of heegaard floer homology

In order to define Heegaard Floer Homology for a connected, closed, oriented 3 manifold, we fix a generic path of nearly symmetric almost complex strucutre $J_s$ over $Sym^g(\Sigma)$. By ...
4
votes
2answers
301 views

contactomorphism of $S^{2n+1}$ for n>1

Is there some result about contactomorphism groups of $S^{2n+1}$ or $T^{2n-1}$ for n>1? For example, do we know the rank of $\pi_{i}(Cont(S^{2n+1})) \otimes \mathbb{Q}?$ where "Cont" means the ...
2
votes
1answer
185 views

density of lagrangian grassmannian in usual grassmannian.

Consider the canonical symplectic structure $(\omega, J)$ on $\mathbb{R}^{2n}$. (i) What can be said about the density of the lagrangian grassmannian $L$ (i.e. those rank $n$ totally isotropic linear ...
16
votes
2answers
738 views

Can a Lagrangian submanifold of ${\mathbb R}^{2n}$ be dense ($n>1$)?

I'm betting `yes, sure!', but don't see it. Could someone please point me toward, or construct for me, a Lagrangian submanifold immersed in standard symplectic ${\mathbb R}^{2n}$ for $n > 1$, ...
8
votes
0answers
431 views

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 ...
2
votes
1answer
546 views

How to calculate Dr. Curt McMullen's expanding eigenvalues for totally degenerate groups?

What is required in order to derive the expanding eigenvalues of Dr. Curt McMullen's torus orbifold bundles over the circle and the corresponding totally degenerate groups, as presented in Section 3.7 ...
10
votes
1answer
572 views

Where are $+$, $-$ and $\infty$ in bordered Heegaard-Floer theory?

Here goes my first MO-question. I've just read Lipshitz, Ozsváth and Thurston's recently updated "A tour of bordered Floer theory". To set the stage let me give two quotes from this paper. ...
30
votes
4answers
2k views

Can cotangent bundles see exotic smooth structures?

I have two questions that are inspired by a couple of questions here on MO (referenced below), as well as by a conversation with some other grad students at a summer school. Caveat: I'm not a ...
3
votes
3answers
712 views

Is there a version of Seiberg-Witten-Floer or Heegard-Floer homology for 3-manifolds with boundary?

Recently, the Seiberg-Witten-Floer homology created by Kronheimer and Mrowka has important applications in Taubes' proofs of Weinstein conjecture and Arnold Chord Conjecture. Also, Cagatay Kutluhan, ...
7
votes
1answer
375 views

Symplectic structures on a homotopy complex projective space

For $n>2$, there are infinitely many differentiable structures on the homotopy type of $\mathbb{C}P^n$. I want to know which differentiable structures support a symplectic form. My question is as ...
4
votes
1answer
467 views

Negative intersection of symplectic submanifolds

For a symplectic 4-manifold, it is possible for two symplectic submanifolds to intersect negatively? Actually it is an exercise in the 4-manifold book by Gompf and Stipsicz to find symplectic planes ...
7
votes
1answer
514 views

Interpreting Witten's Asymptotic Expansion of the WRT invariant.

Witten's asymptotic expansion conjecture as described in "Problems on invariants of knots and 3-manifolds" in Geometry and Topology Monographs, Volume 4 states that ...
2
votes
1answer
472 views

isotropic deformation retract of Weinstein manifolds?

I found the following paragraph in the paper " Intro to symplectic field theory " which I don't understand what does it mean precisely? Suppose W is a symplectic (or Kahler) manifold. D, smooth ...
2
votes
1answer
272 views

Homotopy classes of complex bundle maps and isotropic immersions into contact manifolds

This is a follow-up question to my previous one where I was trying to understand the classes of Legendrian immersions of circles into contact manifolds. I'm interested in classifying isotropic ...
10
votes
2answers
1k views

Classification of symplectic surfaces

Is there a classification of symplectic surfaces, i.e. of surfaces equipped with an area form? Symplectic topology references like McDuff-Salamon seem to start their discussion of open questions with ...
2
votes
2answers
512 views

Why (and whether) is any smooth embedded torus in R^4 isotopic to an embedded Lagrangian torus?

The question is pretty self-explanatory; we are dealing with the standard symplectic structure on ℝ4. Some background: I'm reading the thesis "Lagrangian Unknottedness of Tori in Certain ...