Questions tagged [contact-geometry]
Contact manifolds, contact structures, contact forms, Reeb dynamics, Legendrian knots, contact homology, symplectic field theory
66
questions with no upvoted or accepted answers
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Is tightness decidable?
Given a contact structure on a three-manifold, is there an algorithm to decide whether or not it tight?
For concreteness' sake, let's agree to represent the given contact three-manifold via an open ...
12
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1
answer
869
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On a corollary of a paper by Colin and Honda
The question is about the last sentence of the last corollary of Stabilizing the monodromy of an open book decomposition by Vicent Colin and Ko Honda. This question is also related to this other ...
9
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546
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Schoenflies and symplectic topology
The final report from a workshop on Morse theory in low-dimensional and symplectic topology includes the following question, posed by Michael Hutchings: Can we apply symplectic geometry to solve the ...
7
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286
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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 ...
7
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444
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Correct notion of chain homotopy for linearized homology of augmented DGAs?
$\require{AMScd}$
Preliminaries: Let $(A,\partial)$ be a differential graded $k$-algebra with an augmentation $\epsilon$. That is, $\epsilon$ is a DGA map $\epsilon:(A,\partial) \to k$ where $k$ is ...
7
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205
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When do geodesics reconverge?
Say I stand at the north pole and talk; in sufficiently frictionless conditions, one imagines that someone standing at the south pole could listen.
More generally, say $M$ is a compact Riemannian ...
7
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201
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Biholomorphic neighborhoods of the boundary of Stein domains
Let $(X_1,J_1)$ and $(X_2,J_2)$ be Stein domains with the same contact boundary $(Y,\xi)$. Under what conditions does there exist a biholomorphism between a neighborhood of their respective boundaries ...
6
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586
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Gompf's invariant of $2$-plane fields
I am interested in low dimensional contact topology. These days I read "Handlebody construction of Stein surfaces" written by R. E. Gompf, and study an invariant $\theta (\xi)$. This invariant is ...
5
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231
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Overtwisted contact forms on open manifolds
I tried first at Math Stack Exchange but got no answers, so I thought maybe this question belongs here.
It is known that on closed $3$-manifolds the Reeb vector field of any contact form inducing an ...
5
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0
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245
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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 $...
5
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375
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Is there any known relationship between sutured contact homology and Legendrian contact homology?
On one hand, Colin-Ghiggini-Honda-Hutchings' construction (https://arxiv.org/abs/1004.2942) provides an invariant of a Legendrian $L$ in a closed contact manifold $(M,\xi)$ via the sutured contact ...
5
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196
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Contact geometry: approximation of Legendrian mappings
Let $\alpha$ be a standard contact form on $\mathbb{R}^{2n+1}$. We say that a map $f:\mathbb{R}^k\to\mathbb{R}^{2n+1}$ contact if $f^*\alpha=0$.
Question 1. Is it true that a $C^1$-contact ...
5
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Is there a simply connected contact manifold, "non-exactly" fillable, cappable, such that the whole thing is symplectically aspherical?
Is there an example of a simply connected contact manifold W, with a non exact symplectic filling $M_1$, (that is, $M_1$ is a symplectic manifold, with contact boundary $W$ and a Liouville vector ...
4
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125
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Overtwisted contact structures on $S^3$
All the isotopy classes of overtwisted contact structures are classified by the Hopf invariant. Are any of these contact structures contactomorphic?
Suppose $d_{3}(\xi_{n}) = n$, then my guess is that ...
4
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240
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Maximal Thurston--Bennequin number of boundary knot classes in contact handlebodies
Let $H$ be a contact handlebody. In other words, $H$ is a small regular neighborhood of a Legendrian graph in a contact $3$-manifold (wlog $\mathbb R^3$). Equivalently, $H=(\Sigma\times[0,1],dt+\...
4
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132
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Sheaves with specified singular support at infinity coming from hyperplane arrangements
Given a manifold $M$, we consider its cotangent bundle $T^*M$, and its cocircle bundle $T^\infty M$, quotienting out by the scaling action of the positive reals. Given a Legendrian submanifold $\...
4
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138
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Are these two arguments incompatible?
I want to understand why (if so) these two arguments are not incompatible. And if that's the case, which one is wrong.
First we have this paper (by Honda, Kazez and Matic). We look at the last Lemma ...
4
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123
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Bound on critical points of Lefschetz fibration over the disk with prescribed monodromy
Let $\phi$ be a right-veering diffeomorphism of a surface $\Sigma$ of genus $g$ and $r$ boundary components. Suppose that the diffeomorphism is freely periodic so if $M$ is the associated open book ...
4
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184
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Right-veering periodic automorphisms of surfaces that are not composition of right-handed Dehn twists
Basically the title of the question. For the sake of completeness I state an introduction to the question.
In "Right-veering diffeomorphisms of compact surfaces with boundary I" and II, the authors ...
4
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79
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Neighborhood of (singular) Legendrian with convex boundary
Let $(M, \xi)$ be a contact manifold, and $\Lambda \subset M$ a closed connected Legendrian with a tubular open neighborhood $W$. The question is that, can one find a smaller tubular neighborhood $U \...
4
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235
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Contact manifolds and pseudodifferential operators
By way of background, I am currently trying to understand the theory of pseudodifferential operators in the context of contact geometry. I have some knowledge of pseudodifferential operators on ...
4
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459
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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, ...
3
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Characterization of contact vector fields
Let $H$ be a subbundle of the tangent bundle $TM$ of a smooth manifold $M$.
A vector field $K$ on $M$ is contact if its flow $\Phi_K^t$ preserves $H$.
I found in many references the following ...
3
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115
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Smooth handle attachment vs Weinstein handle attachment
Given a closed smooth manifold $M$ of dimension $n$, to which we attach a $k$-handle $H_k$.
Take $T^{\ast} M$, can one realize $T^{\ast} (M\cup H_k)$ as a result of symplectic or Weinstein handle ...
3
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142
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Chekanov-Eliashberg Legendrian DGA with positive grading?
I was just looking back to some notes that I took a few years ago, when I was reading Etnyre's notes on Legendrian Contact Homology in $\mathbb R^3$ and I happened upon the following question that I ...
3
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317
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boundary connect sum of Ganatra-Pardon-Shende
In Section 3.4 of https://arxiv.org/pdf/1809.03427.pdf, Ganatra-Pardon-Shende define the boundary connnect sum of two exact conical Lagrangians in a Liouville domain. In particular, in Figure 10, they ...
3
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104
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Reference for "holomorphic contact geometry"
Just like holomorphic symplectic geometry is a complexification of real symplectic geometry, I am wondering is there any good survey paper or book talking about holomorphic version of real contact ...
3
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102
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Contact 3-manifolds with hyperkahler Stein fillings?
Is there any classification result on (homeomorphism type) of contact 3-manifolds $\Sigma$ that have Stein filling $W$ that is
1. Hyperkahler (s.t. Stein structure is the Kahler part of it)
2. not ...
3
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236
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Where can I find good surveys on Symplectic and Contact geometry
Are there any good survey articles in symplectic and contact geometry, which focus on the "big picture", i.e how this discipline fits into the mathematical world ?
In the symplectic case : I am ...
3
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217
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Lutz twist and open book decompositions
Let $M^3$ be a closed oriented 3-manifold, endowed with an open book decomposition. Consider a section of the open book, that is a knot $K \subset M$ disjoint from the binding and meeting every page ...
3
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149
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The isotopy class of a Boothby-Wang contact structure
Let $p:\Sigma\to M$ be a non-trivial principal $S^1$-bundle over a closed orientable surface $M$. Let's call $V$ the vector field generating the $S^1$-action. It is known that there exists a Boothby-...
2
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103
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Almost contact structures on spheres
I will write $(M,\xi)=\mathbf{OB}(P,\phi)$ to denote that $M$ admits an open book decomposition with page $P$ and monodromy $\phi$ supporting a contact structure $\xi$. I will focus on the case where $...
2
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79
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Weinstein fillings of a unit cotangent bundle
Given a closed, orientable manifold M, and its unit cotangent bundle $ST^{\ast}M$. I wonder under which conditions $ST^{\ast}M$ admits a subcritical Weinstein filling?
2
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94
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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-...
2
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124
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Convex surfaces with transverse boundary (contact geometry)
Suppose I have a compact surface $\Sigma$ in a contact 3-manifold, where the boundary $\partial\Sigma$ is transverse to the contact structure. Am I able to perturb $\Sigma$ rel boundary so that it is ...
2
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1
answer
188
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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\...
2
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0
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66
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«Euclidean» local systems
The moduli space of G-local systems on a surface is a fundamental object in mathematics. The cases $G=SU(2)$ and $G=SL_2(\mathbb{R})$ are of particular interest. Consider the group $E$ of isometries ...
2
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182
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Stabilizing an open book with Anosov piece
It was proven by Colin and Honda in Stabilizing the monodromy of an open book decomposition that any diffeomorphism can be made pseudo-Anosov and right-veering after a series of positive ...
2
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0
answers
308
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First Chern Class of Contact Structure which is not Torsion
Let $(M,\xi)$ be a closed connected $3-$dimensional contact manifold with contact structure $\xi$. It is known that the first Chern class $c_{1}(\xi)$ defines an element in $H^{2}(M;\mathbb{Z})$ and ...
2
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210
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contact structure on double branched covers of $S^3$
We can assign a natural contact structure to double branched cover of a transverse knot $k$ in the $3$-sphere with its standard contact structure $(S^3,\xi_{st})$, as described in:http://arxiv.org/pdf/...
2
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86
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Extendability of Contact Structures; Foliations of $S^2$
I am currently reading Eliashberg's paper on the classification of overtwisted contact structures (http://bogomolov-lab.ru/G-sem/eliashberg-tight-overtwisted.pdf). In it, there is the following ...
2
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164
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Homotoping a 2-plane field on a closed orientable 3-manifold to a contact structure
I am a beginner in Contact Geometry.
To prove that the inclusion $\text{Cont}(M)\hookrightarrow \text{Dist}(M)$ induces surjection at $\pi_0$ level, the closest I got was based on Ko Honda's notes.
...
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0
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63
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Looking for examples of 3rd-order contact transformations
In the Herglotz Lectures on Contact Transformations and Hamiltonian Systems, after going through contact transformations of the form $X=X(x,y,p)$, $Y=Y(x,y,p)$, $P=P(x,y,p)$ it is stated:
As a final ...
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vote
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57
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Lagrangian cobordisms from a Legendrian knot to its scaled version
Having a Legendrian knot L in $\mathbb R^3$ and its scale aL (the length of Reeb chords of it are scaled by a>0), are these two Legendrians Legendrian isotopic? Maybe weaker, is there an exact ...
1
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0
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159
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Does the blow-up preserve symplectic structure?
Let $X \subset \mathbb P(\mathfrak g)$ be an adjoint variety for a simple complex Lie algebra $\mathfrak g$ appearing in the last line of the Freudenthal Magic Square, that is $\mathfrak g \in \{F_4,...
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67
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In a contact Lie algebra, when is the Reeb vector a semisimple element?
Below is a question I have come across in my research, and it seems like a question that has been answered (or at least asked) in the past; however, I have been unable to find any references that ...
1
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0
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67
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Isotopy of open book supporting same contact structure
In dimension 3, the Giroux correspondence gives us a bijection between contact structures (up to isotopy) and open book decompositions (up to positive stabilisation). Moreover, Giroux shows that two ...
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164
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Maximal dimension guaranteed for integral manifolds of hyperplane distributions
To KSackel and anyone else has viewed this: I'm sorry my edits have been all over the place. I've tried to cut it down to my remaining curiosities, so there's less to wade through (and hopefully fewer ...
1
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0
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210
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Genericity of contact structures all of whose closed Reeb orbits are nondegenerate
First, a contact form $\alpha$ on $M$ with Reeb vector field $R$ is said to be non-degenerate if, for any point $p$ such that $\phi_T^R(p) = p$, we have $\det{(\textrm{id}_{T_pM} - (d \phi_T^R)_p)} \...
1
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How does the knot contact homology augmentation polynomial change under a surgery on the base manifold?
So for every knot $K \in S^3$, there is a knot contact homology of $K$, and we can find the augmentation variety for this homology. The defining polynomial is known as the augmentation polynomial $A(K)...