All Questions
Tagged with floer-homology sg.symplectic-geometry
81 questions
6
votes
0
answers
208
views
Does this pseudo-holomorphic triangle contribute to the product $\mu_2$ in Lagrangian Floer cohomology?
I'm computing the product map $$\mu_2 : CF(L_0,V)\otimes CF(V,L_1)\to CF(L_0,L_1)$$ in Seidel's exact triangle for this specific case:
This is a genus 2 surface, and I color-coded the three (...
- 2,018
1
vote
0
answers
92
views
Lagrangian Floer theory for negative monotone symplectic manifolds and Lagrangians
In the paper "Floer cohomology of Lagrangian intersections and pseudo-holomorphic Disks I", Oh shows that for a compact monotone Lagrangian $L$ in a closed monotone symplectic manifold $M$ ...
- 791
2
votes
0
answers
82
views
Why should we restrict the multiplicitiy of hyperbolic orbit to be one in Embedded contact homology?
Embedded contact homology(abbreviated by ECH) is a Floer type theory specially designed for three dimensional contanct manifolds(or generally, manifold with stable Hamiltonian structure) invented by ...
- 285
2
votes
0
answers
60
views
Confusion about proof of $C^0$ bounds for Floer curves on cotangent bundles
I have trouble understanding the proof of theorem 5.4 from Cielibak's article "Pseudo-holomorphic curves and periodic orbits on cotangent bundles". At the bottom of page 267 he defines a ...
- 21
6
votes
0
answers
225
views
Is Heegaard-Floer homology the Lagrangian Floer homology of $M=\text{Sym}^g(\Sigma), L_0=\mathbb{T}_{\alpha},L_1=\mathbb{T}_{\beta}$?
Is Heegaard Floer homology the Lagrangian Floer homology of $M=\text{Sym}^g(\Sigma), L_0=\mathbb{T}_{\alpha},L_1=\mathbb{T}_{\beta}$? I am interested in the relationship between the theories
...
- 131
2
votes
0
answers
107
views
Product structures in Rabinowitz Floer homology
Let $(M,d\lambda)$ be a compact exact symplectic manifold and $\overline{M}$ its symplectic completion. For simplicity we can think of $\overline{M}$ has a cotangent bundle and $\partial M$ the sphere ...
- 791
3
votes
0
answers
320
views
Bubbling off a sphere in a splitting/stretching manifold
This question is related to my old question Bubbling off of a pseudo holomorphic sphere on surface with cylindrical ends about the bubbling off argument in Seidel's paper The symplectic Floer homology ...
- 2,018
1
vote
0
answers
90
views
Index of Floer operator for Hamiltonian vs Lagrangian Floer Homology
I am trying to see if there is a way to translate the computation of the index of the Floer operator for Hamiltonian Floer to Lagrangian Floer. Hamiltonian Floer homology is a theory that counts (...
- 111
1
vote
0
answers
119
views
Seidel's calculation of the Floer cohomology of a cotangent fibre and its Dehn twist
I am reading Seidel's paper on exact Lagrangian submanifolds in $T^*S^n$ and the graded Kronecker quiver, and in Lemma 2 (2) he claims the following fact: if $F_0$ is a cotangent fibre and $F_1$ is $\...
- 143
7
votes
1
answer
680
views
Relation between symplectic (co)homology and Hochschild (co)homology and deformations
A very fluffy question in which I'm ignorant of homology/cohomology, grading etc:
The open-closed and closed-open string maps relating the symplectic (co)homology and Hochschild (co)homology of the ...
- 73
4
votes
1
answer
227
views
Choice of a family of almost complex structures when defining Floer Homology
Consider a $1$-periodic Hamiltonian $H:S^{1}\times M\rightarrow \mathbb{R}$ defined on a compact symplectic manifold $M$. Let's suppose $M$ is nice enough so that we can develop Floer theory on it. ...
- 791
2
votes
1
answer
181
views
Generic choice of non-degenerate Hamiltonians $H$ in Floer theory
When developing floer theory for an Hamiltonian $H:M\times S^{1}\rightarrow \mathbb{R}$ we will want $H$ to satisfy a non-degenerancy condition, that is, for every $x\in \mathcal{P}(H)$, a periodic ...
- 791
1
vote
0
answers
241
views
Definition of Floer complex in Floer's "Morse theory for Lagrangian intersections"
I am moving the first steps into Lagrangian Floer theory and I am trying to understand the construction, as in the original paper, for the field $\mathbb{Z}_2$ (no orientations) and $\pi_2(P,L) = 0$.
...
- 178
2
votes
1
answer
246
views
Linearization of the Floer equation
In Floer Homology we want to prove that the Moduli spaces $\mathcal{M}(x^{-},x^{+})$ are finite dimensional manifolds. This is done by expressing them as the zero set of a Fredholm map. First one ...
- 791
1
vote
0
answers
94
views
Gluing of Morse-type trajectories in "Floer Homology of Cotangent bundles"
In the paper by A. Abbondadolo and M.Schwarz, "On the Floer homology of cotangent bundles" arXiv link, to prove the desired isomorphism between the Floer homology and the Morse homology of ...
- 791
1
vote
0
answers
79
views
Gluing of hybrid trajectories in Floer homology
In the paper by A. Abbondadolo and M.Schwarz, "On the Floer homology of cotangent bundles" arXiv link, to prove the desired isomorphism between the Floer homology and the Morse homology of ...
- 791
2
votes
0
answers
154
views
Motivation behind the usual setting of the (weak) Arnold conjecture for fixed points of an hamiltonian diffeomorphism
I'm trying to find out the motivations that led V. Arnold to formulate his famous conjecture (I guess theorem by now) in the following form:
Let $(M,\omega)$ be a closed symplectic manifold (add ...
- 2,018
3
votes
1
answer
227
views
Influence of symplectic invariants of the complement on being superheavy
Let $(M,\omega)$ be a symplectic manifold.
I'm trying to show that a compact subset $K\subset M$ is $1$-superheavy [1, Definition 1.3] where $1=PD([M])$ is the unit in $QH^0(M)$.
My question is: How ...
- 186
1
vote
0
answers
121
views
Cup product and PSS map
Let $(M,\omega)$ be a symplectic manifold and let $H$ a Hamiltonian function. If $M$ is not closed we consider $H$ to be linear at infinity to ensure that $HF^*(H)$ is well-defined (I'm particularly ...
- 186
3
votes
0
answers
114
views
Continuation principle for solutions of Floer's equation in $\mathbb{R}\times [0,1]$ and transversality
Consider $(M,\omega)$ a symplectic manifold and $J$ a compatible almost complex structure. For me it's well known that if we consider 2 solutions $u,v:\mathbb{R}\times S^1\rightarrow M$ of Floer's ...
- 791
4
votes
0
answers
198
views
Writting the Floer map in local coordinates using the exponential chart
Following Salamon's Notes in Floer Homology , consider the Floer equation
$$\mathcal{F}(u):=\partial_su+J_t(\partial_tu+\nabla H_t(u))=0$$
Then we can write in local coordinates
$$\mathcal \Phi_u^{-1}(...
- 791
4
votes
2
answers
448
views
Dismissing pseudoholomorphic curves in embedded contact homology
In the papers
The periodic Floer homology of a Dehn twist,
Rounding corners of polygons and the embedded contact homology of $T^3$,
and Combinatorial embedded contact homology for toric contact ...
- 191
2
votes
1
answer
232
views
Associativity of orientations of determinant bundles in Floer homology
I have been reading the paper "Coherent orientations for periodic orbits problems in symplectic geometry" by Floer and Hofer, trying to understand how we can orient the moduli spaces that ...
- 791
2
votes
0
answers
163
views
Compactness of Moduli spaces in Lagrangian Floer Cohomology
I have been reading Denis Aurox lecture notes on Fukaya Categories https://arxiv.org/pdf/1301.7056.pdf , and in page $9$ he starts to discuss the compactness properties of the moduli spaces and how we ...
- 791
6
votes
0
answers
268
views
Computation of the Fredhom index in Floer theory
I have been reading Salamon's lecture notes on Floer homology, and to compute the Fredholm index of operators they use the fact that the spectral flow of $A(s)$ is the Fredholm index. Now in the proof ...
- 791
4
votes
0
answers
385
views
Some clarifications on the PSS isomorphism in Hamiltonian Floer cohomology
I'm looking for some help in understanding the PSS isomorphism map in the context of Hamiltonian Floer cohomology and Morse cohomology with universal Novikov coefficients $\Lambda_{\omega}$ (à la ...
- 2,018
3
votes
1
answer
282
views
Computing the Fredholm index in Floer theory
In Salamon's notes on Floer homology, it's claimed that under some non-degenerancy assumptions the operator $$D:= \partial_s+J_0\partial_t+S(s,t): W^{1,p}(\mathbb{R}\times S^1,\mathbb{R}^{2n})\...
- 791
1
vote
0
answers
96
views
Compactness properties in floer homology of cotangent bundles in the non-periodic case
Following the paper https://arxiv.org/pdf/math/0408280.pdf I have been interested in studying the case of solutions $x:[0,1] \rightarrow T^*M$ such that $x(0)\in T_{q_0}^*M$ and $x(1)\in T_{q_1}^*M$ ...
- 791
1
vote
0
answers
119
views
Examples and classification of holomorphic strips in $(\mathbb{C}\mathbb{P}^n,\mathbb{R}\mathbb{P}^n)$
Consider an exact isotopy $\phi_t$ of $\mathbb{C}\mathbb{P}^n$ such that $\phi_1(\mathbb{R}\mathbb{P}^n)\pitchfork \mathbb{R}\mathbb{P}^n$. When trying to compute the Lagrangian Floer cohomology of $(\...
- 791
1
vote
0
answers
145
views
Bubbling of disks when proving compactness properties in Lagrangian floer cohomology
When defining lagrangian floer cohomology , as it's done in the paper "Floer cohomology of lagrangian Intersections and pseudo holomorphic disks I,II" we will need to look into the ...
- 791
2
votes
0
answers
71
views
Using the removal of singularities theorem in $\mathbb{C}\mathbb{P}^1-\{0,\infty\}$ with lagrangian boundary conditions
Reading the paper "Floer Cohomology of Lagrangian intersections" the authors construct a map $f: \mathbb{R}^n \times [0,2^N]\rightarrow \mathbb{C}\mathbb{P}^n$ such that $f(\tau,0)=f(\tau,2^...
user174565
1
vote
1
answer
247
views
Maslov index equal to $2$ implies that the disk is not multiply covered
In the paper Floer cohomology of lagrangian intersections and pseudo-holomorphic disks in page $1009$ the authors claim that a map $w:D^2\rightarrow S^2$ with $w|_{\partial D^2}\subset L$, where $L$ ...
user174565
2
votes
0
answers
105
views
Linearization of $\bar \partial_J$ in the paper Floer cohomology of Lagrangian intersecitons and pseudo-holomorphic discks 2
Reading the paper Floer cohomology of Lagrangian intersecitons and pseudo-holomorphic discks 2, in page $1004$ the authors want to prove that the linearization of $\bar \partial_J$ is surjective for ...
user174565
2
votes
1
answer
167
views
$\mathbb{R}\mathbb{P}^n$ is monotone in $\mathbb{C}\mathbb{P}^n$, Lagrangian floer cohomology
Following the paper "Floer cohomology of lagrangian intersections and Pseudo-Holomoprhic discks 2" by OH, it is mentioned that $\mathbb{R}\mathbb{P}^n$ is monotone in $\mathbb{C}\mathbb{P}^n$...
user174565
6
votes
0
answers
175
views
Composition of coproduct and product in Lagrangian Floer (co)homology
Let's take a Riemann surface $\Sigma$ and three Lagrangians $L_0,L_1,L_2$ in general position. let's assume that we can set up Lagrangian Floer (co)homology - Here I'm being vague because I don't want ...
- 2,018
4
votes
0
answers
135
views
Choice of almost complex structure in Seidel's Symplectic Floer Homology of a Dehn twist
I'm looking for a clarification of a construction done in Seidel's Symplectic Floer Homology of a Dehn twist: I don't get why his choice of almost complex structure on $\Sigma$ is a valid one for ...
- 2,018
1
vote
0
answers
189
views
Relating the Morse index with the Maslov index
In the following paper https://arxiv.org/pdf/math/0408280.pdf there is created an isomorphism between the Floer Homology of an hamiltonian functional $H$ in the cotangent bundle and the the Morse ...
- 791
4
votes
1
answer
359
views
Lagrangian Floer (co)homology, Novikov coverings and exact symplectic manifolds
I started reading the book "Lagrangian intersection Floer theory anomaly and obstruction", and there are a couple of details and assumptions in the definition of the Novikov covering that I ...
- 791
16
votes
1
answer
2k
views
Why is embedded contact homology so powerful?
The Embedded Contact Homology (ECH), introduced by M. Hutchings, is an invariant of (contact) three-manifolds. Since its introduction, well-known conjectures in symplectic/contact topology in ...
- 915
5
votes
0
answers
98
views
Lagragian floer homology vs homology of $\Omega(L_0,L_1)$
I'm very new to this subject, so apologies for a very naive question and probably many mistakes. Let $M$ be some compact sympletic manifold with $L_0,L_1$ Lagrangian submanifolds which intersects ...
- 2,356
3
votes
0
answers
102
views
Continuation map interpolating two quadratic Hamiltonians with respect to different contact boundaries
Let $(M,\lambda)$ be a Liouville manifold. Consider two different contact boundaries $\partial_{\infty}^1M$ and $\partial_{\infty}^2M$ with respect to the same Liouville flow $Z$. Each of them ...
- 461
2
votes
0
answers
69
views
Definition of signs of isomorphisms $c_u : o(x_1) \to o(x_0)$ in the definition of Floer cohomology via Seidel's book
I'm reading Paul Seidel's book "Fukaya Categories and Picard-Lefschetz Theory", chapter 12, and I'm currently trying to understand the differential on Floer cohomology in terms of ...
- 281
10
votes
0
answers
897
views
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. ...
3
votes
0
answers
200
views
Conley Zehnder index for Floer homology of a symplectomorphism
I'm trying to get some intuition for the Conley-Zehnder index in the setting of Floer homology of a symplectomorphism $\phi : (M,\omega) \to (M,\omega)$. Let's assume that $\phi$ only has non-...
- 2,018
6
votes
0
answers
276
views
Is there an symplectic field theory compactness theorem applicable in the context of Floer cohomology of a symplectomorphism?
Is there any reference in the literature about results regarding symplectic field theory (SFT) compactness for a neck-stretch in the context of Floer homology of a symplectomorphism $\phi \colon (M,\...
- 2,018
6
votes
1
answer
718
views
Bubbling off of a pseudo holomorphic sphere on surface with cylindrical ends
I need some clarification about the reason why we have a sphere bubbling off in the situation described by Seidel in The Symplectic Floer Homology of a Dehn Twist.
I’ll try to summarize to the best ...
- 2,018
7
votes
1
answer
790
views
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 $L$...
- 2,533
3
votes
0
answers
233
views
Locality in Floer theory
There appears to be a dearth of resources and references for the question of 'locality' in Floer theory. In particular, I cannot seem to find any complete statement of what people refer to as '...
- 31
4
votes
0
answers
105
views
How is the instanton Floer homology of Seifert fibrations related to that of a trivial fibration
My question centers around the relationship of the Chern-Simons theories of a Seifert fibration and the trivial product space $\Sigma_g \times S^1$, and its implication for instanton Floer homology. ...
- 1,155
1
vote
1
answer
395
views
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 ...
- 755