# Questions tagged [floer-homology]

The floer-homology tag has no usage guidance.

97
questions

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votes

**1**answer

94 views

### Dismissing pseudoholomorphic curves in embedded contact homology

In the papers
https://arxiv.org/abs/math/0410059,
https://arxiv.org/abs/math/0410061,
https://arxiv.org/abs/1608.07988;
Hutchings, Sullivan and Choi use a interesting trick to dismiss the existence of ...

**2**

votes

**0**answers

33 views

### Associativity of orientations on Moduli spaces 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 ...

**2**

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**0**answers

68 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 ...

**6**

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194 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 ...

**3**

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116 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**

votes

**1**answer

130 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})\...

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86 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 $x(1)\in T_{q_1}^*M$ ...

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59 views

### Constant holomorphic strips in Lagrangian floer cohomology

Let $(P,\omega)$ be a symplectic manifold , when defining the lagrangian floer cohomology of a pair $(L_0,L_1)$ of lagrangian submanifolds with $\Sigma_{L}\geq 3$ we will want to look at the space $\...

**2**

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99 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 $(\...

**2**

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96 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 ...

**3**

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57 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^...

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50 views

### Definition of multiply covered $J$-holomorphic disk

In the paper Floer cohomology of lagrangian intersections and pseudo-holomorphic disks in page $1009$ it's mentioned that a $J-$holomorphic disk $w:D^2\rightarrow S^2$ with $\mu(w)=2$ cannot be ...

**1**

vote

**1**answer

125 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 $L$ ...

**2**

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89 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 ...

**1**

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33 views

### Question about index formula for $k$-cusp trajectories

In the paper Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks $1$, the following is claimed :
Let $(J_{\alpha}, L_0^{\alpha}, L_1^{\alpha})\rightarrow (J,L_0,L_1) $ be a ...

**2**

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382 views

### $\phi_1^{2^{N-1}}u$ cannot be a translation of $u$ in $\mathcal{M}_{J,\phi}(x,y)$

I have been reading and trying to understand the paper Floer cohomology of lagrangian intersections and pseudo-holomorphic disks II , but there is a part where I am not understanding a justification ...

**2**

votes

**1**answer

144 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 $\mathbb{C}\mathbb{P}^n$...

**6**

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130 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 ...

**5**

votes

**1**answer

328 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
...

**4**

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**0**answers

113 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 ...

**1**

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**0**answers

88 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 ...

**3**

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94 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 ...

**3**

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**1**answer

181 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 ...

**2**

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84 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 \...

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89 views

### Action functional for the definition of Lagrangian Floer homology

I have been starting to learn about Lagrangian Floer homology using notes by A. Pedroza (arXiv link).
Consider $(M,\omega)$ a symplectic manifold that is symplectic aspherical and $L_0,L_1$ two ...

**15**

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**1**answer

1k 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 ...

**5**

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81 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 ...

**3**

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71 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 ...

**4**

votes

**2**answers

320 views

### Heegard diagrams for three-manifolds

I have a basic question about the Heegaard diagrams involved in providing a framework
for calculation of Floer-Homology of three-manifolds.
Typically such diagrams look like Figure 1 and Figure 2 here ...

**1**

vote

**0**answers

49 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 ...

**9**

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350 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. ...

**6**

votes

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223 views

### Spectral flow of Dirac operator twisted by instanton

Suppose $E$ is a $SU(2)$-bundle over a closed three manifold $M$ and $S$ is the spinor bundle over $M$. Also assume $D_{A(t)}:\Gamma(S\otimes_{\mathbb c} E)\to \Gamma(S\otimes_{\mathbb c} E)$ is a ...

**3**

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136 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-...

**6**

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213 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,\...

**5**

votes

**1**answer

468 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 ...

**7**

votes

**1**answer

369 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 $L$...

**6**

votes

**2**answers

618 views

### The Floer Equation is Elliptic

Let $(M,\omega)$ be a symplectic manifold and $H \in C^\infty(M \times \mathbb{S}^1)$. Furthermore, let $J$ be an $\omega$-compatible almost complex structure on $M$. The Floer equation is the ...

**2**

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88 views

### Differential of the Rabinowitz Action Functional

On an exact Hamiltonian system $(M,d\alpha,H)$ define the Rabinowitz action functional
$$\mathcal{A}^H \colon C^\infty(\mathbb{S}^1,M) \times (0,+\infty) \to \mathbb{R}$$
by
$$\mathcal{A}^H(\gamma,\...

**5**

votes

**2**answers

709 views

### Background needed to understand modern research on knot homology theories

I am a student of mathematics, and have some background in
Algebraic Topology (Hatcher, Bott-Tu, Milnor-Stasheff),
Differential Geometry (Lee, Kobayashi-Nomizu),
Riemannian Geometry (Do Carmo),
...

**3**

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**0**answers

189 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 '...

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90 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. ...

**7**

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193 views

### $A_{\infty}$ multiplications on Morse cochain complex

Can the higher order $A_{\infty}$ multiplications defined by Fukaya be made trivial(by perturbing gradient trees) when Morse cochain complex is isomorphic to Morse cohomology, in which case the cup ...

**1**

vote

**1**answer

279 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 ...

**6**

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**1**answer

410 views

### Intuition about bubbling off a ghost bubble

I'm trying to improve my intuition about the bubbling phenomenon for $J$-holomorphic curves $\Sigma \to (M,\omega)$, where $\Sigma$ is a compact Riemann surface with possibly boundary. I assume that ...

**4**

votes

**1**answer

655 views

### The singular cohomology embeds into the symplectic cohomology

Viterbo's theorem on cotangent bundles $M=T^*N$ tells you in particular that singular cohomology $H^*(M)$ gets embedded in $SH^*(M)$ via the $c^*$ map. Having a Weinstein manifold (or more generally ...

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61 views

### Reference Request: Central Curvature "Fix"

Context: In Lagrangian-Floer theory, the (an) $\mathbf{A}_\infty$-algebra of a Lagrangian is curved. However, the curvature is central. One consequence of this is that you can get an uncurved $\mathbf{...

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284 views

### 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 ...

**18**

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**1**answer

1k views

### How is Chern-Simons theory related to Floer homology?

Chern-Simons theory (say, with gauge group $G$) is the quantum theory of the Chern-Simons functional
$$CS(A)=\frac{k}{8\pi^2}\int_M \text{Tr}\left(A\wedge dA + \frac{2}{3}A\wedge A \wedge A\right)$$
...

**5**

votes

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82 views

### Morse theory for pairs of submanifolds of complementary dimension

If you have a closed monotone symplectic manifold $M$, then to any pair of closed monotone Lagrangian submanifolds $L_1$, $L_2$ you can associate (modulo some bubbling assumptions) a $\mathbb{Z}_N$-...

**10**

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454 views

### Floer cohomology from mapping spaces of $\infty$ categories

There's a meta-observation (of Urs Schreiber, who attributes it to Ken Brown and Lurie) that 'cohomology theories come from mapping spaces of $(\infty,1)$ categories'. This is described in detail at ...