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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 (...
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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 $M$ ...
Someone's user avatar
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2 votes
0 answers
68 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 ...
ChoMedit's user avatar
  • 265
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 ...
Rbr's user avatar
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194 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 ...
contingent's user avatar
1 vote
0 answers
79 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 ...
Someone's user avatar
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3 votes
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290 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 ...
Riccardo's user avatar
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69 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 (...
Y.H. Chan's user avatar
  • 111
1 vote
0 answers
89 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 $\...
B. S.'s user avatar
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1 answer
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Algebraic variations of the full knot Floer complex

In Hom's paper (arXiv link), p.20, Section 3.3 ends with "There are other algebraic modifications one may consider, such as setting $U^n = 0$ or $UV = 0$", referring to the knot Floer ...
horned-sphere's user avatar
7 votes
1 answer
565 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 ...
86846515312's user avatar
3 votes
0 answers
96 views

Why is this special symplectic loop contractible? (Floer Homology)

In the construction of Floer homology, one shows a formula that connects the Maslov index/ Conley–Zehnder index $\mu$ with the dimension of the moduli spaces of connecting gradient flow lines: $$\dim \...
Martin's user avatar
  • 149
4 votes
1 answer
202 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. ...
Someone's user avatar
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3 votes
1 answer
136 views

Perturbation of vector fields in Morse Homology

Recently I have been reading on Morse Homology. Suppose we have a compact manifold $M$ and a smooth function $f:M \rightarrow \mathbb{R}$ and a Morse vector field $X$ such that we can do Morse ...
Someone's user avatar
  • 781
2 votes
1 answer
144 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 ...
Someone's user avatar
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1 vote
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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 $\pi_2(P,L) = 0$. ...
EmarJ's user avatar
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0 answers
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Gluing maps in Floer Homology and boundary conditions

Recently, I have been trying to a construction of a gluing map regarding the Lagrangian Floer Homology of two fibers in the cotangent bundle $T^*M$ of a manifold , in order to prove that the map $\...
Someone's user avatar
  • 781
2 votes
1 answer
223 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 ...
Someone's user avatar
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3 votes
1 answer
242 views

Moduli space of flat connection over homology 3-sphere

I'm trying to understand the space of flat connections of the trivial $\mathrm{SU}(2)$-bundle over a closed, oriented homology three-sphere (for the purpose of understanding the instanton Floer ...
Lamda8's user avatar
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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 ...
Someone's user avatar
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1 vote
0 answers
73 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 ...
Someone's user avatar
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1 vote
0 answers
77 views

Banach manifold structure on the moduli space of hybrid trajectories

I am reading the paper "On the Floer homology of cotangent bundles", (arXiv link) , by Abbondandolo and Schwarz and in page $35$ to define the isomorphism between the Morse complex and the ...
Someone's user avatar
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2 votes
0 answers
140 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 ...
Riccardo's user avatar
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5 votes
1 answer
156 views

Transversality and $C^l$, $C^{\infty}$ spaces of almost complex structures

Recently I have been trying to get a grip on transversality results in Floer homology. That is suppose we the section $\partial_{J,H}: W^{1,p}(u^*(TM))\rightarrow L^{p}(u^*(TM))$ and we want to prove ...
Someone's user avatar
  • 781
3 votes
1 answer
218 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 ...
bas's user avatar
  • 186
1 vote
0 answers
101 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 ...
bas's user avatar
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3 votes
0 answers
103 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 ...
Someone's user avatar
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4 votes
0 answers
191 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}(...
Someone's user avatar
  • 781
4 votes
2 answers
413 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 ...
kvicente's user avatar
  • 181
2 votes
1 answer
217 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 ...
Someone's user avatar
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2 votes
0 answers
123 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 ...
Someone's user avatar
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6 votes
0 answers
261 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 ...
Someone's user avatar
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4 votes
0 answers
301 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 ...
Riccardo's user avatar
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3 votes
1 answer
258 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})\...
Someone's user avatar
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1 vote
0 answers
91 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 $x(1)\in T_{q_1}^*M$ ...
Someone's user avatar
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1 vote
0 answers
117 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 $(\...
Someone's user avatar
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1 vote
0 answers
133 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 ...
Someone's user avatar
  • 781
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^...
user avatar
1 vote
1 answer
215 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 $L$ ...
user avatar
2 votes
0 answers
101 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 ...
user avatar
2 votes
1 answer
165 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 $\mathbb{C}\mathbb{P}^n$...
user avatar
6 votes
0 answers
159 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 ...
Riccardo's user avatar
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5 votes
1 answer
468 views

Invariance of morse homology, doubt in proof in book "Morse Theory and Floer homology"

I am reading the book "Morse theory and Floer Homology" by Michele Audin and Mihai Damian. Now I am reading the proof of the following theorem. Link to the statement of the theorem ...
Luis Carlos 's user avatar
4 votes
0 answers
130 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 ...
Riccardo's user avatar
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1 vote
0 answers
163 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 ...
Someone's user avatar
  • 781
3 votes
0 answers
110 views

Properties of $I_{\mu}$ for Lagrangian Floer Homology in the Cotangent bundle

Following the notation of the book "Lagrangian intersection Floer theory anomaly and obstruction" suppose we have that our symplectic manifold is a cotangent bundle $T^*M$ with the canonical ...
Someone's user avatar
  • 781
4 votes
1 answer
322 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 ...
Someone's user avatar
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2 votes
0 answers
110 views

Higher genus (Hamiltonian perturbed) holomorphic curves in cotangent bundle of S^1

Consider $T^*S^1$ as symplectic manifold, with hamiltonian function $H(x,y) = y^2$ (y is the fiber direction, I know this is morse bott but it can be perturbed). consider the set of maps $u: \Sigma \...
Yuan Yao's user avatar
  • 113
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 ...
Shaoyun Bai's user avatar
5 votes
0 answers
94 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 ...
davik's user avatar
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